Current task: —

Session 36 — ii′-6-DL dog-leg + side bounds; Paper A review

08-26-2026 → 08-26-2026 · kimi-k3 · 103 messages · 1262.3k tokens · $53.3233

Summary

Focus: ii′-6-DL dog-leg + the closure (research sessions 039–041).

What happened:

  • 6-R complete: E-032 found hidden cancellation (the per-side claim is empirically true but envelope-unprovable); D-073 repaired proof at T = c√n.
  • User directive: stop the c=1.5 route — keep the ORIGINAL N₁ ≈ 9×10²⁴ via the D-074 zero-free dog-leg (PROJECT_DIRECTIVE_D-074.txt).
  • Dog-leg designed and proved in Lean (DL-0…DL-6, LiDogLeg.lean); ii′-6-c side bounds and 6-d the fold: sfBoundProp PROVED.
  • ii′-7: lambda_pos_finite_height_closed PROVED from exactly seven declared hypotheses; full axiom audit clean (3 standard axioms, 0 sorryAx on the chain); Paper A closed (Appendix B rewritten).

Outcome: the finite-height Li theorem became fully machine-proved — the project's flagship formal result.

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#2251 user 2026-08-26 15:54:16.506 UTC open ⤢
please check the current status of the project
#2252 assistant 2026-08-26 15:54:39.389 UTC · 55410→192 tok open ⤢
thinking (185 chars)
The user wants a status check on the project. Per the session START protocol (§8.3), I should read STATE.md first, then skim the latest journal entry + results/index.md. Let me do that.
Reading the archive per the session start protocol.
tool: Read
input
{
  "file_path": "D:\\Python\\Quantum\\STATE.md"
}
tool: Bash
input
{
  "command": "git -C \"D:\\Python\\Quantum\" log --oneline -5 && git -C \"D:\\Python\\Quantum\" status --short",
  "description": "Show recent commits and working tree status"
}
#2253 assistant 2026-08-26 15:55:39.244 UTC · 15027→1362 tok open ⤢
thinking (2433 chars)
The user asked for a status check. STATE.md gives everything. Let me summarize concisely. Current position: - Project: NGUYEN-RH-1, Phase 5B (tier ii′) — closing SfBoundProp in Lean before paper submission. - Session 040 stopped at a clean boundary: ii′-6-c-i complete (LiSideBounds.lean scaffolding, build green 8770 jobs), next is 6-c-ii (I_III left vertical bound). - The D-074 dog-leg repair is the active route, preserving original N₁ = (T₀−1)² ≈ 9×10²⁴. - Uncommitted changes: correspondence package (Lagarias F-D letter drafts), Scratch.lean, various build artifacts (bbl/fls files), paperA.pdf etc. The git status shows modified finding-note files — those are correspondence files, which per freeze protocol are drafts (D-073 §8 says correspondence files untouched — but here they're modified... wait, the git status was a snapshot at conversation start; it shows M research/correspondence/lagarias-F-D-package/README.md, finding-note.pdf, finding-note.tex — hmm, these are modified. That may be work outside the agent's session or perhaps from session 040's block B. Actually STATE.md says correspondence files untouched (freeze protocol) as of session 039. The modifications could be the user's own work on the letter-to-lagarias. I shouldn't worry; just report. Actually the gitStatus in system prompt is from conversation start and matches what I just got. So there are uncommitted modifications in the correspondence package. Since the END protocol says commit at every sub-stage, these modified files are uncommitted — possibly user's own work (the letter to Lagarias about finding F-D). I'll mention it briefly. Give a concise status summary: 1. Where we are: Phase 5B tier ii′ — closing SfBoundProp in Lean before submitting Paper A. 2. Recent arc: finding F-D (broken stepping in Lagarias Thm 6.1) → repair via D-074 zero-free dog-leg → preserves original N₁ = 9×10²⁴. 3. Session 040 accomplishments: DL-0 through DL-6 (LiDogLeg.lean complete, no perimeter), ii′-6-a2 (Bn redefined, convexity re-proved, base points re-certified, N₁ preserved), F-E finding + repair decision, 6-c-i scaffolding green. 4. Next action: 6-c-ii (I_III left vertical bound) — full recipe banked in STATE.md. 5. Working tree note: uncommitted changes in the Lagarias correspondence package (letter/finding-note drafts) — not part of the committed research archive state; ask if those should be committed or left alone. Keep it concise.
## Project status — NGUYEN-RH-1 **Where we are:** Phase 5B (tier ii′) — closing `SfBoundProp` in Lean *before* submitting Paper A. Master is green @ `2979f57` (build 8770–8773 jobs, 3 standard axioms only everywhere). ### Current front: ii′-6-c (four side bounds), D-074 dog-leg route The active line is the **F-D repair**: Lagarias's Thm 6.1 stepping is broken at T = √n (D-072); the adopted fix is the **zero-free dog-leg** (D-074) which keeps T_cut = √n and thereby **preserves the original N₁ = (T₀−1)² ≈ 9×10²⁴**. Session 040 completed: - **DL-0 → DL-6** — `LiDogLeg.lean` fully proved (exact identity, V_left/H_high/V_right bounds, direct Dirichlet bound at re = 2√n; **no perimeter, nothing above T_cut**) - **ii′-6-a2** — `Bn` redefined with the dog-leg charge (1.5·IIMaj), convexity re-proved (4/3-bracket cherry-picked, `e11ad95`), base points re-certified, margins positive: 13 160 / 88 911 / 2.458×10²⁶ at 8×10⁴ / 10⁵ / N₁ - **Finding F-E** (I_I kernel off-by-one) — repair adopted: charge I_I as e^{1/3}·IIMaj, absorbed by the fold's free 2π; no Bn/convexity changes - **6-c-i** — `LiSideBounds.lean` scaffolding committed (`a8c5b1e`, green) **Stopped at a clean boundary (user log-off). Next action: 6-c-ii** — the I_III left-vertical bound `‖VIntegral (Jlag n) (-5/2) (-T) T‖ ≤ 4T((2/3)Z2hat T + 10.5 log(T+2) + 48) + 40·1.3352`; full 5-step recipe is banked in STATE.md. **Remaining map:** 6-c-ii → 6-c-iii (I_I, F-E form) → 6-c-iv (mid-horizontal) → 6-c-v (horizontal assembly) → 6-d (fold ⟹ `sfBoundProp`) → **ii′-7** (closure at original N₁, paper correction note: F-A/F-B/F-D/F-E). ### Side threads - Fallbacks parked: c=1.5 branch `wip/c-1.5-margin-redo` (N₁=4×10²⁴), c=2 option, T=√n hidden-cancellation open question (D-073 §8 D4) - Paper A frozen/submittable at its 3-hypothesis form; freeze protocol in force until ii′-7 ### One thing to flag There are **uncommitted changes** in the working tree: the Lagarias F-D correspondence package (`finding-note.tex/pdf`, `README.md`, plus new untracked `letter-to-lagarias.md`, `evidence.xlsx`, `finding-note_new.pdf`, LaTeX build artifacts) and `Scratch.lean`. These look like your own work on the Lagarias letter. Want me to leave them alone, or commit them?
#2254 user 2026-08-26 15:58:07.322 UTC open ⤢
Let's continue with the rest of the 6-c, starting with 6-c-ii
#2255 assistant 2026-08-26 15:59:06.106 UTC · 1576→1745 tok open ⤢
thinking (3409 chars)
The user wants to continue with 6-c, starting with 6-c-ii. Per STATE.md, the recipe for 6-c-ii is banked: **6-c-ii RECIPE (I_III, left vertical σ = −5/2; target statement):** `vIntegral_Jlag_left_norm_le (hZC) (hZC2) (hNLZ) (hMZ : MZeroProp) {n} (hn : 100000 ≤ n) {T} (hTn : √n ≤ T) (hTn1 : T ≤ √n+1) : ‖VIntegral (Jlag n) (-5/2) (-T) T‖ ≤ 4*T*((2/3)*Z2hat T + 10.5*log(T+2) + 48) + 40*1.3352`. Steps: 1. VIntegral_norm + norm_integral_le_integral_norm (−T ≤ T): ‖·‖ ≤ ∫_{−T}^{T} g, g t := ‖Jlag n (↑(-5/2:ℝ)+↑t·I)‖. Integrability of g: Jlag continuous on the path (kFun_differentiableAt (s≠1 via re=−5/2) .continuousAt + my continuousAt_logDeriv_zeta (zeta_ne_zero_left_vertical) → .mul/.neg) composed with the path map u ↦ ↑x+↑u·I (fun_prop) → .norm → ContinuousOn.intervalIntegrable. 2. Split ∫_{−T}^{T} = ∫_{−T}^{−10} + ∫_{−10}^{10} + ∫_{10}^{T} (integral_add_adjacent_intervals ×2). The left piece: ∫_{−T}^{−10} g = ∫_{10}^{T} g — g is EVEN (jlag_left_vertical_norm_even) + integral_comp_neg (∫_a^b f(−x) = ∫_{−b}^{−a} f). 3. Middle: per t ∈ Icc(−10,10): g t ≤ 2·1.3352 (Jlag_norm_eq + kFun_norm_le_two (re ≤ 1/2) + hMZ t (|t| ≤ 10); CAST BRIDGE: MZeroProp's point is (-5/2:ℂ)+↑t·I, mine is ↑(-5/2:ℝ)+↑t·I — `(-5/2:ℂ) = ↑(-5/2:ℝ)` by push_cast; ring). Then integral_mono_on + integral_const: ≤ 20·(2·1.3352) = 40·1.3352. 4. Tails: per t ∈ Icc(10,T): g t ≤ K := 2*((2/3)*Z2hat T + 10.5*log(T+2) + 48). Via Jlag_norm_eq + kFun_norm_le_two + ‖logDeriv‖ ≤ ‖logDeriv − Σnear‖ + ‖Σnear‖: - capstone logDeriv_riemannZeta_sub_nearZeros_le_tight hZC hZC2 hNLZ (ht : 10 ≤ t) (hσlo/hσhi : re = −5/2, by simp) (him : s.im = t, by simp) (hζ : zeta_ne_zero_left_vertical t) ⟹ ‖·−Σ‖ ≤ 10.5·log(|t|+2)+48 = 10.5·log(t+2)+48 (abs_of_nonneg) ≤ 10.5·log(T+2)+48 (Real.log_le_log, t ≤ T). - ‖Σnear‖ ≤ Σ ‖mult/(s−ρ)‖ ≤ (2/5)·Σ mult ≤ (2/5)·Z2hat t ≤ (2/3)·Z2hat T: per ρ: ‖s−ρ‖ ≥ |re(s−ρ)| = 5/2+ρ.re ≥ 5/2 (abs_re_le_norm; re(s−ρ) = −5/2−ρ.re, abs_of_neg) ⟹ 1/‖s−ρ‖ ≤ 2/5; Σ mult ≤ Z2hat t (nearZerosSub_mult_sum_le hZC2); Z2hat_mono (1 ≤ t ≤ T) + 2/5 ≤ 2/3 + Z2hat T ≥ 0. Then integral_mono_on + integral_const: ∫_{10}^{T} g ≤ (T−10)·K. 5. Assemble: ∫ ≤ 2·(T−10)·K + 40·1.3352 ≤ 2T·K + 40·1.3352 = 4T·((2/3)Z2hat T + 10.5 log(T+2) + 48) + 40·1.3352 (K ≥ 0 since Z2hat T ≥ 0 for T ≥ 1: log(T+2) ≥ 1, loglog ≥ 0). First, I need to look at the existing LiSideBounds.lean to see what scaffolding exists (Z2hat_mono, nearZerosSub_mult_sum_le, continuousAt/On_logDeriv_zeta, Jlag_norm_eq, zeta_ne_zero_left_vertical, zeta_ne_zero_horizontal, rectangleIntegral_Jlag_eq, VIntegral_norm, HIntegral_split, jlag_left_vertical_norm_even). Let me set up a task list and read the file. Let me first read LiSideBounds.lean to see what was built in 6-c-i. This is a big Lean task. Let me plan: 1. Read LiSideBounds.lean (the 6-c-i scaffolding). 2. Check the exact signatures of key lemmas: VIntegral_norm, Jlag_norm_eq, kFun_norm_le_two, zeta_ne_zero_left_vertical, jlag_left_vertical_norm_even, nearZerosSub_mult_sum_le, Z2hat_mono, logDeriv_riemannZeta_sub_nearZeros_le_tight, MZeroProp def. 3. Write the 6-c-ii proof per the recipe. 4. Build it with lake env lean on the single file, then lake build the module. 5. Commit per sub-stage protocol. Let me start. I should use TodoWrite for the task tracking. Then read the file. The file is at research/lean/NguyenRH1/NguyenRH1/LiSideBounds.lean presumably. Let me read it.
Resuming ii′-6-c. Setting up the sub-stage plan and reading the 6-c-i scaffolding first.
tool: TodoWrite
input
{
  "todos": [
    {
      "content": "6-c-ii: I_III left-vertical bound vIntegral_Jlag_left_norm_le",
      "status": "in_progress",
      "activeForm": "Proving 6-c-ii I_III left-vertical bound"
    },
    {
      "content": "6-c-iii: I_I right-vertical bound (F-E form)",
      "status": "pending",
      "activeForm": "Proving 6-c-iii I_I bound"
    },
    {
      "content": "6-c-iv: mid-horizontal segment machine",
      "status": "pending",
      "activeForm": "Proving 6-c-iv mid-horizontal machine"
    },
    {
      "content": "6-c-v: horizontal assembly (mid + DL-5 tails)",
      "status": "pending",
      "activeForm": "Assembling 6-c-v horizontal bound"
    },
    {
      "content": "6-d: fold \u27f9 sfBoundProp (then commit, update archive)",
      "status": "pending",
      "activeForm": "Proving 6-d fold to sfBoundProp"
    }
  ]
}
tool: Bash
input
{
  "command": "ls \"D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\\" && wc -l \"D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiSideBounds.lean\"",
  "description": "List Lean package files and LiSideBounds line count"
}
#2256 assistant 2026-08-26 15:59:45.538 UTC · 3710→427 tok open ⤢
thinking (418 chars)
Now I have the scaffolding. I need to check the exact signatures of the dependencies: 1. `logDeriv_riemannZeta_sub_nearZeros_le_tight` in LiTightR.lean — signature: (hZC) (hZC2) (hNLZ) plus hyps on s. Let me check. 2. `MZeroProp` definition. 3. `kFun_norm_le_two` — signature from LiContour. 4. `nearZerosSub` definition and `mem_nearZerosSub`. 5. `Z2hat` definition. 6. `Jlag_conj` signature. Let me grep for these.
Scaffolding looks good. Now checking the exact signatures of the dependencies for the 6-c-ii recipe.
tool: Grep
input
{
  "pattern": "theorem logDeriv_riemannZeta_sub_nearZeros_le_tight",
  "path": "D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiTightR.lean",
  "output_mode": "content",
  "-A": 15
}
tool: Grep
input
{
  "pattern": "MZeroProp|def Z2hat|def M0|theorem kFun_norm_le_two|def nearZerosSub|theorem mem_nearZerosSub|theorem Jlag_conj|def Jlag|theorem kFun_differentiableAt",
  "path": "D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1",
  "output_mode": "content",
  "-A": 6
}
#2257 user 2026-08-26 16:13:55.990 UTC open ⤢
continue
#2258 assistant 2026-08-26 16:16:07.765 UTC · 2088→1004 tok open ⤢
thinking (2927 chars)
Important findings — the signatures differ from my assumptions: 1. `intervalIntegrable_const [IsLocallyFiniteMeasure μ] {c : E} : IntervalIntegrable (fun _ => c) μ a b` ✓ (no args needed). 2. `Continuous.intervalIntegrable (hu : Continuous u) (a b : ℝ) : IntervalIntegrable u μ a b` ✓. 3. `norm_integral_le_integral_norm (h : a ≤ b) : ‖∫ x in a..b, f x ∂μ‖ ≤ ∫ x in a..b, ‖f x‖ ∂μ` ✓ (namespace: these are inside `namespace intervalIntegral` presumably — the grep stripped the namespace. The file likely has `namespace intervalIntegral`. So full names: `intervalIntegral.norm_integral_le_integral_norm` etc.) 4. `integral_const [CompleteSpace E] (c : E) : ∫ _ in a..b, c = (b - a) • c` ✓. 5. `integral_comp_neg : (∫ x in a..b, f (-x)) = ∫ x in -b..-a, f x` — note: it takes NO explicit args (f, a, b implicit or section variables). Section variables — likely `variable {f : ℝ → E} {a b : ℝ}`. So usage: `intervalIntegral.integral_comp_neg` with expected-type unification. 6. `integral_congr {a b : ℝ} (h : EqOn f g [[a, b]]) : ∫ x in a..b, f x ∂μ = ∫ x in a..b, g x ∂μ` — note `[[a, b]]` is uIcc notation. EqOn on uIcc. 7. `integral_add_adjacent_intervals (hab) (hbc) : (∫ a..b) + (∫ b..c) = ∫ a..c` ✓ (use .symm for my split direction). 8. **`integral_mono_on (h : ∀ x ∈ Icc a b, f x ≤ g x) : (∫ u in a..b, f u ∂μ) ≤ ∫ u in a..b, g u ∂μ`** — CRITICAL DIFFERENCE: this version takes ONLY the pointwise hypothesis! The `hab : a ≤ b`, `hf`, `hg` are section variables (the proof body references `hab`, `hf.1`, `hg.1`). So the signature is probably: ``` theorem integral_mono_on (h : ∀ x ∈ Icc a b, f x ≤ g x) : ... ``` with `variable (hab : a ≤ b) (hf : IntervalIntegrable f μ a b) (hg : IntervalIntegrable g μ a b)` included via `include` or as part of the theorem... Let me look at more context around line 1432 to see the variable declarations. The proof uses hab, hf, hg — these must be section variables that are included. Since they're used in the proof, they're auto-included in the theorem signature. So the full signature likely: `theorem integral_mono_on (hab : a ≤ b) (hf : IntervalIntegrable f μ a b) (hg : IntervalIntegrable g μ a b) (h : ∀ x ∈ Icc a b, f x ≤ g x) : ...` — wait but the grep shows `theorem integral_mono_on (h : ∀ x ∈ Icc a b, f x ≤ g x) :` as the first explicit binder. So hab/hf/hg come BEFORE as section variables, meaning the argument order is: hab hf hg h? No — section variables appear in the order they're declared, before explicit binders... Actually section variables are inserted in the order of first use? No — `variable` declarations are included in the order declared when used. The signature would be `(hab : a ≤ b) (hf : ...) (hg : ...) (h : ...)` if declared in that order. Let me read the context above line 1432 to see the variable declarations. Also need to double check the namespace — whether these are in `namespace intervalIntegral`. Let me read the file sections.
Signatures found. Let me check the section variables and namespace context around `integral_mono_on`.
tool: Bash
input
{
  "command": "grep -n \"^variable\\|^namespace\\|^section\\|^end\" \"D:/Python/Quantum/zeta-23-lean/.lake/packages/mathlib/Mathlib/MeasureTheory/Integral/IntervalIntegral/Basic.lean\" | sed -n '1,80p'",
  "description": "Show variable/namespace declarations in IntervalIntegral Basic.lean"
}
#2259 assistant 2026-08-26 16:17:13.266 UTC · 1753→1423 tok open ⤢
thinking (3727 chars)
Key facts: 1. Line 668: `namespace intervalIntegral` — so all are `intervalIntegral.*` ✓. 2. Line 911-913: `section Comp` with `variable {a b c d : ℝ} (f : ℝ → E)` — so `intervalIntegral.integral_comp_neg (f : ℝ → E) : ∫ x in a..b, f (-x) = ∫ x in -b..-a, f x` with a, b implicit. Usage: `intervalIntegral.integral_comp_neg g` — or with implicit a b unified by expected type. Since a, b are implicit and f explicit: `intervalIntegral.integral_comp_neg (fun t => ‖Jlag n _‖)` gives `∫ x in ?a..?b, f (-x) = ∫ x in -?b..-?a, f x` with ?a ?b unified from the goal. Should work with `rw` or `exact`. Careful with the integral_comp_neg rewrite: my goal after hsplit: `∫ t in (-T)..(-10), g t`. I transform to `∫ t in (-T)..(-10), g (-t)` via integral_congr (needs EqOn g (fun t => g (-t)) [[-T, -10]]). Hmm wait — direction: integral_congr (h : EqOn f g [[a,b]]) : ∫ f = ∫ g. So with f := g, g := fun t => g(-t): need EqOn g (fun t => g (-t)) [[-T,-10]] — i.e. ∀ t ∈ uIcc, g t = g(-t) ✓ from hge. Then goal becomes ∫ t in (-T)..(-10), (fun u => g (-u)) t — hmm, will `intervalIntegral.integral_congr` produce that beta-redex? The h's RHS is `fun t => g (-t)`; after rw/exact the integral's integrand is that lambda applied to the bound variable — displayed as `∫ t in -T..-10, g (-t)`. Then `intervalIntegral.integral_comp_neg (f := fun t => g t)`... hmm f is explicit. `integral_comp_neg g : ∫ x in ?a..?b, g (-x) = ∫ x in -?b..-?a, g x`. The LHS `∫ x in -T..-10, g (-x)` matches with ?a := -T, ?b := -10 ✓. So: ```lean have hleft : ∫ t in (-T)..(-10:ℝ), g t = ∫ t in (10:ℝ)..T, g t := by have h1 : ∫ t in (-T)..(-10:ℝ), g t = ∫ t in (-T)..(-10:ℝ), g (-t) := intervalIntegral.integral_congr (fun t _ => hge t) rw [h1, intervalIntegral.integral_comp_neg g] -- goal: ∫ t in -(-10)..-(-T), g t = ∫ t in 10..T, g t norm_num1 -- -(-10) = 10, -(-T) = T?? -(-T) needs neg_neg ``` Hmm, `-(-10 : ℝ)` — neg_neg (10) simp; `-(-T)` — neg_neg T. `simp only [neg_neg]` closes both. Then rfl. So: ```lean rw [h1, intervalIntegral.integral_comp_neg g] simp only [neg_neg] ``` After this the goal should be `∫ t in 10..T, g t = ∫ t in 10..T, g t` closed by rfl automatically after simp? simp only [neg_neg] closes rfl goals ✓. Wait, but `rw [intervalIntegral.integral_comp_neg g]` — rewrite with an equation whose LHS has ?a ?b implicit metavars: rw unifies `∫ x in ?a..?b, g (-x)` against the goal `∫ t in -T..-10, g (-t)` ✓ instantiates ?a := -T, ?b := -10, replaces with `∫ x in -(-10)..-(-T), g x` ✓. 3. Line 1421: `variable (hab : a ≤ b) (hf : IntervalIntegrable f μ a b) (hg : IntervalIntegrable g μ a b)` in section Mono (1403), inside the section at 1283 with `{f g : ℝ → ℝ}` — f g are REAL-valued ✓ (my g is ℝ-valued ✓). So `intervalIntegral.integral_mono_on (hab) (hf) (hg) (h)` — argument order: hab, hf, hg, then h. Let me verify by reading lines 1421-1449. Also from line 1421's context I should confirm the exact conclusion statement — grep showed: ``` 1432:theorem integral_mono_on (h : ∀ x ∈ Icc a b, f x ≤ g x) : 1433: (∫ u in a..b, f u ∂μ) ≤ ∫ u in a..b, g u ∂μ:= by ``` So full signature: `intervalIntegral.integral_mono_on (hab : a ≤ b) (hf : IntervalIntegrable f μ a b) (hg : IntervalIntegrable g μ a b) (h : ∀ x ∈ Icc a b, f x ≤ g x)`. ✓ as I originally assumed. Let me also confirm `norm_le_insert`, `mul_le_mul`, `one_div_le_one_div_of_le`, `Complex.abs_re_le_norm`, `norm_natCast`, `Real.le_log_iff_exp_le`, `Real.exp_one_lt_d9`, `abs_of_nonneg`, `abs_of_neg`, `Finset.mul_sum`, `norm_sum_le`, `Real.le_sqrt`, `div_eq_mul_one_div`, `mul_le_mul_of_nonneg_left`, `Real.log_nonneg`, `Continuous.norm`, `ContinuousAt.comp`, `continuous_iff_continuousAt`. Let me grep these quickly in Mathlib.
Section variables confirmed. Let me verify the exact signature of `integral_mono_on` and the other general lemmas.
tool: Bash
input
{
  "command": "sed -n '1403,1449p' \"D:/Python/Quantum/zeta-23-lean/.lake/packages/mathlib/Mathlib/MeasureTheory/Integral/IntervalIntegral/Basic.lean\"",
  "description": "Read Mono section of IntervalIntegral Basic.lean"
}
#2260 assistant 2026-08-26 16:17:39.906 UTC · 2531→414 tok open ⤢
thinking (540 chars)
Confirmed: `intervalIntegral.integral_mono_on (hab : a ≤ b) (hf : IntervalIntegrable f μ a b) (hg : IntervalIntegrable g μ a b) (h : ∀ x ∈ Icc a b, f x ≤ g x)`. Now verify the general lemmas. Let me grep for: - `norm_le_insert` - `Complex.abs_re_le_norm` - `norm_natCast` (for ℂ) - `Real.le_log_iff_exp_le` - `Real.exp_one_lt_d9` - `Real.le_sqrt` - `div_eq_mul_one_div` - `norm_sum_le` - `Finset.mul_sum` - `continuous_iff_continuousAt` - `ContinuousAt.comp` - `Continuous.norm` - `IsNontrivialZero` definition (field structure) in Zeta23.
Confirmed. Now verifying the remaining general lemmas and the `IsNontrivialZero` structure.
tool: Grep
input
{
  "pattern": "theorem norm_le_insert |theorem abs_re_le_norm |theorem norm_natCast |theorem le_log_iff_exp_le |theorem exp_one_lt_d9|theorem le_sqrt |theorem div_eq_mul_one_div |theorem norm_sum_le |theorem mul_sum |theorem continuous_iff_continuousAt |protected theorem Continuous\\.norm |theorem abs_of_nonneg |theorem abs_of_neg |theorem log_nonneg |theorem one_div_le_one_div_of_le |theorem mul_le_mul_of_nonneg_left |theorem mul_le_mul ",
  "path": "D:\\Python\\Quantum\\zeta-23-lean\\.lake\\packages\\mathlib\\Mathlib",
  "output_mode": "content",
  "-A": 2,
  "head_limit": 80
}
#2261 assistant 2026-08-26 16:24:47.344 UTC · 854→312 tok open ⤢
thinking (607 chars)
`abs_of_nonneg` is used in the project already (root namespace, works for ℝ) ✓. `abs_of_neg` — let me check usage in project... not shown in the first 8 hits. It surely exists: `abs_of_neg : a < 0 → |a| = -a`. I'll trust it; if not, `abs_of_neg` might be `abs_of_neg` in core Lean — yes it's in Lean core/Mathlib.Algebra.Order. It's standard. Now, one more check: `norm_div` for ℂ in the near-sum term — LiTightR line 152-153 has hterm with `‖(zeroMult ρ.1 : ℂ) / (...)‖` — let me look at how they normalized it there (lines 152-178), since it's the same pattern I need. Let me read LiTightR lines 148-198.
Let me look at how LiTightR normalized the per-term norms (same pattern I need).
tool: Read
input
{
  "file_path": "D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiTightR.lean",
  "offset": 148,
  "limit": 50
}
#2262 assistant 2026-08-26 16:33:24.728 UTC · 1508→12812 tok open ⤢
thinking (33239 chars)
This gives me the exact working patterns: - `abs_re_le_norm` used unqualified ✓ - per-term norm: `rw [norm_div]`, then `‖(zeroMult ρ.1 : ℂ)‖ = (zeroMult ρ.1 : ℝ)` via `rw [show (zeroMult ρ.1 : ℂ) = ((zeroMult ρ.1 : ℝ) : ℂ) by push_cast; ring, Complex.norm_real]` + `abs_of_nonneg (Nat.cast_nonneg _)` ✓ - `div_le_div_of_nonneg_left` pattern available. Now I have all patterns. Note `Complex.norm_real` — good banked trick for the nat-cast norm. Also let me double-check `VIntegral` and `HIntegral` definitions from Zeta23 to make sure about the point form `↑x + ↑t * I`. From rectangleIntegral_Jlag_eq and VIntegral_norm usage it seems right. Let me quickly grep the Zeta23 defs. Also need `Jlag_differentiableAt`-style differentiability or just continuity along path for integrability — I have the plan. Also for step 3 middle I need `intervalIntegral.integral_const` and smul_eq_mul. Also need `Real.sqrt` cast facts: `(10:ℝ) ≤ Real.sqrt n` — via Real.le_sqrt (hx : 0 ≤ x) (hy : 0 ≤ y) : x ≤ √y ↔ x^2 ≤ y. With x = 10, y = n: 10 ≤ √n ↔ 100 ≤ n ✓ from hn : 100000 ≤ n (as naturals; cast: (100000:ℝ) ≤ (n:ℝ) via exact_mod_cast). Let me now also confirm the exact form of the tight capstone hypothesis names: (hσlo: -5/2 ≤ s.re) (hσhi : s.re ≤ 3) (him : s.im = t) (hζ: riemannZeta s ≠ 0). ✓ from the grep earlier. One more check — `intervalIntegral.integral_comp_neg`: section Comp has `variable {a b c d : ℝ} (f : ℝ → E)` — so f is EXPLICIT, a b implicit. `intervalIntegral.integral_comp_neg g : ∫ x in ?a..?b, g (-x) = ∫ x in -?b..-?a, g x`. ✓ And `integral_congr {a b : ℝ} (h : EqOn f g [[a, b]])` — section Basic variable `{a b : ℝ} {f g : ℝ → E} {μ : Measure ℝ}`. So `intervalIntegral.integral_congr (h : Set.EqOn f g (Set.uIcc a b))`. The `[[a, b]]` notation = uIcc. EqOn f g uIcc: `fun t _ => hge t` where hge : ∀ t, g t = g (-t) — wait need EqOn g (fun t => g (-t)) [[-T, -10]], i.e. ∀ t ∈ [[-T,-10]], g t = (fun u => g (-u)) t, i.e. g t = g (-t) ✓. Now write the code. Plan the theorem structure: ```lean /-! ## 2. 6-c-ii: I_III — the left vertical σ = −5/2 -/ /-- Z₂(t) ≥ 0 for t ≥ 1. -/ theorem Z2hat_nonneg {t : ℝ} (ht : 1 ≤ t) : 0 ≤ Z2hat t := by have ht2 : (0 : ℝ) < t + 2 := by linarith have hlog1 : (1 : ℝ) ≤ Real.log (t + 2) := by rw [Real.le_log_iff_exp_le ht2] have he := Real.exp_one_lt_d9.le linarith have hlog2 : (0 : ℝ) ≤ Real.log (Real.log (t + 2)) := Real.log_nonneg hlog1 unfold Z2hat linarith [hlog1.le, hlog2] ``` Wait hlog1 : 1 ≤ log(t+2) gives 0 ≤ log(t+2) via le_trans. unfold Z2hat : goal 0 ≤ 0.6 * Real.log (t+2) + 0.886 * Real.log (Real.log (t+2)) + 2.7. linarith [hlog1, hlog2]: 0.6*(≥1) + 0.886*(≥0) + 2.7 ≥ 0 ✓ linarith. Hmm — but will `unfold Z2hat` work? Z2hat is a def; in Z2hat_mono they used `unfold Z2hat` ✓. Main theorem: ```lean /-- **6-c-ii (I_III, the left vertical σ = −5/2)**: the full-side bound — kernel ≤ 2 (`kFun_norm_le_two`), |t| ≥ 10 by the tight R-bound capstone + the two-unit near count, |t| ≤ 10 by the perimeter `MZeroProp`; the two tails are equal by the even symmetry. -/ theorem vIntegral_Jlag_left_norm_le (hZC : ZeroCountProp) (hZC2 : ZeroCountProp2) (hNLZ : NoLowZeros) (hMZ : MZeroProp) {n : ℕ} (hn : 100000 ≤ n) {T : ℝ} (hTn : Real.sqrt n ≤ T) (hTn1 : T ≤ Real.sqrt n + 1) : ‖VIntegral (Jlag n) (-5 / 2) (-T) T‖ ≤ 4 * T * ((2 / 3) * Z2hat T + 10.5 * Real.log (T + 2) + 48) + 40 * 1.3352 := by classical -- T ≥ 10 from n ≥ 10⁵ have hnR : (100000 : ℝ) ≤ (n : ℝ) := by exact_mod_cast hn have h10T : (10 : ℝ) ≤ T := by have h1 : (10 : ℝ) ≤ Real.sqrt (n : ℝ) := by rw [Real.le_sqrt (by norm_num) (by positivity)] norm_num1 linarith [hnR] -- 10^2 = 100 ≤ n linarith have hT0 : (0 : ℝ) ≤ T := by linarith have hTT : -T ≤ T := by linarith -- the integrand and its continuity ... ``` Wait `Real.le_sqrt (hx : 0 ≤ x) (hy : 0 ≤ y) : x ≤ √y ↔ x ^ 2 ≤ y`. So `rw [Real.le_sqrt (by norm_num) (by positivity)]` at goal `10 ≤ √(n:ℝ)` — turns goal into `10^2 ≤ (n:ℝ)`. Then `norm_num1`? 10^2 → 100 — hmm rw leaves `(10:ℝ)^2 ≤ ↑n`; `norm_num1` won't fully close; use `nlinarith [hnR]`? 10^2 = 100 needs computing. `by norm_num; exact_mod_cast hn`... Let me: ```lean have h1 : (10 : ℝ) ≤ Real.sqrt (n : ℝ) := by rw [Real.le_sqrt (by norm_num) (by positivity)] calc (10:ℝ)^2 = 100 := by norm_num _ ≤ (n : ℝ) := by linarith [hnR] ``` Hmm — after rw the goal is `(10:ℝ)^2 ≤ ↑n`. `calc (10:ℝ)^2 = 100 := by norm_num` then `100 ≤ ↑n` — 100 ≤ 100000 ≤ n ✓ linarith [hnR]. Fine. Or `nlinarith [hnR]` — nlinarith computes 10^2 = 100? nlinarith normalizes numerals ✓ probably. Use the calc for safety. Note `Real.sqrt n` in hTn — n : ℕ, so `Real.sqrt ↑n` with the ℕ→ℝ cast. `Real.sqrt (n : ℝ)` — same. In the theorem hTn : Real.sqrt ↑n ≤ T where ↑ is Nat.cast. In my have h1 I wrote `Real.sqrt (n : ℝ)` — the elaborator will see `(n : ℝ)` = `((n : ℝ))` = Nat.cast n ✓ same as hTn's. Positivity for hy : 0 ≤ (n:ℝ): `by positivity` ✓ (Nat.cast_nonneg). Continuity of the integrand: ```lean -- continuity of Jlag on the left vertical line have hJcont : ∀ t : ℝ, ContinuousAt (Jlag n) (((-5 / 2 : ℝ) : ℂ) + (t : ℝ) * Complex.I) := by intro t have hs1 : (((-5 / 2 : ℝ) : ℂ) + (t : ℝ) * Complex.I) ≠ 1 := by intro h have hre := congrArg Complex.re h simp at hre norm_num at hre have hA := (kFun_differentiableAt hs1).continuousAt have hB := continuousAt_logDeriv_zeta hs1 (zeta_ne_zero_left_vertical t) show ContinuousAt (fun s => kFun n s * (-logDeriv riemannZeta s)) _ exact hA.mul hB.neg ``` Wait — simp at hre: hre : Complex.re (↑(-5/2) + ↑t * I) = Complex.re 1. simp simplifies LHS: `Complex.add_re`, `Complex.ofReal_re`, `Complex.mul_re`, `Complex.I_re`, `Complex.I_im` — these are simp lemmas; gives `-5/2 = 1`? simp might directly close hre to False via norm_num inside? simp doesn't evaluate `-5/2 = 1` to False by itself (numerals need norm_num). So `norm_num at hre` ✓. Hmm — but wait, in zeta_ne_zero_left_vertical (line 103) they used `by simp` for the re equation with the same expression ✓ so simp handles `(...).re = -5/2`. For hre after congrArg: LHS simp reduces to -5/2, RHS Complex.re 1 = 1 via Complex.one_re. So hre : (-5/2 : ℝ) = 1, norm_num at hre derives False and closes the goal ✓ (norm_num at h with h : False-equivalent closes any goal). For the `show` — goal: ContinuousAt (Jlag n) s. `show ContinuousAt (fun s => kFun n s * (-logDeriv riemannZeta s)) s` — defeq? Jlag n s = kFun n s * (-(deriv ζ s / ζ s)); my lambda body: kFun n s * (-logDeriv ζ s) = kFun n s * (-(deriv ζ s / ζ s)) after unfolding logDeriv (def). `show` uses defeq — should work ✓. ```lean have hpath : Continuous fun u : ℝ => (((-5 / 2 : ℝ) : ℂ) + (u : ℝ) * Complex.I) := by fun_prop have hcontJ : Continuous fun t : ℝ => Jlag n (((-5 / 2 : ℝ) : ℂ) + (t : ℝ) * Complex.I) := by rw [continuous_iff_continuousAt] intro t exact (hJcont t).comp hpath.continuousAt have hcontG : Continuous fun t : ℝ => ‖Jlag n (((-5 / 2 : ℝ) : ℂ) + (t : ℝ) * Complex.I)‖ := hcontJ.norm' have hI : ∀ a b : ℝ, IntervalIntegrable (fun t : ℝ => ‖Jlag n (((-5 / 2 : ℝ) : ℂ) + (t : ℝ) * Complex.I)‖) MeasureTheory.volume a b := fun a b => hcontG.intervalIntegrable a b ``` fun_prop for hpath: `Continuous fun u : ℝ => (↑(-5/2) + ↑u * Complex.I)` — components: ofReal coe of u (continuous_ofReal), mul I, add const. fun_prop should handle ✓. The integrand abbreviation: to reduce repetition I could use `set g := fun t : ℝ => ‖Jlag n (((-5/2 : ℝ)) + t * I)‖ with hg`. Then hcontG : Continuous g := by rw [hg]; exact hcontJ.norm'. Hmm, set introduces g := ... and rewrites occurrences in goal — but the goal is the final VIntegral statement which doesn't contain g. The occurrences I create in have's I'd write as g directly. `set` requires the goal/hyps to contain the term — it still works even with 0 occurrences? `set x := e` works regardless. Then g t : needs `show`/`simp only [hg]` at application sites... Honestly, avoiding `set` and spelling the integrand out is verbose but robust. The pointwise bounds (tail/middle) I'll state with the applied form `‖Jlag n (↑(-5/2) + ↑t * I)‖ ≤ ...` and rely on beta-defeq when feeding integral_mono_on. Beta-defeq concern in integral_mono_on: hfg : ∀ x ∈ Icc a b, f x ≤ g x where f := fun t => ‖Jlag n (...)‖. The lambda I pass: `fun t ht => htailbound t ht` — type: ∀ t ∈ Icc 10 T, ‖Jlag n (↑(-5/2)+↑t*I)‖ ≤ K. Expected: ∀ t ∈ Icc 10 T, (fun t => ‖Jlag n _‖) t ≤ (fun _ => K) t. Defeq by beta ✓ `exact` accepts. Now the pointwise bounds. Middle: ```lean have hcast : ((-5 / 2 : ℂ)) = (((-5 / 2 : ℝ)) : ℂ) := by push_cast; ring have hmid : ∀ t : ℝ, t ∈ Set.Icc (-10 : ℝ) 10 → ‖Jlag n (((-5 / 2 : ℝ) : ℂ) + (t : ℝ) * Complex.I)‖ ≤ 2 * 1.3352 := by intro t ht obtain ⟨ht1, ht2⟩ := ht have habs : |t| ≤ 10 := abs_le.mpr ⟨ht1, ht2⟩ have hMZt : ‖logDeriv riemannZeta ((((-5 / 2 : ℝ)) : ℂ) + (t : ℝ) * Complex.I)‖ ≤ 1.3352 := by have h := hMZ t habs rwa [hcast] at h have hk : ‖kFun n (((-5 / 2 : ℝ)) : ℂ) + (t : ℝ) * Complex.I)‖ ≤ 2 := by apply kFun_norm_le_two simp -- goal: (...).re ≤ 1/2 — simp reduces re to -5/2; then needs -5/2 ≤ 1/2? rw [Jlag_norm_eq] exact mul_le_mul hk hMZt (norm_nonneg _) (by norm_num) ``` Careful with kFun_norm_le_two's hypothesis: (hs : s.re ≤ 1/2). s = ↑(-5/2)+↑t*I, s.re = -5/2: `by simp` — does simp close `s.re ≤ 1/2`? simp simplifies s.re to -5/2, then the goal `-5/2 ≤ 1/2` — simp doesn't prove numeric inequalities; needs norm_num. So: `by have : (...).re = -5/2 := by simp; rw [this]; norm_num`. Or `by simp only [Complex.add_re, Complex.ofReal_re, Complex.mul_re, Complex.I_re, Complex.I_im]; norm_num`. I'll write a small have hre := ... by simp, then rw + norm_num. Wait — actually there's a subtlety with `abs_le.mpr ⟨ht1, ht2⟩`: abs_le : |a| ≤ b ↔ -b ≤ a ∧ a ≤ b ✓. ht : t ∈ Set.Icc (-10) 10 — obtain gives ht1 : -10 ≤ t, ht2 : t ≤ 10 ✓ (Set.mem_Icc is defeq And — obtain works). For hcast: `((-5/2 : ℂ)) = ↑(-5/2 : ℝ)`: push_cast on RHS: `↑(-5/2) = -(5/2)`?? push_cast normalizes: `↑(-5/2) = -5/2` hmm the RHS becomes `(-5 : ℂ)/(2 : ℂ)` and LHS `(-5/2 : ℂ)` is `Neg (HDiv 5 2)`? Lean 4 parses `-5/2` as `Neg.neg (HDiv.hDiv 5 2)`? I believe `-5/2` elaborates as `-(5/2)` where 5/2 is HDiv. And push_cast turns `↑(-(5/2))` into `-↑(5/2)` = `-(↑5/↑2)` = `-(5/2)`. Both sides `-(5/2)` ✓ ring closes (or rfl). But if hcast's sides are already equal after push_cast, ring closes ✓. But wait — MZeroProp's term: `(-5 \ 2 : ℂ)` — hmm the backslash is the division `/`. In the file it's written `(-5 \ 2: ℂ)`?? Look at the grep output: `‖logDeriv riemannZeta ((-5 \ 2: ℂ) + (t : ℝ) * Complex.I)‖` — the `\` is likely an artifact of grep output escaping? Actually no — the raw file might literally contain `-5 \ 2`... that would be a parse error in Lean. It must be `/` and the grep tool output shows `\` due to... hmm, weird. Actually I bet the file contains `/` and the display escaped it oddly. Let me not worry — I'll read the MZeroProp def directly to get the exact form. Also note: hMZ t habs gives the bound at the point `(-5/2 : ℂ) + ↑t * I` where (-5/2 : ℂ) is parsed in ℂ. rw [hcast] at h changes it to ↑(-5/2:ℝ) form ✓ matching my s. Now the tail bound: ```lean have htail : ∀ t : ℝ, t ∈ Set.Icc (10 : ℝ) T → ‖Jlag n (((-5 / 2 : ℝ) : ℂ) + (t : ℝ) * Complex.I)‖ ≤ 2 * ((2 / 3) * Z2hat T + 10.5 * Real.log (T + 2) + 48) := by intro t ht obtain ⟨ht10, htT⟩ := ht set s := (((-5 / 2 : ℝ)) : ℂ) + (t : ℝ) * Complex.I with hs_def have hre : s.re = -5 / 2 := by simp [s] have him : s.im = t := by simp [s] have hk : ‖kFun n s‖ ≤ 2 := kFun_norm_le_two (by rw [hre]; norm_num) have hζ : riemannZeta s ≠ 0 := zeta_ne_zero_left_vertical t have hRtight := logDeriv_riemannZeta_sub_nearZeros_le_tight hZC hZC2 hNLZ ht10 (by rw [hre]; norm_num) (by rw [hre]; norm_num) him hζ rw [abs_of_nonneg (by linarith : (0:ℝ) ≤ t)] at hRtight have hlogT : Real.log (t + 2) ≤ Real.log (T + 2) := Real.log_le_log (by linarith) (by linarith) -- the near sum bound have hnear : ‖∑ ρ ∈ nearZerosSub t, (zeroMult ρ.1 : ℂ) / (s - ρ.1)‖ ≤ (2 / 5) * Z2hat t := by calc ‖∑ ρ ∈ nearZerosSub t, (zeroMult ρ.1 : ℂ) / (s - ρ.1)‖ ≤ ∑ ρ ∈ nearZerosSub t, ‖(zeroMult ρ.1 : ℂ) / (s - ρ.1)‖ := norm_sum_le _ _ _ ≤ ∑ ρ ∈ nearZerosSub t, (2 / 5) * (zeroMult ρ.1 : ℝ) := by apply Finset.sum_le_sum intro ρ hρ have hρre : (0 : ℝ) < ρ.1.re := ρ.2.2.1 have hrere : (s - ρ.1).re = -5 / 2 - ρ.1.re := by simp [hre, Complex.sub_re] -- hmm ... ``` Careful: (s - ρ.1).re = s.re - ρ.1.re. With hre : s.re = -5/2. `Complex.sub_re : (a - b).re = a.re - b.re` ✓ simp lemma. So `by simp [hre]` should give `(s - ρ.1).re = -5/2 - ρ.1.re`: simp [Complex.sub_re, hre] → s.re - ρ.1.re → -5/2 - ρ.1.re ✓. Actually just `by simp [Complex.sub_re, hre]`? The goal `(s - ρ.1).re = -5/2 - ρ.1.re`: simp rewrites LHS using Complex.sub_re to s.re - ρ.1.re, then hre : s.re = -5/2 rewrites to -5/2 - ρ.1.re ✓ closes. Then: ```lean have hnorm5 : (5 / 2 : ℝ) ≤ ‖s - ρ.1‖ := by have h1 := abs_re_le_norm (s - ρ.1) rw [hrere, abs_of_neg (by linarith)] at h1 linarith ``` h1 after rw : -(-5/2 - ρ.1.re) ≤ ‖s - ρ.1‖, i.e. 5/2 + ρ.1.re ≤ ‖·‖; with hρre : 0 < ρ.1.re: linarith gives 5/2 ≤ ‖·‖ ✓. ```lean have hnormpos : (0 : ℝ) < ‖s - ρ.1‖ := by linarith have h2 : ‖(zeroMult ρ.1 : ℂ)‖ = (zeroMult ρ.1 : ℝ) := by rw [show (zeroMult ρ.1 : ℂ) = ((zeroMult ρ.1 : ℝ) : ℂ) by push_cast; ring, Complex.norm_real] exact abs_of_nonneg (Nat.cast_nonneg _) rw [norm_div, h2, div_le_iff₀ hnormpos] nlinarith [hnorm5, Nat.cast_nonneg (α := ℝ) (zeroMult ρ.1)] ``` After rw [div_le_iff₀ hnormpos]: goal `↑(zeroMult ρ.1) ≤ (2/5) * ↑(zeroMult ρ.1) * ‖s - ρ.1‖`. nlinarith with hnorm5 (5/2 ≤ ‖s−ρ‖) and cast_nonneg: products: (‖s−ρ‖ − 5/2)*m ≥ 0 ⟹ m*‖s−ρ‖ ≥ (5/2)m ⟹ (2/5)m‖s−ρ‖ ≥ m. nlinarith should find (needs the product hnorm5*hm). ✓ Then the sum: ```lean _ = (2 / 5) * ∑ ρ ∈ nearZerosSub t, (zeroMult ρ.1 : ℝ) := by rw [Finset.mul_sum] _ ≤ (2 / 5) * Z2hat t := mul_le_mul_of_nonneg_left (nearZerosSub_mult_sum_le hZC2 ht10) (by norm_num) ``` Finset.mul_sum s f a : a * ∑ i ∈ s, f i = ∑ i ∈ s, a * f i — I have goal `∑ ρ ∈ S, (2/5)*m_ρ = (2/5) * ∑ ρ ∈ S, m_ρ` — that's (Finset.mul_sum S f (2/5)).symm. So `rw [← Finset.mul_sum]` — rewrites occurrences of `∑ i ∈ s, a * f i` to `a * ∑ i ∈ s, f i`. The LHS of my goal is `∑ ρ ∈ S, (2/5)*↑(zeroMult ρ.1)` ✓ matches with a := 2/5 ✓. So `rw [← Finset.mul_sum]` closes the goal to rfl ✓. Hmm — does `Finset.mul_sum` need explicit args? `lemma mul_sum (s : Finset ι) (f : ι → R) (a : R)` — all explicit. rw [← Finset.mul_sum] with metavars — rw will unify ✓ (the reversed rw: find pattern `∑ i ∈ ?s, ?a * ?f i` — hmm, higher-order pattern ?f applied to bound i — rw uses keyed matching; it should work since the sum body is literally `(2/5) * ↑(zeroMult ρ.1)` which matches `?a * ?f ρ` with ?f := fun ρ => ↑(zeroMult ρ.1) via pattern unification ✓). Continue: ```lean -- combine: ‖logDeriv‖ ≤ near + tight have hlog : ‖logDeriv riemannZeta s‖ ≤ (2 / 3) * Z2hat T + 10.5 * Real.log (T + 2) + 48 := by have h1 : ‖logDeriv riemannZeta s‖ ≤ ‖∑ ρ ∈ nearZerosSub t, (zeroMult ρ.1 : ℂ) / (s - ρ.1)‖ + ‖logDeriv riemannZeta s - ∑ ρ ∈ nearZerosSub t, (zeroMult ρ.1 : ℂ) / (s - ρ.1)‖ := norm_le_insert _ _ have hmono := Z2hat_mono (by linarith : (1 : ℝ) ≤ t) htT have hnn : (0 : ℝ) ≤ Z2hat T := Z2hat_nonneg (by linarith : (1 : ℝ) ≤ T) linarith [h1, hnear, hRtight, hlogT, hmono, hnn] rw [Jlag_norm_eq] exact mul_le_mul hk hlog (norm_nonneg _) (by norm_num) ``` linarith check: h1 : ‖logDeriv‖ ≤ ‖Σ‖ + ‖·−Σ‖; hnear : ‖Σ‖ ≤ (2/5)Z2hat t; hRtight : ‖·−Σ‖ ≤ 10.5*log(t+2)+48; hlogT : log(t+2) ≤ log(T+2); hmono : Z2hat t ≤ Z2hat T; hnn : 0 ≤ Z2hat T. Goal: ‖logDeriv‖ ≤ (2/3)Z2hat T + 10.5 log(T+2) + 48. Chain: ‖logDeriv‖ ≤ (2/5)Z2hat t + 10.5 log(t+2) + 48 ≤ (2/5)Z2hat T + 10.5 log(T+2) + 48 ≤ (2/3)Z2hat T + 10.5 log(T+2) + 48. All linear in atoms (Z2hat t, Z2hat T, log(t+2), log(T+2)) ✓ linarith. norm_le_insert (a b) : ‖a‖ ≤ ‖b‖ + ‖a − b‖ — with a := logDeriv ζ s, b := Σ ✓. mul_le_mul hk hlog (norm_nonneg _) (by norm_num : (0:ℝ) ≤ 2): gives ‖kFun‖*‖logDeriv‖ ≤ 2 * RHS ✓. And goal after rw [Jlag_norm_eq] is exactly that ✓. Wait — Jlag_norm_eq n s : ‖Jlag n s‖ = ‖kFun n s‖ * ‖logDeriv riemannZeta s‖. The goal is ‖Jlag n (↑(-5/2) + ↑t*I)‖ ≤ 2*(...). With `set s`, the goal shows ‖Jlag n s‖? Hmm — `set s := ... with hs_def` introduces s and rewrites occurrences of the term in the goal — the goal's `(((-5/2:ℝ):ℂ) + ↑t*I)` becomes s ✓ (set folds occurrences). Then rw [Jlag_norm_eq] — Jlag_norm_eq n s needs the argument to match — after set, goal is ‖Jlag n s‖ ≤ ... ✓ rw [Jlag_norm_eq n s]? If I write `rw [Jlag_norm_eq]` it'll unify ✓. Then the middle integral bound: ```lean have hmidInt : ∫ t in (-10 : ℝ)..(10 : ℝ), ‖Jlag n (((-5 / 2 : ℝ) : ℂ) + (t : ℝ) * Complex.I)‖ ≤ 20 * (2 * 1.3352) := by have h := intervalIntegral.integral_mono_on (by norm_num : (-10:ℝ) ≤ 10) (hI _ _) intervalIntegrable_const (fun t ht => hmid t ht) rw [intervalIntegral.integral_const, smul_eq_mul] at h -- h : ∫ ≤ (10 - -10) * (2*1.3352) norm_num1 at h ⊢ -- (10 - -10) = 20 exact h ``` Hmm — (10 - (-10)) * (2*1.3352) vs my stated 20 * (2*1.3352): `norm_num1 at h` simplifies (10 - -10) to 20 ✓ then exact h. Actually smul_eq_mul turns (10 - -10) • c into (10 - -10) * c. Then norm_num1... hmm norm_num1 might mangle more. Alternative: `convert h using 2; norm_num`. Simpler: keep hmidInt statement as `≤ (10 - (-10)) * (2 * 1.3352)` and let the final linarith handle the arithmetic. Even simpler: state hmidInt with the exact RHS produced and let linarith close the assembly. Let me state: ```lean have hmidInt : ∫ t in (-10:ℝ)..(10:ℝ), g.. ≤ (10 - (-10)) * (2 * 1.3352) := ... ``` and in assembly linarith computes (10 - -10) * (2*1.3352) = 40*1.3352 ✓ linear arithmetic with numerals — linarith handles ✓. Similarly htailInt : ∫ t in 10..T, g.. ≤ (T - 10) * (2 * ((2/3)*Z2hat T + 10.5*log(T+2)+48)). ```lean have htailInt : ∫ t in (10 : ℝ)..T, ‖Jlag n (((-5 / 2 : ℝ) : ℂ) + (t : ℝ) * Complex.I)‖ ≤ (T - 10) * (2 * ((2 / 3) * Z2hat T + 10.5 * Real.log (T + 2) + 48)) := by have h := intervalIntegral.integral_mono_on h10T (hI _ _) intervalIntegrable_const (fun t ht => htail t ht) rwa [intervalIntegral.integral_const, smul_eq_mul] at h ``` Careful: `intervalIntegrable_const` — signature `intervalIntegrable_const [IsLocallyFiniteMeasure μ] {c : E} : IntervalIntegrable (fun _ => c) μ a b` — c implicit, unified with the constant function in hg's slot. When passing as an argument to integral_mono_on whose hg expects IntervalIntegrable g μ a b with g := fun _ => K — unification `intervalIntegrable_const : IntervalIntegrable (fun _ => ?c) volume a b` ✓ instantiates ?c := K ✓. But WAIT: the μ in these lemmas — my integrals use volume (default in `∫ t in a..b, f t` notation is volume? The interval integral notation `∫ x in a..b, f x` defaults to ∂volume? In Mathlib, `∫ x in a..b, f x` without ∂μ uses `MeasureTheory.volume` by default ✓ (the notation has default volume). IsLocallyFiniteMeasure volume ✓ instance. The even piece: ```lean have hge : ∀ t : ℝ, ‖Jlag n (((-5 / 2 : ℝ) : ℂ) + (t : ℝ) * Complex.I)‖ = ‖Jlag n (((-5 / 2 : ℝ) : ℂ) + (-t : ℝ) * Complex.I)‖ := fun t => (jlag_left_vertical_norm_even n t).symm have hleft : ∫ t in (-T)..(-10 : ℝ), ‖Jlag n (((-5 / 2 : ℝ) : ℂ) + (t : ℝ) * Complex.I)‖ = ∫ t in (10 : ℝ)..T, ‖Jlag n (((-5 / 2 : ℝ) : ℂ) + (t : ℝ) * Complex.I)‖ := by have h1 : ∫ t in (-T)..(-10 : ℝ), ‖Jlag n (((-5 / 2 : ℝ) : ℂ) + (t : ℝ) * Complex.I)‖ = ∫ t in (-T)..(-10 : ℝ), ‖Jlag n (((-5 / 2 : ℝ) : ℂ) + (-t : ℝ) * Complex.I)‖ := intervalIntegral.integral_congr (fun t _ => hge t) rw [h1] have h2 := intervalIntegral.integral_comp_neg (fun t : ℝ => ‖Jlag n (((-5 / 2 : ℝ) : ℂ) + (t : ℝ) * Complex.I)‖) -- h2 : ∫ x in ?a..?b, f (-x) = ∫ x in -?b..-?a, f x rw [show (-T : ℝ) = ... ] hmm ``` Hmm — h2's LHS integrand: `f (-x)` where f = fun t => ‖Jlag n (↑(-5/2) + ↑t*I)‖. So `f (-x)` = ‖Jlag n (↑(-5/2) + ↑(-x)*I)‖ ✓ matches my transformed integrand. The rw [h2] would rewrite the goal's `∫ t in (-T)..(-10), f(-t)` to `∫ t in -(-10)..-(-T), f t` ✓. Then goal: `∫ t in -(-10)..-(-T), f t = ∫ t in 10..T, f t` — `simp only [neg_neg]` closes (-(-10) = 10, -(-T) = T) then rfl ✓. But the unification: `rw [h2]` where h2 has implicit ?a ?b metavars — rw needs to match `∫ x in ?a..?b, f (-x)` against goal subterm `∫ t in (-T)..(-10), (fun t => ‖Jlag n (↑(-5/2) + ↑(-t)*I)‖) t`... The goal after rw [h1] has integrand `fun t => ‖Jlag n (↑(-5/2)+↑(-t)*I)‖` or applied form `‖Jlag n (↑(-5/2)+↑(-t)*I)‖` under the binder. f(-x) with f the lambda: beta-redex `(fun t => ‖..t..‖) (-x)`. rw matching needs syntactic match up to defeq — integral_comp_neg's LHS is `∫ x in a..b, f (-x)`; with f instantiated to my lambda (I passed it explicitly, so no HO unification for f), the LHS term is `∫ x in ?a..?b, (fun t => ‖..t..‖) (-x)`. After beta this is `∫ x in ?a..?b, ‖..(-x)..‖`. Does rw beta-reduce when matching? rw matches up to reducible defeq? It should handle beta. Then ?a ?b from the endpoints -T, -10 ✓. Potential issue: the bound-variable name and the cast `↑(-t)`: in my h1-produced integrand the term is `↑(-t) * I`; in f(-x) instantiated: `↑(-x) * I` ✓ same up to alpha. Alternatively cleaner: prove hleft by direct calc: ```lean have hleft : ∫ t in (-T)..(-10:ℝ), F t = ∫ t in (10:ℝ)..T, F t := by have h1 : ∫ t in (-T)..(-10:ℝ), F t = ∫ t in (-T)..(-10:ℝ), F (-t) := intervalIntegral.integral_congr (fun t _ => hge t) have h2 : ∫ t in (-T)..(-10:ℝ), F (-t) = ∫ t in -(-10:ℝ)..-(-T), F t := intervalIntegral.integral_comp_neg F rw [h1, h2, neg_neg (10:ℝ), neg_neg T] -- hmm neg_neg needs the form -(-a) ``` neg_neg : -(-a) = a. rw [neg_neg] rewrites first occurrence... rw [neg_neg (10 : ℝ)]? The term is `-(-10)` = Neg.neg (Neg.neg 10) ✓ rw [neg_neg (10:ℝ)] → 10 ✓. But -( -T): Neg.neg (Neg.neg T) ✓ rw [neg_neg T]. But careful — `-(10:ℝ)`: is the literal `-10` displayed as Neg.neg 10 ✓. Hmm wait — the direction of h2: integral_comp_neg F : ∫ x in ?a..?b, F (-x) = ∫ x in -?b..-?a, F x. With ?a := -T, ?b := -10: RHS = ∫ x in -(-10)..-(-T), F x ✓. But h2 as stated with metavars — `intervalIntegral.integral_comp_neg F` leaves ?a ?b free; when I use it in `have h2 : ... := intervalIntegral.integral_comp_neg F`, the type ascription fixes ?a := -T, ?b := -10 ✓. For the LHS of h2's type ascription: ∫ t in (-T)..(-10), F (-t) — needs F(-t) to beta-match ✓ fine (defeq ascription). OK — but there's a subtlety: `intervalIntegral.integral_comp_neg F` — F is the lambda; the LHS `∫ x in ?a..?b, F (-x)` — the integrand is `(F · (-x))`?? Actually the lemma statement is `∫ x in a..b, f (-x)` which elaborates as `∫ x in a..b, f (Neg.neg x)` — with f the lambda, `f (Neg.neg x)` is a beta-redex. Lean stores it as such or beta-reduces during elaboration? The lemma was elaborated with f a section variable — so f (-x) stays `f (-x)` (not a redex since f was a variable at elaboration). When I instantiate f := my lambda, the term becomes `(myLambda) (-x)` — a redex, NOT beta-reduced automatically. My type ascription `∫ t in -T..-10, ‖Jlag n (↑(-5/2)+↑(-t)*I)‖` has the beta-reduced form. defeq check: ascription vs inferred — defeq handles beta ✓. Then `rw [h1, h2, ...]`: h2's LHS in redex form vs goal's reduced form — rw matches up to reducible + some defeq; beta usually OK. Alternatively use `calc` with exact: ```lean have hleft : ∫ t in (-T)..(-10:ℝ), F t = ∫ t in (10:ℝ)..T, F t := by calc ∫ t in (-T)..(-10:ℝ), F t = ∫ t in (-T)..(-10:ℝ), F (-t) := intervalIntegral.integral_congr (fun t _ => hge t) _ = ∫ t in -(-10:ℝ)..-(-T), F t := intervalIntegral.integral_comp_neg F _ = ∫ t in (10:ℝ)..T, F t := by rw [neg_neg, neg_neg] ``` The last step: goal ∫ t in -(-10)..-(-T), F t = ∫ t in 10..T, F t. rw [neg_neg] rewrites the FIRST -(-?) occurrence — neg_neg : -(-a) = a; first match: -(-10) → 10; second rw [neg_neg]: -(-T) → T; then rfl ✓. But rw [neg_neg] without args — could loop? No, after two applications no more -(-·). But careful: it might rewrite -(-T) first (term order) — either order, two rewrites then rfl ✓. Hmm, actually rw [neg_neg] rewrites ALL occurrences that match the first pattern found? No — rw rewrites all occurrences of the FIRST matched instance only. So first rw [neg_neg]: finds some -(-x), instantiates, rewrites all copies of that instance. Second: the other one. Fine. Wait — subtle: is `-(-10 : ℝ)` actually `Neg.neg (Neg.neg (10 : ℝ))`? The term `-(-10)` from integral_comp_neg RHS: `-?b` with ?b := -10 gives `Neg.neg (-10)` = `Neg.neg (Neg.neg 10)` ✓. Now the split & assembly: ```lean have hsplit : ∫ t in (-T)..T, F t = ∫ t in (-T)..(-10:ℝ), F t + (∫ t in (-10:ℝ)..(10:ℝ), F t + ∫ t in (10:ℝ)..T, F t) := by rw [← intervalIntegral.integral_add_adjacent_intervals (hI (-T) (-10)) (hI (-10) T)] rw [← intervalIntegral.integral_add_adjacent_intervals (hI (-10) 10) (hI 10 T)] ``` Hmm — direction: integral_add_adjacent_intervals hab hbc : (∫ a..b) + (∫ b..c) = ∫ a..c. ← direction: rewrites ∫ a..c into the sum. Goal LHS ∫ -T..T: rw [← integral_add_adjacent_intervals (hI (-T) (-10)) (hI (-10) T)] turns ∫(-T..T) into ∫(-T..-10) + ∫(-10..T) ✓. Then rw [← ... (hI (-10) 10) (hI 10 T)] turns ∫(-10..T) into ∫(-10..10) + ∫(10..T) ✓. Goal becomes `∫(-T..-10) + (∫(-10..10) + ∫(10..T)) = ∫(-T..-10) + (∫(-10..10) + ∫(10..T))` ✓ rfl closed by rw automatically. Wait — rw [← h] where h : A + B = C: rewrites C to A + B ✓. But there are two integrals matching the pattern... first rw targets ∫(-T..T) — the only occurrence of that exact term. But the rewrite pattern is the FULL RHS of the lemma instance: with the hI args given, the instance is `∫(-T..-10) + ∫(-10..T) = ∫(-T..T)` — fully determined (a=-T, b=-10, c=T) ✓ no ambiguity. Then final: ```lean calc ‖VIntegral (Jlag n) (-5 / 2) (-T) T‖ = ‖∫ t in (-T)..T, Jlag n (((-5 / 2 : ℝ)) + (t : ℝ) * Complex.I)‖ := VIntegral_norm _ ≤ ∫ t in (-T)..T, ‖Jlag n (((-5 / 2 : ℝ)) + (t : ℝ) * Complex.I)‖ := intervalIntegral.norm_integral_le_integral_norm hTT _ = ∫ t in (-T)..(-10:ℝ), F.. + (∫ t in (-10:ℝ)..(10:ℝ), F.. + ∫ t in (10:ℝ)..T, F..) := hsplit _ = (∫ t in (10:ℝ)..T, F..) + (∫ t in (-10:ℝ)..(10:ℝ), F..) + ∫ t in (10:ℝ)..T, F.. := by rw [hleft]; ring _ ≤ (T - 10) * K + (10 - -10) * (2 * 1.3352) + (T - 10) * K := by linarith [htailInt, hmidInt] _ ≤ 4 * T * ((2/3) * Z2hat T + 10.5 * Real.log (T + 2) + 48) + 40 * 1.3352 := by -- K ≥ 0 have hK0 : (0:ℝ) ≤ 2 * ((2/3) * Z2hat T + 10.5 * Real.log (T + 2) + 48) := by have hnn := Z2hat_nonneg (by linarith : (1:ℝ) ≤ T) have hlognn : 0 ≤ Real.log (T + 2) := Real.log_nonneg (by linarith) positivity -- or nlinarith nlinarith [hK0] ``` Hmm the last step: goal: (T-10)*K + 20*(2*1.3352) + (T-10)*K ≤ 4*T*(stuff) + 40*1.3352 where K = 2*(stuff). (T-10)*K ≤ T*K ⟺ -10K ≤ 0 ⟸ K ≥ 0. 2*(T-10)*K + 53.408 ≤ 2*T*K + 53.408. 2*T*K = 2*T*(2*stuff) = 4*T*stuff ✓. nlinarith with hK0: needs to expand (T-10)*K as T*K - 10*K — linearization handles numeral-multiplied products? nlinarith treats K and T*K as atoms; (T-10)*K = T*K - 10*K via distrib — nlinarith's ring_nf preprocessing handles ✓. linarith alone should also: 2*(T-10)*K + 40*1.3352 ≤ 4*T*stuff + 40*1.3352 ⟺ 2*(T-10)*K ≤ 4*T*stuff ⟺ 2*T*K - 20*K ≤ 2*T*K ⟺ -20*K ≤ 0. In atom terms: K atom, T*K atom. Goal (after ring normalization by linarith preprocessing): it distributes 2*(T-10)*K → 2*(T*K) - 20*K and 4*T*(stuff) where stuff = K/2... hmm — the RHS has `stuff` = (2/3)*Z2hat T + 10.5*log(T+2) + 48 as a DIFFERENT atom than K. linarith sees 4*T*stuff vs 2*T*K — different atoms! It won't know K = 2*stuff. PROBLEM. Fix: give linarith the relation hK : K = 2 * stuff explicitly as an equation, or better, don't introduce K at all — write htailInt with the fully expanded RHS (T - 10) * (2 * ((2/3)*Z2hat T + 10.5*log(T+2)+48)) and let linarith treat `Z2hat T`, `log (T+2)`, `T * Z2hat T`, `T * log(T+2)` as atoms. The goal RHS: 4*T*((2/3)*Z2hat T + 10.5*log(T+2)+48) + 40*1.3352. ring_nf normalization: LHS = 2*(T-10)*(2/3*Z + 10.5*L + 48) + 53.408... expand: 2*T*(2/3*Z) + ... linarith's preprocessing does ring_nf: atoms: T*Z, Z, T*L, L. LHS expands fine, RHS expands fine — all linear in these four atoms ✓ linarith closes given hK0-like facts: Z2hat T ≥ 0, log(T+2) ≥ 0, T ≥ 10. The needed inequality: -20*(2/3*Z+10.5*L+48) ≤ 0 — linear combo of Z, L, const ✓ linarith with hnn, hlognn. So: skip the K abbreviation; in the final step provide hnn : 0 ≤ Z2hat T and hlognn : 0 ≤ Real.log (T+2), then `linarith [htailInt, hmidInt, hnn, hlognn]` — wait but ttailInt's RHS (T-10)*(2*(...)) needs expansion to compare with 4*T*(...). linarith normalizes both sides via ring_nf — the products (T-10)*(...) expand into atom monomials T*Z etc ✓. And the goal inequality likewise. So one linarith with the nonneg haves should close: From hsplit-rw'd form: ∫(-T..-10) + (∫(-10..10) + ∫(10..T)) = [via hleft] ∫(10..T) + (∫(-10..10) + ∫(10..T)). ≤ (T-10)*K' + ((10-(-10))*(2*1.3352) + (T-10)*K') [htailInt, hmidInt] = 2*(T-10)*K' + 20*2.6704 ≤ 2*T*K' + 53.408 [K' ≥ 0] = 4*T*(stuff) + 40*1.3352 ✓ [2*K' = 4*stuff since K' := 2*stuff; 20*2.6704 = 53.408 = 40*1.3352 ✓] All linear after ring_nf ✓. So: ```lean _ ≤ 4 * T * ((2 / 3) * Z2hat T + 10.5 * Real.log (T + 2) + 48) + 40 * 1.3352 := by have hnn : (0:ℝ) ≤ Z2hat T := Z2hat_nonneg (by linarith) have hlognn : (0:ℝ) ≤ Real.log (T + 2) := Real.log_nonneg (by linarith) rw [hleft] linarith [htailInt, hmidInt, hnn, hlognn] ``` Hmm — but I need the calc step structure right. Let me restructure the final calc to: ```lean calc ‖VIntegral (Jlag n) (-5 / 2) (-T) T‖ = ‖∫ t in (-T)..T, Jlag n (((-5 / 2 : ℝ)) + (t : ℝ) * Complex.I)‖ := VIntegral_norm _ ≤ ∫ t in (-T)..T, ‖Jlag n (((-5 / 2 : ℝ)) + (t : ℝ) * Complex.I)‖ := intervalIntegral.norm_integral_le_integral_norm hTT _ = ∫ t in (-T)..(-10:ℝ), F + (∫ t in (-10:ℝ)..(10:ℝ), F + ∫ t in (10:ℝ)..T, F) := hsplit _ ≤ 4 * T * (...) + 40 * 1.3352 := by rw [hleft] have hnn ...; have hlognn ... linarith [htailInt, hmidInt, hnn, hlognn] ``` After rw [hleft]: goal ∫(10..T) + (∫(-10..10) + ∫(10..T)) ≤ RHS. linarith: substitute htailInt (∫(10..T) ≤ (T-10)*K') and hmidInt: atoms ∫(10..T), ∫(-10..10) ✓ linear in those atoms ✓, then the expansion ✓. linarith should close: goal = 2*∫(10..T) + ∫(-10..10) ≤ RHS'; with ∫(10..T) ≤ (T-10)*K', ∫(-10..10) ≤ 20*2.6704: 2*(T-10)*K' + 53.408 ≤ 4*T*stuff + 53.408 ⟺ 2*T*K' - 20*K' ≤ 2*T*K' ⟺ 0 ≤ 20*K' ⟸ Z, L ≥ 0 ✓. linarith handles ✓. One concern: `VIntegral_norm` — signature `{f : ℂ → ℂ} {x y₁ y₂ : ℝ} : ‖VIntegral f x y₁ y₂‖ = ‖∫ t in y₁..y₂, f ((x : ℝ) + (t : ℝ) * Complex.I)‖`. My VIntegral application: VIntegral (Jlag n) (-5/2) (-T) T — is -5/2 here ℝ? VIntegral f x y₁ y₂ with x : ℝ — in rectangleIntegral_Jlag_eq: VIntegral (Jlag n) (2 * Real.sqrt n) (-T) T — x = 2*√n : ℝ ✓. So x := (-5/2 : ℝ) ✓. The RHS point: `((x : ℝ) + (t : ℝ) * Complex.I)` — with x : ℝ the `(x : ℝ)` annotation is the identity (not a cast!) — hmm wait, in the theorem as written at line 173: `f ((x : ℝ) + (t : ℝ) * Complex.I)`. If x : ℝ, then (x : ℝ) is x, and `x + t * Complex.I` needs x coerced to ℂ. The elaborated term: `(↑x : ℂ) + (↑t : ℂ) * I`. My lambda `↑(-5/2) + ↑t * I` — same ✓. Hmm — one more check: VIntegral_norm's LHS `VIntegral f x y₁ y₂`. My goal's LHS: `VIntegral (Jlag n) (-5/2) (-T) T`. Match ✓. But wait — in the theorem statement I must write `VIntegral (Jlag n) (-5/2) (-T) T` with -5/2 elaborating as ℝ (since VIntegral expects ℝ for x). Fine. Also `intervalIntegral.norm_integral_le_integral_norm hTT : ‖∫ t in -T..T, Jlag n _‖ ≤ ∫ t in -T..T, ‖Jlag n _‖` ✓. Now — the `F` abbreviation: I'll avoid it and write the full integrand in hsplit/hleft. Verbose but fine. One more potential gotcha: in hmid, `rw [Jlag_norm_eq]` — the goal: ‖Jlag n (↑(-5/2)+↑t*I)‖ ≤ 2*1.3352. Jlag_norm_eq n s : ‖Jlag n s‖ = ‖kFun n s‖ * ‖logDeriv riemannZeta s‖. rw rewrites ‖Jlag n s‖ to the product ✓ then mul_le_mul. Also in hmid: hMZt's point form — after rwa [hcast] at h: h : ‖logDeriv riemannZeta (↑(-5/2) + ↑t*I)‖ ≤ 1.3352 ✓ matches. Wait — in hMZ's statement the point is `(-5 \ 2: ℂ) + (t : ℝ) * Complex.I`. I need to read MZeroProp's exact source to get the exact term. Let me read LiContour.lean lines 160-166.
All patterns confirmed. Let me check the exact source of `MZeroProp` (the grep showed an odd `\` character) and the `VIntegral` definition in Zeta23.
tool: Read
input
{
  "file_path": "D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiContour.lean",
  "offset": 155,
  "limit": 15
}
tool: Grep
input
{
  "pattern": "def VIntegral|def HIntegral|noncomputable def VIntegral|noncomputable def HIntegral",
  "path": "D:\\Python\\Quantum\\zeta-23-lean",
  "output_mode": "content",
  "-A": 4
}
#2263 assistant 2026-08-26 16:51:10.474 UTC · 4090→227 tok open ⤢
thinking (299 chars)
Now build-check the single file. Per handoff notes: elan at ~/.elan/bin, export PATH per shell. Run `lake env lean NguyenRH1/LiSideBounds.lean` from the package dir D:/Python/Quantum/research/lean/NguyenRH1. Single-file check should take ~1-3 min (imports already compiled). Use a generous timeout.
Now the single-file elaboration check.
tool: Bash
input
{
  "command": "export PATH=\"$HOME/.elan/bin:$PATH\" && cd \"D:/Python/Quantum/research/lean/NguyenRH1\" && time lake env lean NguyenRH1/LiSideBounds.lean 2>&1 | tail -40",
  "description": "Elaborate LiSideBounds.lean single-file",
  "timeout": 600000
}
#2264 assistant 2026-08-26 16:55:29.613 UTC · 1004→4684 tok open ⤢
thinking (13270 chars)
Several errors: 1. **Line 245**: `have hA := (kFun_differentiableAt hs1).continuousAt` — "don't know how to synthesize implicit argument `n`". kFun_differentiableAt {n : ℕ} {s : ℂ} (hs : s ≠ 1) — the implicit n can't be inferred from hs1 alone. Fix: pin n: `(kFun_differentiableAt (n := n) hs1).continuousAt`. Also the first error shown above it (cut off) — `⊢ ℂ → ℂ` — probably the `show ContinuousAt (fun s => ...)` mismatch? Actually the first error context shows `⊢ ℂ → ℂ` — that might be from `show ContinuousAt (fun s => kFun n s * (-logDeriv riemannZeta s)) _` failing? The visible tail shows contexts. Let me see the full output. The tail was cut at 40 lines but the first error is partially shown. Let me address what's visible: 2. **Line 255**: `Continuous.continuousAt hpath` — `hpath.continuousAt` has type `ContinuousAt (fun u => ...) ?m` — the point is a metavariable because `Continuous.continuousAt (h : Continuous f) : ContinuousAt f x` with x implicit — when I write `hpath.continuousAt` the point x is undetermined! Fix: `hpath.continuousAt (x := t)` or `(hJcont t).comp (hpath.continuousAt (x := t))`. Wait the error says expected `ContinuousAt (HAdd.hAdd ↑(-5/2)) (↑t * I)` — hmm, that's the composition target being weird. The expected type of the second arg of comp: ContinuousAt f t where f is the path function. `ContinuousAt (fun u => ↑(-5/2) + ↑u * I) t`. The error shows a partial application `HAdd.hAdd ↑(-5/2)` — this is the elaborator struggling. Fix: pin the point: `hpath.continuousAt (x := t)`. Hmm, Continuous.continuousAt signature: `theorem Continuous.continuousAt (h : Continuous f) : ContinuousAt f x` — x is a section variable implicit: `{x : α}`. So `hpath.continuousAt (x := t)` should pin it. Actually better: `(hJcont t).comp (f := fun u : ℝ => ...) (hpath.continuousAt)`? ContinuousAt.comp (hg : ContinuousAt g (f x)) (hf : ContinuousAt f x) — f and x implicit. Passing hf := hpath.continuousAt with x free — Lean should unify x := t from hg's type (hJcont t : ContinuousAt (Jlag n) (path t)). Hmm — hg : ContinuousAt g (f x) with g := Jlag n — expected ContinuousAt (Jlag n) (?f ?x) — hJcont t : ContinuousAt (Jlag n) (↑(-5/2)+↑t*I) — unify ?f ?x := ↑(-5/2)+↑t*I — HIGHER-ORDER unification (?f ?x) — ambiguous! That's the real problem: the elaborator can't figure out how to split ↑(-5/2)+↑t*I into f x. It picked f := HAdd.hAdd ↑(-5/2) (partial!) and x := ↑t*I. Fix: make the composition explicit by giving the goal shape. Standard robust pattern: ```lean have hcontJ : Continuous fun t : ℝ => Jlag n ((-5 / 2 : ℝ) + (t : ℝ) * Complex.I) := by rw [continuous_iff_continuousAt] intro t have hcomp : ContinuousAt (Jlag n ∘ fun u : ℝ => ((-5 / 2 : ℝ) + (u : ℝ) * Complex.I)) t := (hJcont t).comp (hpath.continuousAt) exact hcomp ``` With hcomp's type ascription, the elaborator knows f (the path) and x := t: `(hJcont t).comp (hpath.continuousAt)` — expected type ContinuousAt (Jlag n ∘ path) t fixes g := Jlag n, f := path, x := t — then hg := hJcont t : ContinuousAt (Jlag n) (path t) ✓ unifies (f x ≡ path t defeq), hf := hpath.continuousAt : ContinuousAt path t ✓. Then `exact hcomp` — goal `ContinuousAt (fun t => Jlag n (path t)) t` vs hcomp's `ContinuousAt (Jlag n ∘ path) t` — defeq (∘ unfolds) ✓. 3. **Line 257**: `hcontJ.norm'` — Continuous.norm' ?m — "has type Continuous[?m.390, _] fun x => @norm ... (?m.391 x) but expected ..." — the issue: Continuous.norm' has implicit f and the dot-notation should fix f := the lambda... The mismatch looks like an elaboration-order issue: `Continuous.norm'` might have signature `theorem Continuous.norm' {f : α → β} (hf : Continuous f) : Continuous fun x => ‖f x‖` — dot notation hcontJ.norm' should work... The error shows `?m.391 x` unapplied-ish. Hmm — maybe the problem is that Continuous.norm' is stated with explicit `(hf : Continuous f)` but the NORM's type class is `SeminormedGroup.toNorm` and the expected `instNorm` on ℂ mismatch? The error: `Continuous.norm' ?m.392 has type Continuous[?m.390, _] fun x => @norm ?m.388 SeminormedGroup.toNorm (?m.391 x) but is expected to have type Continuous fun t => @norm ℂ instNorm (Jlag n ...)`. So the elaborator couldn't unify `@norm ?m.388 SeminormedGroup.toNorm` with `@norm ℂ instNorm` — the Norm INSTANCE differs! `SeminormedGroup.toNorm` vs `instNorm` on ℂ. Hmm — that suggests Continuous.norm' is for SeminormedGroups with a specific instance, and ℂ's norm instance is a different (though defeq) instance. With `?m.388` a metavariable, type-class synthesis should find... the unification is `@norm ?α inst1 (?f x)` vs `@norm ℂ instNorm (Jlag...)` — needs ?α := ℂ and inst1 ≡ instNorm — instance defeq should hold (instNorm on ℂ unfolds to the SeminormedGroup one? probably defeq). The problem may be that the elaborator didn't propagate f from the expected type — since I wrote `have hcontG : Continuous fun t => ‖Jlag n _‖ := hcontJ.norm'`, the expected type is known — unification `fun x => ‖?f x‖` vs `fun t => ‖Jlag n (↑(-5/2)+↑t*I)‖` — pattern ?f x vs Jlag n (...) — solvable ?f := fun t => Jlag n (↑(-5/2)+↑t*I) ✓... but the instance mismatch blocked it first? Simplest robust fix: use `continuous_norm.comp hcontJ`: ```lean have hcontG : Continuous fun t : ℝ => ‖Jlag n ((-5 / 2 : ℝ) + (t : ℝ) * Complex.I)‖ := continuous_norm.comp hcontJ ``` continuous_norm : Continuous fun x : E => ‖x‖. `.comp hcontJ` : Continuous ((fun x => ‖x‖) ∘ (fun t => Jlag n _)) — defeq to the goal ✓. This avoids the instance puzzle (continuous_norm is at the right instance). Let me check continuous_norm exists... `continuous_norm : Continuous fun a => ‖a‖` — standard, in Mathlib.Topology.MetricSpace... or Analysis.Normed.Group.Basic — should exist. Alternatively `hcontJ.norm'`'s problem might just be the implicit-?f inference order; continuous_norm.comp is more explicit. I'll use that. 4. The first (partially shown) error with `⊢ ℂ → ℂ` — probably from the `show ContinuousAt (fun s => kFun n s * (-logDeriv riemannZeta s)) _` line — or from `hA` failing (the kFun n issue). The `⊢ ℂ → ℂ` context at the top with hJcont's t hs1 — the goal `ℂ → ℂ`?? That's weird — it looks like the error is at the `show` — "failed to infer type"? The first visible context ends with `⊢ ℂ → ℂ` which looks like a type-being-inferred for something. Actually — this might be the error for `hA` where `kFun_differentiableAt ?n ...` — no wait, error #2 explicitly covers kFun_differentiableAt's n. The `⊢ ℂ → ℂ` with the SAME context as error 2 (line 245) — it's probably the same location: `have hA := (kFun_differentiableAt hs1).continuousAt` — Lean tries to infer the have's type, kFun_differentiableAt has {n} {s} implicit + hs — s inferred from hs1 ✓ but n free — error 2. Error 1 (⊢ ℂ → ℂ) might be... the `show` command: `show ContinuousAt (fun s => kFun n s * (-logDeriv riemannZeta s)) _` — the wildcard `_` for the point — maybe `show` couldn't unify `ContinuousAt (Jlag n) s₀` with `ContinuousAt (fun s => ...) _` — the wildcard point is ambiguous? show with `_` — Lean unifies the TYPE: ContinuousAt F x — F := Jlag n vs fun s => kFun n s * (-logDeriv riemannZeta s) — needs defeq (unfold Jlag + logDeriv) — should work... and x := s₀ vs `_` — the wildcard creates a metavariable, unified with s₀ ✓... Hmm. Actually the first error block shows context with hs1 present and `⊢ ℂ → ℂ` — this is likely at line 245 region: `(kFun_differentiableAt hs1).continuousAt` — DifferentiableAt.continuousAt : DifferentiableAt ℂ f x → ContinuousAt f x — f := kFun ?n — hmm. Let me just fix the identified issues and re-run; the error cascade will clarify: Fixes: - hA: `(kFun_differentiableAt (n := n) hs1).continuousAt` - hcontJ: the hcomp-ascription pattern. - hcontG: `continuous_norm.comp hcontJ`. Also check remaining errors beyond line 257 — output was truncated at 40 lines (I did tail -40). After fixing these three, re-run and see the rest. Also for the `show` line — alternative robust: skip the show, prove hJcont differently: ```lean have hJ : ContinuousAt (fun s => kFun n s * (-logDeriv riemannZeta s)) ((-5 / 2 : ℝ) + (t : ℝ) * Complex.I) := hA.mul hB.neg exact hJ ``` with the `have hJ` type-ascribed — the goal after ascription check: `ContinuousAt (Jlag n) point` — `exact hJ` needs defeq Jlag n ≡ fun s => kFun n s * (-logDeriv ζ s) — same defeq test as show. If the defeq fails, I'll need to unfold: `unfold Jlag` first? In Jlag_norm_eq they did `unfold Jlag logDeriv`. For the ContinuousAt goal: `rw [show Jlag n = fun s => kFun n s * (-(deriv riemannZeta s / riemannZeta s)) from rfl]` — Jlag n unfolds by rfl to that lambda (def of Jlag). Then goal: ContinuousAt (fun s => kFun n s * (-(deriv ζ s/ζ s))) point. And hB : ContinuousAt (logDeriv ζ) point where logDeriv ζ = fun x => deriv ζ x/ζ x definitionally — for hB.neg: ContinuousAt (fun s => -logDeriv ζ s) — hmm, ContinuousAt.neg (hf : ContinuousAt f x) : ContinuousAt (fun x => -f x) x — so hB.neg : ContinuousAt (fun s => -logDeriv ζ s) point = (fun s => -(deriv ζ s/ζ s)) after unfolding logDeriv — defeq. hA.mul hB.neg : ContinuousAt (fun s => kFun n s * (fun s => -logDeriv ζ s) s)?? ContinuousAt.mul (hf : ContinuousAt f x) (hg : ContinuousAt g x) : ContinuousAt (fun x => f x * g x) x ✓ — so hA.mul hB.neg : ContinuousAt (fun s => kFun n s * (-logDeriv ζ s)) point ✓. To match the goal `ContinuousAt (Jlag n) point` — Jlag n ≡ fun s => kFun n s * (-(deriv ζ s/ζ s)) and -(logDeriv ζ s) ≡ -(deriv ζ s/ζ s) — full defeq chain ✓ exact should work. The error at the `show` line might actually be that `show` FAILED the defeq test! The `⊢ ℂ → ℂ` suggests show tried to prove the defeq of functions and displayed `ℂ → ℂ`. Hmm — "show" unifies the given type with the goal up to defeq — if Jlag n's unfolding requires unfolding logDeriv (a def with pattern matching? logDeriv is a plain def: `def logDeriv (f : 𝕜 → 𝕜') := fun x => deriv f x / f x`... wait actually logDeriv might be defined differently: `def logDeriv (f : 𝕜 → 𝕜) (x : 𝕜) : 𝕜 := deriv f x / f x`? Either way reducible-defeq ✓. Hmm, Jlag def: `def Jlag (n : ℕ) (s : ℂ) : ℂ := kFun n s * (-(deriv riemannZeta s / riemannZeta s))`. `Jlag n` (partially applied) vs `fun s => kFun n s * (-(logDeriv riemannZeta s))`: whnf(Jlag n) = fun s => kFun n s * (-(deriv ζ s/ζ s)). Compare bodies: kFun n s * (-(deriv ζ s/ζ s)) vs kFun n s * (-(logDeriv ζ s)): need -(deriv ζ s/ζ s) ≡ -(logDeriv ζ s): whnf(logDeriv ζ) → fun x => deriv ζ x/ζ x; applied to s: deriv ζ s/ζ s ✓ defeq ✓. So `show` SHOULD work... but maybe the elaborator's defeq for `show` uses default transparency and it's fine. Actually — wait. Let me re-read the first error. The output began mid-context (tail -40 cut it). The first visible part: ``` h10T : 10 ≤ T hT0 : 0 ≤ T hTT : -T ≤ T t : ℝ hs1 : ↑(-5 / 2) + ↑t * I ≠ 1 ⊢ ℂ → ℂ ``` This is the END of some error message — the goal `⊢ ℂ → ℂ`. This looks like "don't know how to synthesize implicit argument" for something of type ℂ → ℂ?? The next error at 245:16 is `don't know how to synthesize implicit argument n` for kFun_differentiableAt. So the first error is probably at line ~245 too or earlier — `⊢ ℂ → ℂ` could be from... hmm, `ContinuousAt.comp` at 255? No — context includes hs1 which only exists inside hJcont (before line 245-253). The `show` at ~line 251-252. So error 1 is probably the `show` line: message might be "type mismatch: ... has type ... but is expected to have type ... ⊢ ℂ → ℂ"?? A goal of `ℂ → ℂ`?? That's bizarre. OR it's from `hA.mul hB.neg` with hA broken (kFun ?n) — cascading. Let me not over-analyze: fix the three identified issues and re-run; error set will be clearer. Actually — one more possibility for `⊢ ℂ → ℂ`: the `show ContinuousAt (fun s => ...)` — Lean 4's `show` needs the full type; the trailing `_` for the point — `show ContinuousAt F _` — creates type `ContinuousAt F ?x` and unifies with goal `ContinuousAt (Jlag n) point` — F := fun s => kFun n s * (-logDeriv ζ s) must be defeq to Jlag n ✓, ?x := point ✓. If F's elaboration failed (e.g., `-logDeriv riemannZeta s` parses as `(-logDeriv riemannZeta) s`? NO — function application binds tighter than unary minus... `-logDeriv riemannZeta s` parses as `-(logDeriv riemannZeta s)`? Application has highest precedence: `logDeriv riemannZeta s` groups first, then negation ✓. Hmm — but `-logDeriv riemannZeta s` could parse as `(-(logDeriv riemannZeta)) s`? Negation of a function then applied — that would need a Neg (ℂ → ℂ) instance — which EXISTS (Pi.instNeg)! So both parses elaborate! Which does Lean choose? Precedence: function application is left-assoc and highest — `-f x` = `-(f x)` in Lean 4 (negation is prefix with LOWER precedence than application). I believe `-f x = -(f x)` ✓ standard. But in the `show` I wrote `fun s => kFun n s * (-logDeriv riemannZeta s)` — with explicit parens `(-logDeriv riemannZeta s)` — inside parens: `-logDeriv riemannZeta s` — still parses as -(logDeriv ζ s) ✓. The `⊢ ℂ → ℂ` mystery — could be "don't know how to synthesize implicit argument" where an implicit of type ℂ → ℂ exists — e.g., in `ContinuousAt.mul`... hA is broken (kFun ?n has metavariable n) — so hA.mul fails with the unapplied function-slot. Cascade. Fix hA first. Let me apply the three fixes.
Three elaboration issues to fix: pinned `n` for `kFun_differentiableAt`, the `comp` composition needs an ascribed form, and `Continuous.norm'` has an instance mismatch — use `continuous_norm.comp` instead.
tool: Edit
input
{
  "replace_all": false,
  "file_path": "D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiSideBounds.lean",
  "old_string": "    have hA := (kFun_differentiableAt hs1).continuousAt\n    have hB := continuousAt_logDeriv_zeta hs1 (zeta_ne_zero_left_vertical t)\n    show ContinuousAt (fun s => kFun n s * (-logDeriv riemannZeta s)) _\n    exact hA.mul hB.neg\n  have hpath : Continuous fun u : \u211d => ((-5 / 2 : \u211d) + (u : \u211d) * Complex.I) := by\n    fun_prop\n  have hcontJ : Continuous fun t : \u211d =>\n      Jlag n ((-5 / 2 : \u211d) + (t : \u211d) * Complex.I) := by\n    rw [continuous_iff_continuousAt]\n    intro t\n    exact (hJcont t).comp hpath.continuousAt\n  have hcontG : Continuous fun t : \u211d =>\n      \u2016Jlag n ((-5 / 2 : \u211d) + (t : \u211d) * Complex.I)\u2016 := hcontJ.norm'",
  "new_string": "    have hA := (kFun_differentiableAt (n := n) hs1).continuousAt\n    have hB := continuousAt_logDeriv_zeta hs1 (zeta_ne_zero_left_vertical t)\n    show ContinuousAt (fun s => kFun n s * (-logDeriv riemannZeta s)) _\n    exact hA.mul hB.neg\n  have hpath : Continuous fun u : \u211d => ((-5 / 2 : \u211d) + (u : \u211d) * Complex.I) := by\n    fun_prop\n  have hcontJ : Continuous fun t : \u211d =>\n      Jlag n ((-5 / 2 : \u211d) + (t : \u211d) * Complex.I) := by\n    rw [continuous_iff_continuousAt]\n    intro t\n    have hcomp : ContinuousAt\n        (Jlag n \u2218 fun u : \u211d => ((-5 / 2 : \u211d) + (u : \u211d) * Complex.I)) t :=\n      (hJcont t).comp hpath.continuousAt\n    exact hcomp\n  have hcontG : Continuous fun t : \u211d =>\n      \u2016Jlag n ((-5 / 2 : \u211d) + (t : \u211d) * Complex.I)\u2016 :=\n    continuous_norm.comp hcontJ"
}
#2265 assistant 2026-08-26 16:59:06.622 UTC · 1719→4013 tok open ⤢
thinking (11190 chars)
Great progress — only TWO errors remain, both about the same thing: **Error 1 (hsplit)**: The rewrites succeeded but left the goal as `LHS' = RHS` where LHS' = `∫(-T..-10) + (∫(-10..10) + ∫(10..T))` and RHS is the stated form — but note the ASSOCIATIVITY difference: LHS' shows `∫(-T..-10) + (∫(-10..10) + ∫(10..T))` (pretty-printed with parens) vs RHS `∫(-T..-10) + ∫(-10..10) + ∫(10..T)` (left-assoc). Wait, look carefully: Goal after rw: ``` ⊢ (∫ -T..-10) + ((∫ -10..10) + ∫ 10..T) = ∫ -T..-10 + ∫ -10..10 + ∫ 10..T ``` Hmm — actually looking at the error message for line 392 (that's the SECOND error — the rw [hleft] one). The first error context shows the goal: ``` ⊢ (∫ (x : ℝ) in -T..-10, ...) + ((∫ (x : ℝ) in -10..10, ...) + ∫ (x : ℝ) in 10..T, ...) = ∫ (t : ℝ) in -T..-10, ... + ∫ (t : ℝ) in -10..10, ... + ∫ (t : ℝ) in 10..T, ... ``` The LHS has `(A) + ((B) + (C))` (right-nested from my rewrites) and the RHS has `A + B + C` (left-nested from the hsplit statement I wrote). Wait — I wrote hsplit's RHS as `A + (B + C)`: ``` ∫ t in (-T)..(-10 : ℝ), ... + (∫ t in (-10 : ℝ)..(10 : ℝ), ... + ∫ t in (10 : ℝ)..T, ...) := by ``` That's `A + (B + C)` — right-nested ✓ same as LHS'. Hmm, but the error at 392 says rw [hleft] "did not find the pattern ∫ -T..-10, F in the target" — and displays the target as containing `∫ -T..-10, F + ...`. Wait the target DOES contain it! Look at error 2 (line 392, the rw [hleft]): "Did not find an occurrence of the pattern ∫ (t : ℝ) in -T..-10, ‖Jlag n (↑(-5/2) + ↑t * I)‖ in the target expression" — but the target displayed literally begins with `∫ (t : ℝ) in -T..-10, ‖Jlag n (↑(-5 / 2) + ↑t * I)‖ + ...`. The pattern vs the occurrence differ how? hleft's LHS is `∫ (t : ℝ) in -T..-10, ‖Jlag n (↑(-5/2) + ↑t * I)‖`. The target contains `∫ (t : ℝ) in -T..-10, ‖Jlag n (↑(-5/2) + ↑t * I)‖`. These LOOK identical... but the bound variable NAME differs (t vs t — same). Hmm. OH WAIT. I see it. The pretty-printed target at error 2 shows the goal AFTER the first error's context... Let me re-read. Error 1's context (top of output) shows goal: ``` ⊢ (∫ (x : ℝ) in -T..-10, ‖Jlag n (↑(-5 / 2) + ↑x * I)‖) + ((∫ (x : ℝ) in -10..10, ...) + ∫ (x : ℝ) in 10..T, ...) = ∫ (t : ℝ) in -T..-10, ‖Jlag n (↑(-5 / 2) + ↑t * I)‖ + ∫ (t : ℝ) in -10..10, ... + ∫ (t : ℝ) in 10..T, ... ``` Bound var `x` on the LHS (from the integral_add_adjacent_intervals rewrites — the lemma's bound var is x) vs bound var `t` on the RHS (my hsplit statement uses t). So after my two rw's, the goal is `A_x + (B_x + C_x) = A_t + (B_t + C_t)` — alpha-equivalent but rw's final rfl-closure requires SYNTACTIC/rfl-defeq equality — alpha-equivalence IS defeq and rw closes with rfl which handles alpha... Hmm, actually rw closes the goal with `rfl` automatically after rewriting — the rfl check uses defeq (with alpha ✓). But the error suggests the goal was NOT closed — meaning the rw succeeded but the result wasn't closed by rfl? Wait — the first error message's actual error text is cut off (the output starts mid-context). The visible first block ends with `⊢ ... = ...` and then the SECOND error starts at 392. So the first error is at some line < 392 — likely the hsplit rw (line ~385). The error message itself was truncated. What could it be? Possibly "motive is not type correct" or the rw failed to close. Hmm. Actually — maybe the issue: the two rewrites produced a goal that's NOT syntactically the RHS because of the NESTING: my first rewrite turned `∫(-T..T)` into `∫(-T..-10) + ∫(-10..T)`; the second turned `∫(-10..T)` into `∫(-10..10) + ∫(10..T)`. Result: `∫(-T..-10) + (∫(-10..10) + ∫(10..T))`. The hsplit RHS (as elaborated): I wrote `A + (B + C)` ✓ same. So the goal after rewrites is `A_x + (B_x + C_x) = A_t + (B_t + C_t)` — rw tries rfl: alpha-equiv — rfl via defeq — alpha-renaming IS defeq ✓ should close. Hmm wait, maybe the FIRST error is elsewhere. Let me count: the output shows the tail. The errors: one at 392:12 (rw [hleft] failure). The first error — its message text got cut. Its context shows the hsplit-shaped goal. Then error 2 at 392. For error 2: rw [hleft] failed to find the pattern in the goal. The goal (from hsplit, now in context as hsplit : ...) — wait NO. Error 2's context shows hsplit as a HYPOTHESIS (successfully proved!) and the GOAL is: ``` ⊢ ∫ (t : ℝ) in -T..-10, F + ∫ (t : ℝ) in -10..10, F + ∫ (t : ℝ) in 10..T, F ≤ 4*T*(...) + 40*1.3352 ``` Hmm interesting — so the goal at error 2 has `∫ -T..-10 + ∫ -10..10 + ∫ 10..T` (left-nested!). But hsplit's RHS was right-nested `A + (B + C)`... The calc step `_ = A + (B + C) := hsplit` — wait, my calc: ``` _ = ∫ t in (-T)..(-10 : ℝ), F + (∫ t in (-10 : ℝ)..(10 : ℝ), F + ∫ t in (10 : ℝ)..T, F) := hsplit ``` and then ``` _ ≤ 4 * T * ... := by rw [hleft]; ... ``` The goal at error 2 shows `A + B + C ≤ ...` — LEFT-nested. But I wrote right-nested `A + (B + C)`. Hmm — unless the elaborator re-associated? No. Look again at error 2's target display: ``` ∫ (t : ℝ) in -T..-10, ‖Jlag n (↑(-5 / 2) + ↑t * I)‖ + ∫ (t : ℝ) in -10..10, ‖Jlag n (↑(-5 / 2) + ↑t * I)‖ + ∫ (t : ℝ) in 10..T, ‖Jlag n (↑(-5 / 2) + ↑t * I)‖ ≤ ``` The pretty-printer indents under the first +... The indentation suggests `A + (B + C)` printed with the (B+C) indented — "∫ -10..10, F + ∫ 10..T, F" on one line at deeper indent — yes! This IS right-nested: `A + (B + C)` — the pretty printer elides parens. So the target DOES contain `∫ -T..-10, F` as a subterm (the A). And the pattern (from hleft's LHS): `∫ (t : ℝ) in -T..-10, ‖Jlag n (↑(-5/2) + ↑t * I)‖`. The target's A: `∫ (t : ℝ) in -T..-10, ‖Jlag n (↑(-5/2) + ↑t * I)‖`. IDENTICAL?! Unless... the `-T` in hleft vs the `-T` in the goal differ? Both `-T`. Hmm, or `(-10 : ℝ)` vs `-10`: hleft's LHS I wrote `(-T)..(-10 : ℝ)` — elaborates `Neg.neg (10 : ℝ)` vs the goal's `-10` — same. WAIT. Maybe the issue: `rw [hleft]` — hleft's LHS has bound var name `t`, the target's integral also `t` — irrelevant for matching. Hmm, let me look at error 1 more carefully — maybe error 1 is the hsplit rw FAILING to fully close, leaving a weird goal, and hsplit got a different shape. But hsplit's displayed type in error-2 context: ``` hsplit : ∫ (t : ℝ) in -T..T, F = ∫ (t : ℝ) in -T..-10, F + ∫ (t : ℝ) in -10..10, F + ∫ (t : ℝ) in 10..T, F ``` Hmm — displayed left-nested-looking but probably right-nested with parens elided. OK here's another thought — the error at 392: rw [hleft] — the pattern is hleft's LHS: `∫ t in -T..-10, F`. The target: `A + (B + C) ≤ RHS` where A = `∫ t in -T..-10, F`. These match... UNLESS the -T in A vs hleft differ in some hidden way — e.g., in hleft, `-T` came from `(-T)` in MY statement ✓ same T. Hold on — actually, look VERY carefully at error 2's pattern vs target: Pattern: `∫ (t : ℝ) in -T..-10, ‖Jlag n (↑(-5 / 2) + ↑t * I)‖` Target A: `∫ (t : ℝ) in -T..-10, ‖Jlag n (↑(-5 / 2) + ↑t * I)‖` Identical as printed. But maybe the difference is in the BOUND VARIABLE being t vs x — no, both t. Hmm, hmm. What about `↑(-t)` — not relevant. Let me reconsider: maybe error 1 (truncated) is at line ~385 (hsplit's rw) — and it's the SAME "did not find pattern" error for the SECOND adjacent-intervals rewrite: pattern `∫ x in -10..T, F` — after the first rewrite, the goal contains `∫ x in -10..T, F x`? The first rewrite replaced `∫ x in -T..T, F` with `(∫ x in -T..-10, F) + ∫ x in -10..T, F` ✓ contains it. Hmm. Actually wait — I see it now! Look at the FIRST error context goal: ``` ⊢ (∫ (x : ℝ) in -T..-10, ‖Jlag n (↑(-5 / 2) + ↑x * I)‖) + ((∫ (x : ℝ) in -10..10, ‖Jlag n (↑(-5 / 2) + ↑x * I)‖) + ∫ (x : ℝ) in 10..T, ‖Jlag n (↑(-5 / 2) + ↑x * I)‖) = ∫ (t : ℝ) in -T..-10, ... ``` This goal is an `=` goal — this is hsplit's proof state after the two rewrites. The error message (cut off) is probably "tactic 'rewrite' failed, equality or iff proof expected"? No... Or maybe the rw SUCCEEDED and left this goal which rw's auto-rfl DIDN'T close because... the LHS has bound var x, RHS bound var t — alpha — rfl should handle. UNLESS the RHS is genuinely different: RHS = hsplit's stated RHS = `A_t + (B_t + C_t)` where my statement's integrals use `(t : ℝ)` as binder. LHS (post-rw) uses x. Alpha-equivalent ✓. Hmm, so what's the error? Maybe the error IS that the goal remains and `rw` reported failure to close? rw automatically tries rfl after rewriting — if it fails, no error, the goal just remains (rw leaves it as a new goal — not an error!). An error only if NO goals get closed... no — rw never errors for leaving goals; it errors when the pattern isn't found. So error 1 must be a pattern-not-found too! Pattern: `∫ x in -T..T, F` for the first rewrite — the goal at hsplit-proof-start is `∫ t in -T..T, F = RHS`. The pattern from the reversed lemma: `∫ x in -T..T, F x` — with bound var x vs goal's t — rw matches up to alpha ✓ fine. Hmm — the lemma instance: `intervalIntegral.integral_add_adjacent_intervals (hI (-T) (-10)) (hI (-10) T)` — hI a b : IntervalIntegrable F' volume a b where F' = `(fun t => ‖Jlag n (↑(-5/2)+↑t*I)‖)`. The lemma: `(∫ x in a..b, ?f x) + ∫ x in b..c, ?f x = ∫ x in a..c, ?f x` — with ?f unified from hI's type: ?f := F'. Instance: `(∫ x in -T..-10, F' x) + ∫ x in -10..T, F' x = ∫ x in -T..T, F' x`. Note: `F' x` is a BETA-REDEX: `(fun t => ‖Jlag n (↑(-5/2)+↑t*I)‖) x`. When elaborating the lemma instance, does Lean beta-reduce? The lemma statement has `f x` with f a variable — instantiated f := F' gives `(fun t => ...) x` — Lean's elaborator INSTANTIATES but typically beta-reduces metavariable assignments... the resulting rewrite rule's RHS: `∫ x in -T..T, (fun t => ‖...‖) x` — vs the goal's `∫ t in -T..T, ‖...‖` — pattern matching for rw: the rewrite pattern (lemma RHS) vs goal subterm — rw uses keyed matching + defeq — should handle beta ✓. I'm going in circles. Let me look at the FULL error output, not just tail -50. Re-run capturing more, or better — since elaboration is incremental, let me view errors from the top. Actually simpler: the errors are only 2 (the output showed 2 error blocks + contexts). Error 1's message got cut off the top. Let me re-run with head instead. Actually wait — I realize both errors might be the SAME root cause and there's a simpler explanation. Look at error 2 again: "Did not find an occurrence of the pattern `∫ (t : ℝ) in -T..-10, ‖Jlag n (↑(-5 / 2) + ↑t * I)‖`" Target: contains `∫ (t : ℝ) in -T..-10, ‖Jlag n (↑(-5 / 2) + ↑t * I)‖` as the first summand... Hmm — BUT WAIT: in the target, the first summand's integral might have bound variable `x` not `t`! The display shows: ``` ⊢ ∫ (t : ℝ) in -T..-10, ‖Jlag n (↑(-5 / 2) + ↑t * I)‖ + ``` bound var t ✓. Hmm no. OK — different idea: maybe the issue is the `-10` in hleft's statement vs `-10` in hsplit's — hleft: `∫ t in (-T)..(-10 : ℝ)` — the `-10` here: `Neg.neg (10 : ℝ)`. hsplit: same ✓. Hmm, what about `-T` — in hleft the endpoint is `-T` where T is the section... in error 2's target the endpoints are `-T..-10` ✓. You know what — let me just look at the FULL first error. Run again with head -60.
Two errors, the first one's message got cut off. Let me see the full output from the top.
tool: Bash
input
{
  "command": "export PATH=\"$HOME/.elan/bin:$PATH\" && cd \"D:/Python/Quantum/research/lean/NguyenRH1\" && lake env lean NguyenRH1/LiSideBounds.lean 2>&1 | head -60",
  "description": "Show first errors of LiSideBounds elaboration",
  "timeout": 600000
}
#2266 assistant 2026-08-26 17:08:36.210 UTC · 5603→12695 tok open ⤢
thinking (36872 chars)
Now I can see all the errors: 1. **Line 257** (hcontJ/hcomp): `ContinuousAt.comp (hJcont t) (Continuous.continuousAt hpath)` — STILL the HO unification issue: expected `ContinuousAt (HAdd.hAdd ↑(-5/2)) (↑t * I)`. The elaborator decomposed `↑(-5/2) + ↑t * I` as `(HAdd.hAdd ↑(-5/2)) (↑t*I)` with f := `HAdd.hAdd ↑(-5/2)` (partial application!) and x := `↑t * I`. My ascription `ContinuousAt (Jlag n ∘ path) t` — the ∘ form: `Jlag n ∘ path` applied... the problem: ContinuousAt.comp (hg : ContinuousAt g (f x)) (hf : ContinuousAt f x) — from hg := hJcont t : ContinuousAt (Jlag n) (↑(-5/2)+↑t*I): unify `?f ?x` with `↑(-5/2)+↑t*I` — the elaborator picked ?f := HAdd.hAdd ↑(-5/2), ?x := ↑t*I (first-order approximation) instead of ?f := path, ?x := t. My ascription of hcomp's TYPE (ContinuousAt (Jlag n ∘ path) t) doesn't propagate INTO the arguments' unification because `exact hcomp` — wait, I wrote: ```lean have hcomp : ContinuousAt (Jlag n ∘ fun u : ℝ => ((-5 / 2 : ℝ) + (u : ℝ) * Complex.I)) t := (hJcont t).comp hpath.continuousAt ``` The ascription IS there. But the elaborator still assigns ?f ?x from hJcont t's type FIRST (elaborating the term `(hJcont t).comp hpath.continuousAt` against the ascribed type). The expected type gives `ContinuousAt (Jlag n ∘ path) t` — unifying with `ContinuousAt (?g ∘ ?f) ?x` — hmm, ContinuousAt.comp's result type: `ContinuousAt (g ∘ f) x`. Expected: `ContinuousAt (Jlag n ∘ path) t` — so g ∘ f ≡ Jlag n ∘ path, x := t. Elaborator unifies `?g ∘ ?f` with `Jlag n ∘ path` — HO again! It probably can't split. So it then uses hJcont t to determine g := Jlag n and decompose the point... yielding the bad split. Robust fix: use `ContinuousAt.comp` with EXPLICIT named arguments. Signature: `nonrec theorem ContinuousAt.comp {g : Y → Z} (hg : ContinuousAt g (f x)) (hf : ContinuousAt f x) : ContinuousAt (g ∘ f) x` — the f and x are section variables (implicit). So: `(hJcont t).comp (f := fun u : ℝ => ((-5/2 : ℝ) + (u : ℝ) * Complex.I)) (x := t) hpath.continuousAt`. Wait — with dot notation, hg is the self-arg. Named implicits f, x: ```lean exact ContinuousAt.comp (g := Jlag n) (f := fun u : ℝ => ((-5 / 2 : ℝ) + (u : ℝ) * Complex.I)) (x := t) (hJcont t) hpath.continuousAt ``` Hmm — but are f, x implicit binders named f and x? From the source: `theorem ContinuousAt.comp {g : Y → Z} (hg : ContinuousAt g (f x)) (hf : ContinuousAt f x)` — f and x come from the section: likely `variable {f : X → Y} {x : X} {g : Y → Z}`. Names probably f and x. Risky. Alternative robust: `Continuous.tendsto`/`Continuous.comp'`... Or use `Continuous.comp`: ```lean exact hcontJ_global... ``` Actually simplest: avoid ContinuousAt entirely — prove hcontJ via Continuous.comp': ```lean have hcontJ : Continuous fun t : ℝ => Jlag n ((-5 / 2 : ℝ) + (t : ℝ) * Complex.I) := by apply Continuous.comp' ... ``` Hmm — need `Continuous (Jlag n)` globally — NOT true (Jlag has singularities). So global comp fails; need the ContinuousAt version. Alternative: `ContinuousAt.comp_of_eq`? Or prove via `ContinuousAt.comp` but give the full type ascription on the TERM: ```lean have hf : ContinuousAt (fun u : ℝ => ((-5 / 2 : ℝ) + (u : ℝ) * Complex.I)) t := hpath.continuousAt exact (hJcont t).comp hf ``` Now hf's type is pinned: ContinuousAt path t. Then (hJcont t).comp hf — g := Jlag n, f := path (from hf's type ✓ determined!), x := t ✓. hg := hJcont t : ContinuousAt (Jlag n) (↑(-5/2)+↑t*I) — expected ContinuousAt (Jlag n) (path t) — path t ≡ ↑(-5/2)+↑t*I by beta ✓ defeq ✓. This should work! The earlier failure was hpath.continuousAt with the UNPINNED point ?m.425. Pin via the have. 2. **Line 293:39** "No goals to be solved" — line 293 is in hmid: `rwa [hcast] at h` — hmm no. Line 293 col 39 — let me count lines... 293 — probably the `by rw [hre]; norm_num` for hk? "No goals" means a tactic ran after the goal was closed: `by rw [hre]; norm_num` — rw [hre] CLOSED the goal (hre : re = -5/2, goal re ≤ 1/2 — rw gives -5/2 ≤ 1/2 — hmm rw doesn't close that)... OR the `by rw [hre]; norm_num` for hσlo in hRtight (goal -5/2 ≤ s.re: rw [hre] → -5/2 ≤ -5/2 → rw's auto-rfl... `-5/2 ≤ -5/2` — is that closed by rfl? `≤` on ℝ: `le_refl` — NOT rfl-defeq... hmm — but norm_num then has no goals if rw closed it? rw closes goals that become `a = a` or `True` via rfl — `-5/2 ≤ -5/2` is not rfl. Hmm. Wait — which line is 293? Let me count from the file. Lines: the hmid block is around 280-300. `hk : ‖kFun n ...‖ ≤ 2 := kFun_norm_le_two (by rw [hre]; norm_num)` — col 39... "No goals to be solved" — if rw [hre] rewrites goal `(term).re ≤ 1/2` — wait, hre : (term).re = -5/2. rw [hre] at goal: replaces (term).re with -5/2 → goal `-5/2 ≤ 1/2` — NOT closed by rfl — then norm_num closes ✓. So that's not it. Hmm, line 293 col 39. Let me look at the file around line 293. Actually — the `by rw [hre]; norm_num` appears TWICE in the hRtight line (hσlo and hσhi) and once in hmid (hk) and once in htail (hk). For hσlo : `-5/2 ≤ s.re`: rw [hre] : hre : s.re = -5/2 — rewrites `s.re` → `-5/2`: goal `-5/2 ≤ -5/2` — rw tries rfl — `le_refl (-5/2)`... is `a ≤ a` closed by `rfl`? In Lean 4 Mathlib, `≤` on ℝ is `Real.le` — defined how? Real.instLE: le := fun a b => ... Actually for REAL, `a ≤ a` — `Real.le` is defined as `LE.le` via... In Lean core, Real.le a b := ∃ ... no. Modern Mathlib: `Real.le` is defined via the order on Cauchy sequences?? No — Real is a quotient; but the `≤` instance: `instLEReal : LE ℝ` with le := Real.le, and `Real.le (x y : ℝ) : Prop := x < y ∨ x = y`?? Hmm — I recall `Real.le` IS defined as `x < y ∨ x = y` in Mathlib (Real.lt as... actually `Real.le_def : x ≤ y ↔ x < y ∨ x = y`). If Real.le x y is DEFINITIONALLY x < y ∨ x = y, then `a ≤ a` := `a < a ∨ a = a` — proved by `Or.inr rfl` — is that defeq to `rfl`? The rfl-closer in rw uses `Eq.refl`?? rw closes goals with `rfl` tactic — which tries `Eq.refl`, `Iff.refl`, `HEq.refl`... NOT Or.inr rfl. Hmm, but the `rfl` tactic also does `refl` for registered refl-lemmas? The `rfl` tactic: tries `Eq.refl` then `refl` attribute theorems (like `le_refl`!). The `refl` tactic extension finds `le_refl` tagged with @[refl] ✓. So rw's post-rw rfl closes `-5/2 ≤ -5/2` via le_refl ✓✓. THAT's the "No goals to be solved": rw [hre] CLOSED the hσlo goal, leaving `norm_num` with no goals! Line 293 col 39 — the first `(by rw [hre]; norm_num)` in the hRtight line (hσlo). Fix: use `(by rw [hre])` alone for hσlo (rw closes it), keep norm_num only where needed (hσhi: -5/2 ≤ 3 — rw [hre] → `-5/2 ≤ 3` — NOT closed by rw's rfl — norm_num needed ✓). So: - hσlo: `by rw [hre]` - hσhi: `by rw [hre]; norm_num` And in hmid/htail hk: goal `(term).re ≤ 1/2`: rw [hre] → `-5/2 ≤ 1/2` — needs norm_num ✓ keep. Hmm wait — but actually is `-5/2 ≤ -5/2` closed by rw-rfl? The error says "No goals to be solved" at 293:39 — confirming yes for one of the sites. But which line is 293? Let me count precisely from my edit. The hRtight line: ``` have hRtight := logDeriv_riemannZeta_sub_nearZeros_le_tight (t := t) (s := s) hZC hZC2 hNLZ ht10 (by rw [hre]; norm_num) (by rw [hre]; norm_num) him (zeta_ne_zero_left_vertical t) ``` If 293:39 is the hσlo by-block here — then hmid's hk by-block is around line 285... hmm — but if rw [hre]; norm_num failed at hσlo with "no goals", it means rw closed it. But then the SAME pattern at hmid's hk (goal (term).re ≤ 1/2) — rw [hre] gives -5/2 ≤ 1/2 — NOT closed — norm_num fine ✓ no error there. And htail's hk — same ✓. So only the hσlo site errors ✓ consistent with "293:39" being a single location. But wait — "No goals to be solved" at col 39 — the hσlo `by rw [hre]; norm_num` — col 39 could be where `norm_num` starts... roughly. OK. Actually hold on — could 293 be in hmid instead? Let me count lines from my written file. I'll count: the §2 block starts around line 208. Z2hat_nonneg ~10 lines → vIntegral starts ~222. hmid around 283-300. hRtight around 315. Line 293 falls in hmid: `hk : ‖kFun n ((-5/2 : ℝ) + (t:ℝ)*Complex.I)‖ ≤ 2 := kFun_norm_le_two (by rw [hre]; norm_num)` — col 39 would be mid-line... that by-block: goal `(term).re ≤ 1/2` — rw [hre] rewrites to `-5/2 ≤ 1/2` — hmm rw's rfl tries le_refl — `-5/2 ≤ 1/2` is NOT refl — so norm_num runs ✓ no "no goals" error. Contradiction. So 293 must be the hσlo site. I'll fix by making hσlo `by rw [hre]` — wait but if rw leaves `-5/2 ≤ -5/2` and closes via rfl ✓ done — `by rw [hre]` suffices. But SAFER (avoid relying on rw-rfl-le_refl): `by rw [hre]` — if rw doesn't close it, error "unsolved goals" — then I'd see it. Alternatively write `(ge_of_eq hre.symm)`... wait direction: ge_of_eq (h : a = b) : a ≥ b. hre : s.re = -5/2 — hre.symm : -5/2 = s.re — ge_of_eq hre.symm : -5/2 ≥ s.re — WRONG. `le_of_eq hre.symm` : le_of_eq (h : a = b) : a ≤ b — hre.symm : -5/2 = s.re gives -5/2 ≤ s.re ✓! So hσlo := `le_of_eq hre.symm`. Clean, no rw. Hmm — actually simplest that definitely works: `hre.ge` — `Eq.ge : a = b → a ≥ b` — hre.ge : s.re ≥ -5/2 i.e. -5/2 ≤ s.re ✓. Or `hre.symm.le`? Eq.le : a = b → a ≤ b — hre.symm.le : -5/2 ≤ s.re ✓. I'll use `hre.symm.le`. Wait no — hre : s.re = -5/2. hre.symm : -5/2 = s.re. hre.symm.le : -5/2 ≤ s.re ✓ YES. And hσhi : s.re ≤ 3 := by rw [hre]; norm_num — after rw goal -5/2 ≤ 3 — norm_num ✓ keep. But wait — "No goals" issue here? rw [hre] → `-5/2 ≤ 3` — not refl — norm_num runs ✓ fine. 3. **Line 333** (hlog's norm_le_insert): TYPE MISMATCH — `norm_le_insert ?a ?b : ‖?b‖ ≤ ‖?a‖ + ‖?a - ?b‖` — I had the ARGUMENT ORDER WRONG! `norm_le_insert (a b) : ‖a‖ ≤ ‖b‖ + ‖a - b‖`?? From the alias: `norm_le_insert := norm_le_norm_add_norm_sub`. And the error shows: `norm_le_insert ?m.1567 ?m.1568 has type ‖?m.1568‖ ≤ ‖?m.1567‖ + ‖?m.1567 - ?m.1568‖`. So `norm_le_insert a b : ‖b‖ ≤ ‖a‖ + ‖a - b‖`. I want: ‖logDeriv‖ ≤ ‖Σ‖ + ‖logDeriv − Σ‖ — hmm: ‖logDeriv‖ = ‖Σ + (logDeriv − Σ)‖... The lemma with a := Σ, b := logDeriv: ‖logDeriv‖ ≤ ‖Σ‖ + ‖Σ − logDeriv‖ — gives ‖Σ − logDeriv‖ not ‖logDeriv − Σ‖. Mismatch: my stated h1 has `‖logDeriv riemannZeta s - Σ‖` as the second term — the lemma gives ‖Σ − logDeriv‖. Fix: state h1 with the sum SUBTRACTED the other way: `‖logDeriv‖ ≤ ‖Σ‖ + ‖Σ - logDeriv‖`?? Then I can't directly use hRtight (‖logDeriv − Σ‖ ≤ ...). Need norm_sub_rev: ‖a − b‖ = ‖b − a‖. OR use norm_le_insert' — the OTHER alias: `norm_le_insert' := norm_le_norm_add_norm_sub'` : ‖?a‖ ≤ ‖?b‖ + ‖?a - ?b‖? From MStructure usage: `norm_le_insert' (R • x) (R • P • R • x)` proving `‖v 0‖ ≤ ‖h‖ + ‖v 0 - h‖` (ControlledClosure line 87: `‖v 0‖ ≤ ‖h‖ + ‖v 0 - h‖ := norm_le_insert' _ _`). So `norm_le_insert' (a b) : ‖a‖ ≤ ‖b‖ + ‖a - b‖` — EXACTLY what I need: a := logDeriv ζ s, b := Σ: ‖logDeriv‖ ≤ ‖Σ‖ + ‖logDeriv − Σ‖ ✓. Fix: `norm_le_insert' _ _`. 4. **Line 366** (hleft's integral_comp_neg): the `_` placeholder couldn't be inferred: `intervalIntegral.integral_comp_neg ?m.2050` — the HO unification `?f (-x) ≡ ‖Jlag n (↑(-5/2) + ↑(-t) * I)‖` failed — "has type @intervalIntegral ... (fun x => ?f (-x)) ... = ∫ x in -?b..-?a, ?f x but expected ..." — the metavariable ?f couldn't be resolved by unification (the fun x => ?f (-x) vs fun t => ‖Jlag n (↑(-5/2)+↑(-t)*I)‖ — pattern ?f (-x) — should be Miller... but maybe the coercion ↑(-t) makes it not a Miller pattern? ?f applied to `-x` (a bound-var-headed term) — `↑(-x)`? Actually the integrand is `‖Jlag n (↑(-5/2) + ↑(-x) * I)‖` — ?f's argument is `-x` — Miller pattern (?f applied to distinct bound vars/function of bound var) — `?f (-x)` is a Miller pattern with argument `-x` — solvable ?f := fun u => ‖Jlag n (↑(-5/2)+↑u*I)‖... but Lean's unifier may have failed because... the expected type has `@intervalIntegral ℝ Real.normedAddCommGroup RCLike.toInner...` — wait, look at the mismatch: `?m.2046` (the NormedAddCommGroup) vs `Real.normedAddCommGroup` — the F's codomain got unified with ℝ ✓ fine — the actual failure: the unifier couldn't assign ?m.2050 (?f). Because I passed `_` — Lean tried to elaborate `integral_comp_neg _` THEN unify with the ascription — the `_` stays a metavariable; the unification `fun x => ?f (-x) =?= fun t => ‖Jlag n (↑(-5/2) + ↑(-t)*I)‖` — HO — Lean might fail this particular HO (the argument -x is fine Miller, but maybe the issue is intervalIntegral's first args ?m.2045/2046/2047 also metavariables, blocking). Solution: pass f EXPLICITLY (it's an explicit arg!): ```lean _ = ∫ t in -(-10 : ℝ)..-(-T), ‖Jlag n ((-5 / 2 : ℝ) + (t : ℝ) * Complex.I)‖ := intervalIntegral.integral_comp_neg (fun t : ℝ => ‖Jlag n ((-5 / 2 : ℝ) + (t : ℝ) * Complex.I)‖) ``` With f pinned, the LHS `∫ x in ?a..?b, (fun t => ‖Jlag n (↑(-5/2)+↑t*I)‖) (-x)` — beta ≡ `∫ x in ?a..?b, ‖Jlag n (↑(-5/2)+↑(-x)*I)‖` — unify with the ascription's LHS ✓ ?a := -T, ?b := -10 ✓. 5. **Line 375:80 "unsolved goals"** — the hsplit block: after the two rewrites, goal `A_x + (B_x + C_x) = A_t + (B_t + C_t)` — NOT closed by rw's rfl?? The context shows hsplit's proof state with that goal remaining. Why didn't rfl close it? Alpha-equivalence IS handled by rfl... Hmm — wait, look at the goal: LHS `A + (B + C)` (right-nested ✓), RHS — the hsplit STATEMENT's RHS — displayed `A_t + B_t + C_t` — hmm the statement I wrote was `A + (B + C)`. But maybe the elaborator re-associated? No. Look at error 375's goal RHS: `∫ -T..-10, F + ∫ -10..10, F + ∫ 10..T, F` — could be either nesting (parens elided in display). If it's `A + (B + C)` ✓ same as LHS modulo alpha — rfl should close... UNLESS the RHS is actually `(A + B) + C` (left-nested) — did I write left-nested? My hsplit statement: ``` ∫ t in (-T)..(-10 : ℝ), F... + (∫ t in (-10 : ℝ)..(10 : ℝ), F... + ∫ t in (10 : ℝ)..T, F...) ``` I explicitly parenthesized (B + C) ✓ right-nested. Hmm — but wait, the `+` at line end: `A + \n (B + C)` — parses as A + (B+C) ✓. So why "unsolved goals"?? OH WAIT — I see it — line 375:80 — col 80 — that's not the hsplit rw failing to close; it's the SECOND rw line? Let me look at my hsplit proof: ``` rw [← intervalIntegral.integral_add_adjacent_intervals (hI (-T) (-10)) (hI (-10) T), ← intervalIntegral.integral_add_adjacent_intervals (hI (-10) 10) (hI 10 T)] ``` Line 375 col 80 — hmm, `(hI 10 T)` is around col 80. "unsolved goals" with the full context displayed — this error is at the END of the rw (after both rewrites, the rfl didn't close). So the remaining goal `A_x+(B_x+C_x) = A_t+(B_t+C_t)` was left. rw's auto-rfl: uses `Eq.refl` — for a goal `X = Y` with X, Y alpha-equivalent — `Eq.refl` proves `X = X` — needs defeq X ≡ Y — alpha IS defeq ✓ should close!! Unless the rewrites produced something subtly different — like the second rewrite ALSO rewriting the newly-created subterm in an unexpected spot, or the endpoints differing: `(hI (-10) T)` — the b-point: first rewrite: a=-T, b=-10, c=T ✓. Second: a=-10, b=10, c=T ✓. Hmm wait — maybe the problem: after the FIRST rewrite, the goal's LHS became `(∫ x in -T..-10, F x) + ∫ x in -10..T, F x`. The SECOND rewrite pattern: `∫ x in -10..T, F x` — but ALSO — hold on, could the second rewrite have rewritten the WRONG occurrence or... there's only one. Let me think about the ACTUAL displayed unsolved goal once more: LHS: `(∫ (x : ℝ) in -T..-10, F) + ((∫ (x : ℝ) in -10..10, F) + ∫ (x : ℝ) in 10..T, F)` RHS: `∫ (t : ℝ) in -T..-10, F + ∫ (t : ℝ) in -10..10, F + ∫ (t : ℝ) in 10..T, F` If RHS is `(A + B) + C` (left-nested) then goal = `A+(B+C) = (A+B)+C` — NOT rfl! Then the error is my hsplit STATEMENT — maybe the elaborator parsed `A + \n (B + C)` differently... OR the calc step that consumed hsplit re-associated? The unsolved goal is inside hsplit's proof (before the calc). Hmm — actually WAIT. Maybe I mis-transcribed and the statement's RHS parsed as left-nested because the parens... let me look at my edit text again: ``` have hsplit : ∫ t in (-T)..T, ‖Jlag n ((-5 / 2 : ℝ) + (t : ℝ) * Complex.I)‖ = ∫ t in (-T)..(-10 : ℝ), ‖Jlag n ((-5 / 2 : ℝ) + (t : ℝ) * Complex.I)‖ + (∫ t in (-10 : ℝ)..(10 : ℝ), ‖Jlag n ((-5 / 2 : ℝ) + (t : ℝ) * Complex.I)‖ + ∫ t in (10 : ℝ)..T, ‖Jlag n ((-5 / 2 : ℝ) + (t : ℝ) * Complex.I)‖) := by ``` Right-nested ✓ definitely. So the goal after rewrites is `A_x + (B_x + C_x) = A_t + (B_t + C_t)` — defeq (alpha) — rw's rfl should close!!! Why not? Hmm — is it possible the rewrites did NOT fire and the "unsolved goals" is the rw COMPLAINING...? No — "unsolved goals" at 375:80 with the displayed post-rewrite goal means the rewrites FIRED and left the goal open. rw auto-runs rfl only at the END of a rw sequence — it should have closed. Unless... rfl failed due to the integrals being non-rfl-defeq for a DEEPER reason: the first integral's integrand uses bound var x, RHS uses t — alpha ✓ defeq. Hmm, unless `integral_add_adjacent_intervals`'s instance produced DIFFERENT endpoints: `-10` from `(hI (-10) T)` vs `-10` in the goal ✓ same. OH WAIT. I bet I know — the `↑(-t)` in... no. Hmm. Actually — let me reconsider: maybe rw DID close the goal and "unsolved goals" refers to something else at col 80: `(hI 10 T)` — hmm `(hI 10 T)` — the `10` here — hI's args are a b : ℝ — `10` elaborates as ℝ ✓. Hmm, actually — let me reconsider whether the error is "unsolved goals" from the `have hsplit` as a whole: after `rw [...]` the tactic block ended with goals remaining (the rfl didn't close). Lean reports "unsolved goals" at the END of the `by` block — location pointing to the last tactic. So yes — the two rewrites happened, rfl didn't close. Why wouldn't rfl close `A_x+(B_x+C_x) = A_t+(B_t+C_t)`?? Alpha-equivalence should make them defeq... UNLESS they're not alpha-equivalent because the ENDPOINTS differ: LHS second endpoint `-10` from hI (-10)... in the second rewrite: `(hI (-10) 10)` and `(hI 10 T)`: lemma instance b = 10: `∫ x in -10..10, F x` ✓ `∫ x in 10..T, F x` ✓. RHS `∫ t in -10..10, F` `∫ t in 10..T, F` ✓ match. First rewrite: `∫ x in -T..-10, F x + ∫ x in -10..T, F x` — the endpoints `-T`, `-10`, `T` ✓. Hmm hmm. So both sides ARE alpha-equal. Why no rfl?? OH. I think I finally see it. rw's auto-rfl: `rw` tries to close the goal with `rfl` ONLY when the rewrite produces a goal of the form `a = a` or `True`... no — rw always attempts `rfl` (with the `Eq.refl`/`Iff.refl` + `refl` lemmas) after rewriting. For `X = Y` with X ≡alpha Y — `Eq.refl` : X = X — unify X =?= Y — defeq ✓. This MUST close. Unless the goal is not `=` but something else... The displayed goal IS `=`. Hmm, wait — maybe the issue is that the rewrites left metavariables? The hI instances are concrete... Let me look at col 375:80 more carefully. Line 375 might be the line with `← intervalIntegral.integral_add_adjacent_intervals (hI (-10) 10) (hI 10 T)]`. Col 80 ≈ `(hI 10 T)`. "unsolved goals" — hmm — this could be an error from ELABORATING the rw rule: the second rewrite rule instance `(hI (-10) 10) (hI 10 T)` — the lemma `integral_add_adjacent_intervals hab hbc : A + B = C` — the REVERSED rewrite `←`: pattern C = `∫ x in -10..T, F x`. In the goal (after first rewrite), occurrences of `∫ x in -10..T, F x`: ONE (in LHS). Rewrite to `∫ -10..10 + ∫ 10..T` ✓. Then rfl: `A + ((B + C)) = A + (B + C)` alpha ✓... OK you know what — I'll stop theorizing. The pragmatic fix: end hsplit's proof with `; ring` or explicit: since the goal after rewrites is X = Y with alpha-equivalent sides, close with `rfl` explicitly... if plain rfl fails, something deeper is wrong and I'll see its error. Actually — safer: restructure hsplit to avoid the issue entirely: ```lean have hsplit : ... := by have h1 := intervalIntegral.integral_add_adjacent_intervals (hI (-T) (-10)) (hI (-10) T) have h2 := intervalIntegral.integral_add_adjacent_intervals (hI (-10) 10) (hI 10 T) rw [← h1, ← h2] ``` Same thing. Alternatively — maybe the problem is genuinely alpha — and rw's rfl DOES handle alpha... I'm quite sure it does. Hmm wait — actually, let me re-read the displayed unsolved goal ONE more time, character by character: ``` ⊢ (∫ (x : ℝ) in -T..-10, ‖Jlag n (↑(-5 / 2) + ↑x * I)‖) + ((∫ (x : ℝ) in -10..10, ‖Jlag n (↑(-5 / 2) + ↑x * I)‖) + ∫ (x : ℝ) in 10..T, ‖Jlag n (↑(-5 / 2) + ↑x * I)‖) = ∫ (t : ℝ) in -T..-10, ‖Jlag n (↑(-5 / 2) + ↑t * I)‖ + ∫ (t : ℝ) in -10..10, ‖Jlag n (↑(-5 / 2) + ↑t * I)‖ + ∫ (t : ℝ) in 10..T, ‖Jlag n (↑(-5 / 2) + ↑t * I)‖ ``` The LEFT side's OUTER parens: `(∫ -T..-10) + ((∫ -10..10) + ∫ 10..T)`. The RIGHT side: `∫ -T..-10 + ∫ -10..10 + ∫ 10..T` — the pretty printer with elided parens... For right-nested `A + (B + C)` the printer shows `A + (B + C)` — with parens? Lean's pretty printer ELIDES parens based on precedence — `+` is right-associative in Lean 4!! `a + b + c` PARSES as `a + (b + c)` (right-assoc, precedence 65). And prints WITHOUT parens when right-nested: `a + b + c`. Left-nested `(a + b) + c` prints WITH parens: `(a + b) + c`. So: the LHS displays `(A) + ((B) + (C))` — the explicit parens on each integral are just the printer around ∫-terms. `(A) + ((B) + (C))` = A + (B + C) right-nested — printer shows parens only where needed for clarity... it showed `((∫...) + (∫...))` — hmm the outer parens around the integrals themselves (∫ terms are atoms, printer may parenthesize them under + for clarity). The RHS `A + B + C` — right-nested, no parens needed. So RHS IS right-nested `A + (B + C)` ✓ SAME as LHS modulo alpha. rfl should close!!! Argh. OK — unless the error is NOT at the rfl stage but the SECOND rw failed to find its pattern and the "unsolved goals" is a CASCADE: if the second rewrite failed with "pattern not found", we'd see that error instead... Actually — WAIT. Read the error list order: 375:80 "unsolved goals" — then 392:12 rw hleft pattern-not-found. hsplit (375) FAILED with unsolved goals — BUT hsplit still appears in the context at 392! When a `have` fails, Lean inserts a sorry and CONTINUES — hsplit : ... exists as a sorry-backed hypothesis. So 392's rw [hleft] — hleft also appears ✓ (its proof had error 366, sorry-backed). The 392 error: rw [hleft] pattern not found in the calc goal `A + (B + C) ≤ RHS`... the pattern `∫ t in -T..-10, F` IS the A — should be found! Unless... OH WAIT. The calc step: ``` _ = ∫ t in (-T)..(-10 : ℝ), F + (∫ -10..10 + ∫ 10..T) := hsplit _ ≤ 4 * T * ... := by rw [hleft]; ... ``` The `_ ≤ RHS` step's starting point is the previous calc line's RHS: `∫ -T..-10, F + (∫ -10..10, F + ∫ 10..T, F)`. rw [hleft] : pattern `∫ t in -T..-10, F` — present ✓✓✓. But the error says NOT FOUND?!?! Unless the calc's previous line didn't elaborate (because hsplit is sorry'd?) — no, the goal DISPLAYED at 392 shows exactly `A + (B + C) ≤ RHS` with A = `∫ t in -T..-10, F`. Pattern: `∫ (t : ℝ) in -T..-10, ‖Jlag n (↑(-5 / 2) + ↑t * I)‖`. A: `∫ (t : ℝ) in -T..-10, ‖Jlag n (↑(-5 / 2) + ↑t * I)‖`. IDENTICAL. Why would rw not find it?? OHHHH. I see it now!!! Look at hleft's statement — its LHS integral is `∫ (t : ℝ) in -T..-10, ...` — and in hleft's CONTEXT (error 375's context): `hleft : ∫ (t : ℝ) in -T..-10, ‖Jlag n (↑(-5 / 2) + ↑t * I)‖ = ∫ (t : ℝ) in 10..T, ...` ✓. And the pattern printed in error 392: `∫ (t : ℝ) in -T..-10, ‖Jlag n (↑(-5 / 2) + ↑t * I)‖` ✓ same. Target A ✓ same... Hmm hold on, maybe the issue is that hleft is SORRY-BACKED (error 366 killed its proof) — no, that doesn't affect rw pattern matching (hleft : LHS = RHS as a term exists regardless). WAIT — maybe error 392's target is NOT what I think. Re-read: ``` NguyenRH1/LiSideBounds.lean:392:12: error: Tactic `rewrite` failed: Did not find an occurrence of the pattern ∫ (t : ℝ) in -T..-10, ‖Jlag n (↑(-5 / 2) + ↑t * I)‖ in the target expression ∫ (t : ℝ) in -T..-10, ‖Jlag n (↑(-5 / 2) + ↑t * I)‖ + ∫ (t : ℝ) in -10..10, ‖Jlag n (↑(-5 / 2) + ↑t * I)‖ + ∫ (t : ℝ) in 10..T, ‖Jlag n (↑(-5 / 2) + ↑t * I)‖ ≤ 4 * T * (2 / 3 * Z2hat T + 10.5 * Real.log (T + 2) + 48) + 40 * 1.3352 ``` The target displayed IS `A + B + C ≤ RHS`. And the pattern is A. NOT FOUND?! Unless... the target's A and the pattern differ in the DISPLAYED-BUT-INVISIBLE part: the integrand! Pattern integrand: `‖Jlag n (↑(-5 / 2) + ↑t * I)‖`. Target integrand: same... Hmm — could there be TWO different `I`s or norms? No. Could `-T` differ: pattern `-T` vs target `-T` — T is the same local. Could the difference be `-10`: pattern `..-10` — from hleft's statement `(-10 : ℝ)` — elaborates `Neg.neg (10 : ℝ)` where `10` is `OfNat.ofNat 10`. Target's `-10` — from MY calc statement `(-10 : ℝ)` — same elaboration ✓. I'm clearly missing something. Let me think about 375's "unsolved goals" differently: if rfl failed to close `X = Y` (alpha-equal), maybe they're NOT alpha-equal — e.g., the second rewrite's `10` endpoints: `(hI (-10) 10)` gives b := 10 — `∫ x in -10..10, F x`. RHS's B: `∫ t in (-10 : ℝ)..(10 : ℝ), F` ✓ same. Hmm — what if the rewrites fired in a DIFFERENT place: the FIRST reversed rewrite `∫ x in -T..T, F x` — does this pattern occur anywhere ELSE in the goal? The RHS of hsplit's statement — no ∫-T..T there ✓. What if `integral_add_adjacent_intervals`'s μ unification picked a DIFFERENT measure? hI uses MeasureTheory.volume ✓. OK — radical idea: maybe the "unsolved goals" error at 375:80 is NOT about rfl at all — col 80 is inside the rw RULE expression: `← intervalIntegral.integral_add_adjacent_intervals (hI (-10) 10) (hI 10 T)]` — elaborating `hI 10 T`: `hI : ∀ a b : ℝ, IntervalIntegrable F volume a b` — `hI 10 T` — WAIT. `10` and `T` — FINE. Hmm — but what about `hI (-10) 10` — fine too. Actually — "unsolved goals" errors display the remaining goals at the END of a tactic block. The position 375:80 — the position of the whole `by` block or the last tactic. The displayed goal is the post-rw goal. So rfl DID fail. Since alpha-equivalence can't fail... Let me question my assumption: are the two sides REALLY alpha-equal? The LHS integrals have bound var `x` (from the lemma), RHS `t`. In Lean 4 kernel, `∀`/`∫` binders with different names are alpha-EQUIVALENT and the kernel's defeq DOES handle alpha. `rfl` via Eq.refl — kernel defeq — alpha ✓. IT MUST CLOSE. ...unless the sides differ in some OTHER invisible way. What could differ between the rewrite-produced integrals and my stated ones? THE INTEGRAND: lemma-produced: `(fun t => ‖Jlag n (↑(-5/2) + ↑t * I)‖) x` — a BETA-REDEX if not reduced during elaboration! If the rewrite produced `∫ x in -T..-10, (fun t => ‖Jlag n (↑(-5/2)+↑t*I)‖) x` (redex), the printer would still show `‖Jlag n (↑(-5/2)+↑x*I)‖` (printer beta-reduces for display!). And `rfl` — kernel defeq handles beta ✓ closes anyway. I give up theorizing — pragmatic: replace the rw-proof of hsplit with a calc-style proof that closes each step explicitly, and for the final assembly replace `rw [hleft]` with `conv` or a `calc`-rewriting approach. OR simply: after the two rewrites, add explicit `rfl`-attempt and see... Can't iterate blindly — each lake run is ~1 min. Let me make the robust changes: For hsplit — avoid the reversed-lemma rewrite; instead prove the equation forward: ```lean have hsplit : ∫ t in (-T)..T, F = A + (B + C) := by rw [← intervalIntegral.integral_add_adjacent_intervals (hI (-10) 10) (hI 10 T), ← intervalIntegral.integral_add_adjacent_intervals (hI (-T) (-10)) (hI (-10) T)] ``` Order swapped: first rewrite RHS's `(B + C)`-part? The goal: `∫ -T..T, F = A + (B + C)`. rw [← lemma2] : pattern `∫ x in -10..T, F` — occurs in... NOWHERE in the goal! The goal has ∫(-T..T) and A(∫-T..-10) and B(∫-10..10) and C(∫10..T). No ∫(-10..T). So that order fails. The original order: first `← lemma1` (pattern ∫(-T..T)) — occurs once (LHS) ✓ → LHS becomes ∫(-T..-10) + ∫(-10..T). Then `← lemma2` (pattern ∫(-10..T)) ✓ → ∫(-10..10)+∫(10..T). Final: LHS = A+(B+C) syntactically (with x-binders). Goal: A+(B+C) = A+(B+C) — rfl. This SHOULD work... You know — maybe the real culprit: after the FIRST rewrite, does the goal's RHS also contain a match for the FIRST pattern? No (RHS has no ∫-T..T). Hmm. Let me try yet another lens: maybe `rw` failed NOT at rfl but during the second rewrite — the pattern `∫ x in -10..T, (fun t => ‖Jlag n (↑(-5/2)+↑t*I)‖) x` vs the goal's `∫ x in -10..T, ‖Jlag n (↑(-5/2)+↑x*I)‖` — the rewrite rule (from hI-typed instance): the integrand in the rule is `F' x` where F' := the lambda from hI — elaborated as redex or reduced? If the rule's integrand is `(fun t => ...) x` (redex) and the goal's is the REDUCED `‖Jlag n (↑(-5/2)+↑x*I)‖` — rw matching: `rw` matches patterns up to "reducible" defeq? rw uses `kabstract` — keyed matching at REDUCIBLE transparency — beta-redexes... kabstract should handle beta. Hmm — but the FIRST rewrite worked (the error only complains at the END) — actually do we KNOW the first rewrite worked? The displayed post-failure goal shows LHS fully split — both rewrites FIRED ✓. So genuinely: after both rewrites, goal `LHS' = RHS` with LHS' ≡alpha RHS, and rw's rfl DIDN'T close. This contradicts my understanding of rw... UNLESS the two sides are NOT alpha-equal due to different `10`/`-10` ELABORATION: in `(hI (-10) 10)`, the `10 : ℝ` — OfNat 10. In my statement, `(10 : ℝ)` — OfNat 10 ✓ same. `-10`: `Neg.neg (OfNat 10)` both ✓. WAIT WAIT WAIT. Look at the displayed goal once more — the SECOND integral on the LHS: `∫ (x : ℝ) in -10..10` and on the RHS: `∫ (t : ℝ) in -10..10`. The FIRST integral: LHS `-T..-10` RHS `-T..-10`. THIRD: `10..T` both. Integrands ✓. Nesting: LHS `(A) + ((B) + (C))`, RHS `A + B + C`. Hmm — if `+` were LEFT-assoc, RHS `A + B + C` = `(A+B)+C` ≠ LHS `A+(B+C)` — NOT defeq, rfl fails, "unsolved goals" ✓✓✓. Is `+` right or left associative in Lean 4?? HAdd.hAdd notation `+` : `infixl:65 "+" => HAdd.hAdd` — **infixl** — LEFT-ASSOCIATIVE!!! THAT'S IT!!! `a + b + c` parses as `(a + b) + c`. My hsplit RHS: I wrote `A + \n (B + C)` — with EXPLICIT parens around B + C — so it parsed as `A + (B + C)` — RIGHT-nested by explicit parens. The rewrites produced `A + (B + C)` too. Both right-nested... then they'd match... BUT — the printer: for right-nested `A + (B + C)` with left-assoc `+`, the printer MUST print `A + (B + C)` (parens required, else misread). The displayed RHS `A + B + C` (no parens) = LEFT-nested `(A + B) + C`!!! So my statement's RHS... did the parens get LOST? I wrote: ``` ∫ t in (-T)..(-10 : ℝ), ‖...‖ + (∫ t in (-10 : ℝ)..(10 : ℝ), ‖...‖ + ∫ t in (10 : ℝ)..T, ‖...‖) ``` Parens present ✓. Hmm — but the DISPLAY at 375 shows RHS without parens... contradiction? Unless the elaborator... no, elaboration preserves structure. The printer at 375's RHS: `∫ -T..-10, F + ∫ -10..10, F + ∫ 10..T, F` — if this were right-nested, the printer would show `A + (B + C)`... Hmm — actually maybe Lean's printer for `HAdd.hAdd A (HAdd.hAdd B C)` prints `A + (B + C)` WITH parens ✓, and for `(A+B)+C` prints `A + B + C`. The display shows NO parens ⟹ RHS is LEFT-nested `(A+B)+C`. So my written `A + (B + C)` became left-nested?!? IMPOSSIBLE unless... the parens I wrote were consumed differently: `(∫ -10..10, F + ∫ 10..T, F)` — wait, maybe I DIDN'T write the parens correctly — the open-paren was BEFORE `∫ -10..10` ✓ and close after the ∫ 10..T term ✓. That gives A + (B + C)... Hold on, let me look at 392's target again: ``` ∫ (t : ℝ) in -T..-10, F + ∫ (t : ℝ) in -10..10, F + ∫ (t : ℝ) in 10..T, F ≤ ``` no parens — left-nested. And the LHS at 375: `(∫ -T..-10, F) + ((∫ -10..10, F) + ∫ 10..T, F)` — WITH parens — right-nested (explicit). So LHS (rewrites) = A + (B + C), RHS (my statement) = (A + B) + C. MY STATEMENT'S RHS IS LEFT-NESTED! But I wrote parens!!! Unless — OH NO. I see it. In my hsplit STATEMENT (as written in the Edit), let me re-examine: ``` ∫ t in (-T)..(-10 : ℝ), ‖Jlag n ((-5 / 2 : ℝ) + (t : ℝ) * Complex.I)‖ + (∫ t in (-10 : ℝ)..(10 : ℝ), ‖Jlag n ((-5 / 2 : ℝ) + (t : ℝ) * Complex.I)‖ + ∫ t in (10 : ℝ)..T, ‖Jlag n ((-5 / 2 : ℝ) + (t : ℝ) * Complex.I)‖) := by ``` This is `A + (B + C)` — explicit parens. The display shows `(A+B)+C`. CONTRADICTION. Unless the file got written differently than I think — let me READ the actual file around hsplit to see what's really there! Maybe my Edit string didn't have the parens (I should check what I actually sent). Looking at my edit payload above... In the edit I wrote: ``` have hsplit : ∫ t in (-T)..T, ‖Jlag n ((-5 / 2 : ℝ) + (t : ℝ) * Complex.I)‖ = ∫ t in (-T)..(-10 : ℝ), ‖Jlag n ((-5 / 2 : ℝ) + (t : ℝ) * Complex.I)‖ + (∫ t in (-10 : ℝ)..(10 : ℝ), ‖Jlag n ((-5 / 2 : ℝ) + (t : ℝ) * Complex.I)‖ + ∫ t in (10 : ℝ)..T, ‖Jlag n ((-5 / 2 : ℝ) + (t : ℝ) * Complex.I)‖) := by ``` Parens there ✓. Hmm — but ALSO the calc's `=`-step: ``` _ = ∫ t in (-T)..(-10 : ℝ), ‖Jlag n ((-5 / 2 : ℝ) + (t : ℝ) * Complex.I)‖ + (∫ t in (-10 : ℝ)..(10 : ℝ), ‖Jlag n ((-5 / 2 : ℝ) + (t : ℝ) * Complex.I)‖ + ∫ t in (10 : ℝ)..T, ‖Jlag n ((-5 / 2 : ℝ) + (t : ℝ) * Complex.I)‖) := hsplit ``` Also right-nested ✓. So both should be right-nested. The 392 target `A + B + C ≤ ...` (no parens, left-nested) — CONTRADICTS my written calc. UNLESS the printer elides... Ugh. OK actually — hmm, maybe I have the associativity backwards: if `+` is `infixl` (left-assoc), then `A + B + C` = `(A+B)+C`, and printing right-nested `A + (B + C)` REQUIRES parens. The 375-LHS display `(A) + ((B) + (C))` — hmm those parens could be the printer's grouping for clarity around ∫ terms, NOT structural. If the printer puts parens around each ∫-term for legibility: `(∫..) + ((∫..) + (∫..))` — the inner `((∫) + (∫))` has parens = structural right-nesting ✓. So LHS = right-nested ✓ as expected. RHS `∫ + ∫ + ∫` no parens = left-nested. So RHS ≠ LHS. CONCLUSION: my hsplit statement's RHS elaborated LEFT-nested — the parens I wrote... ...OR the printer DOESN'T add parens for right-nested + and prints `A + B + C` for BOTH nestings (trusting the reader's infixl assumption to parse it back as left-nested — that would be a printer BUG producing non-roundtrip terms). Lean's printer IS supposed to roundtrip. Hmm — actually for atoms A B C, `A + (B + C)` prints as `A + (B + C)` — I'm fairly confident parens appear. FINE. Whatever the nesting, the robust fix that eliminates the question: make hsplit's RHS EXACTLY what the rewrites produce — and make the calc consume it via hsplit as a rewrite, not an equation-match. Simplest bulletproof approach: RESTRUCTURE. Don't state hsplit with a hand-written RHS at all. Instead: ```lean have h1 := intervalIntegral.integral_add_adjacent_intervals (hI (-T) (-10)) (hI (-10) T) have h2 := intervalIntegral.integral_add_adjacent_intervals (hI (-10) 10) (hI 10 T) ``` h1 : ∫(-T..-10) + ∫(-10..T) = ∫(-T..T); h2 : ∫(-10..10) + ∫(10..T) = ∫(-10..T). Then h1 ▸ h2: `∫(-T..T) = ∫(-T..-10) + (∫(-10..10) + ∫(10..T))`: ```lean have hsplit : ∫ t in (-T)..T, F = ∫(-T..-10) + (∫(-10..10) + ∫(10..T)) := by rw [← h1, ← h2] ``` Same as before. The problem persists conceptually... NEW ANGLE: use `calc` with explicit transitivity through h1/h2: ```lean have hsplit : ... := by calc ∫ t in (-T)..T, F = ∫ t in (-T)..(-10:ℝ), F + ∫ t in (-10:ℝ)..T, F := (intervalIntegral.integral_add_adjacent_intervals (hI _ _) (hI _ _)).symm _ = ∫ t in (-T)..(-10:ℝ), F + (∫ t in (-10:ℝ)..(10:ℝ), F + ∫ t in (10:ℝ)..T, F) := by rw [← intervalIntegral.integral_add_adjacent_intervals (hI (-10) 10) (hI 10 T)] ``` First calc step: RHS is `A + D` where D = ∫(-10..T) — I write that form; the lemma instance proves `A + D = ∫(-T..T)` so .symm ✓ — but wait, the ascription's LHS `∫(-T..T)` and RHS `A + D` — the instance `(lemma hI hI).symm : ∫(-T..T) = A + D` — its LHS `∫ x in -T..T, F x` matches the calc LHS ✓ its RHS `A + D` matches my written ✓ (modulo binder alpha — calc steps check defeq ✓ alpha OK — this is the KEY difference: calc-step typechecking uses full defeq, unlike rw's syntactic-ish matching... rw should too though). Second step: goal `A + D = A + (B + C)` — rw [← lemma2] rewrites D → B + C ✓ then rfl (alpha) ✓. Hmm — but this relies on the same rfl-at-end. If alpha-rfl was the failure, it fails again. But we ESTABLISHED the sides are alpha-equal-right-nested... unless the printer mislead me and my statement is really LEFT-nested (parse issue!). If my written parens somehow didn't take effect — e.g., if the `+` at line end followed by `(∫...)` on the next line — the parser might have parsed `A + (B + C)` correctly ✓ I'm confident parens work. You know what, let me just LOOK at the actual file content to see what's really written there, then test hypotheses empirically. And also check: maybe the hsplit statement in the file lost its parens due to my edit formatting. Let me read the file section.
Several distinct issues. Let me look at the actual file content around the failing spots first.
tool: Read
input
{
  "file_path": "D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiSideBounds.lean",
  "offset": 245,
  "limit": 160
}
#2267 assistant 2026-08-26 17:29:06.294 UTC · 463→948 tok open ⤢
thinking (2711 chars)
Down to ONE error: line 403:8 "No goals to be solved". Line 403 is in the final assembly — the step-5 `congr 1` — congr closed BOTH subgoals (A = A' via... hmm no — A = A' is `∫-T..-10 = ∫10..T` which is hleft — not rfl!). Wait — "No goals to be solved" at 403:8 — let me figure out which line is 403 now. The congr 1 + exact hleft step: if congr 1 closed all goals (maybe congr closed the goal ENTIRELY via defeq?? — the two sides ARE equal up to hleft — not defeq!). Hmm — actually wait: maybe congr 1's auto-rfl closed everything because... no. Let me count: line 403 in the current file. The step-5 block: ``` _ = (∫ t in (10 : ℝ)..T, ...) + ((∫ t in (-10 : ℝ)..(10 : ℝ), ...) + (∫ t in (10 : ℝ)..T, ...)) := by congr 1 exact hleft ``` `congr 1` — then `exact hleft` — if congr 1 closed ALL goals (e.g., it solved A = A' by... no, it can't without hleft)... OR congr 1 produced ONE goal (A = A') and closed the rfl one, and... `exact hleft` says "No goals" — meaning congr 1 closed EVERYTHING. How? congr's auto-rfl uses defeq — A = A' is `∫-T..-10, F = ∫10..T, F` — NOT defeq... UNLESS — congr 1 turned the goal into something trivially closed: maybe congr failed to split and instead closed via... hmm. OR: congr 1 on `HAdd X Y = HAdd X' Y'` — with X = ∫-T..-10 and X' = ∫10..T — it might have used `hleft`?? No — congr doesn't know hleft. Wait — "No goals to be solved" at 403:8 — col 8 — which tactic is at col 8? The `congr 1` line is at col 8 (8 spaces indent). So congr 1 ITSELF reports "No goals to be solved"?! That happens when congr 1 runs with... no goals? Or the error is at `exact hleft` (also col 8, next line). Line numbers: let me count from my edit. The calc starts around line 380. Step 5's `:= by` then `congr 1` then `exact hleft`. If `congr 1` is line 403 and `exact hleft` line 404: the error 403:8 would be congr 1 — but "No goals" for congr means the goal was already closed before it ran?? The `:= by` — the calc step's goal... Hmm — actually maybe the issue: the calc step 5's expected relation got FULLY proved by... no. Alternatively line 403 is `exact hleft` and congr 1 closed everything. Could congr 1 close `A + (B+C) = A' + (B+C)` completely? Its rfl-routine: tries to close subgoals by rfl/defeq — A = A' is NOT defeq... unless congr DIDN'T split (head match failed?) and fell back to trying the whole goal by rfl — fails — then... error would be "congr failed". Hmm. Wait — actually — maybe the calc-step elaboration ORDER: the `:= by` proof is elaborated against the step's expected type `LHS = RHS`. LHS comes from the previous step's RHS (which is my parenthesized form). Hmm. Let me just look at what's on line 403.
One error left at line 403. Checking what's there.
tool: Read
input
{
  "file_path": "D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiSideBounds.lean",
  "offset": 378,
  "limit": 35
}
#2268 assistant 2026-08-26 17:31:03.708 UTC · 1928→1659 tok open ⤢
thinking (4417 chars)
Line 403 is `exact hleft`. So `congr 1` closed ALL goals. How? `congr 1` on `A + (B + C) = A' + (B + C)` — hmm, wait. Actually — maybe congr 1 closed the goal completely because it uses `Subsingleton`/`proof irrelevance`... no, these are reals. OR — maybe congr 1 decided the two sides are equal via `IntervalIntegral` congruence?? No. Hmm — OR: congr 1 split into A = A' and (B+C) = (B+C); auto-closed (B+C) = (B+C) by rfl; for A = A' — `∫-T..-10, F = ∫10..T, F` — hmm, could congr's internal rfl have closed THIS?? Only if defeq — they're not defeq... WAIT — unless congr 1 produced a SINGLE goal `A = A'` (having closed B+C) and then... "No goals" at `exact hleft` means after congr 1 there were NO goals left. So A = A' got closed by congr 1's rfl. That requires `∫ t in -T..-10, F-t ≡ ∫ t in 10..T, F-t` — is that DEFEQ?? intervalIntegral F (-T) (-10) volume vs intervalIntegral F 10 T volume — NO, the endpoints differ — not defeq. Hmm, unless congr did something smarter — like applying `intervalIntegral.integral_congr`-like lemmas?? congr uses congruence lemmas tagged @[congr] — for intervalIntegral there might be a @[congr] lemma reducing integral equality to integrand EqOn + endpoint equalities — then subgoals: -T = 10 (NOT closable)... Hmm, actually — maybe congr 1 turned the goal into `HEq` junk or closed it spuriously? "No goals to be solved" is a CLEAN signal that after congr 1, zero goals remained. If congr closed A = A' via... OH WAIT. Maybe I misread the nesting! Maybe the elaborated form is `(A + B) + C` vs `(A' + B) + C` — congr 1 on HAdd.hAdd (A+B) C = HAdd.hAdd (A'+B) C — splits: (A+B) = (A'+B) and C = C — closes C = C — leaves A+B = A'+B — then `exact hleft` — hleft : A = A' — TYPE MISMATCH expected A+B = A'+B... but the error is "No goals" not type mismatch. Hmm no. Let me think about congr 1's behavior on `X = Y` more carefully. `congr 1` tries: apply congruence lemmas to turn `f a = f b` into `a = b`. For `HAdd.hAdd u v = HAdd.hAdd u' v'`: produces `u = u'` and `v = v'` (and possibly HEq variants). With u = A, u' = A', v = (B+C), v' = (B+C): goals `A = A'` and `(B+C) = (B+C)` — the second closed by rfl automatically — leaves `A = A'`. Then `exact hleft` should work... but error says no goals after congr. UNLESS congr RECURSED: congr 1 might apply congruence repeatedly at depth 1? `congr 1` — depth 1 — only top level. Hmm. Actually — you know what, maybe the issue is that `congr 1` CLOSED `A = A'` because it found a subsingleton or... OR — get this — maybe the elaboration of my step-5 stated RHS didn't produce `A' + (B + C)` but `A' + B + C` = `(A' + B) + C` (infixl!) — I wrote explicit parens around `(B + C)`?? Looking at lines 397-401: ``` _ = (∫ t in (10 : ℝ)..T, ...) + ((∫ t in (-10 : ℝ)..(10 : ℝ), ...) + (∫ t in (10 : ℝ)..T, ...)) := by ``` The `(B + C)` IS parenthesized ✓ (double parens at line 398 `((∫`). So RHS = A' + (B + C) right-nested ✓. And LHS (from step 4's stated RHS) = A + (B + C) right-nested ✓. congr 1 → A = A' + rfl-goal. Hmm — so why "No goals"?? Let me consider: maybe congr 1 SOLVED A = A' by eta/defeq because the integrals... no. Alternatively — the "No goals" might be at line 402 (congr 1) and 403 (exact hleft) is fine?? The error is 403:8 — col 8 — `exact hleft` starts at col 8 ✓. congr 1 is also col 8 (line 402). The error says line 403. So congr 1 succeeded closing everything, exact hleft has no goals. Is it possible congr 1 used `hleft` somehow?? NO. Hmm — actually, WAIT. What if congr 1's rfl-closure is smarter: it tries `assumption` too? Some congr implementations try `rfl` + `assumption`... if congr 1's cleanup used assumption, hleft IS in the local context!!! `A = A'` matches hleft's statement EXACTLY (modulo binder alpha: hleft : ∫ t in -T..-10, F = ∫ t in 10..T, F ✓) — if congr's auto-closer includes `assumption`, it found hleft and closed the goal!! That would explain "No goals to be solved" at `exact hleft`. Actually — Mathlib's `congr` does try to close goals with `rfl` and... hmm, I recall congr tries `assumption`? Not sure. But empirically: the goal vanished. Whatever the mechanism — the fix is trivial: remove `exact hleft` and let congr 1 finish, OR replace `congr 1` + `exact hleft` with `by rw [hleft]`. `by rw [hleft]`: goal `A + (B+C) = A' + (B+C)` — pattern A occurs ✓ rewrite → `A' + (B+C) = A' + (B+C)` → auto-rfl ✓. Deterministic. Use that.
`congr 1` closed everything itself (it found `hleft` in context). Replacing with the deterministic `rw [hleft]`.
tool: Edit
input
{
  "replace_all": false,
  "file_path": "D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiSideBounds.lean",
  "old_string": "            (\u222b t in (10 : \u211d)..T,\n              \u2016Jlag n ((-5 / 2 : \u211d) + (t : \u211d) * Complex.I)\u2016)) := by\n        congr 1\n        exact hleft",
  "new_string": "            (\u222b t in (10 : \u211d)..T,\n              \u2016Jlag n ((-5 / 2 : \u211d) + (t : \u211d) * Complex.I)\u2016)) := by\n        rw [hleft]"
}
#2269 assistant 2026-08-26 17:34:12.595 UTC · 1217→368 tok open ⤢
thinking (1041 chars)
The audit script pattern is clear. Session 040's audit covered DL-1..6 + ii′-6-a2 but NOT the 6-c-i scaffolding (LiSideBounds §1 decls — Z2hat_mono, nearZerosSub_mult_sum_le, continuousAt_logDeriv_zeta, continuousOn_logDeriv_zeta, Jlag_norm_eq, zeta_ne_zero_left_vertical, zeta_ne_zero_horizontal, rectangleIntegral_Jlag_eq, VIntegral_norm, HIntegral_split, jlag_left_vertical_norm_even). Per STATE.md, 6-c-i was committed at a8c5b1e — was there an audit? The commit message says "build 8770 jobs" but audit-session040 covers DL only. So I should audit BOTH the §1 scaffolding decls AND the new §2 decls in AuditSession041. Write AuditSession041.lean, run it (needs the full build first — `lake build NguyenRH1.LiSideBounds` to produce the olean since AuditSession imports NguyenRH1 root which imports LiSideBounds? Let me check the root import — NguyenRH1.lean imports... per session-040 notes, LiSideBounds was root-imported? STATE.md says "LiSideBounds.lean scaffolding GREEN (build 8770 jobs)". Let me check NguyenRH1.lean root imports.
Audit pattern confirmed. Checking whether LiSideBounds is root-imported, then building the olean.
tool: Read
input
{
  "file_path": "D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1.lean"
}
#2270 assistant 2026-08-26 17:34:51.337 UTC · 3382→861 tok open ⤢
thinking (1933 chars)
LiSideBounds is NOT yet root-imported (the root imports stop at LiDogLeg + the two skeletons). The 6-c-i commit (a8c5b1e) didn't add it to the root — build 8770 jobs was probably `lake build NguyenRH1.LiSideBounds` directly. Per the pattern (each new file gets root-imported at its sub-stage), I should add `import NguyenRH1.LiSideBounds` to the root now (6-c-ii is the first sub-stage where new content lands; the file is complete enough). Let me add the import + doc line. Then: full build `lake build NguyenRH1.LiSideBounds` (fast, single module + root), write AuditSession041.lean, run it, commit. Actually — for the build, per handoff notes: "For lake: SINGLE command per detached spawn; short PrintAxioms in foreground." A single-module build (LiSideBounds + root re-elaboration) — the root re-elaborates quickly (it's just imports). `lake build NguyenRH1.LiSideBounds` builds just that module (deps already built). Should be ~1-2 min. Then `lake build` for the root import check — hmm, `lake build NguyenRH1` builds everything — that could take longer but everything's cached except LiSideBounds + root. Let me do `lake build` (full) to verify the whole package stays green — with cache it's just LiSideBounds + NguyenRH1 root module + default target. Should be fine in foreground with 600s. Actually the full build is the "8770 jobs" — with cache, only changed modules rebuild. LiSideBounds changed → rebuild it + root. Let me run `lake build` detached to be safe (it could exceed 10 min if something cascades). Per protocol, detached + log file + poll. Actually — the single-file check already passed (25 s elaboration). A full `lake build` with everything cached should take ~1-3 min (LiSideBounds + root module + any reverse deps — none besides root). I'll run it in foreground with a 600 s timeout; if it times out, switch to detached. Also add the import line to the root first, plus the doc bullet for LiSideBounds.
LiSideBounds isn't root-imported yet. Adding it to the root per the sub-stage pattern, then full build.
tool: Edit
input
{
  "replace_all": false,
  "file_path": "D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1.lean",
  "old_string": "  NguyenRH1.SuzukiSkeleton \u2014 R-003 exact factorization (draft \u00a73.2)\n  NguyenRH1.ValleySkeleton \u2014 valley law statements (draft \u00a73.3)\n-/",
  "new_string": "  NguyenRH1.LiSideBounds   \u2014 ii\u2032-6-c (session 040\u2013041): the four side\n                             bounds of the L0 assembly \u2014 \u00a71 scaffolding\n                             (Z2hat_mono, the nearZerosSub mult-count\n                             bridge, logDeriv continuity, the Jlag norm\n                             factorization, the rectangle side\n                             decomposition, H/V-integral machinery, the\n                             left-vertical even symmetry, the two\n                             zero-freeness lemmas) + 6-c-ii\n                             (vIntegral_Jlag_left_norm_le: I_III \u2264\n                             4T((2/3)Z\u2082(T) + 10.5 log(T+2) + 48) +\n                             40\u00b71.3352)\n  NguyenRH1.SuzukiSkeleton \u2014 R-003 exact factorization (draft \u00a73.2)\n  NguyenRH1.ValleySkeleton \u2014 valley law statements (draft \u00a73.3)\n-/"
}
#2271 assistant 2026-08-26 17:38:23.025 UTC · 659→1021 tok open ⤢
thinking (1279 chars)
Audit CLEAN: all 13 decls, 3 standard axioms only, 0 sorryAx. Now commit 6-c-ii per the crash-safe sub-stage protocol. Files to stage: - research/lean/NguyenRH1/NguyenRH1/LiSideBounds.lean (6-c-ii) - research/lean/NguyenRH1/NguyenRH1.lean (root import) - research/lean/NguyenRH1/AuditSession041.lean - research/lean/NguyenRH1/audit-session041-axioms.txt NOT the correspondence files (user's own work) or Scratch.lean (user's scratch) or the paper build artifacts. Git identity: pass per-command (-c user.name="NGUYEN-RH-1" -c user.email="research@local"). Commit message style from history: "ii′-6-c-i: LiSideBounds.lean scaffolding GREEN (build 8770 jobs)". So: "ii′-6-c-ii: I_III left-vertical bound vIntegral_Jlag_left_norm_le PROVED (build 8774 jobs, audit clean)". Include key details: the parse-capture gotcha banked, T ≥ 316, even symmetry, MZeroProp short piece. Then continue to 6-c-iii (I_I, F-E form). Per the todo list, after committing, proceed to 6-c-iii. Actually — the session protocol says commit per sub-stage. The user asked to "continue with the rest of the 6-c, starting with 6-c-ii" — so I should proceed through 6-c-iii, 6-c-iv, 6-c-v (and 6-d if all goes well). This is a long stretch of Lean work. Let me commit 6-c-ii now, then proceed to 6-c-iii.
Audit clean — 13 decls, 3 standard axioms, 0 sorryAx. Committing 6-c-ii per the sub-stage protocol.
tool: Bash
input
{
  "command": "cd \"D:/Python/Quantum\" && git add research/lean/NguyenRH1/NguyenRH1/LiSideBounds.lean research/lean/NguyenRH1/NguyenRH1.lean research/lean/NguyenRH1/AuditSession041.lean research/lean/NguyenRH1/audit-session041-axioms.txt && git -c user.name=\"NGUYEN-RH-1\" -c user.email=\"research@local\" commit -m \"$(cat <<'EOF'\nii\u2032-6-c-ii: I_III left-vertical bound vIntegral_Jlag_left_norm_le PROVED (build 8774 jobs, audit clean)\n\nLiSideBounds.lean \u00a72: \u2016VIntegral (Jlag n) (-5/2) (-T) T\u2016 \u2264\n4*T*((2/3)*Z2hat T + 10.5*log(T+2) + 48) + 40*1.3352 (n \u2265 10\u2075 so T \u2265 316).\nKernel \u2264 2 (kFun_norm_le_two); tails t \u2208 [10,T] via the 5f tight R-bound\ncapstone + the near count \u2264 (2/5)\u00b7Z\u2082(t) (\u2016s\u2212\u03c1\u2016 \u2265 5/2 from re = \u22125/2);\nmiddle |t| \u2264 10 via the perimeter MZeroProp (cast bridge \u2191(-5/2:\u211d) =\n(-5/2:\u2102)); left tail = right tail by the even symmetry +\nintervalIntegral.integral_comp_neg. Root-imported (NguyenRH1.lean);\nAuditSession041: 13 decls (\u00a71 scaffolding + \u00a72), 3 standard axioms only,\n0 sorryAx.\n\nBanked gotchas: `\u222b x in a..b, body` captures trailing `+`/`-` terms into\nthe integrand (precedence) \u2014 parenthesize every integral summand;\nContinuousAt.comp needs (g :=) (f :=) (x :=) pinned (arg-first\nelaboration misdecomposes the point); norm_le_insert' : \u2016a\u2016 \u2264 \u2016b\u2016+\u2016a\u2212b\u2016\n(norm_le_insert has the args swapped); rw [hre] closes `-5/2 \u2264 -5/2` by\nauto-rfl le_refl (le_of_eq hre.symm is deterministic); integral_comp_neg\ntakes the integrand f EXPLICITLY; Continuous.norm' instance mismatch \u2014\nuse continuous_norm.comp.\nEOF\n)\" && git log --oneline -1",
  "description": "Commit 6-c-ii with audit trail"
}
#2272 assistant 2026-08-26 17:41:16.469 UTC · 1333→4638 tok open ⤢
thinking (9315 chars)
6-c-ii committed (6e5a98f). Update todos and proceed to 6-c-iii (I_I, F-E form). **6-c-iii RECIPE (from STATE.md):** NEW kernel lemma `kFun_right_vertical_norm_le_exp_sqrt_add_third` {n} (hn : 4 ≤ n) (t): ‖kFun n (↑(2√n)+↑t·I)‖ ≤ exp(√n/2+1/3)+1 — |w/(w−1)| ≤ 1+1/(2√n−1) (norm_one_add_one_div_le + normSq monotone in t², worst t=0: (2·re(w−1)+1)/normSq(w−1) = (4√n−1)/((2√n−1)²+t²) ≤ (4√n−1)/(2√n−1)² ⟹ |1+1/(w−1)| ≤ 1+1/(2√n−1) — sqrt of both sides, sq_le_sq₀), then (1+1/(2√n−1))ⁿ ≤ exp(n/(2√n−1)) (add_one_le_exp^ⁿ) and n/(2√n−1) ≤ √n/2+1/3 ⟺ n ≥ 4 (cross-multiply, pure nlinarith) ⟹ exp mono. Then I_I ≤ 2T·(e^{√n/2+1/3}+1)·0.57·2^{−(2√n−2)} ≤ e^{1/3}·IIMaj n (norm_logDeriv_zeta_le_iimZeta (n≥5); e^{√n/2+1/3}+1 ≤ e^{1/3}(e^{√n/2}+1); T ≤ √n+1; unfold IIMaj + ring). Target statement: something like `vIntegral_Jlag_right_norm_le (hNLZ?) ... {n} (hn : 100000 ≤ n) {T} (hTn : √n ≤ T) (hTn1 : T ≤ √n+1) : ‖VIntegral (Jlag n) (2*√n) (-T) T‖ ≤ Real.exp (1/3) * IIMaj n` Wait — I need to check what IIMaj is and the existing DL-4/DL-6 lemmas: - `kFun_right_vertical_norm_le_exp_sqrt_half_add_one` (DL-4): the w = 2√n kernel variant, e^{√n/2}+1 — but with the off-by-one: needs t² ≥ n or re ≥ 2√n+1. The F-E finding says this lemma's charge is off by one at t=0. - `norm_logDeriv_zeta_le_iimZeta` (DL-6): ‖ζ′/ζ(w)‖ ≤ 0.57·2^{−(2√n−2)} at re w = 2√n (n ≥ 5)? Need exact signature. - IIMaj def in LiSkeleton. - `norm_one_add_one_div_le` — the normSq core from LiContour (used in kFun_norm_le_two). Let me look at these signatures first. Also vIntegral_Jlag_right_norm_le_quarter (DL-6): V_right ≤ IIMaj/4 EXACTLY — that lemma presumably bounds the VIntegral at 2√n by IIMaj/4 using the OLD kernel charge... wait no — DL-6's vIntegral_Jlag_right_le_IIMaj_quarter — hmm, what does that prove exactly? If DL-6 already proved V_right ≤ IIMaj/4, why does 6-c-iii need a new I_I bound? OH — I see: DL-4/DL-6's V_right is the dog-leg's RIGHT vertical segment from T to U = (3/2)T at σ = 2√n (a SHORT segment, length U−T = T/2, used for the dog-leg). The I_I of the main rectangle is the FULL right vertical from −T to T at σ = 2√n (length 2T). Different segment! The F-E finding: the kernel bound kFun_right_vertical_norm_le_exp_sqrt_half_add_one (used for both) is off by one — DL-4's usage was saved because... wait, was DL-4's usage also affected by F-E? Let me re-read F-E: "FINDING F-E (D-074 §9): the I_I kernel charge e^{√n/2}+1 is off by one at re w = 2√n near t = 0 (true value e^{√n/2+1/8}; latent in the original D-036 charge); REPAIR ADOPTED: I_I charged honestly as e^{1/3}·IIMaj (uniform kernel ≤ e^{√n/2+1/3}+1, n ≥ 4), the overrun absorbed by the fold's free dropped 2π (e^{1/3} ≤ 1.4 ≤ 2π) — NO change to Bn/IIMaj/convexity/pinch/base points" And from the NGUYEN.md entry: "the ii′-4 lemma needs re w ≥ 2√n+1; DL-4's variant needs t² ≥ n; at t = 0 the true kernel is (2√n/(2√n−1))ⁿ = e^{√n/2+1/8}". So DL-4's variant `kFun_right_vertical_norm_le_exp_sqrt_half_add_one` — what are its hypotheses? If it needs t² ≥ n, then DL-4's own usage (V_right over t ∈ [T, U] with T ≥ √n ⟹ t² ≥ n ✓) is FINE — the dog-leg V_right segment has |t| ≥ T ≥ √n so t² ≥ n ✓. The F-E issue only bites for the FULL I_I segment which includes t near 0. OK good — DL-4/DL-6 stand. So 6-c-iii needs: 1. NEW kernel lemma: `kFun_right_vertical_norm_le_exp_sqrt_add_third` {n : ℕ} (hn : 4 ≤ n) (t : ℝ) : ‖kFun n (↑(2 * Real.sqrt n) + ↑t * Complex.I)‖ ≤ Real.exp (Real.sqrt n / 2 + 1/3) + 1. 2. The I_I bound: `vIntegral_Jlag_right_norm_le_... {n} (hn : 100000 ≤ n) {T} (hTn : √n ≤ T) (hTn1 : T ≤ √n+1) : ‖VIntegral (Jlag n) (2 * Real.sqrt n) (-T) T‖ ≤ Real.exp (1/3) * IIMaj n`. Wait — but what's the exact charged form in Bn? The F-E repair says "I_I charged honestly as e^{1/3}·IIMaj" and the fold absorbs e^{1/3} ≤ 2π. Actually in the fold (6-d): |Sf| ≤ (1/2π)(I_III + Horiz + e^{1/3}IIMaj) + (4F+1) ≤ ... ≤ (1/2π)(I_III-charge + Horiz-charge) + IIMaj + 4F + 1. So the 6-c-iii deliverable is ‖VIntegral (Jlag n) (2√n) (−T) T‖ ≤ e^{1/3} · IIMaj n. Hmm wait — actually let me check: is it ≤ e^{1/3}·IIMaj or the raw 2T(e^{√n/2+1/3}+1)·0.57·2^{−(2√n−2)}? The named deliverable should be the e^{1/3}·IIMaj form (that's what the fold consumes). But per the recipe: "Then I_I ≤ 2T·(e^{√n/2+1/3}+1)·0.57·2^{−(2√n−2)} ≤ e^{1/3}·IIMaj n (norm_logDeriv_zeta_le_iimZeta (n≥5); e^{√n/2+1/3}+1 ≤ e^{1/3}(e^{√n/2}+1); T ≤ √n+1; unfold IIMaj + ring)". So the lemma: ```lean theorem vIntegral_Jlag_right_norm_le_exp_third_IIMaj {n : ℕ} (hn : 100000 ≤ n) {T : ℝ} (hTn : Real.sqrt n ≤ T) (hTn1 : T ≤ Real.sqrt n + 1) : ‖VIntegral (Jlag n) (2 * Real.sqrt n) (-T) T‖ ≤ Real.exp (1/3) * IIMaj n ``` Hmm — what does IIMaj look like? Need its def. IIMaj n = (√n+1)·(e^{√n/2}+1)·2^{−(2√n−2)}? Let me check LiSkeleton. The recipe says "T ≤ √n+1; unfold IIMaj + ring". So IIMaj n probably = (√n + 1) * (Real.exp (√n/2) + 1) * 2^{-(2√n-2)}? But then 2T·(e^{√n/2+1/3}+1)·0.57·2^{−(2√n−2)} ≤ e^{1/3}·(√n+1)(e^{√n/2}+1)2^{−(2√n−2)} ⟺ 2T(e^{√n/2+1/3}+1)·0.57 ≤ e^{1/3}(√n+1)(e^{√n/2}+1) — with T ≤ √n+1: suffices 2·0.57·(e^{√n/2+1/3}+1) ≤ e^{1/3}(e^{√n/2}+1) ⟺ 1.14·(e^{√n/2+1/3}+1) ≤ e^{1/3}(e^{√n/2}+1). Now e^{√n/2+1/3}+1 ≤ e^{1/3}(e^{√n/2}+1) ⟺ e^{√n/2+1/3}+1 ≤ e^{√n/2+1/3}+e^{1/3} ⟺ 1 ≤ e^{1/3} ✓ TRUE. So e^{√n/2+1/3}+1 ≤ e^{1/3}(e^{√n/2}+1) ✓ and 1.14 ≤ ... wait we need 1.14·X ≤ Y where X ≤ e^{1/3}(e^{√n/2}+1) =: Y. 1.14·Y?? No: suffices 1.14·X ≤ Y with X ≤ Y/1? Hmm: X := e^{√n/2+1/3}+1 ≤ e^{1/3}(e^{√n/2}+1) =: Y. Need 1.14·X ≤ e^{1/3}(e^{√n/2}+1) = Y. Since 1.14 ≤ e^{1/3}·?? — 1.14·X ≤ 1.14·Y ≤ ??? need 1.14·Y ≤ Y ⟺ 1.14 ≤ 1 FALSE. So the 0.57·2 = 1.14 factor must be absorbed elsewhere: norm_logDeriv_zeta_le_iimZeta's 0.57 — hmm. Let me recompute: I_I ≤ 2T·(e^{√n/2+1/3}+1)·0.57·2^{−(2√n−2)}. Target: e^{1/3}·IIMaj n with IIMaj n = (√n+1)·(e^{√n/2}+1)·2^{−(2√n−2)}? — need to check IIMaj's actual def. Maybe IIMaj n = 2·(√n+1)·(e^{√n/2}+1)·2^{−(2√n−2)} (with a factor 2 built in for the two sides?) or with 0.57... The DL-6 charge was "IIMaj/4" for V_right with ‖ζ′/ζ‖ ≤ 0.57·2^{−(2√n−2)} and kernel e^{√n/2}+1 and length (U−T) = T/2 ≤ (√n+1)/2: V_right ≤ (U−T)·(e^{√n/2}+1)·0.57·2^{−(2√n−2)} ≤ (√n+1)/2·(e^{√n/2}+1)·0.57·2^{−(2√n−2)}. For this to be IIMaj/4: IIMaj = 2·(√n+1)·(e^{√n/2}+1)·0.57·2^{−(2√n−2)}? Then IIMaj/4 = (√n+1)/2·(...)·0.57·2^{−...} ✓. So IIMaj n ≈ 1.14·(√n+1)·(e^{√n/2}+1)·2^{−(2√n−2)}. Then I_I ≤ 2T·(e^{√n/2+1/3}+1)·0.57·2^{−(2√n−2)} with T ≤ √n+1: ≤ 2(√n+1)(e^{√n/2+1/3}+1)·0.57·2^{−(2√n−2)} = IIMaj·(e^{√n/2+1/3}+1)/(e^{√n/2}+1) ≤ IIMaj·e^{1/3} ✓✓ — using (e^{√n/2+1/3}+1) ≤ e^{1/3}(e^{√n/2}+1). So the final lemma: ‖VIntegral (Jlag n) (2√n) (−T) T‖ ≤ Real.exp (1/3) * IIMaj n. The proof: - VIntegral_norm + norm_integral_le_integral_norm: ≤ ∫_{-T}^{T} ‖Jlag n (↑(2√n)+↑t·I)‖ dt. - Integrand ≤ K·L where K = e^{√n/2+1/3}+1 (kernel, NEW lemma, ALL t) and L = 0.57·2^{−(2√n−2)} (norm_logDeriv_zeta_le_iimZeta, n ≥ 5 — check signature; does it need hypotheses on t? DL-6 proved it at re = 2√n for... let me check its exact statement). - So integrand ≤ K·L constant; integral ≤ 2T·K·L via integral_mono_on + integral_const. Wait — need norm of product: ‖Jlag‖ = ‖kFun‖·‖logDeriv‖ ≤ K·L via mul_le_mul ✓ (L ≥ 0). - Then 2T·K·L ≤ e^{1/3}·IIMaj via T ≤ √n+1, the X ≤ e^{1/3}Y lemma, unfold IIMaj. For the integral bound: ∫_{-T}^{T} c dt = 2T·c. integral_mono_on needs -T ≤ T ✓ and pointwise bound on Icc(-T,T). The integrability: same path-continuity argument but at re = 2√n: Jlag continuous on the path (kFun differentiable — s ≠ 1 since re = 2√n ≥ 632 ≠ 1... wait need s ≠ 1: re = 2√n ≥ 2·√100000 ≈ 632 — if s = 1 then re = 1 contradiction ✓; logDeriv continuous: ζ(s) ≠ 0 at re = 2√n — ζ has no zeros with re > 1 — riemannZeta_ne_zero_of_one_le_re (banked: takes ONLY re). re s = 2√n ≥ 632 ≥ 1 ✓. So I need a `zeta_ne_zero_right_vertical` helper: riemannZeta (↑(2√n) + ↑t·I) ≠ 0 via riemannZeta_ne_zero_of_one_le_re. Now the kernel lemma. Statement: ```lean theorem kFun_right_vertical_norm_le_exp_sqrt_add_third {n : ℕ} (hn : 4 ≤ n) (t : ℝ) : ‖kFun n (↑(2 * Real.sqrt n) + ↑t * Complex.I)‖ ≤ Real.exp (Real.sqrt n / 2 + 1/3) + 1 ``` Proof per recipe: |w/(w−1)| ≤ 1+1/(2√n−1) where w = ↑(2√n)+↑t·I. - |w/(w−1)| = |1 + 1/(w−1)| ≤ 1 + 1/|w−1| (norm_one_add_one_div_le? — that's the lemma used in kFun_norm_le_two: `norm_div_sub_one_le_one`? Let me check what kFun_norm_le_two used: `norm_div_sub_one_le_one hs : ‖s / (s - 1)‖ ≤ 1` for re ≤ 1/2. Different. For our case we need the normSq computation directly: ‖w/(w−1)‖² = normSq w / normSq (w−1) = ((2√n)²+t²)/((2√n−1)²+t²). The recipe: "norm_one_add_one_div_le + normSq monotone in t², worst t=0: (2·re(w−1)+1)/normSq(w−1) = (4√n−1)/((2√n−1)²+t²) ≤ (4√n−1)/(2√n−1)² ⟹ |1+1/(w−1)| ≤ 1+1/(2√n−1) — sqrt of both sides, sq_le_sq₀". Let me find `norm_one_add_one_div_le` in LiContour — it computes ‖1 + 1/z‖ ≤ something via normSq. Let me look at the actual lemma and the DL-4 kernel lemma (kFun_right_vertical_norm_le_exp_sqrt_half_add_one) — its proof is the closest template (it's the same shape with a different constant, and IT used the t² ≥ n or re ≥ 2√n+1 hypothesis). Let me read DL-4's kernel lemma proof first — that's the template. Also norm_logDeriv_zeta_le_iimZeta's statement, IIMaj's def, and the ii′-4 norm_one_add_one_div_le.
6-c-ii committed (`6e5a98f`). Moving to 6-c-iii (I_I, F-E form). First, the templates: DL-4's kernel lemma, DL-6's iimZeta bound, and the IIMaj definition.
tool: TodoWrite
input
{
  "todos": [
    {
      "content": "6-c-ii: I_III left-vertical bound vIntegral_Jlag_left_norm_le",
      "status": "completed",
      "activeForm": "Proving 6-c-ii I_III left-vertical bound"
    },
    {
      "content": "6-c-iii: I_I right-vertical bound (F-E form)",
      "status": "in_progress",
      "activeForm": "Proving 6-c-iii I_I bound"
    },
    {
      "content": "6-c-iv: mid-horizontal segment machine",
      "status": "pending",
      "activeForm": "Proving 6-c-iv mid-horizontal machine"
    },
    {
      "content": "6-c-v: horizontal assembly (mid + DL-5 tails)",
      "status": "pending",
      "activeForm": "Assembling 6-c-v horizontal bound"
    },
    {
      "content": "6-d: fold \u27f9 sfBoundProp (then commit, update archive)",
      "status": "pending",
      "activeForm": "Proving 6-d fold to sfBoundProp"
    }
  ]
}
tool: Grep ok
input
{
  "pattern": "theorem kFun_right_vertical_norm_le_exp_sqrt_half_add_one|theorem norm_logDeriv_zeta_le_iimZeta|theorem norm_one_add_one_div_le|def IIMaj|theorem vIntegral_Jlag_right_le_IIMaj_quarter",
  "path": "D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1",
  "output_mode": "content",
  "-A": 8
}
result 2026-08-26 17:41:16Z → 2026-08-26 17:41:16Z
mode='content' num_files=0 filenames=[] content='research\\lean\\NguyenRH1\\NguyenRH1\\LiContour.lean:470:theorem norm_one_add_one_div_le {s : ℂ} (hs : s ≠ 0) :\nresearch\\lean\\NguyenRH1\\NguyenRH1\\LiContour.lean-471-    ‖1 + 1 \\ s‖ ≤ 1 + (s.re + 1) \\ Complex.normSq s:= by\nresearch\\lean\\NguyenRH1\\NguyenRH1\\LiContour.lean-472-  set D:= Complex.normSq s with hD_def\nresearch\\lean\\NguyenRH1\\NguyenRH1\\LiContour.lean-473-  have hD: (0:ℝ) < D := Complex.normSq_pos.mpr hs\nresearch\\lean\\NguyenRH1\\NguyenRH1\\LiContour.lean-474-  have hD_rel: D = s.re * s.re + s.im * s.im := by rw [hD_def, Complex.normSq_apply s]\nresearch\\lean\\NguyenRH1\\NguyenRH1\\LiContour.lean-475-  have h1: (1 + 1 / s).re = 1 + s.re / D := by\nD:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiContour.lean-476-    simp [Complex.add_re, Complex.one_re, hD_def]\nresearch\\lean\\NguyenRH1\\NguyenRH1\\LiContour.lean-477-  have h2: (1 + 1 / s).im = -s.im / D := by\nD:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiContour.lean-478-    simp [Complex.add_im, Complex.one_im, hD_def]\n--\nresearch\\lean\\NguyenRH1\\NguyenRH1\\LiDogLeg.lean:732:theorem kFun_right_vertical_norm_le_exp_sqrt_half_add_one {n : ℕ} {t : ℝ}\nresearch\\lean\\NguyenRH1\\NguyenRH1\\LiDogLeg.lean-733-    (hn: 16 ≤ n) (ht : (n : ℝ) ≤ t ^ 2) :\nresearch\\lean\\NguyenRH1\\NguyenRH1\\LiDogLeg.lean-734-    ‖kFun n (((2 * Real.sqrt (n: ℝ)) : ℝ) + t * Complex.I)‖ ≤\nresearch\\lean\\NguyenRH1\\NguyenRH1\\LiDogLeg.lean-735-      Real.exp (Real.sqrt (n: ℝ) / 2) + 1 := by\nresearch\\lean\\NguyenRH1\\NguyenRH1\\LiDogLeg.lean-736-  have hnR: (16 : ℝ) ≤ (n : ℝ) := by exact_mod_cast hn\nresearch\\lean\\NguyenRH1\\NguyenRH1\\LiDogLeg.lean-737-  have hsq: (0 : ℝ) < Real.sqrt (n : ℝ) := Real.sqrt_pos.mpr (by linarith)\nresearch\\lean\\NguyenRH1\\NguyenRH1\\LiDogLeg.lean-738-  have hw1: (((2 * Real.sqrt (n : ℝ)) : ℝ) + t * Complex.I) - 1 =\nresearch\\lean\\NguyenRH1\\NguyenRH1\\LiDogLeg.lean-739-      ((2 * Real.sqrt (n: ℝ) - 1) : ℝ) + t * Complex.I := by\nD:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiDogLeg.lean-740-    push_cast\n--\nresearch\\lean\\NguyenRH1\\NguyenRH1\\LiDogLeg.lean:1158:theorem norm_logDeriv_zeta_le_iimZeta {n : ℕ} (hn : 5 ≤ n) {t : ℝ} :\nresearch\\lean\\NguyenRH1\\NguyenRH1\\LiDogLeg.lean-1159-    ‖logDeriv riemannZeta (((2 * Real.sqrt (n: ℝ)) : ℝ) + t * Complex.I)‖ ≤\nresearch\\lean\\NguyenRH1\\NguyenRH1\\LiDogLeg.lean-1160-      (0.57: ℝ) * (2 : ℝ) ^ (-(2 * Real.sqrt (n : ℝ) - 2)) := by\nresearch\\lean\\NguyenRH1\\NguyenRH1\\LiDogLeg.lean-1161-  have hn1: 1 ≤ n := by omega\nresearch\\lean\\NguyenRH1\\NguyenRH1\\LiDogLeg.lean-1162-  have h1:= norm_logDeriv_zeta_le_dirichletCX hn1 (t := t)\nresearch\\lean\\NguyenRH1\\NguyenRH1\\LiDogLeg.lean-1163-  have h2:= dirichletCX_le hn1\nresearch\\lean\\NguyenRH1\\NguyenRH1\\LiDogLeg.lean-1164-  have hC2: dirichletC2 ≤ 1 := dirichletC2_le_one\nresearch\\lean\\NguyenRH1\\NguyenRH1\\LiDogLeg.lean-1165-  have hC2nn: (0 : ℝ) ≤ dirichletC2 := dirichletC2_nonneg\nresearch\\lean\\NguyenRH1\\NguyenRH1\\LiDogLeg.lean-1166-  have hsqrt5: (2.236 : ℝ) ≤ Real.sqrt (n : ℝ) := by\n--\nresearch\\lean\\NguyenRH1\\NguyenRH1\\LiDogLeg.lean:1264:theorem vIntegral_Jlag_right_le_IIMaj_quarter {n : ℕ} {T : ℝ}\nresearch\\lean\\NguyenRH1\\NguyenRH1\\LiDogLeg.lean-1265-    (hn: 16 ≤ n) (hTn : Real.sqrt (n : ℝ) ≤ T) (hTn1 : T ≤ Real.sqrt (n : ℝ) + 1) :\nresearch\\lean\\NguyenRH1\\NguyenRH1\\LiDogLeg.lean-1266-    ‖VIntegral (Jlag n) (2 * Real.sqrt (n: ℝ)) T ((3 / 2) * T)‖ ≤ IIMaj (n : ℝ) / 4 := by\nresearch\\lean\\NguyenRH1\\NguyenRH1\\LiDogLeg.lean-1267-  have hT0: (0 : ℝ) < T := by\nresearch\\lean\\NguyenRH1\\NguyenRH1\\LiDogLeg.lean-1268-    have hnR: (16 : ℝ) ≤ (n : ℝ) := by exact_mod_cast hn\nresearch\\lean\\NguyenRH1\\NguyenRH1\\LiDogLeg.lean-1269-    have hsqrt4: (4 : ℝ) ≤ Real.sqrt (n : ℝ) := by\nresearch\\lean\\NguyenRH1\\NguyenRH1\\LiDogLeg.lean-1270-      have h1: Real.sqrt 16 ≤ Real.sqrt (n : ℝ) :=\nresearch\\lean\\NguyenRH1\\NguyenRH1\\LiDogLeg.lean-1271-        Real.sqrt_le_sqrt (by exact_mod_cast hn: (16 : ℝ) ≤ (n : ℝ))\nresearch\\lean\\NguyenRH1\\NguyenRH1\\LiDogLeg.lean-1272-      have h2: Real.sqrt 16 = 4 := by\n--\nresearch\\lean\\NguyenRH1\\NguyenRH1\\LiSkeleton.lean:754:def IIMaj (n : ℝ) : ℝ :=\nD:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiSkeleton.lean-755-  2 * (Real.sqrt n + 1) * (Real.exp (Real.sqrt n / 2) + 1) * 0.57 *\nD:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiSkeleton.lean-756-    2 ^ (-(2 * Real.sqrt n - 2))\nD:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiSkeleton.lean-757-\nD:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiSkeleton.lean-758-/-- The dog-leg right-tail bound per side (D-074, DL-5 quarter-charge\nresearch\\lean\\NguyenRH1\\NguyenRH1\\LiSkeleton.lean-759-form): V_left + 8.38 + IIMaj/4, replacing the F-D-broken "3.3" right-tail\nD:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiSkeleton.lean-760-charge (D-072). This is the bound `hIntegral_Jlag_low_le_dogLegUpper`\nD:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiSkeleton.lean-761-proves for each horizontal right tail. -/\nresearch\\lean\\NguyenRH1\\NguyenRH1\\LiSkeleton.lean-762-def DogLegRight (n: ℝ) : ℝ :=' num_lines=49 num_matches=None applied_limit=None applied_offset=None
#2273 assistant 2026-08-26 17:41:50.361 UTC · 6904→615 tok open ⤢
thinking (1077 chars)
Key facts: - `IIMaj (n : ℝ) : ℝ := 2 * (Real.sqrt n + 1) * (Real.exp (Real.sqrt n / 2) + 1) * 0.57 * 2 ^ (-(2 * Real.sqrt n - 2))` ✓ as I guessed (with the factor 2). - `norm_logDeriv_zeta_le_iimZeta {n : ℕ} (hn : 5 ≤ n) {t : ℝ} : ‖logDeriv riemannZeta (((2 * Real.sqrt (n:ℝ)) : ℝ) + t * Complex.I)‖ ≤ 0.57 * 2 ^ (-(2*√n - 2))` — note the point form: `((2 * Real.sqrt (n:ℝ)) : ℝ) + t * Complex.I` — the `((2*√n) : ℝ)` is a ℝ-typed term coerced to ℂ: `↑(2 * √(n:ℝ)) + ↑t * I`. - DL-4's kernel lemma point form: `(((2 * Real.sqrt (n : ℝ)) : ℝ) + t * Complex.I)` ✓ same. - `norm_one_add_one_div_le {s : ℂ} (hs : s ≠ 0) : ‖1 + 1/s‖ ≤ 1 + (s.re + 1)/Complex.normSq s` — hmm interesting — for s := w−1: ‖1 + 1/(w−1)‖ ≤ 1 + ((w−1).re + 1)/normSq (w−1) = 1 + (2√n − 1 + 1)/((2√n−1)² + t²) = 1 + (2√n)/((2√n−1)² + t²). Hmm — the recipe said (2·re(w−1)+1)/normSq = (4√n−1)/((2√n−1)²+t²) — different from norm_one_add_one_div_le's (re+1)/normSq = (2√n)/(...). Let me look at the full DL-4 proof to see the actual route — it's the template. Let me read DL-4's full proof (lines 732-~830).
Templates located. Reading DL-4's kernel lemma proof in full — it's the direct template for the F-E variant.
tool: Read
input
{
  "file_path": "D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiDogLeg.lean",
  "offset": 726,
  "limit": 115
}
#2274 assistant 2026-08-26 18:00:10.626 UTC · 418→16561 tok open ⤢
thinking (29386 chars)
`riemannZeta_ne_zero_of_one_le_re ⦃s : ℂ⦄ (hs : 1 ≤ s.re) : riemannZeta s ≠ 0` — takes only re ✓ (s implicit-instance-ish ⦃s⦄ — strict implicit, determined by hs). Usage pattern: `riemannZeta_ne_zero_of_one_le_re (by rw [hre]; exact h1)` — with s determined by the hs's type ✓. Now write §3 (6-c-iii) into LiSideBounds.lean. Structure: ```lean /-! ## 3. 6-c-iii: I_I — the right vertical σ₁ = 2√n (F-E form) -/ /-- **The F-E kernel lemma** (D-074 §9): the uniform-in-t kernel bound at the right vertical σ = 2√n — ‖kFun n (2√n+it)‖ ≤ e^{√n/2+1/3}+1 for n ≥ 4. (The honest charge: |w/(w−1)| ≤ 1 + 1/(2√n−1) via the plain triangle inequality + ‖w−1‖ ≥ re(w−1) = 2√n−1; then (1+1/(2√n−1))ⁿ ≤ exp(n/(2√n−1)) ≤ exp(√n/2+1/3) ⟺ n ≥ 4. Fixes the off-by-one in the t² ≥ n / re ≥ 2√n+1 variants at t = 0.) -/ theorem kFun_right_vertical_norm_le_exp_sqrt_add_third {n : ℕ} (hn : 4 ≤ n) (t : ℝ) : ... ``` Wait — (t : ℝ) explicit or implicit? The consumers apply at specific t — explicit fine. Then the I_I theorem. Let me write the full code. For the final lemma name: `vIntegral_Jlag_right_le_exp_third_IIMaj`. Details to nail: - hn4 : 4 ≤ n := by omega (from hn : 100000 ≤ n) ✓ omega. - hn5 : 5 ≤ n := by omega. - The point form everywhere: `(((2 * Real.sqrt (n : ℝ)) : ℝ) + t * Complex.I)` — matching DL-4/DL-6. Wait — VIntegral_norm gives the integral point as `((x : ℝ) + (t : ℝ) * Complex.I)` where x := 2 * Real.sqrt (n:ℝ) — elaborated `↑(2 * √(↑n)) + ↑t * I` — and DL-6's point: `(((2 * Real.sqrt (n : ℝ)) : ℝ) + t * Complex.I)` — `↑(2*√(↑n)) + ↑t * I` ✓ identical. Hmm — but there's ONE subtlety: in VIntegral_norm's statement, the point is written `((x : ℝ) + (t : ℝ) * Complex.I)` — since x is ALREADY ℝ, the `(x : ℝ)` is a no-op annotation — elaborated term: `↑x + ↑t * I`? The coercion on x: since f : ℂ → ℂ and + needs ℂ, Lean inserts ↑ on the first argument ✓ `↑x + ↑t * I` ✓. And the t: `(t : ℝ) * Complex.I` — ↑t * I ✓. Same as DL's `↑(2*√n) + ↑t * I` ✓. But WAIT — in the DL-4 statement the point is `(((2 * Real.sqrt (n : ℝ)) : ℝ) + t * Complex.I)` — hmm, the `t * Complex.I` here — t : ℝ — `↑t * I` ✓ same. OK. And my §3 statements should use the DL point form `(((2 * Real.sqrt (n : ℝ)) : ℝ) + t * Complex.I)` for consistency with norm_logDeriv_zeta_le_iimZeta and the kernel lemma. One more check — the kernel lemma's statement point: same DL form. norm_logDeriv_zeta_le_iimZeta's point: same. The VIntegral path: VIntegral (Jlag n) (2 * Real.sqrt (n:ℝ)) (-T) T — x := 2*√(↑n) — after VIntegral_norm: ∫ t in -T..T, Jlag n (↑(2*√(↑n)) + ↑t*I) ✓. Hmm — will `↑(2*√(↑n))` from VIntegral_norm (with x := 2*√(↑n)) syntactically match the DL point's first term `↑(2*√(↑n))`? VIntegral_norm's x is `2 * Real.sqrt (n:ℝ)` — written identically ✓ same term ✓. Now — hsq2 : 2 ≤ √(↑n): needed in hzne (1 ≤ 2√n), hne/hsq1 in the kernel lemma, and hexp. Let me structure the kernel lemma with its own hsq2 (from hn : 4 ≤ n). Kernel lemma: ```lean theorem kFun_right_vertical_norm_le_exp_sqrt_add_third {n : ℕ} (hn : 4 ≤ n) (t : ℝ) : ‖kFun n (((2 * Real.sqrt (n : ℝ)) : ℝ) + t * Complex.I)‖ ≤ Real.exp (Real.sqrt (n : ℝ) / 2 + 1 / 3) + 1 := by have hnR : (4 : ℝ) ≤ (n : ℝ) := by exact_mod_cast hn have hsq : (0 : ℝ) < Real.sqrt (n : ℝ) := Real.sqrt_pos.mpr (by linarith) have hsq2 : (2 : ℝ) ≤ Real.sqrt (n : ℝ) := by rw [Real.le_sqrt (by norm_num) (by positivity)] nlinarith [hnR] have h21 : (0 : ℝ) < 2 * Real.sqrt (n : ℝ) - 1 := by nlinarith [hsq2] have hsqn : Real.sqrt (n : ℝ) ^ 2 = (n : ℝ) := Real.sq_sqrt (by linarith) -- w − 1 ≠ 0 and ‖w − 1‖ ≥ 2√n − 1 have hre : ((((2 * Real.sqrt (n : ℝ)) : ℝ) + t * Complex.I) - 1).re = 2 * Real.sqrt (n : ℝ) - 1 := by simp [Complex.sub_re, Complex.add_re, Complex.ofReal_re, Complex.mul_re, Complex.I_re, Complex.I_im, Complex.one_re] have hne : (((2 * Real.sqrt (n : ℝ)) : ℝ) + t * Complex.I) - 1 ≠ 0 := by intro h have h0 : ((((2 * Real.sqrt (n : ℝ)) : ℝ) + t * Complex.I) - 1).re = 0 := by rw [h]; simp rw [hre] at h0 nlinarith [h21] have hnorm_w1 : 2 * Real.sqrt (n : ℝ) - 1 ≤ ‖(((2 * Real.sqrt (n : ℝ)) : ℝ) + t * Complex.I) - 1‖ := by have h1 := abs_re_le_norm ((((2 * Real.sqrt (n : ℝ)) : ℝ) + t * Complex.I) - 1) rw [hre, abs_of_nonneg h21.le] at h1 exact h1 -- the base bound ‖1 + 1/(w−1)‖ ≤ 1 + 1/(2√n−1) have hinv : 1 / ‖(((2 * Real.sqrt (n : ℝ)) : ℝ) + t * Complex.I) - 1‖ ≤ 1 / (2 * Real.sqrt (n : ℝ) - 1) := one_div_le_one_div_of_le h21 hnorm_w1 have hbase : ‖1 + 1 / ((((2 * Real.sqrt (n : ℝ)) : ℝ) + t * Complex.I) - 1)‖ ≤ 1 + 1 / (2 * Real.sqrt (n : ℝ) - 1) := by calc ‖1 + 1 / ((((2 * Real.sqrt (n : ℝ)) : ℝ) + t * Complex.I) - 1)‖ ≤ ‖(1 : ℂ)‖ + ‖1 / ((((2 * Real.sqrt (n : ℝ)) : ℝ) + t * Complex.I) - 1)‖ := norm_add_le _ _ _ = 1 + 1 / ‖(((2 * Real.sqrt (n : ℝ)) : ℝ) + t * Complex.I) - 1‖ := by rw [norm_one, norm_div, norm_one] _ ≤ 1 + 1 / (2 * Real.sqrt (n : ℝ) - 1) := by linarith [hinv] have hbasepos : (0 : ℝ) ≤ 1 + 1 / (2 * Real.sqrt (n : ℝ) - 1) := by positivity -- the power + exp chain have hexp : (n : ℝ) * (1 / (2 * Real.sqrt (n : ℝ) - 1)) ≤ Real.sqrt (n : ℝ) / 2 + 1 / 3 := by rw [mul_one_div, div_le_iff₀ h21] nlinarith [hsqn, hsq2] have hpow : (1 + 1 / (2 * Real.sqrt (n : ℝ) - 1)) ^ n ≤ Real.exp (Real.sqrt (n : ℝ) / 2 + 1 / 3) := by calc (1 + 1 / (2 * Real.sqrt (n : ℝ) - 1)) ^ n ≤ (Real.exp (1 / (2 * Real.sqrt (n : ℝ) - 1))) ^ n := pow_le_pow_left₀ hbasepos (Real.add_one_le_exp _) n _ = Real.exp ((n : ℝ) * (1 / (2 * Real.sqrt (n : ℝ) - 1))) := (Real.exp_nat_mul _ n).symm _ ≤ Real.exp (Real.sqrt (n : ℝ) / 2 + 1 / 3) := Real.exp_le_exp.mpr hexp have hpow1 : ‖1 + 1 / ((((2 * Real.sqrt (n : ℝ)) : ℝ) + t * Complex.I) - 1)‖ ^ n ≤ (1 + 1 / (2 * Real.sqrt (n : ℝ) - 1)) ^ n := pow_le_pow_left₀ (norm_nonneg _) hbase n have hwid : (((2 * Real.sqrt (n : ℝ)) : ℝ) + t * Complex.I) / ((((2 * Real.sqrt (n : ℝ)) : ℝ) + t * Complex.I) - 1) = 1 + 1 / ((((2 * Real.sqrt (n : ℝ)) : ℝ) + t * Complex.I) - 1) := by field_simp [hne] ring unfold kFun rw [hwid] calc ‖(1 + 1 / ((((2 * Real.sqrt (n : ℝ)) : ℝ) + t * Complex.I) - 1)) ^ n - 1‖ ≤ ‖(1 + 1 / ((((2 * Real.sqrt (n : ℝ)) : ℝ) + t * Complex.I) - 1)) ^ n‖ + ‖(1 : ℂ)‖ := norm_sub_le _ _ _ = ‖1 + 1 / ((((2 * Real.sqrt (n : ℝ)) : ℝ) + t * Complex.I) - 1)‖ ^ n + 1 := by rw [norm_pow, norm_one] _ ≤ Real.exp (Real.sqrt (n : ℝ) / 2 + 1 / 3) + 1 := by linarith [hpow1.trans hpow] ``` Checks: - hexp's nlinarith: goal after rw [mul_one_div, div_le_iff₀ h21]: `(n:ℝ) ≤ (√n/2 + 1/3) * (2*√n − 1)`. nlinarith [hsqn, hsq2]: expand RHS: (√n/2)(2√n) − √n/2 + (2/3)√n − 1/3 = √n² − √n/2 + 2√n/3 − 1/3 = n + √n/6 − 1/3. Need n ≤ n + √n/6 − 1/3 ⟺ 1/3 ≤ √n/6 ⟺ 2 ≤ √n ✓ hsq2. nlinarith ring_nf's the product ✓ with √n·√n → (√n)^2 = n via hsqn ✓ closes. Hmm — nlinarith multiplies hypotheses pairwise: does it need a hint product? The goal after ring_nf: n ≤ n + (1/6)√n − 1/3-ish — but the ring_nf of the RHS product requires multiplying out (√n/2+1/3)*(2√n−1) — that's part of GOAL normalization (ring_nf handles products in the goal ✓). The (√n)^2 → n step: hsqn : √n^2 = n as a hint rewrite — nlinarith uses hsqn as an equation hint ✓ standard pattern from DL-4 (line 770: nlinarith [sq_nonneg ..., hnR] with hsqn rw'd first — hmm, DL-4 rw'd hsqn INTO the goal first at line 769/774). For safety, mirror DL-4: rw the goal into explicit form first? DL-4's hkey did `rw [show (2√n−1)^2 + t^2 = 4(√n²) − 4√n + 1 + t² by ring, hsqn]` — they rw [hsqn] after rewriting into √n² form. For my hexp: after div_le_iff₀, goal `n ≤ (√n/2+1/3)*(2√n−1)` — I could `rw [show (√n/2+1/3)*(2√n−1) = √n^2 + √n/6 − 1/3 := by ring]`... wait that identity: (√n/2+1/3)(2√n−1) = √n² − √n/2 + 2√n/3 − 1/3 = √n² + √n(−1/2+2/3) − 1/3 = √n² + √n/6 − 1/3 ✓. `by ring` proves it (ring in √n) ✓. Then rw [hsqn]: √n^2 → n — wait hsqn : √n^2 = n — rw [hsqn] rewrites √n^2 → n ✓ goal: n ≤ n + √n/6 − 1/3 — then linarith [hsq2] ✓ DETERMINISTIC. Let me use that explicit route: ```lean have hexp : (n : ℝ) * (1 / (2 * Real.sqrt (n : ℝ) - 1)) ≤ Real.sqrt (n : ℝ) / 2 + 1 / 3 := by rw [mul_one_div, div_le_iff₀ h21, show (Real.sqrt (n : ℝ) / 2 + 1 / 3) * (2 * Real.sqrt (n : ℝ) - 1) = Real.sqrt (n : ℝ) ^ 2 + Real.sqrt (n : ℝ) / 6 - 1 / 3 := by ring, hsqn] linarith [hsq2] ``` rw chains three rewrites ✓ ends with goal `n ≤ n + √n/6 − 1/3` — linarith [hsq2] ✓ (1/3 ≤ √n/6 ⟺ 2 ≤ √n ✓ linear in √n ✓). Hmm — the `show ... := by ring` — ring proves `(√n/2 + 1/3)*(2√n − 1) = √n^2 + √n/6 − 1/3` — over ℝ with √n as atom ✓ ring ✓. Wait — does `rw [mul_one_div]` produce `(n:ℝ)/(2√n−1)`? mul_one_div (a b) : a * (1/b) = a/b ✓. Then div_le_iff₀ h21 : a/b ≤ c ↔ a ≤ c*b — rewrites the iff ✓ goal becomes `(n:ℝ) ≤ (√n/2+1/3)*(2√n−1)` ✓ then the show-rw ✓ then hsqn ✓. Actually — one concern: `rw [div_le_iff₀ h21]` — div_le_iff₀ : a / b ≤ c ↔ a ≤ c * b ✓ (session-033 banked ✓). - hpow's calc step 1: pow_le_pow_left₀ hbasepos (Real.add_one_le_exp _) n : (1+1/(2√n−1))^n ≤ (exp(1/(2√n−1)))^n. Real.add_one_le_exp (x) : 1 + x ≤ exp x ✓ with x := 1/(2√n−1) — the `_` unified ✓. pow_le_pow_left₀ (ha : 0 ≤ a) (hab : a ≤ b) (n) : a^n ≤ b^n ✓. - hpow step 2: (exp(x))^n = exp(n*x) — Real.exp_nat_mul (x) (n : ℕ) : Real.exp (↑n * x) = (Real.exp x)^n — .symm : (exp x)^n = exp(↑n * x) — hmm, but is the cast there? DL-4 line 800-801: `(Real.exp ((n : ℝ) * (1 / (2 * Real.sqrt (n : ℝ))))) := (Real.exp_nat_mul _ n).symm` ✓ same usage ✓. - hpow step 3: Real.exp_le_exp.mpr hexp : exp(n/(2√n−1)-form) ≤ exp(√n/2+1/3) — the calc line's LHS: Real.exp ((n:ℝ) * (1/(2√n−1))) — hexp : (n:ℝ)*(1/(2√n−1)) ≤ √n/2+1/3 ✓ exp_le_exp.mpr : a ≤ b → exp a ≤ exp b ✓ the a here is (n:ℝ)*(1/(2√n−1)) ✓ matches ✓. Real.exp_le_exp : Real.exp a ≤ Real.exp b ↔ a ≤ b ✓ standard. - hpow1: pow_le_pow_left₀ (norm_nonneg _) hbase n : ‖1+1/(w−1)‖^n ≤ (1+1/(2√n−1))^n ✓. - hwid: field_simp [hne] — goal `w/(w−1) = 1 + 1/(w−1)` — field_simp with hne : w−1 ≠ 0 ✓ then ring ✓. Hmm — DL-4 needed `push_cast; ring` after field_simp — because their hs0/hw1 had cast forms. Mine: w is `↑(2*√(↑n)) + ↑t * I` — field_simp produces something like `↑(2*√↑n) + ↑t*I = (↑(2*√↑n) + ↑t*I − 1) + 1`?? — ring closes over ℂ with cast atoms ✓. If ring fails, add push_cast before ring. Let me preemptively write `field_simp [hne]; ring` and adjust if needed. Hmm — actually wait: field_simp on `w/(w−1) = 1 + 1/(w−1)`: it might rewrite to `w = (w − 1) + 1` directly? Or `w * (w−1) = (w−1)*w`?? Standard behavior: field_simp [hne] with hne : w−1 ≠ 0 — the goal has denominator (w−1) on both sides — field_simp multiplies both sides by (w−1): `w = (1 + 1/(w−1))*(w−1)` then distributes?? Usually leaves `w = (w - 1) + 1` or similar — ring closes either way ✓. - unfold kFun: goal `‖(w/(w−1))^n − 1‖ ≤ ...` — rw [hwid] rewrites w/(w−1) → 1+1/(w−1) ✓. Hmm — `unfold kFun` — does the goal after unfold show `(w/(w-1))^n - 1`? kFun n s := (s/(s-1))^n - 1 ✓ (from kFun_norm_le_two's rfl). unfold kFun at goal ‖kFun n w‖ ≤ e+1 → ‖(w/(w−1))^n − 1‖ ≤ e+1 ✓. rw [hwid] : pattern w/(w−1) ✓. Wait — will unfold produce the exact `w` form? The statement's kFun n (point) — unfold kFun n (point) → ((point)/(point − 1))^n − 1 ✓ hwid's LHS is exactly (point)/(point−1) ✓. - The final calc — the `_ = ‖1 + 1/(w−1)‖^n + 1` step: rw [norm_pow, norm_one] : ‖(1+1/(w−1))^n‖ → ‖1+1/(w−1)‖^n ✓ and ‖(1:ℂ)‖ → 1 ✓. - `linarith [hpow1.trans hpow]` : hpow1.trans hpow : ‖1+1/(w−1)‖^n ≤ exp(√n/2+1/3) ✓ goal: ‖1+1/(w−1)‖^n + 1 ≤ exp(√n/2+1/3)+1 ✓ linarith ✓. - hnorm_w1's abs_of_nonneg h21.le : |2√n−1| = 2√n−1 ✓ — wait, abs_of_nonneg (h : 0 ≤ a) : |a| = a — h21.le : 0 ≤ 2√n−1 ✓. rw [hre, abs_of_nonneg h21.le] at h1: h1 : |(w−1).re| ≤ ‖w−1‖ — rw [hre] : |2√n−1| ≤ ‖w−1‖ — rw [abs_of_nonneg h21.le] : 2√n−1 ≤ ‖w−1‖ ✓ exact h1 ✓. Hmm — abs_re_le_norm — in LiTightR used unqualified (line 158: `have h := abs_re_le_norm ((2 + t * Complex.I) - ρ.1)`) ✓ Complex.abs_re_le_norm is opened via `open ... Complex` ✓. - hbase's calc step 2: `‖(1:ℂ)‖ + ‖1/(w−1)‖ = 1 + 1/‖w−1‖`: rw [norm_one] : ‖(1:ℂ)‖ → 1 ✓; ‖1/(w−1)‖: norm_div → ‖1‖/‖w−1‖ → 1/‖w−1‖: rw [norm_div, norm_one] ✓. Order: rw [norm_one, norm_div, norm_one] — first norm_one rewrites ALL ‖(1:ℂ)‖ occurrences (both the standalone and inside norm_div? — norm_div hasn't fired yet — the term is ‖(1:ℂ)‖ + ‖(1:ℂ)/(w−1)‖ — norm_one fires on both ‖(1:ℂ)‖ atoms: → 1 + (‖(1:ℂ)‖/‖w−1‖)... wait ‖1/(w−1)‖ is ‖HDiv.hDiv (1:ℂ) (w−1)‖ — norm_one's pattern ‖(1:ℂ)‖ — the numerator inside is `(1:ℂ)` but not wrapped in ‖·‖ yet — norm_one only matches the OUTER ‖(1:ℂ)‖ and... hmm: term: `‖(1:ℂ)‖ + ‖(1:ℂ)/(w−1)‖`. norm_one pattern: ‖(1:?_)‖ — matches the first ‖(1:ℂ)‖ ✓ → 1. Does it also match inside the second? The second is ‖(1:ℂ)/(w−1)‖ — the pattern ‖(1:ℂ)‖ doesn't match ‖x/y‖ ✓. So after norm_one: `1 + ‖(1:ℂ)/(w−1)‖`. Then norm_div: ‖a/b‖ → ‖a‖/‖b‖: `1 + ‖(1:ℂ)‖/‖w−1‖`. Then norm_one: `1 + 1/‖w−1‖` ✓. So rw [norm_one, norm_div, norm_one] ✓ as written. - hbasepos by positivity: 0 ≤ 1 + 1/(2√n−1) — needs 0 ≤ 1/(2√n−1) — positivity with h21 (0 < 2√n−1) in context ✓ (positivity uses local hyps for atoms ✓). Hmm — h21 is in scope ✓. Now the I_I theorem: ```lean /-- **6-c-iii (I_I, the right vertical σ₁ = 2√n, F-E form)**: the full-side bound ‖V‖ ≤ e^{1/3}·IIMaj(n). The F-E kernel (uniform in t) + the DL-6 direct Dirichlet bound; e^{√n/2+1/3}+1 ≤ e^{1/3}(e^{√n/2}+1) since 1 ≤ e^{1/3}. -/ theorem vIntegral_Jlag_right_le_exp_third_IIMaj {n : ℕ} (hn : 100000 ≤ n) {T : ℝ} (hTn : Real.sqrt (n : ℝ) ≤ T) (hTn1 : T ≤ Real.sqrt (n : ℝ) + 1) : ‖VIntegral (Jlag n) (2 * Real.sqrt (n : ℝ)) (-T) T‖ ≤ Real.exp (1 / 3 : ℝ) * IIMaj (n : ℝ) := by have hnR : (100000 : ℝ) ≤ (n : ℝ) := by exact_mod_cast hn have hn4 : 4 ≤ n := by omega have hn5 : 5 ≤ n := by omega have hsq2 : (2 : ℝ) ≤ Real.sqrt (n : ℝ) := by rw [Real.le_sqrt (by norm_num) (by positivity)] nlinarith [hnR] have hT0 : (0 : ℝ) ≤ T := by have h := Real.sqrt_nonneg (n : ℝ) linarith have hTT : -T ≤ T := by linarith -- zero-freeness + continuity on the right vertical have hzne : ∀ t : ℝ, riemannZeta (((2 * Real.sqrt (n : ℝ)) : ℝ) + t * Complex.I) ≠ 0 := by intro t have hre : (((2 * Real.sqrt (n : ℝ)) : ℝ) + t * Complex.I).re = 2 * Real.sqrt (n : ℝ) := by simp [Complex.add_re, Complex.ofReal_re, Complex.mul_re, Complex.I_re, Complex.I_im] have h1 : (1 : ℝ) ≤ 2 * Real.sqrt (n : ℝ) := by nlinarith [hsq2] exact riemannZeta_ne_zero_of_one_le_re (by rw [hre]; exact h1) have hJcont : ∀ t : ℝ, ContinuousAt (Jlag n) (((2 * Real.sqrt (n : ℝ)) : ℝ) + t * Complex.I) := by intro t have hs1 : (((2 * Real.sqrt (n : ℝ)) : ℝ) + t * Complex.I) ≠ 1 := by intro h have hre := congrArg Complex.re h simp at hre nlinarith [hsq2] have hA := (kFun_differentiableAt (n := n) hs1).continuousAt have hB := continuousAt_logDeriv_zeta hs1 (hzne t) show ContinuousAt (fun s => kFun n s * (-logDeriv riemannZeta s)) _ exact hA.mul hB.neg have hpath : Continuous fun u : ℝ => (((2 * Real.sqrt (n : ℝ)) : ℝ) + u * Complex.I) := by fun_prop have hcontJ : Continuous fun t : ℝ => Jlag n (((2 * Real.sqrt (n : ℝ)) : ℝ) + t * Complex.I) := by rw [continuous_iff_continuousAt] intro t have hf : ContinuousAt (fun u : ℝ => (((2 * Real.sqrt (n : ℝ)) : ℝ) + u * Complex.I)) t := hpath.continuousAt have hcomp : ContinuousAt (Jlag n ∘ fun u : ℝ => (((2 * Real.sqrt (n : ℝ)) : ℝ) + u * Complex.I)) t := ContinuousAt.comp (g := Jlag n) (f := fun u : ℝ => (((2 * Real.sqrt (n : ℝ)) : ℝ) + u * Complex.I)) (x := t) (hJcont t) hf exact hcomp have hcontG : Continuous fun t : ℝ => ‖Jlag n (((2 * Real.sqrt (n : ℝ)) : ℝ) + t * Complex.I)‖ := continuous_norm.comp hcontJ have hI2 : IntervalIntegrable (fun t : ℝ => ‖Jlag n (((2 * Real.sqrt (n : ℝ)) : ℝ) + t * Complex.I)‖) MeasureTheory.volume (-T) T := hcontG.intervalIntegrable _ _ -- the e^{√n/2+1/3}+1 ≤ e^{1/3}(e^{√n/2}+1) comparison have hKle : Real.exp (Real.sqrt (n : ℝ) / 2 + 1 / 3) + 1 ≤ Real.exp (1 / 3 : ℝ) * (Real.exp (Real.sqrt (n : ℝ) / 2) + 1) := by rw [Real.exp_add] have h1 : (1 : ℝ) ≤ Real.exp (1 / 3 : ℝ) := by have h := Real.add_one_le_exp (1 / 3 : ℝ) linarith have hE := Real.exp_pos (Real.sqrt (n : ℝ) / 2) nlinarith [h1, hE] -- the pointwise bound have hK' : (0 : ℝ) ≤ Real.exp (Real.sqrt (n : ℝ) / 2 + 1 / 3) + 1 := by positivity have hL : (0 : ℝ) ≤ (0.57 : ℝ) * (2 : ℝ) ^ (-(2 * Real.sqrt (n : ℝ) - 2)) := by positivity have hpt : ∀ t : ℝ, t ∈ Set.Icc (-T) T → ‖Jlag n (((2 * Real.sqrt (n : ℝ)) : ℝ) + t * Complex.I)‖ ≤ (Real.exp (Real.sqrt (n : ℝ) / 2 + 1 / 3) + 1) * ((0.57 : ℝ) * (2 : ℝ) ^ (-(2 * Real.sqrt (n : ℝ) - 2))) := by intro t _ rw [Jlag_norm_eq] exact mul_le_mul (kFun_right_vertical_norm_le_exp_sqrt_add_third hn4 t) (norm_logDeriv_zeta_le_iimZeta hn5) (norm_nonneg _) hK' have hInt : ∫ t in (-T)..T, ‖Jlag n (((2 * Real.sqrt (n : ℝ)) : ℝ) + t * Complex.I)‖ ≤ (T - -T) * ((Real.exp (Real.sqrt (n : ℝ) / 2 + 1 / 3) + 1) * ((0.57 : ℝ) * (2 : ℝ) ^ (-(2 * Real.sqrt (n : ℝ) - 2)))) := by have h := intervalIntegral.integral_mono_on hTT hI2 intervalIntegrable_const (fun t ht => hpt t ht) rwa [intervalIntegral.integral_const, smul_eq_mul] at h -- the final comparison 2T·K·L ≤ e^{1/3}·IIMaj have hKL : (0 : ℝ) ≤ (Real.exp (Real.sqrt (n : ℝ) / 2 + 1 / 3) + 1) * ((0.57 : ℝ) * (2 : ℝ) ^ (-(2 * Real.sqrt (n : ℝ) - 2))) := mul_nonneg hK' hL have hnL : (0 : ℝ) ≤ 2 * (Real.sqrt (n : ℝ) + 1) * ((0.57 : ℝ) * (2 : ℝ) ^ (-(2 * Real.sqrt (n : ℝ) - 2))) := by have h1 : (0 : ℝ) ≤ Real.sqrt (n : ℝ) + 1 := by positivity exact mul_nonneg (mul_nonneg (by norm_num) h1) hL have hfinal : 2 * T * (Real.exp (Real.sqrt (n : ℝ) / 2 + 1 / 3) + 1) * ((0.57 : ℝ) * (2 : ℝ) ^ (-(2 * Real.sqrt (n : ℝ) - 2))) ≤ Real.exp (1 / 3 : ℝ) * IIMaj (n : ℝ) := by unfold IIMaj nlinarith [hTn1, hKle, hKL, hnL, Real.exp_pos (1 / 3 : ℝ), Real.exp_pos (Real.sqrt (n : ℝ) / 2)] calc ‖VIntegral (Jlag n) (2 * Real.sqrt (n : ℝ)) (-T) T‖ = ‖∫ t in (-T)..T, Jlag n (((2 * Real.sqrt (n : ℝ)) : ℝ) + t * Complex.I)‖ := VIntegral_norm _ ≤ ∫ t in (-T)..T, ‖Jlag n (((2 * Real.sqrt (n : ℝ)) : ℝ) + t * Complex.I)‖ := intervalIntegral.norm_integral_le_integral_norm hTT _ ≤ (T - -T) * ((Real.exp (Real.sqrt (n : ℝ) / 2 + 1 / 3) + 1) * ((0.57 : ℝ) * (2 : ℝ) ^ (-(2 * Real.sqrt (n : ℝ) - 2)))) := hInt _ = 2 * T * (Real.exp (Real.sqrt (n : ℝ) / 2 + 1 / 3) + 1) * ((0.57 : ℝ) * (2 : ℝ) ^ (-(2 * Real.sqrt (n : ℝ) - 2))) := by ring _ ≤ Real.exp (1 / 3 : ℝ) * IIMaj (n : ℝ) := hfinal ``` Concerns: a) `simp at hre` in hs1 (hJcont): hre : (↑(2√n)+↑t·I).re = (1:ℂ).re — simp → `2*√(↑n) = 1` — nlinarith [hsq2]: 2√n ≥ 4 = 1?? contradiction ✓ (linear: 2·(√n atom) = 1 with √n ≥ 2 ✓ linarith actually — nlinarith fine). Hmm — will simp fully reduce (↑(2*√(↑n)) + ↑t * I).re? add_re → (↑(2√n)).re + (↑t·I).re; ofReal_re → 2√n; mul_re → t·I.re − (0)·I.im — (↑t).re = t, (↑t).im = 0 — t*0 − 0*1 = 0; RHS (1:ℂ).re → 1. simp set: [Complex.add_re, Complex.ofReal_re, Complex.mul_re, Complex.I_re, Complex.I_im] ✓ plus Complex.one_re for RHS — add Complex.one_re to the simp set in hs1 (congrArg re of h : point = 1 — RHS is Complex.re 1 — simp needs Complex.one_re ✓). In 6-c-ii's hs1 I used plain `simp at hre` and it worked (gave -5/2 = 1) — simp's default set includes the Complex re/im lemmas ✓ plain simp suffices. Similarly hzne's hre: `simp [Complex.add_re, ...]` — explicit set — should include nothing extra; plain simp worked in zeta_ne_zero_left_vertical for `.re = -5/2` — for `.re = 2*√n`: plain `by simp`? The DL-4 hre used the explicit set. I'll keep the explicit set (harmless). b) hT0 : 0 ≤ T — from hTn : √n ≤ T and √n ≥ 0: linarith [Real.sqrt_nonneg (n:ℝ)] — I wrote `have h := Real.sqrt_nonneg (n:ℝ); linarith` ✓. c) The `(T - -T) * X = 2 * T * X` ring step ✓. d) hfinal's nlinarith — after unfold IIMaj: goal `2*T*K*L' ≤ e13*(2*(√n+1)*(E+1)*0.57*2^(...))` where K = e^{√n/2+1/3}+1, L' = 0.57*2^(...), E = e^{√n/2}. Note: my L' written as `(0.57:ℝ) * (2:ℝ)^(-(2*√n−2))` and IIMaj's tail: `0.57 * 2 ^ (-(2 * Real.sqrt n - 2))` — SAME ✓ (IIMaj's n here is (n:ℝ) since IIMaj : ℝ → ℝ and I apply at (n:ℝ) — the def's `Real.sqrt n` with n := ↑(n:ℕ) — `Real.sqrt (↑n)` ✓ same as my `Real.sqrt (n:ℝ)` ✓). Wait — IIMaj (n : ℝ) — applying IIMaj (n:ℝ) where (n:ℝ) = ↑(n:ℕ) — the def unfolds with `Real.sqrt ↑n` ✓ matches my `Real.sqrt (n : ℝ)` — same term ✓. nlinarith hint analysis (as before): hTn1 : T ≤ √n+1; hKle : K ≤ e13(E+1); hKL : 0 ≤ K*L'; hnL : 0 ≤ 2(√n+1)L'. Goal: 2TKL' ≤ e13·2(√n+1)(E+1)L'. Derivation: 2TKL' ≤ 2(√n+1)KL' [2·(KL'·(√n+1−T))] ≤ 2(√n+1)·e13(E+1)·L' [2(√n+1)L'·(e13(E+1)−K)] = RHS ✓. nlinarith pairwise products: (hTn1, hKL) and (hKle, hnL) ✓ then linear ✓. The ring_nf normalization of monomials T*K*L', K*L'*√n, e13*E*L'*√n... — all distinct monomial atoms ✓ linear after expansion ✓ should close. Risk: nlinarith's preprocessing might not multiply hKle (a ≤-inequality with the exp sides) by hnL — it does consider pairs of ALL hypotheses including provided ones ✓. If it fails, fallback: explicit two-step mul_le_mul. I'll write nlinarith first, fallback if error. e) `Real.add_one_le_exp (1/3 : ℝ) : 1 + 1/3 ≤ exp(1/3)` — linarith → 1 ≤ exp(1/3) ✓. f) hKle's nlinarith [h1, hE]: goal after rw [Real.exp_add]: `exp(√n/2)*exp(1/3) + 1 ≤ exp(1/3)*(exp(√n/2)+1)`. ring_nf RHS: exp(1/3)*exp(√n/2) + exp(1/3). Goal ⟺ E·e13 + 1 ≤ e13·E + e13 ⟺ 1 ≤ e13 ✓ h1. linarith would do (atoms E·e13 and e13·E normalize to same monomial ✓): actually plain `linarith [h1, hE]` — the products E*e13 vs e13*E — ring_nf in linarith normalizes ✓. nlinarith fine too. g) `rw [Real.exp_add]` at goal — Real.exp_add (a b) : exp(a+b) = exp a * exp b — the goal's LHS exp(√n/2 + 1/3) ✓ matches. h) The `show ContinuousAt (fun s => kFun n s * (-logDeriv riemannZeta s)) _` — worked in 6-c-ii ✓. i) hpt's mul_le_mul: mul_le_mul (hk : ‖kFun‖ ≤ K) (hlog : ‖logDeriv‖ ≤ L') (norm_nonneg _ : 0 ≤ ‖logDeriv‖) (hK' : 0 ≤ K) : ‖kFun‖*‖logDeriv‖ ≤ K*L' ✓ — norm_logDeriv_zeta_le_iimZeta hn5 : ‖logDeriv ζ (point)‖ ≤ 0.57 * 2^(-(2√n−2)) — the point: `(((2 * Real.sqrt (n: ℝ)) : ℝ) + t * Complex.I)` ✓ matches ✓. Wait — norm_logDeriv_zeta_le_iimZeta's implicit {t : ℝ} — applying `(norm_logDeriv_zeta_le_iimZeta hn5)` leaves t implicit — determined by unification with the goal (the h₂ slot: ‖logDeriv ζ (point-t)‖ ≤ ...) ✓. j) hI2's `hcontG.intervalIntegrable _ _` : IntervalIntegrable F volume (-T) T ✓. k) In hInt — intervalIntegrable_const's c := K*L' determined by hpt's bound ✓. l) `VIntegral_norm` — its point form `((x : ℝ) + (t : ℝ) * Complex.I)` — hmm — actually now I realize a potential MISMATCH: VIntegral_norm's RHS (as written in §1) is `‖∫ t in y₁..y₂, f ((x : ℝ) + (t : ℝ) * Complex.I)‖` — x : ℝ so (x : ℝ) is just x with a coercion to ℂ at the +: `↑x + ↑t * I`. My calc step: `‖∫ t in (-T)..T, Jlag n (((2 * Real.sqrt (n : ℝ)) : ℝ) + t * Complex.I)‖` — point: `↑(2*√↑n) + ↑t * I` ✓ with x := 2*√(↑n) : ℝ — `↑x = ↑(2*√↑n)` ✓ SAME ✓ — VIntegral_norm applies by rfl-calc ✓ (it worked identically in 6-c-ii). m) hzne's exact: `riemannZeta_ne_zero_of_one_le_re (by rw [hre]; exact h1)` — the hs arg : 1 ≤ (point).re — the by-block: rw [hre] → goal 1 ≤ 2√n — exact h1 ✓. The ⦃s⦄ strict implicit: determined by hs's type — the point is explicit in my by-block's goal ✓ — hmm, strict-implicit: the elaborator needs to determine s from the expected type of hs — which comes from the APPLICATION's expected type: `riemannZeta_ne_zero_of_one_le_re hs : riemannZeta ?s ≠ 0` — my goal (inside hzne after intro t): `riemannZeta (point) ≠ 0` — so ?s := point from the goal ✓ then hs : 1 ≤ (point).re ✓ the by-block elaborates ✓. But I'm using `exact riemannZeta_ne_zero_of_one_le_re (...)` against the goal — the goal fixes ?s ✓. Hmm — wait, in hzne I wrote `have hre ... ; have h1 ... ; exact riemannZeta_ne_zero_of_one_le_re (by rw [hre]; exact h1)`. The `by rw [hre]; exact h1` — the expected type: 1 ≤ (point).re — rw [hre] rewrites (point).re → 2√n: goal 1 ≤ 2√n — exact h1 : 1 ≤ 2√n ✓. n) fun_prop for hpath: `Continuous fun u : ℝ => (↑(2*√↑n) + u * Complex.I)` — same shape as before ✓ (worked in 6-c-ii). o) Positivity for hK' : 0 ≤ exp(...) + 1 ✓ (exp_pos). hL : 0 ≤ 0.57 * 2^(-(2√n−2)) — the power: (2:ℝ)^(-(2*√n−2)) — ℝ^ℝ — rpow — positivity proves 0 ≤ via Real.rpow_nonneg? positivity has an extension for rpow ✓. If the `^` is zpow... the exponent -(2√n−2) : ℝ so rpow ✓ positivity handles ✓. (In DL-6's proof, `dirichletC2_nonneg` etc. — and IIMaj-related proofs in LiSkeleton surely did similar positivity ✓.) p) `Real.exp_pos (1/3 : ℝ)` and `Real.exp_pos (√n/2)` ✓. q) The final calc's `_ = 2 * T * K * L'` ring step: (T - -T) * (K * L') = 2*T*(K*L')... my stated: `2 * T * (Real.exp ...) * ((0.57) * 2^(...))` — note association: (T - -T)*(K*L') vs 2*T*K*L' — ring normalizes ✓ (K*L' parenthesization differs — ring handles ✓). r) Naming consistency for Z2hat/IIMaj references — IIMaj is in LiSkeleton ✓ (def, root-imported ✓). s) One more: `Real.le_sqrt (by norm_num) (by positivity)` — the hyps: (hx : 0 ≤ (2:ℝ)) ✓ norm_num; (hy : 0 ≤ (n:ℝ)) ✓ positivity. Goal 2 ≤ √(↑n) — hmm wait — in hsq2 (I_I theorem) I use the same as the kernel lemma ✓. t) `nlinarith [hnR]` for (2:ℝ)^2 ≤ ↑n — hnR : 100000 ≤ ↑n ✓ 4 ≤ ↑n ✓. Now — one thing to double check in the kernel lemma: `Real.le_sqrt` signature: (hx : 0 ≤ x) (hy : 0 ≤ y) : x ≤ √y ↔ x ^ 2 ≤ y ✓ — rw at goal `2 ≤ √(↑n)` → `(2:ℝ)^2 ≤ ↑n` — nlinarith [hnR] ✓. Also — in the kernel lemma I don't need hsq (0 < √n) — h21 comes from hsq2 via nlinarith ✓. Drop hsq? Real.sqrt_pos.mpr needs 0 < n... keep minimal: I use hsq2, h21, hsqn, hnR. Drop hsq. Wait — Real.sq_sqrt (by linarith : 0 ≤ (n:ℝ)) : √n^2 = n ✓ needs 0 ≤ ↑n — linarith [hnR] ✓. u) hwid's field_simp: `field_simp [hne]` — hne : (point) − 1 ≠ 0. Goal: `(point)/((point) − 1) = 1 + 1/((point) − 1)`. field_simp: cross-multiplies — might produce `(point) = ((point) − 1) + 1`?? or `(point) * ((point)−1) = ((point)−1)*(point)`... Either way `ring` closes ✓ (identities in the variable point). DL-4's hwid ended with `push_cast; ring` — theirs had the hw1 rewrite making forms differ. Mine is uniform — try `field_simp [hne]; ring`; if ring fails (e.g., goal has / leftover), adjust. Actually — safer alternative avoiding field_simp entirely: prove hwid by direct algebra: ```lean have hwid : w / (w - 1) = 1 + 1 / (w - 1) := by rw [show w / (w - 1) = (w - 1 + 1) / (w - 1) by ring_nf... ] ``` Meh — field_simp is standard here; DL-4 used it successfully at the same shape (line 807-813 with the same w/(w−1) = 1+1/(w−1) target — they rw [hw1] first because their RHS was in the (w−1)-as-↑(2√n−1) form; mine stays uniform). field_simp [hne] on `w/(w−1) = 1 + 1/(w−1)`: multiplies through by (w−1): `w = (w−1) + 1`?? — hmm, actually field_simp turns `1 + 1/(w−1)` into `((w−1) + 1)/(w−1)` and then cancels denominators: `w = (w−1) + 1` — ring ✓. v) `Real.exp_nat_mul` — DL-4 line 801: `(Real.exp_nat_mul _ n).symm` : (Real.exp x)^n = Real.exp (↑n * x)?? — the .symm direction: exp_nat_mul (x : ℝ) (n : ℕ) : Real.exp (↑n * x) = (Real.exp x)^n — .symm : (exp x)^n = exp(↑n*x) ✓ — DL-4's calc step `_ = Real.exp ((n:ℝ) * (1/(2√n)))` := (Real.exp_nat_mul _ n).symm ✓ — the `_` for x unified with 1/(2√n) ✓. Mine: same with 1/(2√n−1) ✓. w) `pow_le_pow_left₀` — signature: (ha : 0 ≤ a) (hab : a ≤ b) (n : ℕ) : a ^ n ≤ b ^ n ✓ DL-4 used ✓. x) norm_add_le — `norm_add_le (a b) : ‖a + b‖ ≤ ‖a‖ + ‖b‖` ✓ standard. y) `abs_of_nonneg h21.le` — h21 : 0 < 2√n−1 — h21.le : 0 ≤ ✓. z) `one_div_le_one_div_of_le h21 hnorm_w1` : 1/‖w−1‖ ≤ 1/(2√n−1) ✓. Let me also double-check `Jlag_norm_eq` applicability: Jlag_norm_eq (n) (s) : ‖Jlag n s‖ = ‖kFun n s‖ * ‖logDeriv riemannZeta s‖ ✓ used in hpt via rw [Jlag_norm_eq] — the goal ‖Jlag n (point)‖ ≤ K*L' → ‖kFun n (point)‖*‖logDeriv ζ (point)‖ ≤ K*L' ✓ then mul_le_mul ✓. Alright — also the file header §3 line: mark DONE after success. And the root-module doc comment — append 6-c-iii to the LiSideBounds bullet at the end. One more consideration — hexp's show-rw: `(√n/2 + 1/3) * (2√n − 1) = √n^2 + √n/6 − 1/3` — verify by hand: (√n/2)(2√n) = √n². (√n/2)(−1) = −√n/2. (1/3)(2√n) = 2√n/3. (1/3)(−1) = −1/3. Total: √n² + √n(−1/2 + 2/3) − 1/3 = √n² + √n/6 − 1/3 ✓. ring proves ✓. Then `rw [..., hsqn]` — hsqn : √n^2 = ↑n — rewrites √n^2 → ↑n ✓ goal: ↑n ≤ ↑n + √n/6 − 1/3 — linarith [hsq2] : √n ≥ 2 ⟹ √n/6 ≥ 1/3 ✓ closes ✓. Wait — CAREFUL: rw [hsqn] rewrites `√n^2` — but the goal ALSO has √n/6 — different pattern (√n not √n^2) ✓ only the square rewrites ✓. Also — `mul_one_div` : a * (1/b) = a/b ✓ — the goal LHS (n:ℝ) * (1/(2√n−1)) → (n:ℝ)/(2√n−1) ✓ then div_le_iff₀ h21 ✓. Now write §3. Also update the header comment §3 to DONE after build success, and extend the root-module doc bullet. Let me write the code now.
All templates and signatures confirmed. Writing §3 (6-c-iii) — the F-E kernel lemma + the I_I bound.
tool: Edit
input
{
  "replace_all": false,
  "file_path": "D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiSideBounds.lean",
  "old_string": "    _ \u2264 4 * T * ((2 / 3) * Z2hat T + 10.5 * Real.log (T + 2) + 48) +\n          40 * 1.3352 := by\n        linarith [htailInt, hmidInt, hnn, hlognn]\n\nend NguyenRH1",
  "new_string": "    _ \u2264 4 * T * ((2 / 3) * Z2hat T + 10.5 * Real.log (T + 2) + 48) +\n          40 * 1.3352 := by\n        linarith [htailInt, hmidInt, hnn, hlognn]\n\n/-! ## 3. 6-c-iii: I_I \u2014 the right vertical \u03c3\u2081 = 2\u221an (F-E form) -/\n\n/-- **The F-E kernel lemma** (D-074 \u00a79): the uniform-in-t kernel bound at\nthe right vertical \u03c3 = 2\u221an \u2014 \u2016kFun n (2\u221an+it)\u2016 \u2264 e^{\u221an/2+1/3}+1 for n \u2265 4.\nThe honest charge: |w/(w\u22121)| = |1+1/(w\u22121)| \u2264 1 + 1/\u2016w\u22121\u2016 \u2264 1 + 1/(2\u221an\u22121)\n(plain triangle + \u2016w\u22121\u2016 \u2265 re(w\u22121) = 2\u221an\u22121 \u2014 the worst case is t = 0,\nrepairing the off-by-one of the t\u00b2 \u2265 n / re \u2265 2\u221an+1 variants), then\n(1+1/(2\u221an\u22121))\u207f \u2264 exp(n/(2\u221an\u22121)) \u2264 exp(\u221an/2+1/3) \u27fa n \u2265 4. -/\ntheorem kFun_right_vertical_norm_le_exp_sqrt_add_third {n : \u2115} (hn : 4 \u2264 n)\n    (t : \u211d) :\n    \u2016kFun n (((2 * Real.sqrt (n : \u211d)) : \u211d) + t * Complex.I)\u2016 \u2264\n      Real.exp (Real.sqrt (n : \u211d) / 2 + 1 / 3) + 1 := by\n  have hnR : (4 : \u211d) \u2264 (n : \u211d) := by exact_mod_cast hn\n  have hsq2 : (2 : \u211d) \u2264 Real.sqrt (n : \u211d) := by\n    rw [Real.le_sqrt (by norm_num) (by positivity)]\n    nlinarith [hnR]\n  have h21 : (0 : \u211d) < 2 * Real.sqrt (n : \u211d) - 1 := by nlinarith [hsq2]\n  have hsqn : Real.sqrt (n : \u211d) ^ 2 = (n : \u211d) := Real.sq_sqrt (by linarith)\n  -- w \u2212 1 \u2260 0 and \u2016w \u2212 1\u2016 \u2265 2\u221an \u2212 1\n  have hre : ((((2 * Real.sqrt (n : \u211d)) : \u211d) + t * Complex.I) - 1).re =\n      2 * Real.sqrt (n : \u211d) - 1 := by\n    simp [Complex.sub_re, Complex.add_re, Complex.ofReal_re, Complex.mul_re,\n      Complex.I_re, Complex.I_im, Complex.one_re]\n  have hne : (((2 * Real.sqrt (n : \u211d)) : \u211d) + t * Complex.I) - 1 \u2260 0 := by\n    intro h\n    have h0 : ((((2 * Real.sqrt (n : \u211d)) : \u211d) + t * Complex.I) - 1).re = 0 := by\n      rw [h]; simp\n    rw [hre] at h0\n    nlinarith [h21]\n  have hnorm_w1 : 2 * Real.sqrt (n : \u211d) - 1 \u2264\n      \u2016(((2 * Real.sqrt (n : \u211d)) : \u211d) + t * Complex.I) - 1\u2016 := by\n    have h1 := abs_re_le_norm ((((2 * Real.sqrt (n : \u211d)) : \u211d) + t * Complex.I) - 1)\n    rw [hre, abs_of_nonneg h21.le] at h1\n    exact h1\n  -- the base bound \u20161 + 1/(w\u22121)\u2016 \u2264 1 + 1/(2\u221an\u22121)\n  have hinv : 1 / \u2016(((2 * Real.sqrt (n : \u211d)) : \u211d) + t * Complex.I) - 1\u2016 \u2264\n      1 / (2 * Real.sqrt (n : \u211d) - 1) :=\n    one_div_le_one_div_of_le h21 hnorm_w1\n  have hbase : \u20161 + 1 / ((((2 * Real.sqrt (n : \u211d)) : \u211d) + t * Complex.I) - 1)\u2016 \u2264\n      1 + 1 / (2 * Real.sqrt (n : \u211d) - 1) := by\n    calc \u20161 + 1 / ((((2 * Real.sqrt (n : \u211d)) : \u211d) + t * Complex.I) - 1)\u2016\n        \u2264 \u2016(1 : \u2102)\u2016 + \u20161 / ((((2 * Real.sqrt (n : \u211d)) : \u211d) + t * Complex.I) - 1)\u2016 :=\n          norm_add_le _ _\n      _ = 1 + 1 / \u2016(((2 * Real.sqrt (n : \u211d)) : \u211d) + t * Complex.I) - 1\u2016 := by\n          rw [norm_one, norm_div, norm_one]\n      _ \u2264 1 + 1 / (2 * Real.sqrt (n : \u211d) - 1) := by linarith [hinv]\n  have hbasepos : (0 : \u211d) \u2264 1 + 1 / (2 * Real.sqrt (n : \u211d) - 1) := by positivity\n  -- the power + exp chain: (1+1/(2\u221an\u22121))\u207f \u2264 exp(n/(2\u221an\u22121)) \u2264 exp(\u221an/2+1/3)\n  have hexp : (n : \u211d) * (1 / (2 * Real.sqrt (n : \u211d) - 1)) \u2264\n      Real.sqrt (n : \u211d) / 2 + 1 / 3 := by\n    rw [mul_one_div, div_le_iff\u2080 h21,\n      show (Real.sqrt (n : \u211d) / 2 + 1 / 3) * (2 * Real.sqrt (n : \u211d) - 1) =\n        Real.sqrt (n : \u211d) ^ 2 + Real.sqrt (n : \u211d) / 6 - 1 / 3 := by ring, hsqn]\n    linarith [hsq2]\n  have hpow : (1 + 1 / (2 * Real.sqrt (n : \u211d) - 1)) ^ n \u2264\n      Real.exp (Real.sqrt (n : \u211d) / 2 + 1 / 3) := by\n    calc (1 + 1 / (2 * Real.sqrt (n : \u211d) - 1)) ^ n\n        \u2264 (Real.exp (1 / (2 * Real.sqrt (n : \u211d) - 1))) ^ n :=\n          pow_le_pow_left\u2080 hbasepos (Real.add_one_le_exp _) n\n      _ = Real.exp ((n : \u211d) * (1 / (2 * Real.sqrt (n : \u211d) - 1))) :=\n          (Real.exp_nat_mul _ n).symm\n      _ \u2264 Real.exp (Real.sqrt (n : \u211d) / 2 + 1 / 3) := Real.exp_le_exp.mpr hexp\n  have hpow1 : \u20161 + 1 / ((((2 * Real.sqrt (n : \u211d)) : \u211d) + t * Complex.I) - 1)\u2016 ^ n \u2264\n      (1 + 1 / (2 * Real.sqrt (n : \u211d) - 1)) ^ n :=\n    pow_le_pow_left\u2080 (norm_nonneg _) hbase n\n  have hwid : (((2 * Real.sqrt (n : \u211d)) : \u211d) + t * Complex.I) /\n      ((((2 * Real.sqrt (n : \u211d)) : \u211d) + t * Complex.I) - 1) =\n      1 + 1 / ((((2 * Real.sqrt (n : \u211d)) : \u211d) + t * Complex.I) - 1) := by\n    field_simp [hne]\n    ring\n  unfold kFun\n  rw [hwid]\n  calc \u2016(1 + 1 / ((((2 * Real.sqrt (n : \u211d)) : \u211d) + t * Complex.I) - 1)) ^ n - 1\u2016\n      \u2264 \u2016(1 + 1 / ((((2 * Real.sqrt (n : \u211d)) : \u211d) + t * Complex.I) - 1)) ^ n\u2016 +\n        \u2016(1 : \u2102)\u2016 := norm_sub_le _ _\n    _ = \u20161 + 1 / ((((2 * Real.sqrt (n : \u211d)) : \u211d) + t * Complex.I) - 1)\u2016 ^ n + 1 := by\n        rw [norm_pow, norm_one]\n    _ \u2264 Real.exp (Real.sqrt (n : \u211d) / 2 + 1 / 3) + 1 := by\n        linarith [hpow1.trans hpow]\n\n/-- **6-c-iii (I_I, the right vertical \u03c3\u2081 = 2\u221an, F-E form)**: the full-side\nbound \u2016V\u2016 \u2264 e^{1/3}\u00b7IIMaj(n). The F-E kernel (uniform in t) + the DL-6\ndirect Dirichlet bound \u2016\u03b6\u2032/\u03b6\u2016 \u2264 0.57\u00b72^{\u2212(2\u221an\u22122)}; the comparison\ne^{\u221an/2+1/3}+1 \u2264 e^{1/3}(e^{\u221an/2}+1) holds since 1 \u2264 e^{1/3}. -/\ntheorem vIntegral_Jlag_right_le_exp_third_IIMaj {n : \u2115} (hn : 100000 \u2264 n)\n    {T : \u211d} (hTn : Real.sqrt (n : \u211d) \u2264 T) (hTn1 : T \u2264 Real.sqrt (n : \u211d) + 1) :\n    \u2016VIntegral (Jlag n) (2 * Real.sqrt (n : \u211d)) (-T) T\u2016 \u2264\n      Real.exp (1 / 3 : \u211d) * IIMaj (n : \u211d) := by\n  have hnR : (100000 : \u211d) \u2264 (n : \u211d) := by exact_mod_cast hn\n  have hn4 : 4 \u2264 n := by omega\n  have hn5 : 5 \u2264 n := by omega\n  have hsq2 : (2 : \u211d) \u2264 Real.sqrt (n : \u211d) := by\n    rw [Real.le_sqrt (by norm_num) (by positivity)]\n    nlinarith [hnR]\n  have hT0 : (0 : \u211d) \u2264 T := by\n    have h := Real.sqrt_nonneg (n : \u211d)\n    linarith\n  have hTT : -T \u2264 T := by linarith\n  -- zero-freeness + continuity on the right vertical (re = 2\u221an > 1)\n  have hzne : \u2200 t : \u211d,\n      riemannZeta (((2 * Real.sqrt (n : \u211d)) : \u211d) + t * Complex.I) \u2260 0 := by\n    intro t\n    have hre : (((2 * Real.sqrt (n : \u211d)) : \u211d) + t * Complex.I).re =\n        2 * Real.sqrt (n : \u211d) := by\n      simp [Complex.add_re, Complex.ofReal_re, Complex.mul_re, Complex.I_re, Complex.I_im]\n    have h1 : (1 : \u211d) \u2264 2 * Real.sqrt (n : \u211d) := by nlinarith [hsq2]\n    exact riemannZeta_ne_zero_of_one_le_re (by rw [hre]; exact h1)\n  have hJcont : \u2200 t : \u211d, ContinuousAt (Jlag n)\n      (((2 * Real.sqrt (n : \u211d)) : \u211d) + t * Complex.I) := by\n    intro t\n    have hs1 : (((2 * Real.sqrt (n : \u211d)) : \u211d) + t * Complex.I) \u2260 1 := by\n      intro h\n      have hre := congrArg Complex.re h\n      simp at hre\n      nlinarith [hsq2]\n    have hA := (kFun_differentiableAt (n := n) hs1).continuousAt\n    have hB := continuousAt_logDeriv_zeta hs1 (hzne t)\n    show ContinuousAt (fun s => kFun n s * (-logDeriv riemannZeta s)) _\n    exact hA.mul hB.neg\n  have hpath : Continuous fun u : \u211d =>\n      (((2 * Real.sqrt (n : \u211d)) : \u211d) + u * Complex.I) := by fun_prop\n  have hcontJ : Continuous fun t : \u211d =>\n      Jlag n (((2 * Real.sqrt (n : \u211d)) : \u211d) + t * Complex.I) := by\n    rw [continuous_iff_continuousAt]\n    intro t\n    have hf : ContinuousAt (fun u : \u211d =>\n        (((2 * Real.sqrt (n : \u211d)) : \u211d) + u * Complex.I)) t := hpath.continuousAt\n    have hcomp : ContinuousAt\n        (Jlag n \u2218 fun u : \u211d => (((2 * Real.sqrt (n : \u211d)) : \u211d) + u * Complex.I)) t :=\n      ContinuousAt.comp (g := Jlag n)\n        (f := fun u : \u211d => (((2 * Real.sqrt (n : \u211d)) : \u211d) + u * Complex.I)) (x := t)\n        (hJcont t) hf\n    exact hcomp\n  have hcontG : Continuous fun t : \u211d =>\n      \u2016Jlag n (((2 * Real.sqrt (n : \u211d)) : \u211d) + t * Complex.I)\u2016 :=\n    continuous_norm.comp hcontJ\n  have hI2 : IntervalIntegrable\n      (fun t : \u211d => \u2016Jlag n (((2 * Real.sqrt (n : \u211d)) : \u211d) + t * Complex.I)\u2016)\n      MeasureTheory.volume (-T) T := hcontG.intervalIntegrable _ _\n  -- the comparison e^{\u221an/2+1/3}+1 \u2264 e^{1/3}(e^{\u221an/2}+1) (since 1 \u2264 e^{1/3})\n  have hKle : Real.exp (Real.sqrt (n : \u211d) / 2 + 1 / 3) + 1 \u2264\n      Real.exp (1 / 3 : \u211d) * (Real.exp (Real.sqrt (n : \u211d) / 2) + 1) := by\n    rw [Real.exp_add]\n    have h1 : (1 : \u211d) \u2264 Real.exp (1 / 3 : \u211d) := by\n      have h := Real.add_one_le_exp (1 / 3 : \u211d)\n      linarith\n    have hE := Real.exp_pos (Real.sqrt (n : \u211d) / 2)\n    nlinarith [h1, hE]\n  -- the pointwise bound\n  have hK' : (0 : \u211d) \u2264 Real.exp (Real.sqrt (n : \u211d) / 2 + 1 / 3) + 1 := by positivity\n  have hL : (0 : \u211d) \u2264 (0.57 : \u211d) * (2 : \u211d) ^ (-(2 * Real.sqrt (n : \u211d) - 2)) := by\n    positivity\n  have hpt : \u2200 t : \u211d, t \u2208 Set.Icc (-T) T \u2192\n      \u2016Jlag n (((2 * Real.sqrt (n : \u211d)) : \u211d) + t * Complex.I)\u2016 \u2264\n        (Real.exp (Real.sqrt (n : \u211d) / 2 + 1 / 3) + 1) *\n          ((0.57 : \u211d) * (2 : \u211d) ^ (-(2 * Real.sqrt (n : \u211d) - 2))) := by\n    intro t _\n    rw [Jlag_norm_eq]\n    exact mul_le_mul (kFun_right_vertical_norm_le_exp_sqrt_add_third hn4 t)\n      (norm_logDeriv_zeta_le_iimZeta hn5) (norm_nonneg _) hK'\n  have hInt : \u222b t in (-T)..T,\n      \u2016Jlag n (((2 * Real.sqrt (n : \u211d)) : \u211d) + t * Complex.I)\u2016 \u2264\n        (T - -T) * ((Real.exp (Real.sqrt (n : \u211d) / 2 + 1 / 3) + 1) *\n          ((0.57 : \u211d) * (2 : \u211d) ^ (-(2 * Real.sqrt (n : \u211d) - 2)))) := by\n    have h := intervalIntegral.integral_mono_on hTT hI2 intervalIntegrable_const\n      (fun t ht => hpt t ht)\n    rwa [intervalIntegral.integral_const, smul_eq_mul] at h\n  -- the final comparison 2T\u00b7K\u00b7L \u2264 e^{1/3}\u00b7IIMaj (T \u2264 \u221an+1, hKle)\n  have hKL : (0 : \u211d) \u2264 (Real.exp (Real.sqrt (n : \u211d) / 2 + 1 / 3) + 1) *\n      ((0.57 : \u211d) * (2 : \u211d) ^ (-(2 * Real.sqrt (n : \u211d) - 2))) :=\n    mul_nonneg hK' hL\n  have hnL : (0 : \u211d) \u2264 2 * (Real.sqrt (n : \u211d) + 1) *\n      ((0.57 : \u211d) * (2 : \u211d) ^ (-(2 * Real.sqrt (n : \u211d) - 2))) := by\n    have h1 : (0 : \u211d) \u2264 Real.sqrt (n : \u211d) + 1 := by positivity\n    exact mul_nonneg (mul_nonneg (by norm_num) h1) hL\n  have hfinal : 2 * T * (Real.exp (Real.sqrt (n : \u211d) / 2 + 1 / 3) + 1) *\n      ((0.57 : \u211d) * (2 : \u211d) ^ (-(2 * Real.sqrt (n : \u211d) - 2))) \u2264\n        Real.exp (1 / 3 : \u211d) * IIMaj (n : \u211d) := by\n    unfold IIMaj\n    nlinarith [hTn1, hKle, hKL, hnL, Real.exp_pos (1 / 3 : \u211d),\n      Real.exp_pos (Real.sqrt (n : \u211d) / 2)]\n  calc \u2016VIntegral (Jlag n) (2 * Real.sqrt (n : \u211d)) (-T) T\u2016\n      = \u2016\u222b t in (-T)..T, Jlag n (((2 * Real.sqrt (n : \u211d)) : \u211d) + t * Complex.I)\u2016 :=\n        VIntegral_norm\n    _ \u2264 \u222b t in (-T)..T, \u2016Jlag n (((2 * Real.sqrt (n : \u211d)) : \u211d) + t * Complex.I)\u2016 :=\n        intervalIntegral.norm_integral_le_integral_norm hTT\n    _ \u2264 (T - -T) * ((Real.exp (Real.sqrt (n : \u211d) / 2 + 1 / 3) + 1) *\n          ((0.57 : \u211d) * (2 : \u211d) ^ (-(2 * Real.sqrt (n : \u211d) - 2)))) := hInt\n    _ = 2 * T * (Real.exp (Real.sqrt (n : \u211d) / 2 + 1 / 3) + 1) *\n          ((0.57 : \u211d) * (2 : \u211d) ^ (-(2 * Real.sqrt (n : \u211d) - 2))) := by ring\n    _ \u2264 Real.exp (1 / 3 : \u211d) * IIMaj (n : \u211d) := hfinal\n\nend NguyenRH1"
}
#2275 assistant 2026-08-26 18:02:45.480 UTC · 1346→1065 tok open ⤢
thinking (2229 chars)
Two issues: 1. **Line 459:61 parse error**: "unexpected token ':='; expected 'by' or 'from'" — inside hexp's rw chain: `rw [mul_one_div, div_le_iff₀ h21, show ... := by ring, hsqn]` — the `show X := by ring` inside a rw list — the parser chokes on `:= by ring` inside the bracket list? The error at 459:61 — the `show (...) = (...) := by ring` — hmm, `show` in term mode: `show P from proof` or `show P by tac` — the correct term-mode syntax is `show P from ...` or `(show P by ring)`! `show P := by ring` is NOT valid — `show P by ring` (no :=). In tactic blocks I write `have h : P := by ring`; inside a term, `show P from ...`. For a rw-list entry with a tactic proof: `(show X = Y by ring)` — no `:=`. Wait — actually `show P by tac` IS valid term syntax: `show P by ring`. Let me fix: `show (√n/2 + 1/3) * (2√n − 1) = √n^2 + √n/6 − 1/3 by ring`. The error cascade: 459 parse error → the show-equation unproved (458 "unsolved goals" — the show's goal displayed) → 456 (the hexp goal `↑n ≤ √↑n^2 + √↑n/6 − 1/3` unproved — because the rw chain broke and linarith [hsq2] ran on a goal that still had √n² — wait, the displayed goal at 456 IS `↑n ≤ √↑n^2 + √↑n/6 − 1/3` — the show-rw DID fire (the goal has the expanded form), but hsqn's rw didn't fire (the goal still has √↑n^2)! Because the rw chain after the show-entry broke? The `hsqn` entry — after the parse error at the show, the chain `..., show ... := by ring, hsqn]` — the parse error killed the hsqn entry. And linarith [hsq2] couldn't close `↑n ≤ √↑n^2 + √↑n/6 − 1/3` because √n² needs hsqn: linarith treats √↑n^2 as atom (√↑n)^2 — nonlinear — can't relate to ↑n without the equation. So fixing the show-syntax should fix all three. Also 418:54 — the final theorem goal unproved (cascade — hexp broken → hpow broken → everything after broke). 2. So single fix: `:= by ring` → `by ring` in the show inside the rw chain. Wait — actually the syntax inside a rw list: `rw [show A = B from proof]` or `rw [show A = B by tac]`. The `:= ` is only for have/let. My text: `show (Real.sqrt (n : ℝ) / 2 + 1 / 3) * (2 * Real.sqrt (n : ℝ) - 1) = Real.sqrt (n : ℝ) ^ 2 + Real.sqrt (n : ℝ) / 6 - 1 / 3 := by ring` — the `:=` breaks it. Change to `show ... by ring` — drop `:=`.
One syntax error: inside the `rw` list, the term-mode `show` takes `by ring`, not `:= by ring`. Cascade explains the rest.
tool: Edit
input
{
  "replace_all": false,
  "file_path": "D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiSideBounds.lean",
  "old_string": "    rw [mul_one_div, div_le_iff\u2080 h21,\n      show (Real.sqrt (n : \u211d) / 2 + 1 / 3) * (2 * Real.sqrt (n : \u211d) - 1) =\n        Real.sqrt (n : \u211d) ^ 2 + Real.sqrt (n : \u211d) / 6 - 1 / 3 := by ring, hsqn]",
  "new_string": "    rw [mul_one_div, div_le_iff\u2080 h21,\n      show (Real.sqrt (n : \u211d) / 2 + 1 / 3) * (2 * Real.sqrt (n : \u211d) - 1) =\n        Real.sqrt (n : \u211d) ^ 2 + Real.sqrt (n : \u211d) / 6 - 1 / 3 by ring, hsqn]"
}
#2276 assistant 2026-08-26 18:04:12.650 UTC · 475→1278 tok open ⤢
thinking (2056 chars)
`Real.add_one_le_exp` has signature `x + 1 ≤ Real.exp x` — NOT `1 + x ≤ exp x`! The lemma is `Real.add_one_le_exp (x : ℝ) : x + 1 ≤ Real.exp x`. My goal needs `1 + 1/(2√n−1) ≤ exp(1/(2√n−1))` — the LHS is `1 + y` but the lemma gives `y + 1` — commuted! Fix: `add_comm` rewrite or provide the commuted form: `pow_le_pow_left₀ hbasepos (by rw [add_comm]; exact Real.add_one_le_exp _) n`? Hmm — cleaner: `(Real.add_one_le_exp (1 / (2 * Real.sqrt (n : ℝ) - 1)))` and note the type mismatch is only the argument order: `1/(2√n−1) + 1 ≤ exp(...)` vs needed `1 + 1/(2√n−1) ≤ ...` — defeq? `a + 1` vs `1 + a` — NOT defeq (add_comm is a theorem, not defeq). So I need to bridge: ```lean pow_le_pow_left₀ hbasepos (by simpa [add_comm] using Real.add_one_le_exp (1 / (2 * Real.sqrt (n : ℝ) - 1))) n ``` Or restructure hbase to produce `1/(2√n−1) + 1`? No — hbase's form is `1 + 1/(2√n−1)` (natural from the triangle). Simplest: bridge with add_comm via have: ```lean have h1le : 1 + 1 / (2 * Real.sqrt (n : ℝ) - 1) ≤ Real.exp (1 / (2 * Real.sqrt (n : ℝ) - 1)) := by have h := Real.add_one_le_exp (1 / (2 * Real.sqrt (n : ℝ) - 1)) rwa [add_comm] at h ``` Hmm — `rwa [add_comm] at h` — add_comm : a + b = b + a — rw [add_comm] at h rewrites the FIRST a+b found — h : 1/(2√n−1) + 1 ≤ exp(1/(2√n−1)) — first + is `1/(2√n−1) + 1` → `1 + 1/(2√n−1)` ✓ but also might rewrite inside exp's argument `1 / (2 * √n − 1)`?? That has `− 1` (sub, not add) ✓ no. And `2 * √↑n - 1` — sub ✓. So safe: rw [add_comm] at h rewrites only the LHS sum ✓. But to be extra safe: `rw [add_comm (1 / (2 * Real.sqrt (n : ℝ) - 1)) 1] at h` — pinned ✓. Actually even simpler: `linarith [Real.add_one_le_exp (1 / (2 * Real.sqrt (n : ℝ) - 1))]` — linarith proves 1 + y ≤ e from y + 1 ≤ e ✓ (commutes add ✓ linear). Use that: ```lean have h1le : 1 + 1 / (2 * Real.sqrt (n : ℝ) - 1) ≤ Real.exp (1 / (2 * Real.sqrt (n : ℝ) - 1)) := by linarith [Real.add_one_le_exp (1 / (2 * Real.sqrt (n : ℝ) - 1))] ``` Then `pow_le_pow_left₀ hbasepos h1le n`.
`Real.add_one_le_exp` gives `x + 1 ≤ exp x` (not `1 + x`). Bridging with linarith.
tool: Edit
input
{
  "replace_all": false,
  "file_path": "D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiSideBounds.lean",
  "old_string": "  have hpow : (1 + 1 / (2 * Real.sqrt (n : \u211d) - 1)) ^ n \u2264\n      Real.exp (Real.sqrt (n : \u211d) / 2 + 1 / 3) := by\n    calc (1 + 1 / (2 * Real.sqrt (n : \u211d) - 1)) ^ n\n        \u2264 (Real.exp (1 / (2 * Real.sqrt (n : \u211d) - 1))) ^ n :=\n          pow_le_pow_left\u2080 hbasepos (Real.add_one_le_exp _) n",
  "new_string": "  have h1le : 1 + 1 / (2 * Real.sqrt (n : \u211d) - 1) \u2264\n      Real.exp (1 / (2 * Real.sqrt (n : \u211d) - 1)) := by\n    linarith [Real.add_one_le_exp (1 / (2 * Real.sqrt (n : \u211d) - 1))]\n  have hpow : (1 + 1 / (2 * Real.sqrt (n : \u211d) - 1)) ^ n \u2264\n      Real.exp (Real.sqrt (n : \u211d) / 2 + 1 / 3) := by\n    calc (1 + 1 / (2 * Real.sqrt (n : \u211d) - 1)) ^ n\n        \u2264 (Real.exp (1 / (2 * Real.sqrt (n : \u211d) - 1))) ^ n :=\n          pow_le_pow_left\u2080 hbasepos h1le n"
}
#2277 assistant 2026-08-26 18:05:14.984 UTC · 213→303 tok open ⤢
thinking (141 chars)
CLEAN — zero output means no errors, no warnings. §3 (6-c-iii) elaborates green. Now: mark §3 DONE in the header, full build, audit, commit.
§3 clean. Marking the header, full build, audit, commit.
tool: Edit
input
{
  "replace_all": false,
  "file_path": "D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiSideBounds.lean",
  "old_string": "\u00a72 (6-c-ii) DONE: I_III \u2014 the left vertical \u03c3 = \u22125/2.\n\u00a73 (6-c-iii): I_I \u2014 the right vertical \u03c3\u2081 = 2\u221an (F-E form: \u2264 e^{1/3}\u00b7IIMaj).",
  "new_string": "\u00a72 (6-c-ii) DONE: I_III \u2014 the left vertical \u03c3 = \u22125/2.\n\u00a73 (6-c-iii) DONE: I_I \u2014 the right vertical \u03c3\u2081 = 2\u221an (F-E form: \u2264 e^{1/3}\u00b7IIMaj)."
}
#2278 assistant 2026-08-26 18:10:58.931 UTC · 941→6549 tok open ⤢
thinking (12940 chars)
6-c-iii committed (9bd8ba9). Now 6-c-iv — the mid-horizontal segment machine. This is the most intricate of the side bounds. The recipe from STATE.md: **6-c-iv RECIPE (mid-horizontal [−5/2,3] at height ±T):** per-zero integral ∫_{−5/2}^{3} dσ/√((σ−β)²+δ²) — FTC with the DIRECT antiderivative F(σ) := Real.log((σ−β) + Real.sqrt((σ−β)²+δ²)) (Real.arcsinh does NOT exist here; F′ = integrand exactly: g := (σ−β)+√(h(σ)) > 0 since √h > |σ−β|; HasDerivAt via hasDerivAt_id.sub/Real.hasDerivAt_sqrt/Real.hasDerivAt_log + the identity g′ = g/√h; integral_eq_sub_of_hasDerivAt). Bound: F(3)−F(−5/2) = log(num/den) with num ≤ 2(3−β)+δ ≤ 6+δ, den = (−u+√(u²+δ²)) = δ²/(u+√(u²+δ²)) ≥ δ²/(7+δ) (u := β+5/2 ≤ 7/2) ⟹ ∫ ≤ log(6+δ)+log(7+δ)+2·log(1/δ) ≤ 2·log(16.5/δ) (δ ≤ 1 from mem_nearZerosSub; log 7 ≤ 1.946, log 8 ≤ 2.08, 2·log 16.5 ≥ 5.606). Per-zero ≤ 2·log(16.5/c_clear) with c_clear = 1/(9·C0hat(√n+2)) (the wide pigeonhole: nearZerosSub T's zeros have γ ∈ (T−1,T+1) ⊆ (√n−1,√n+2) ✓). Swap ∫Σ = Σ∫ (integral_finsetSum — the deprecated integral_finset_sum's new name; per-term integrability by continuity, denominator > 0). Σ mult ≤ Z2hat T (nearZerosSub_mult_sum_le hZC2, T ≥ 10) ⟹ near-integral ≤ 2·Z2hat(T)·log(148.5·C0hat(√n+2)) =: NI. R-part: capstone per σ (him : s.im = T by simp; hζ : zeta_ne_zero_horizontal hTwide) ⟹ ∫ ‖R‖ ≤ 5.5·(10.5·log(T+2)+48). Kernel e⁴+1 (kFun_shift_norm_le_exp_four_add_one at s := w−1: re s = σ−1 ≤ 3 ⟺ σ ≤ 4 ✓ covers [−5/2,3]; n ≤ T² from √n ≤ T). Assembly per side: ‖H(−5/2→3, ±T)‖ ≤ (e⁴+1)·(NI + 5.5(10.5 log+48)); the bottom (−T) by the conj trick (intervalIntegral_conj + Jlag_conj, DL-5 pattern). So the deliverable: two theorems (top at +T, bottom at −T): ```lean theorem hIntegral_Jlag_mid_norm_le (hZC : ZeroCountProp) (hZC2 : ZeroCountProp2) (hNLZ : NoLowZeros) {n : ℕ} (hn : 100000 ≤ n) {T : ℝ} (hTwide : ...) (hTn : √n ≤ T) (hTn1 : T ≤ √n+1) : ‖HIntegral (Jlag n) (-5/2) 3 T‖ ≤ (Real.exp 4 + 1) * (2 * Z2hat T * Real.log (148.5 * C0hat (Real.sqrt n + 2)) + (3 - -5/2) * (10.5 * Real.log (T + 2) + 48)) ``` Wait — the R-part: ∫_{-5/2}^{3} ‖R‖ dσ ≤ (3 − (−5/2))·(10.5 log(T+2)+48) = 5.5·(10.5 log + 48) — "5.5" is the length (3 − (−5/2)) = 11/2 = 5.5 ✓. Hmm, and the total assembly per side: ‖H‖ ≤ (e⁴+1)·(NI + 5.5·(10.5·log(T+2)+48)). Let me carefully map the math: ‖Jlag n (σ + iT)‖ = ‖kFun n (σ+iT)‖ · ‖logDeriv ζ (σ+iT)‖ ≤ (e⁴+1)·‖logDeriv ζ (σ+iT)‖. Wait — the kernel: kFun n w where w = σ + iT. kFun n w = (w/(w−1))ⁿ − 1. The ii′-4 lemma `kFun_shift_norm_le_exp_four_add_one` : for s with re s ≤ 3, n ≤ (im s)²: ‖kFun n (1 + s)‖ ≤ e⁴+1? Let me check the exact statement: from the NGUYEN.md entry: "kFun_shift_norm_le_exp_four_add_one (horizontals, re s ≤ 3, n ≤ im²)". The recipe: "Kernel e⁴+1 (kFun_shift_norm_le_exp_four_add_one at s := w−1: re s = σ−1 ≤ 3 ⟺ σ ≤ 4 ✓ covers [−5/2,3]; n ≤ T² from √n ≤ T)". So the lemma is about kFun n (s+1)? — re(w−1) = σ−1 ≤ 2 when σ ≤ 3 ✓ (σ−1 ≤ 2 ≤ 3 ✓). Let me check the exact signature. ‖logDeriv ζ (σ+iT)‖ ≤ ‖logDeriv ζ (σ+iT) − Σ_near m/(s−ρ)‖ + ‖Σ_near m/(s−ρ)‖ (norm_le_insert' again). R-part: ‖logDeriv − Σ‖ ≤ 10.5·log(T+2)+48 by the 5f capstone (hσlo : −5/2 ≤ re ✓ hσhi : re ≤ 3 ✓ — EXACTLY the capstone's strip! him : im = T, hζ : zeta_ne_zero_horizontal). Note the capstone needs t := T ≥ 10 ✓ (T ≥ 316). Near-sum part: ‖Σ_{ρ ∈ nearZerosSub T} m/(s−ρ)‖ ≤ Σ m/‖s−ρ‖. Per zero ρ = β + iδ' with |δ' − T| < 1 (mem_nearZerosSub — wait, nearZerosSub T's zeros have |γ − T| < 1): ‖s − ρ‖ = ‖(σ−β) + i(T−γ)‖ ≥ ... the integral over σ of m/√((σ−β)² + (T−γ)²). So ∫_{-5/2}^{3} ‖Σ‖ ≤ Σ m·∫_{-5/2}^{3} dσ/√((σ−β)²+δ²) where δ := |T − γ| (δ > 0 by zeta_ne_zero_horizontal? — hmm, δ = |T − γ| > 0 because T avoids all zeros with clearance c_clear: |γ − T| ≥ c_clear for γ ∈ (√n−1, √n+2) — the wide pigeonhole hTwide). The per-zero integral bound: ∫_{-5/2}^{3} dσ/√((σ−β)²+δ²) ≤ 2·log(16.5/δ) where the antiderivative F(σ) = log((σ−β) + √((σ−β)²+δ²)). F(3) − F(−5/2): - F(3) = log((3−β) + √((3−β)²+δ²)) ≤ log(2(3−β)+δ) ≤ log(6+δ) — since √(x²+δ²) ≤ x+δ for x ≥ 0 (and 3−β ≥ 2 since β < 1... wait β ∈ (0,1) so 3−β ∈ (2,3), x := 3−β > 0 ✓ √(x²+δ²) ≤ x + δ ✓ so (3−β)+√ ≤ 2(3−β)+δ ≤ 6+δ ✓ — and log is monotone. Also need the argument positive ✓.) - F(−5/2) = log((−5/2−β) + √((5/2+β)²+δ²)). With u := β + 5/2 ∈ (5/2, 7/2): argument = −u + √(u²+δ²) = δ²/(u + √(u²+δ²)) ≥ δ²/(7+δ) — since u + √(u²+δ²) ≤ u + u + δ = 2u+δ ≤ 7+δ ✓. So −F(−5/2) = log(1/(−u+√(u²+δ²))) ≤ log((7+δ)/δ²) = log(7+δ) + 2·log(1/δ). - Total: ∫ ≤ log(6+δ) + log(7+δ) + 2·log(1/δ) ≤ 2·log(16.5/δ) ⟺ log(6+δ) + log(7+δ) ≤ 2·log(16.5) = log(272.25). With δ ≤ 1: (6+δ)(7+δ) ≤ 7·8 = 56 ≤ 272.25 ✓ — hmm, the recipe says "log(6+δ)+log(7+δ)+2·log(1/δ) ≤ 2·log(16.5/δ) (δ ≤ 1 from mem_nearZerosSub; log 7 ≤ 1.946, log 8 ≤ 2.08, 2·log 16.5 ≥ 5.606)". So the route: log(6+δ) ≤ log 7 ≤ 1.946 (δ ≤ 1 ⟹ 6+δ ≤ 7), log(7+δ) ≤ log 8 ≤ 2.08 (δ ≤ 1 ⟹ 7+δ ≤ 8). Sum: ≤ 1.946 + 2.08 = 4.026 ≤ 5.606 = 2·log(16.5) — need 2·log(16.5) ≥ 5.606 ⟺ log(16.5) ≥ 2.803 ⟺ 16.5 ≥ e^2.803 ≈ 16.49 ✓ (certified). So ∫ ≤ 4.026 + 2·log(1/δ) ≤ 2·log(16.5) + 2·log(1/δ) = 2·log(16.5/δ) ✓. Then with δ ≥ c_clear = 1/(9·C0hat(√n+2)): log(1/δ) ≤ log(9·C0hat(√n+2)) — so per-zero ≤ 2·log(16.5·9·C0hat(√n+2)) = 2·log(148.5·C0hat(√n+2)). Then Σ mult·per-zero ≤ 2·log(148.5·C0hat(√n+2))·Σ mult ≤ 2·Z2hat(T)·log(148.5·C0hat(√n+2)) =: NI. The δ ≤ 1 input: |T − γ| < 1 from mem_nearZerosSub ✓ (ρ ∈ nearZerosSub T ⟹ |ρ.im − T| < 1). The δ ≥ c_clear input: ρ ∈ nearZerosSub T ⟹ |γ − T| < 1 ⟹ γ ∈ (T−1, T+1) ⊆ (√n−1, √n+2) (since √n ≤ T ≤ √n+1: T−1 ≥ √n−1 ✓, T+1 ≤ √n+2 ✓). Then hTwide ρ hnt γlo γhi : |γ − T| ≥ 1/(9·C0hat(√n+2)). Wait — hTwide's statement (from zeta_ne_zero_horizontal's signature): `∀ ρ : ℂ, IsNontrivialZero ρ → Real.sqrt n - 1 < ρ.im → ρ.im < Real.sqrt n + 2 → 1 / (9 * C0hat (Real.sqrt n + 2)) ≤ |ρ.im - T|`. ✓. And δ := |T − γ| = |γ − T| (abs_sub_comm). Now — the near-sum integral swap: ∫_{-5/2}^{3} ‖Σ_{ρ} m/(s−ρ)‖ dσ ≤ ∫ Σ m/‖s−ρ‖ = Σ ∫ m/‖s−ρ‖ dσ (norm_sum_le pointwise + integral mono + integral_finsetSum). The pieces needed: 1. **The per-zero arcsinh-FTC machine**: ∫_{-5/2}^{3} dσ/√((σ−β)²+δ²) = F(3) − F(−5/2) with F(σ) = log((σ−β) + √((σ−β)²+δ²)). This is the long pole of 6-c-iv. Components: - F differentiable everywhere (δ > 0): g(σ) := (σ−β) + √((σ−β)²+δ²) > 0 since √((σ−β)²+δ²) > |σ−β| ≥ −(σ−β) ⟹ g > 0... wait: √((σ−β)²+δ²) > √((σ−β)²) = |σ−β| (δ > 0, strict) ✓ so g = (σ−β) + √(...) > (σ−β) + |σ−β| ≥ 0 ✓. - F′(σ) = g′(σ)/g(σ) = (1 + (σ−β)/√((σ−β)²+δ²))/g(σ) = (√((σ−β)²+δ²) + (σ−β))/(√((σ−β)²+δ²)·g(σ)) = g(σ)/(√(...)·g(σ)) = 1/√((σ−β)²+δ²) ✓. - HasDerivAt F (1/√((σ−β)²+δ²)) σ: via HasDerivAt.log (g's HasDerivAt at σ with derivative 1 + (σ−β)/√h) (g(σ) ≠ 0). g's derivative: d/dσ[(σ−β) + √((σ−β)²+δ²)] = 1 + (2(σ−β))/(2√((σ−β)²+δ²)) = 1 + (σ−β)/√h. HasDerivAt: (hasDerivAt_id σ).sub_const β gives d/dσ(σ−β) = 1; Real.hasDerivAt_sqrt (hx : x ≠ 0)... the sqrt chain: HasDerivAt (fun σ => (σ−β)²+δ²) (2(σ−β)) σ then .sqrt (h ≠ 0) gives derivative (2(σ−β))/(2√h) = (σ−β)/√h. Then .add. Then .log (g ≠ 0). Then the algebra to show the resulting derivative equals 1/√h — the recipe says "the identity g′ = g/√h" — i.e. g′(σ) = 1 + (σ−β)/√h = (√h + (σ−β))/√h = g/√h — field identity. So HasDerivAt g (g σ / √h σ)... hmm, actually the log-derivative: HasDerivAt (log ∘ g) (g' / g σ) — with g' = g σ / √h: (g σ/√h)/g σ = 1/√h ✓. The cleanest Lean route: ```lean have hgder : HasDerivAt g ((g σ) / Real.sqrt (h σ)) σ := ... have hF : HasDerivAt (fun σ => Real.log (g σ)) (1 / Real.sqrt (h σ)) σ := by have := hgder.log (ne_of_gt hgpos) -- this : HasDerivAt (log ∘ g) ((g σ / √h) / (g σ)) rw [div_eq_iff (ne_of_gt hgpos)] at this?... convert ``` Hmm — HasDerivAt.log (hf : HasDerivAt f f' x) (hx : f x ≠ 0) : HasDerivAt (fun x => Real.log (f x)) (f' / f x) x. With f' = g σ / √h σ: f'/f σ = (g σ/√h σ)/g σ = 1/√h σ (field_simp with g σ ≠ 0, √h > 0). So `convert hgder.log hgpos.ne' using 1; field_simp; ring`? The convert target: HasDerivAt (fun x => Real.log (g x)) (1/√h x) x vs produced HasDerivAt (fun x => Real.log (g x)) ((g x/√h x)/g x) x — convert using 1 leaves the derivative equality: 1/√h = (g/√h)/g — field_simp [g≠0, √h≠0]; ring ✓. - FTC: `intervalIntegral.integral_eq_sub_of_hasDerivAt` — signature? Something like: (hderiv : ∀ x ∈ uIcc a b, HasDerivAt F (F' x) x) (hint : IntervalIntegrable F' volume a b) : ∫ x in a..b, F' x = F b - F a. Let me check exact form. There's `intervalIntegral.integral_eq_sub_of_hasDerivAt (hf : ∀ x ∈ Set.uIcc a b, HasDerivAt f (f' x) x) (h'int : IntervalIntegrable f' volume a b) : ∫ x in a..b, f' x = f b - f a`. - Integrability of 1/√h on [−5/2, 3]: continuous (h > 0 ⟹ √h continuous nonzero ⟹ 1/√h continuous) ⟹ Continuous.intervalIntegrable. 2. **The pointwise split** ‖logDeriv‖ ≤ ‖logDeriv − Σ‖ + ‖Σ‖ and the integral decomposition — but CAREFUL: the integral of ‖Jlag‖ splits as ≤ (e⁴+1)·∫‖logDeriv‖, and ∫‖logDeriv‖ ≤ ∫(‖R‖ + ‖Σ‖) = ∫‖R‖ + ∫‖Σ‖ (integral_add of integrable functions + integral_mono). Each piece: - ∫_{-5/2}^{3} ‖R‖ dσ ≤ 5.5·(10.5 log(T+2)+48): pointwise capstone + integral_mono_on + integral_const. Needs ‖R(σ+iT)‖ ≤ 10.5 log(T+2)+48 for σ ∈ [−5/2,3] — capstone (hσlo/hσhi per σ; re(σ+iT) = σ by simp; im = T; ζ ≠ 0 via zeta_ne_zero_horizontal hTwide). - ∫ ‖Σ‖ ≤ Σ mult·∫ dσ/√((σ−β)²+δ²) ≤ Σ mult·2·log(16.5/δ_ρ) ≤ 2·log(148.5·C0hat(√n+2))·Σ mult ≤ 2·Z2hat(T)·log(148.5·C0hat(√n+2)). The per-ρ bound 2·log(148.5·C0hat(√n+2)) — note: I need log(1/δ) ≤ log(9·C0hat(√n+2)) ⟸ 1/δ ≤ 9·C0hat(√n+2) ⟸ δ ≥ 1/(9·C0hat(√n+2)) ✓ hTwide. Then per-zero integral ≤ 4.026 + 2·log(1/δ)... hmm wait — the recipe's final per-zero form is 2·log(16.5/δ) then ≤ 2·log(16.5·9·C0hat(√n+2)). Let me redo: per-zero ≤ log(6+δ) + log(7+δ) + 2 log(1/δ) ≤ 1.946 + 2.08 + 2·log(1/δ) ≤ 2·log(16.5) + 2·log(1/δ) = 2·log(16.5/δ) ✓. Then 16.5/δ ≤ 16.5·9·C0hat(√n+2) = 148.5·C0hat(√n+2) ⟹ log(16.5/δ) ≤ log(148.5·C0hat(√n+2)) ✓. So per-zero ≤ 2·log(148.5·C0hat(√n+2)) — UNIFORM in ρ ✓. Then Σ mult·that = that·2·Σmult... wait: per-zero bound B0 := 2·log(148.5·C0hat(√n+2)); Σ_ρ mult_ρ·B0 = B0·Σmult ≤ B0·Z2hat(T) ✓. So NI := 2·Z2hat(T)·log(148.5·C0hat(√n+2))... Hmm wait — the recipe says "near-integral ≤ 2·Z2hat(T)·log(148.5·C0hat(√n+2)) =: NI". And the E-030-validated fit is "NI + 5.5(10.5 log+48) ≤ 7Ĥ (ratio ≤ 0.40)" — and Bn's horizontal share is 2(e⁴+1)(7Ĥ). So the 6-c-iv per-side deliverable: ‖H(−5/2→3, ±T)‖ ≤ (e⁴+1)·(2·Z2hat(T)·log(148.5·C0hat(√n+2)) + 5.5·(10.5·log(T+2)+48)). 3. **Certified numerics**: log 7 ≤ 1.946, log 8 ≤ 2.08, 2·log 16.5 ≥ 5.606 (i.e., log 16.5 ≥ 2.803). How were these done before? Session-036's LiTightR has "certified numerics (integral-comparison log machines + all certified log/exp/sqrt bounds)". Let me check what's available in LiTightR for log 7, log 8, log 16.5. There might be existing lemmas like `log_seven_le`, `log_eight_le`. Let me grep. 4. **The bottom side (−T)**: the conj trick — clone DL-5's pattern (hIntegral_Jlag_low_neg_norm_le_dogLegUpper). Let me look at how DL-5 did the neg side — it used intervalIntegral_conj + Jlag_conj to reduce the −T side to the +T side. Let me read that. Also — the per-zero bound derivation requires δ_ρ := |T − γ_ρ| with both bounds: 0 < c_clear ≤ δ ≤ 1 — wait δ > 0 is implied by δ ≥ c_clear > 0 ✓. Also the integrand for the FTC is 1/√((σ−β)²+δ²) with δ := |T − γ| — and ‖s − ρ‖ = √((σ−β)² + (T−γ)²) = √((σ−β)² + δ²) — since (T−γ)² = |T−γ|² = δ² ✓. And ‖s−ρ‖ = Complex.norm — ‖s−ρ‖ = √(normSq(s−ρ)) = √((σ−β)² + (T−γ)²) — need the identity ‖s−ρ‖ = Real.sqrt((σ−β)² + (T−γ)²) — Complex.norm_eq_sqrt_normSq... via Complex.normSq_apply + sq. Let me handle: ‖s−ρ‖ = Real.sqrt (Complex.normSq (s−ρ)) (Complex.norm_def?) — `Complex.norm_def : ‖z‖ = √(normSq z)`? Hmm — norm is defined as √normSq? `Complex.norm = fun z => Real.sqrt (normSq z)`? I recall `Complex.norm_def : ‖z‖ = Real.sqrt (normSq z)`... or `Complex.norm_eq_sqrt_normSq`? Let me check. There's `Complex.normSq_eq_norm_sq : normSq z = ‖z‖^2` and `Complex.norm_def : ‖z‖ = Real.sqrt (normSq z)`... something exists. Actually — for the near-sum per-term: m/‖s−ρ‖ — I need ∫ dσ m/‖s−ρ‖ = m·∫ dσ/√((σ−β)²+δ²) — so the identity ‖s−ρ‖ = √((σ−β)²+δ²) is needed. Route: ‖s−ρ‖² = normSq(s−ρ) = (s−ρ).re² + (s−ρ).im² = (σ−β)² + (T−γ)² = (σ−β)² + δ². So ‖s−ρ‖ = √((σ−β)²+δ²) via: congruence on sqrt of the normSq computation + `Real.sqrt_sq_eq_abs`/`Complex.norm_def`. Complex.norm is `Real.sqrt (normSq z)` — check `Complex.norm_def`. OK — this is a BIG sub-stage (the recipe anticipates the segment-integral machine as the heavy part). Let me check the DL-5 neg-side pattern + integral_eq_sub_of_hasDerivAt + certified log bounds available + Real.hasDerivAt_sqrt signature + Complex.norm def lemma + integral_finsetSum name. Let me also check `kFun_shift_norm_le_exp_four_add_one`'s exact signature.
6-c-iii committed (`9bd8ba9`). Now 6-c-iv — the mid-horizontal segment machine (the heaviest side bound). First, signature recon: the DL-5 neg-side pattern, the kernel shift lemma, FTC lemma names, certified log bounds, and the finsetSum integral lemma.
tool: TodoWrite
input
{
  "todos": [
    {
      "content": "6-c-ii: I_III left-vertical bound vIntegral_Jlag_left_norm_le",
      "status": "completed",
      "activeForm": "Proving 6-c-ii I_III left-vertical bound"
    },
    {
      "content": "6-c-iii: I_I right-vertical bound (F-E form)",
      "status": "completed",
      "activeForm": "Proving 6-c-iii I_I bound"
    },
    {
      "content": "6-c-iv: mid-horizontal segment machine",
      "status": "in_progress",
      "activeForm": "Proving 6-c-iv mid-horizontal machine"
    },
    {
      "content": "6-c-v: horizontal assembly (mid + DL-5 tails)",
      "status": "pending",
      "activeForm": "Assembling 6-c-v horizontal bound"
    },
    {
      "content": "6-d: fold \u27f9 sfBoundProp (then commit, update archive)",
      "status": "pending",
      "activeForm": "Proving 6-d fold to sfBoundProp"
    }
  ]
}
tool: Grep
input
{
  "pattern": "theorem kFun_shift_norm_le_exp_four_add_one|theorem hIntegral_Jlag_low_neg_norm_le_dogLegUpper|theorem hIntegral_Jlag_low_le_dogLegUpper",
  "path": "D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1",
  "output_mode": "content",
  "-A": 10
}
tool: Grep ok
input
{
  "pattern": "theorem integral_eq_sub_of_hasDerivAt|theorem integral_finsetSum|theorem intervalIntegral_conj |theorem hasDerivAt_sqrt |theorem norm_def |theorem log_seven|theorem log_eight|log 7 \u2264|log 8 \u2264|Real\\.log 16",
  "path": "D:\\Python\\Quantum",
  "output_mode": "content",
  "-A": 4,
  "head_limit": 60
}
result 2026-08-26 18:10:59Z → 2026-08-26 18:10:59Z
mode='content' num_files=0 filenames=[] content="STATE.md:363:mem_nearZerosSub; log 7 ≤ 1.946, log 8 ≤ 2.08, 2·log 16.5 ≥ 5.606).\nD:\\Python\\Quantum\\STATE.md-364-Per-zero ≤ 2·log(16.5/c_clear) with c_clear = 1/(9·C0hat(√n+2)) (the\nSTATE.md-365-wide pigeonhole: nearZerosSub T's zeros have γ ∈ (T−1,T+1) ⊆\nD:\\Python\\Quantum\\STATE.md-366-(√n−1,√n+2) ✓). Swap ∫Σ = Σ∫ (integral_finsetSum — the deprecated\nD:\\Python\\Quantum\\STATE.md-367-integral_finset_sum's new name; per-term integrability by continuity,\n--\nresearch\\lean\\NguyenRH1\\NguyenRH1\\LiCircleBound.lean:2412:        have h2 : Real.log (16 * R) = Real.log 16 + Real.log R :=\nresearch\\lean\\NguyenRH1\\NguyenRH1\\LiCircleBound.lean-2413-          Real.log_mul (by norm_num: (16 : ℝ) ≠ 0) (by linarith [hR] : R ≠ 0)\nresearch\\lean\\NguyenRH1\\NguyenRH1\\LiCircleBound.lean:2414:        have h3 : Real.log 16 = 4 * Real.log 2 := by\nresearch\\lean\\NguyenRH1\\NguyenRH1\\LiCircleBound.lean-2415-          rw [show (16: ℝ) = 2 ^ 4 by norm_num, Real.log_pow]\nD:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiCircleBound.lean-2416-          push_cast\nD:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiCircleBound.lean-2417-          ring\nresearch\\lean\\NguyenRH1\\NguyenRH1\\LiCircleBound.lean-2418-        have h4:= log_two_le_one" num_lines=13 num_matches=None applied_limit=None applied_offset=None
#2279 assistant 2026-08-26 18:13:22.014 UTC · 1151→388 tok open ⤢
thinking (681 chars)
All signatures confirmed: - `intervalIntegral.integral_eq_sub_of_hasDerivAt (hderiv : ∀ x ∈ uIcc a b, HasDerivAt f (f' x) x) (hint : IntervalIntegrable f' volume a b) : ∫ y in a..b, f' y = f b - f a` ✓ - `HasDerivAt.sqrt (hf : HasDerivAt f f' x) (hx : f x ≠ 0) : HasDerivAt (fun y => √(f y)) (f' / (2 * √(f x))) x` ✓ - `HasDerivAt.log (hf : HasDerivAt f f' x) (hx : f x ≠ 0) : HasDerivAt (fun y => log (f y)) (f' / f x) x` ✓ - `intervalIntegral.integral_finsetSum (h : ∀ i ∈ s, IntervalIntegrable (f i) μ a b) : ∫ x in a..b, ∑ i ∈ s, f i x ∂μ = ∑ i ∈ s, ∫ x in a..b, f i x ∂μ` ✓ Now the DL-5 neg-side conj trick — let me read the hIntegral_Jlag_low_neg proof to clone the pattern.
All signatures confirmed. Now reading DL-5's neg-side conj trick.
tool: Read ok
input
{
  "file_path": "D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiDogLeg.lean",
  "offset": 1367,
  "limit": 75
}
result 2026-08-26 18:13:22Z → 2026-08-26 18:13:22Z
{
  "type": "text",
  "file": {
    "filePath": "D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiDogLeg.lean",
    "content": "theorem hIntegral_Jlag_low_neg_norm_le_dogLegUpper {n : \u2115} {T : \u211d}\n    (hn : 16 \u2264 n) (hTn : Real.sqrt (n : \u211d) \u2264 T) (hTn1 : T \u2264 Real.sqrt (n : \u211d) + 1) :\n    \u2016HIntegral (Jlag n) 3 (2 * Real.sqrt (n : \u211d)) (-T)\u2016 \u2264\n      ((Real.sqrt (n : \u211d) + 1) / 2) * (Real.exp (5 / 2) + 1) / 2 + (8.38 : \u211d) +\n        IIMaj (n : \u211d) / 4 := by\n  have hT0 : (0 : \u211d) < T := by\n    have hnR : (16 : \u211d) \u2264 (n : \u211d) := by exact_mod_cast hn\n    have hsqrt4 : (4 : \u211d) \u2264 Real.sqrt (n : \u211d) := by\n      have h1 : Real.sqrt 16 \u2264 Real.sqrt (n : \u211d) :=\n        Real.sqrt_le_sqrt (by exact_mod_cast hn : (16 : \u211d) \u2264 (n : \u211d))\n      have h2 : Real.sqrt 16 = 4 := by\n        rw [Real.sqrt_eq_iff_eq_sq (by norm_num) (by norm_num)]; norm_num\n      rwa [h2] at h1\n    linarith\n  have hconj : HIntegral (Jlag n) 3 (2 * Real.sqrt (n : \u211d)) (-T) =\n      (starRingEnd \u2102) (HIntegral (Jlag n) 3 (2 * Real.sqrt (n : \u211d)) T) := by\n    unfold HIntegral\n    rw [\u2190 intervalIntegral.intervalIntegral_conj]\n    apply intervalIntegral.integral_congr\n    intro \u03c3 h\u03c3\n    show Jlag n ((\u03c3 : \u211d) + (((-T : \u211d)) : \u2102) * Complex.I) =\n      (starRingEnd \u2102) (Jlag n ((\u03c3 : \u211d) + T * Complex.I))\n    have hid2 : ((\u03c3 : \u211d) + (((-T : \u211d)) : \u2102) * Complex.I) =\n        (starRingEnd \u2102) ((\u03c3 : \u211d) + T * Complex.I) := by\n      simp\n    rw [hid2]\n    exact Jlag_conj n (by\n      intro h\n      have him : (((\u03c3 : \u211d) + T * Complex.I)).im = T := by simp\n      rw [h] at him\n      simp at him\n      linarith [hT0])\n  rw [hconj, Complex.norm_conj]\n  exact hIntegral_Jlag_low_le_dogLegUpper hn hTn hTn1\n\n/-- **THE TWO-SIDE RIGHT-HORIZONTAL BOUND** (the DL capstone for\nii\u2032-6-c): both horizontal right tails bounded by the dog-leg charge. -/\ntheorem dogLegUpper_two_sides {n : \u2115} {T : \u211d}\n    (hn : 16 \u2264 n) (hTn : Real.sqrt (n : \u211d) \u2264 T) (hTn1 : T \u2264 Real.sqrt (n : \u211d) + 1) :\n    \u2016HIntegral (Jlag n) 3 (2 * Real.sqrt (n : \u211d)) T\u2016 +\n      \u2016HIntegral (Jlag n) 3 (2 * Real.sqrt (n : \u211d)) (-T)\u2016 \u2264\n    2 * (((Real.sqrt (n : \u211d) + 1) / 2) * (Real.exp (5 / 2) + 1) / 2 + (8.38 : \u211d) +\n      IIMaj (n : \u211d) / 4) := by\n  rw [show (2 : \u211d) * (((Real.sqrt (n : \u211d) + 1) / 2) * (Real.exp (5 / 2) + 1) / 2 +\n      (8.38 : \u211d) + IIMaj (n : \u211d) / 4) =\n      (((Real.sqrt (n : \u211d) + 1) / 2) * (Real.exp (5 / 2) + 1) / 2 + (8.38 : \u211d) +\n        IIMaj (n : \u211d) / 4) +\n      (((Real.sqrt (n : \u211d) + 1) / 2) * (Real.exp (5 / 2) + 1) / 2 + (8.38 : \u211d) +\n        IIMaj (n : \u211d) / 4) by ring]\n  exact add_le_add (hIntegral_Jlag_low_le_dogLegUpper hn hTn hTn1)\n    (hIntegral_Jlag_low_neg_norm_le_dogLegUpper hn hTn hTn1)\n\n\nend NguyenRH1\n\nend",
    "numLines": 56,
    "startLine": 1367,
    "totalLines": 1422
  }
}
#2280 assistant 2026-08-26 18:23:10.700 UTC · 1750→17240 tok open ⤢
thinking (39187 chars)
The conj trick is clean: `intervalIntegral.intervalIntegral_conj` + `integral_congr` + `Jlag_conj` + `Complex.norm_conj`. The neg-side theorem reduces to the pos-side one. Note: `intervalIntegral.intervalIntegral_conj : ∫ x in a..b, conj (f x) = conj (∫ x in a..b, f x)` — used with `rw [← intervalIntegral.intervalIntegral_conj]`. Now let me plan 6-c-iv's Lean content. This is a big block. Components: **A. The per-zero arcsinh-FTC machine.** The most delicate part. Let me define it carefully. For ρ : ℂ (a nontrivial zero — β := ρ.re ∈ (0,1), γ := ρ.im) and T with δ := |T − γ| > 0: ∫_{-5/2}^{3} dσ/√((σ−β)²+δ²) = F(3) − F(−5/2), F(σ) := log((σ−β) + √((σ−β)²+δ²)). Lean formalization plan (as a standalone lemma): ```lean theorem integral_one_div_sqrt_sq_add_sq_le {β δ : ℝ} (hδ : 0 < δ) : ∫ σ in (-5/2 : ℝ)..(3 : ℝ), 1 / Real.sqrt ((σ - β) ^ 2 + δ ^ 2) ≤ Real.log (6 + δ) + Real.log (7 + δ) + 2 * Real.log (1 / δ) ``` Hmm — actually the deliverable per-zero is the bound 2·log(16.5/δ); but it's cleaner to prove the FTC equality + the F(3)/F(−5/2) estimates as separate lemmas: 1. `integral_one_div_sqrt_sq_add_sq_eq`: the FTC evaluation: ∫ σ in a..b, 1/√((σ−β)²+δ²) = F(b) − F(a) with F σ = log((σ−β)+√((σ−β)²+δ²)). 2. The two endpoint estimates: - F(3) ≤ log(6+δ) [for β < 1: (3−β) + √((3−β)²+δ²) ≤ 2(3−β) + δ ≤ 6+δ; the argument positive; log mono] - −F(−5/2) ≤ log(7+δ) + 2·log(1/δ) [for 0 < β: with u := β+5/2 ∈ (5/2, 7/2): F(−5/2) = log(−u + √(u²+δ²)) and −u+√(u²+δ²) = δ²/(u+√(u²+δ²)) ≥ δ²/(7+δ) > 0; so F(−5/2) ≥ log(δ²/(7+δ)) = 2·log δ − log(7+δ); so −F(−5/2) ≤ log(7+δ) − 2·log δ = log(7+δ) + 2·log(1/δ).] Then ∫ ≤ log(6+δ) + log(7+δ) + 2·log(1/δ). Wait — 2·log(1/δ) vs −2·log δ: log(1/δ) = −log δ (Real.log_div / log_one) — log(1/δ) = log 1 − log δ = −log δ. In Lean: `Real.log_div (by norm_num) (ne_of_gt hδ)` : log(1/δ) = log 1 − log δ = −log δ. Or `Real.log_inv : log x⁻¹ = −log x` — 1/δ = δ⁻¹ (one_div_eq_inv). I'll keep log(1/δ) throughout to avoid sign juggling — the final target uses log(1/δ) anyway. The endpoint algebra: F(3) ≤ log(6+δ): - g3 := (3−β) + √((3−β)²+δ²). Need: g3 ≤ 6+δ and g3 > 0, then Real.log_le_log (g3 > 0) (g3 ≤ 6+δ). - √(x²+δ²) ≤ x + δ for x ≥ 0: Real.sqrt_le_iff? `Real.sqrt_le_iff : √x ≤ y ↔ 0 ≤ y ∧ x ≤ y^2`? Something like: Real.sqrt_le_iff (hy : 0 ≤ y)? — there's `Real.sqrt_le_iff : √x ≤ y ↔ 0 ≤ y ∧ x ≤ y ^ 2`? Hmm — or `Real.sqrt_le' (hy : 0 < y) : √x ≤ y ↔ x ≤ y^2`. Let me use: √((3−β)²+δ²) ≤ (3−β) + δ ⟺ (3−β)²+δ² ≤ ((3−β)+δ)² = (3−β)² + 2(3−β)δ + δ² ⟺ 0 ≤ 2(3−β)δ ✓ (3−β > 0 since β < 1, δ > 0). Via `Real.sqrt_le_iff`/`sq_le_sq`: I'll find the right form: `Real.sqrt_le_iff {x y : ℝ} : √x ≤ y ↔ 0 ≤ y ∧ x ≤ y ^ 2`? Hmm. There's `Real.sqrt_le_iff (hy : 0 ≤ y) : √x ≤ y ↔ x ≤ y ^ 2`?? and `Real.sqrt_le_sqrt (h : x ≤ y) : √x ≤ √y`. And `sq_le_sq₀ (ha : 0 ≤ a) (hb : 0 ≤ b) : a ≤ b ↔ a^2 ≤ b^2`? The banked gotcha mentions sq_le_sq₀ exists. Route: apply (sq_le_sq₀ (Real.sqrt_nonneg _) (by positivity : 0 ≤ (3−β)+δ)).mpr-ish... sq_le_sq₀ : 0 ≤ a → 0 ≤ b → (a ≤ b ↔ a ^ 2 ≤ b ^ 2)? Then √h ≤ (3−β)+δ ⟺ (√h)² ≤ ((3−β)+δ)² — with Real.sq_sqrt (h ≥ 0): h = (3−β)²+δ² ≤ ((3−β)+δ)² — nlinarith [δ > 0, β < 1]. OK. - Then g3 ≤ (3−β) + ((3−β)+δ) = 2(3−β)+δ ≤ 6+δ ⟺ 2(3−β) ≤ 6 ⟺ β ≥ 0 ✓ (0 < β). - g3 > 0: √((3−β)²+δ²) > |3−β| ≥ 0... strictly: √((3−β)²+δ²) > √((3−β)²) = |3−β| (Real.sqrt_lt_sqrt (sq_nonneg) (by nlinarith [hδ]) : (3−β)² < (3−β)²+δ²) — Real.sqrt_lt_sqrt (hx : 0 ≤ x) (h : x < y) : √x < √y ✓. And |3−β| = 3−β ≥ 2 > 0. So √ > 3−β ⟹ g3 = (3−β) + √ > 2(3−β) > 0 ✓. −F(−5/2) ≤ log(7+δ) + 2·log(1/δ): - With u := β + 5/2 (0 < β < 1 ⟹ 5/2 < u < 7/2): - g25 := (−5/2−β) + √((5/2+β)²+δ²) = −u + √(u²+δ²). - g25 > 0: √(u²+δ²) > |u| = u (u > 0) ⟹ −u + √ > 0 ✓. - The rationalization: g25 · (u + √(u²+δ²)) = (√(u²+δ²))² − u² = δ² ⟹ g25 = δ²/(u+√(u²+δ²)). - u + √(u²+δ²) ≤ u + (u + δ) = 2u + δ ≤ 7 + δ (u < 7/2): √(u²+δ²) ≤ u+δ ⟺ u²+δ² ≤ (u+δ)² ⟺ 0 ≤ 2uδ ✓. - So g25 = δ²/(u+√) ≥ δ²/(7+δ) (denominator positive, ≤) — div_le_div_of_nonneg_left (δ² ≥ 0) (0 < u+√) (u+√ ≤ 7+δ): `div_le_div_of_nonneg_left (ha : 0 ≤ a) (hb : 0 < b) (hc : b ≤ c) : a / c ≤ a / b` — wait direction: a/c ≤ a/b ⟸ b ≤ c ✓ — so δ²/(7+δ) ≤ δ²/(u+√) ✓ — i.e. g25 ≥ δ²/(7+δ). - log mono: F(−5/2) = log g25 ≥ log(δ²/(7+δ)) = log(δ²) − log(7+δ) (Real.log_div (δ² ≠ 0) (7+δ ≠ 0)) = 2·log δ − log(7+δ) (Real.log_pow : log(δ^2) = 2·log δ ✓). - So −F(−5/2) ≤ log(7+δ) − 2·log δ = log(7+δ) + 2·log(1/δ) (log(1/δ) = −log δ). The rationalization identity g25 = δ²/(u+√(u²+δ²)): field_simp + the (√h)² = h identity. In Lean: ```lean have hg25id : -u + Real.sqrt (u ^ 2 + δ ^ 2) = δ ^ 2 / (u + Real.sqrt (u ^ 2 + δ ^ 2)) := by rw [div_eq_iff (by positivity... : u + √ ≠ 0)] -- (−u + √) * (u + √) = δ² have hs : (Real.sqrt (u ^ 2 + δ ^ 2)) ^ 2 = u ^ 2 + δ ^ 2 := Real.sq_sqrt (by positivity) nlinarith [hs] ``` `(−u + √)·(u + √) = √² − u² = δ²` — nlinarith [hs]: the product expands: −u·u + √·u − u·√ + √·√ = −u² + √²... nlinarith handles (√·√ → via hs hint) ✓. Wait — div_eq_iff (hb : b ≠ 0) : a / b = c ↔ a = c * b. I want g = δ²/(u+√): rw [div_eq_iff hne] on goal `g = δ²/(u+√)` — hmm, div_eq_iff rewrites δ²/(u+√) = g-form... the goal is `-u + √ = δ^2 / (u+√)`. rw [div_eq_iff hne] : a / b = c ↔ a = c * b — with a := δ², b := u+√, c := -u+√: goal becomes δ² = (−u+√)·(u+√) ✓ then nlinarith [hs]. ✓ - Also g25 > 0 (for the log to be meaningful): as computed ✓ needed for... actually F(−5/2) = log g25 — log of positive — the estimate −F ≤ ... uses log mono log(δ²/(7+δ)) ≤ log g25 ⟸ δ²/(7+δ) ≤ g25 (with δ²/(7+δ) > 0 for Real.log_le_log's positivity side hyps: Real.log_le_log (hx : 0 < x) (h : x ≤ y) : log x ≤ log y — x := δ²/(7+δ) > 0 ✓ h : δ²/(7+δ) ≤ g25 ✓). Then: −F(−5/2) = −log g25 ≤ −log(δ²/(7+δ)) = −(2 log δ − log(7+δ)) = log(7+δ) − 2 log δ. And the final target 2·log(1/δ) — relate −2 log δ = 2·log(1/δ): Real.log_div (1≠0) (δ≠0) : log(1/δ) = log 1 − log δ = −log δ (log_one). So 2·log(1/δ) = −2·log δ ✓. Then the FTC lemma itself: ```lean theorem integral_one_div_sqrt_sq_add_sq (β δ : ℝ) (hδ : 0 < δ) (a b : ℝ) : ∫ σ in a..b, 1 / Real.sqrt ((σ - β) ^ 2 + δ ^ 2) = Real.log ((b - β) + Real.sqrt ((b - β) ^ 2 + δ ^ 2)) - Real.log ((a - β) + Real.sqrt ((a - β) ^ 2 + δ ^ 2)) := by apply intervalIntegral.integral_eq_sub_of_hasDerivAt · intro σ hσ -- HasDerivAt F (1/√h σ) σ · -- integrability of 1/√h ``` The HasDerivAt proof — let me structure it carefully: ```lean have hgpos : ∀ σ : ℝ, 0 < (σ - β) + Real.sqrt ((σ - β) ^ 2 + δ ^ 2) := by intro σ have h1 : |(σ - β)| < Real.sqrt ((σ - β) ^ 2 + δ ^ 2) := by have h2 : (σ - β) ^ 2 < (σ - β) ^ 2 + δ ^ 2 := by nlinarith [hδ] have h3 := Real.sqrt_lt_sqrt (sq_nonneg _) h2 rwa [Real.sqrt_sq_eq_abs] at h3 have h4 : -(σ - β) < Real.sqrt ((σ - β) ^ 2 + δ ^ 2) := by have := neg_abs_le (σ - β) -- -(σ−β) ≤ |σ−β| linarith linarith ``` neg_abs_le : -a ≤ |a| ✓ root. So (σ−β) + √h > (σ−β) + (−(σ−β)) = 0 ✓ linarith. ```lean have hgder : ∀ σ : ℝ, HasDerivAt (fun σ => (σ - β) + Real.sqrt ((σ - β) ^ 2 + δ ^ 2)) (((σ - β) + Real.sqrt ((σ - β) ^ 2 + δ ^ 2)) / Real.sqrt ((σ - β) ^ 2 + δ ^ 2)) σ := by intro σ have hh : (0:ℝ) < (σ - β) ^ 2 + δ ^ 2 := by positivity?? -- needs δ ≠ 0: nlinarith [hδ, sq_nonneg...] ``` h(σ) = (σ−β)²+δ² > 0 since δ² > 0: nlinarith [sq_nonneg (σ−β), hδ] or by positivity with hδ? positivity can't do δ² > 0 from hδ (it can: 0 < δ ⟹ 0 < δ^2? positivity proves 0 ≤ δ^2 not strict; the SUM (σ−β)²+δ² > 0 needs δ² > 0 strict — from hδ : 0 < δ: δ^2 > 0 via pow_pos hδ 2 ✓ then add_pos_of_nonneg_of_pos (sq_nonneg) (pow_pos hδ 2). I'll write: ```lean have hh : (0:ℝ) < (σ - β) ^ 2 + δ ^ 2 := add_pos_of_nonneg_of_pos (sq_nonneg _) (pow_pos hδ 2) ``` pow_pos : 0 < a → 0 < a^n (n ≠ 0): `pow_pos (h : 0 < a) (n : ℕ) : 0 < a ^ n` ✓. The derivative of g(σ) = (σ−β) + √(h(σ)): g′ = 1 + h′/(2√h) where h′ = 2(σ−β). And 1 + 2(σ−β)/(2√h) = 1 + (σ−β)/√h = (√h + (σ−β))/√h = g/√h ✓. ```lean have h1 : HasDerivAt (fun σ => σ - β) 1 σ := hasDerivAt_id.sub_const β have h2 : HasDerivAt (fun σ => (σ - β) ^ 2 + δ ^ 2) (2 * (σ - β)) σ := by have := (hasDerivAt_id.sub_const β).pow 2 -- this : HasDerivAt (fun σ => (σ - β)^2) (2 * (σ-β)^1 * 1) σ have h3 := this.add_const (δ ^ 2) -- convert derivative: 2*(σ-β)^1*1 = 2*(σ-β) convert h3 using 1 ring have h4 : HasDerivAt (fun σ => Real.sqrt ((σ - β) ^ 2 + δ ^ 2)) (2 * (σ - β) / (2 * Real.sqrt ((σ - β) ^ 2 + δ ^ 2))) σ := h2.sqrt hh.ne' have h5 : HasDerivAt (fun σ => (σ - β) + Real.sqrt ((σ - β) ^ 2 + δ ^ 2)) (1 + 2 * (σ - β) / (2 * Real.sqrt ((σ - β) ^ 2 + δ ^ 2))) σ := h1.add h4 -- the derivative simplifies to g/√h convert h5 using 1 have hsqrt0 : Real.sqrt ((σ - β) ^ 2 + δ ^ 2) ≠ 0 := ne_of_gt (Real.sqrt_pos.mpr hh) have hg0 : ((σ - β) + Real.sqrt ((σ - β) ^ 2 + δ ^ 2)) ≠ 0 := (hgpos σ).ne' field_simp ring ``` Check the derivative algebra: need `g/√h = 1 + 2(σ−β)/(2√h)` ⟺ g/√h = 1 + (σ−β)/√h ⟺ g = √h + (σ−β) ✓ (g = (σ−β) + √h ✓). field_simp [hsqrt0] then ring ✓. field_simp on goal `((σ−β)+√h)/√h = 1 + 2*(σ−β)/(2*√h)`: multiplies out: ((σ−β)+√h)·(2√h) = (2√h + 2(σ−β))·√h/√h?? — field_simp clears denominators: LHS·√h·2√h vs... standard: both sides × 2√h: 2√h((σ−β)+√h) = 2√h + 2(σ−β)·... hmm — let me just trust field_simp + ring (banked: "field_simp closes goals (trailing ring errors)"). Actually the identity: ((σ−β)+√h)/√h = 1 + 2(σ−β)/(2√h). Multiply both sides by 2√h: 2((σ−β)+√h) = 2√h + 2(σ−β) ⟺ 2(σ−β) + 2√h = 2√h + 2(σ−β) ✓ ring. `hasDerivAt_id.sub_const β` — hmm: `hasDerivAt_id (x) : HasDerivAt id 1 x`; `.sub_const (c) : HasDerivAt (fun x => f x - c) (f') x` — HasDerivAt.sub_const (hf : HasDerivAt f f' x) (c) : HasDerivAt (fun x => f x - c) f' x ✓. hasDerivAt_id — signature: `hasDerivAt_id (x : 𝕜) : HasDerivAt id 1 x` ✓. So `(hasDerivAt_id σ).sub_const β : HasDerivAt (fun x => x - β) 1 σ` ✓ — note the function is `fun x => x - β` vs my `fun σ => σ - β` ✓ alpha. `(hasDerivAt_id.sub_const β).pow 2` — HasDerivAt.pow (hf) (n : ℕ) : HasDerivAt (fun x => (f x)^n) (n * f x ^ (n-1) * f') x ✓ — gives derivative `2 * (σ−β)^1 * 1` — my convert handles ✓. Hmm — actually the .pow form: `HasDerivAt.pow {f : 𝕜 → 𝕜} {f'} (hf : HasDerivAt f f' x) (n : ℕ) : HasDerivAt (fun y => f y ^ n) ((n : 𝕜) * f x ^ (n - 1) * f') x`? — the derivative: `↑n * (f x)^(n-1) * f'`. For n=2: `↑2 * (σ−β)^1 * 1` — my target derivative `2 * (σ−β)` — convert via ring ✓. But the convert needs the FUNCTION to match too: .pow gives `fun y => (y−β)^2` vs my `fun σ => (σ−β)^2 + δ^2` — the .add_const (δ^2) fixes: HasDerivAt.add_const (hf) (c): `fun y => (y−β)^2 + δ^2` ✓ derivative unchanged ✓. Hmm — `HasDerivAt.add_const (hf : HasDerivAt f f' x) (c : 𝕜) : HasDerivAt (fun x => f x + c) f' x` ✓ (banked gotcha session-011: "HasDerivAt.add_const adds to VALUE (use .comp for precomposition)" — it appends a constant to the function value ✓ what I want). Then h5's convert: `convert h5 using 1` — h5 : HasDerivAt g (1 + 2(σ−β)/(2√h)) σ; goal: HasDerivAt g (g σ/√h) σ — convert using 1: function parts match (same g — need defeq: my g-lambda vs h5's — both `fun σ => (σ−β) + √(h σ)` ✓ syntactically equal), leaves derivative equality: g/√h = 1 + 2(σ−β)/(2√h) — field_simp [hsqrt0]; ring ✓. Then the F derivative: ```lean have hF : HasDerivAt (fun σ => Real.log ((σ - β) + Real.sqrt ((σ - β) ^ 2 + δ ^ 2))) (1 / Real.sqrt ((σ - β) ^ 2 + δ ^ 2)) σ := by have h6 := (hgder σ).log (hgpos σ).ne' -- h6 : HasDerivAt (log ∘ g) ((g σ / √h) / g σ) convert h6 using 1 field_simp [(hgpos σ).ne'] ``` (g/√h)/g = 1/√h ⟺ g·√h... field_simp with g ≠ 0, √h ≠ 0: (g/√h)/g = 1/√h ⟺ multiply by g·√h: g·g?? — (g/√h)/g = g/(√h·g) = 1/√h ✓ field_simp + ring... `field_simp` then the goal might become √h·g = √h·g or similar — closed by ring (or rfl). Might need `ring` after field_simp. Write `field_simp [hsqrt0, (hgpos σ).ne']` then `ring` — but field_simp might fully close. Include a trailing ring attempt: the banked pattern is field_simp closes or leaves ring-closable. I'll write: ```lean convert h6 using 1 field_simp [hsqrt0, (hgpos σ).ne'] ``` If field_simp leaves a goal, add ring. To be safe: ```lean convert h6 using 1 have hg0 : ((σ - β) + Real.sqrt ((σ - β) ^ 2 + δ ^ 2)) ≠ 0 := (hgpos σ).ne' field_simp [hsqrt0, hg0] ring ``` Hmm — but if field_simp closes it, ring errors "no goals". The banked gotcha says field_simp often closes with trailing ring errors — i.e., `field_simp; ring` is a common pattern where field_simp leaves a ring-goal. If field_simp FULLY closes, the trailing ring fails. Risk either way. Alternative deterministic: ```lean convert h6 using 1 rw [div_div, mul_comm, ← div_div, div_self hg0] -- messy ``` Or: `(g σ / √h) / g σ = 1/√h`: ```lean rw [show ((σ - β) + Real.sqrt ((σ - β) ^ 2 + δ ^ 2)) / Real.sqrt ((σ - β) ^ 2 + δ ^ 2) / ((σ - β) + Real.sqrt ((σ - β) ^ 2 + δ ^ 2)) = (Real.sqrt ((σ - β) ^ 2 + δ ^ 2))⁻¹ * (((σ - β) + Real.sqrt ((σ - β) ^ 2 + δ ^ 2)) / ((σ - β) + Real.sqrt ((σ - β) ^ 2 + δ ^ 2))) := by ring, div_self hg0, ...] ``` Overcomplicating. Standard approach: `field_simp [*]; ring` and if field_simp closed it, drop the ring on the error. I'll write `field_simp [hsqrt0, hg0]; ring` and adjust on failure. Actually — cleaner: prove the derivative algebra separately as a have before convert: ```lean have hder : ((σ - β) + Real.sqrt ((σ - β) ^ 2 + δ ^ 2)) / Real.sqrt ((σ - β) ^ 2 + δ ^ 2) / ((σ - β) + Real.sqrt ((σ - β) ^ 2 + δ ^ 2)) = 1 / Real.sqrt ((σ - β) ^ 2 + δ ^ 2) := by field_simp [hsqrt0, hg0] convert h6 using 1 exact hder.symm -- hmm direction ``` convert h6 using 1 leaves goal: (target deriv) = (h6 deriv) i.e. `1/√h = (g/√h)/g`? or the reverse? convert produces equality goals between differing parts: goal `1 / √h = (g σ / √h) / g σ` — then `exact hder.symm` ✓ where hder : (g/√h)/g = 1/√h ✓. But the direction convert presents could be either. Use `exact?`-free robust: `field_simp [hsqrt0, hg0]` directly on the convert goal (whichever direction, field_simp normalizes both sides then ring/closes). Hmm — field_simp on `1/√h = (g/√h)/g`: clears to `√h * g = √h * g`-ish — closed by field_simp itself or needs ring. The banked pattern `div_add_div + div_eq_div_iff + ring as the always-works fraction pattern` (session-038). Let me use div_eq_div_iff: ```lean rw [div_eq_div_iff hsqrt0 hg0] -- hmm ``` div_eq_div_iff (hb : b ≠ 0) (hd : d ≠ 0) : a / b = c / d ↔ a * d = c * b. For `1/√h = (g/√h)/g`: a := 1, b := √h, c := g/√h, d := g: 1·g = (g/√h)·√h ⟺ g = g ✓ (div_mul_cancel₀ g √h ≠ 0)... ring can't do div_mul_cancel. `field_simp` normalizes divisions into num/den form — g/√h/g... OK let me stop overthinking: the pattern `field_simp [hsqrt0, hg0]; ring` — if it fails at ring with "no goals", I drop ring; if it fails with a leftover goal, the error shows it. ONE iteration will settle it. Same for the earlier h5-convert. Then the integrability: 1/√h continuous: ```lean have hint : IntervalIntegrable (fun σ => 1 / Real.sqrt ((σ - β) ^ 2 + δ ^ 2)) volume a b := by apply Continuous.intervalIntegrable apply Continuous.div continuous_const · -- Continuous (fun σ => √((σ−β)²+δ²)) apply Continuous.sqrt exact ((continuous_id.sub continuous_const).pow 2).add continuous_const · intro σ exact ne_of_gt (Real.sqrt_pos.mpr (add_pos_of_nonneg_of_pos (sq_nonneg _) (pow_pos hδ 2))) ``` Continuous.sqrt : `Continuous Real.sqrt` — it's `Real.continuous_sqrt` ✓. `.sqrt` method? There might not be Continuous.sqrt method — use `Real.continuous_sqrt.comp`. Let me write: `(Real.continuous_sqrt).comp (...)`. - continuous_id : Continuous id ✓; .sub continuous_const : Continuous fun σ => σ − β ✓; .pow 2? — `Continuous.pow (hf : Continuous f) (n : ℕ) : Continuous fun x => (f x)^n` ✓; .add continuous_const ✓. - Continuous.div (hf) (hg) (h0 : ∀ x, g x ≠ 0) : Continuous fun x => f x / g x ✓. Then integral_eq_sub_of_hasDerivAt (hderiv) (hint): hderiv : ∀ x ∈ uIcc a b, HasDerivAt F (1/√h x) x — from the per-σ hF ✓ (uIcc membership irrelevant — bound holds everywhere). **B. The endpoint estimates.** Two lemmas: ```lean theorem arcsinh_right_endpoint_le {β δ : ℝ} (hβ : β < 1) (hδ : 0 < δ) : Real.log ((3 - β) + Real.sqrt ((3 - β) ^ 2 + δ ^ 2)) ≤ Real.log (6 + δ) theorem arcsinh_left_endpoint_le {β δ : ℝ} (hβ : 0 < β) (hδ : 0 < δ) : -Real.log ((-5/2 - β) + Real.sqrt ((-5/2 - β) ^ 2 + δ ^ 2)) ≤ Real.log (7 + δ) + 2 * Real.log (1 / δ) ``` Hmm — wait, for the left endpoint, (−5/2−β)² = (5/2+β)² ✓ (neg_sq). I'll phrase with u := 5/2 + β: (−5/2−β) = −(5/2+β) and (−5/2−β)^2 = (5/2+β)^2 via `neg_sq`? `neg_sq (a) : (-a)^2 = a^2` — hmm — neg_sq : (-a) ^ 2 = a ^ 2 ✓. Or ring_nf. Let me phrase the left endpoint directly with the u-substitution inside the proof. **C. The per-zero integral bound**: combine FTC (a := −5/2, b := 3) + the two endpoint estimates: ```lean theorem integral_one_div_sqrt_le {β δ : ℝ} (hβ0 : 0 < β) (hβ1 : β < 1) (hδ : 0 < δ) : ∫ σ in (-5/2 : ℝ)..(3 : ℝ), 1 / Real.sqrt ((σ - β) ^ 2 + δ ^ 2) ≤ Real.log (6 + δ) + Real.log (7 + δ) + 2 * Real.log (1 / δ) ``` by rw [integral_one_div_sqrt_sq_add_sq]; linarith [endpoints]. Hmm wait — F(3) − F(−5/2) ≤ log(6+δ) − F(−5/2) ≤ log(6+δ) + (log(7+δ) + 2 log(1/δ)) ✓ linarith [right_endpoint, left_endpoint]. **D. The log-mono simplification to 2·log(16.5/δ)**: ```lean theorem per_zero_le_two_log {δ : ℝ} (hδ : 0 < δ) (hδ1 : δ ≤ 1) : Real.log (6 + δ) + Real.log (7 + δ) + 2 * Real.log (1 / δ) ≤ 2 * Real.log (16.5 / δ) := by have h1 : Real.log (6 + δ) + Real.log (7 + δ) = Real.log ((6 + δ) * (7 + δ)) := (Real.log_mul (by positivity... ≠0) (by positivity... ≠0)).symm have h2 : Real.log ((6 + δ) * (7 + δ)) ≤ Real.log (16.5 * 16.5) := Real.log_le_log (by positivity) (by nlinarith [hδ, hδ1] : (6+δ)*(7+δ) ≤ 16.5*16.5) have h3 : Real.log (16.5 * 16.5) = 2 * Real.log 16.5 := by rw [show (16.5 : ℝ) * 16.5 = 16.5 ^ 2 by ring, Real.log_pow] ring have h4 : Real.log (16.5 / δ) = Real.log 16.5 + Real.log (1 / δ) := Real.log_div (by norm_num) (ne_of_gt hδ) -- log(16.5/δ) = log 16.5 − log δ ``` Hmm — Real.log_div gives log(16.5/δ) = log 16.5 − log δ. And log(1/δ) = −log δ. So 2·log(16.5/δ) = 2·log16.5 − 2·log δ = 2·log16.5 + 2·log(1/δ). So the goal: log(6+δ)+log(7+δ) + 2·log(1/δ) ≤ 2·log16.5 + 2·log(1/δ) ⟺ log((6+δ)(7+δ)) ≤ 2·log16.5 ✓ via h2 h3. linarith [h1 ▸ h2, h3, h4-rewrites]. Structure it as: ```lean have h5 : 2 * Real.log (16.5 / δ) = 2 * Real.log 16.5 + 2 * Real.log (1 / δ) := by have h6 : Real.log (1 / δ) = -Real.log δ := by rw [Real.log_div one_ne_zero (ne_of_gt hδ), Real.log_one, zero_sub] rw [Real.log_div (by norm_num) (ne_of_gt hδ), h6] ring rw [h5] linarith [h2, h3, h1] ``` Wait — after rw [h5]: goal `log(6+δ) + log(7+δ) + 2·log(1/δ) ≤ 2·log 16.5 + 2·log(1/δ)`. With h1 : LHS-sum = log((6+δ)(7+δ)), h2 : log((6+δ)(7+δ)) ≤ log(16.5·16.5), h3 : log(16.5·16.5) = 2·log 16.5: linarith ✓ (atoms: log(6+δ), log(7+δ), log((6+δ)(7+δ)), log(16.5·16.5), log 16.5, log(1/δ) ✓ linear). Hmm — h1 needs rewrites INSIDE the goal? linarith uses h1 as a linear equation between atoms ✓ (log(6+δ) + log(7+δ) = log((6+δ)(7+δ)) — atom-level equation ✓ linarith handles ✓. Real.log_mul (ha : a ≠ 0) (hb : b ≠ 0) : log (a * b) = log a + log b ✓. So h1 : log(6+δ) + log(7+δ) = log((6+δ)(7+δ)) := (Real.log_mul ...).symm ✓. The positivity side goals: 6+δ ≠ 0 (δ > 0 ⟹ 6+δ > 0 ✓ ne_of_gt, positivity or linarith), 7+δ ≠ 0 ✓, (6+δ)(7+δ) > 0 ✓ (Real.log_le_log's first arg: 0 < (6+δ)(7+δ) — by positivity with hδ in context? positivity: 6+δ > 0 needs hδ — positivity uses hyps ✓ hmm — 6+δ: positivity proves 0 < 6+δ? It needs δ > 0 — from hδ in context ✓ positivity should manage; fallback nlinarith). h2's inequality: (6+δ)(7+δ) ≤ 16.5·16.5 = 272.25: nlinarith [hδ, hδ1] — (6+δ)(7+δ) = 42 + 13δ + δ² ≤ 42+13+1 = 56 ≤ 272.25 ✓ nlinarith (needs δ² ≤ δ or δ² ≤ 1: from hδ1 : δ ≤ 1 and hδ : 0 < δ: δ² ≤ δ·1 = δ ✓ product hint — nlinarith pairwise ✓). **E. The final per-zero bound with the clearance**: per ρ ∈ nearZerosSub T: with δ_ρ := |T − γ_ρ|: - 0 < δ_ρ: from hTwide (δ ≥ c_clear > 0). - δ_ρ ≤ 1: mem_nearZerosSub gives |γ − T| < 1 — hmm |ρ.im − T| < 1 — and δ_ρ = |T − γ| = |γ − T| ✓ (abs_sub_comm) — so δ_ρ < 1 hence ≤ 1 ✓. - The integral bound: ∫ dσ/√((σ−β)²+δ²) ≤ 2·log(16.5/δ) [from C+D combined] ≤ 2·log(148.5·C0hat(√n+2)) [from δ ≥ c_clear]. Combine C+D into one: per_zero_integral_le {β δ} (hβ0) (hβ1) (hδ) (hδ1) : ∫ ≤ 2·log(16.5/δ). Then the clearance step per ρ. **F. The near-sum bound**: ```lean -- per ρ: ‖∫ σ, m/(s σ − ρ)‖ machinery ∫ σ in -5/2..3, ‖Σ_{ρ ∈ nearZerosSub T} (zeroMult ρ.1 : ℂ)/(s σ − ρ.1)‖ dσ ≤ ∫ σ, Σ_ρ ‖(zeroMult ρ.1:ℂ)/(s σ − ρ.1)‖ := integral_mono (norm_sum_le) = ∫ σ, Σ_ρ (zeroMult ρ.1 : ℝ)/‖s σ − ρ.1‖ ≤ ∫ σ, Σ_ρ (zeroMult ρ.1 : ℝ) * (1/√((σ−β_ρ)²+δ_ρ²)) -- ‖s−ρ‖ = √(...) so m/‖s−ρ‖ = m·(1/√(...)) = Σ_ρ (zeroMult ρ.1 : ℝ) * ∫ σ, 1/√(...) -- integral_finsetSum + integral_const_mul ≤ Σ_ρ (zeroMult ρ.1 : ℝ) * (2·log(148.5·C0hat(√n+2))) = (2·log(148.5·C0hat(√n+2))) · Σ_ρ mult ≤ 2·Z2hat(T)·log(148.5·C0hat(√n+2)) ``` The identity ‖s σ − ρ.1‖ = √((σ−β)² + (T−γ)²): ```lean have hnormid : ‖((σ : ℝ) + (T : ℝ) * Complex.I) - ρ.1‖ = Real.sqrt ((σ - ρ.1.re) ^ 2 + (T - ρ.1.im) ^ 2) := by have h1 : Complex.normSq (((σ : ℝ) + (T : ℝ) * Complex.I) - ρ.1) = (σ - ρ.1.re) ^ 2 + (T - ρ.1.im) ^ 2 := by simp [Complex.normSq_apply, Complex.sub_re, Complex.add_re, Complex.ofReal_re, Complex.mul_re, Complex.I_re, Complex.I_im, Complex.sub_im, Complex.add_im, Complex.ofReal_im, Complex.mul_im, Complex.I_re] ring -- ‖z‖ = √(normSq z) rw [← h1] -- hmm need ‖z‖ = √(normSq z) exact (Complex.norm_def ...)? ``` Complex.norm: `Complex.norm z = Real.sqrt (normSq z)`? — `Complex.norm_def : ‖z‖ = Real.sqrt (normSq z)`? Let me check — `Complex.norm_def (z : ℂ) : ‖z‖ = Real.sqrt (normSq z)`? I believe `Complex.norm` is defined via normSq sqrt — there IS `Complex.norm_def : ‖z‖ = √(normSq z)`? or `Complex.norm_eq_sqrt_normSq`? Let me grep. Actually `Complex.normSq_eq_norm_sq : normSq z = ‖z‖ ^ 2` — so ‖z‖ = √(normSq z) via `Complex.norm_eq_sqrt_normSq`?? — hmm, in Mathlib: `Complex.norm_def : ‖z‖ = Real.sqrt (normSq z)`? Let me just grep for it. And (T − γ)² = δ² with δ := |T − γ|: `sq_abs (T − γ) : |T−γ|² = (T−γ)²`? `sq_abs (a) : |a| ^ 2 = a ^ 2` ✓ exists (sq_abs). So (T−γ)² = |T−γ|² = δ² ✓. Then 1/‖s−ρ‖ = 1/√((σ−β)²+δ²) and m/‖s−ρ‖ = m·(1/√(...)) (div_eq_mul_one_div or mul_one_div.symm). **G. The R-part integral**: ```lean ∫ σ in -5/2..3, ‖logDeriv ζ (σ+iT) − Σ_near‖ dσ ≤ (3 − -5/2)·(10.5·log(T+2)+48) ``` pointwise capstone per σ ∈ Icc(−5/2,3): needs ht : 10 ≤ T ✓, hσlo : −5/2 ≤ (σ+iT).re = σ ✓ (σ ≥ −5/2 from Icc), hσhi : σ ≤ 3 ✓, him : im = T, hζ : zeta_ne_zero_horizontal hTwide hn hTn hTn1 σ... wait — zeta_ne_zero_horizontal's statement: riemannZeta ((σ:ℝ) + (T:ℝ)*I) ≠ 0 ✓ EXACTLY the path point ✓ (hTwide) (hn : 1 ≤ n) (hTn) (hTn1). Hmm wait — zeta_ne_zero_horizontal {n T σ} (hTwide ...) (hn : 1 ≤ n) (hTn : √n ≤ T) (hTn1 : T ≤ √n+1) : riemannZeta (↑σ + ↑T * I) ≠ 0 ✓. The capstone bound: ‖logDeriv − Σ‖ ≤ 10.5·log(|T|+2)+48 = 10.5·log(T+2)+48 (|T| = T, T ≥ 0). So per σ ≤ K_R := 10.5·log(T+2)+48 — constant in σ ✓ integral_const: (3 − −5/2)·K_R = 5.5·K_R ✓. Integrability of ‖logDeriv − Σ‖: continuous (logDeriv continuous on the path via zeta_ne_zero_horizontal; the sum continuous — each term (zeroMult)/(s−ρ) continuous in s (denominator nonzero: s−ρ ≠ 0 since s = σ+iT and ρ has im ≠ T... wait — could s = ρ? ρ = β+iγ with γ = T? — ρ ∈ nearZerosSub T has |γ−T| < 1 — if γ = T then s = ρ would need σ = β — possible?! Hmm — s ranges over σ ∈ [−5/2,3] with im T; ρ with im γ: s − ρ = 0 ⟺ σ = β ∧ T = γ. If γ = T, then at σ = β the difference vanishes!! BUT — zeta_ne_zero_horizontal says ζ(σ+iT) ≠ 0 for all σ — and ρ is a zero — so s ≠ ρ always (since ζ(s) ≠ 0 = ζ(ρ) ⟹ s ≠ ρ ✓). But for the CONTINUITY of 1/(s−ρ) along the path, I need s−ρ ≠ 0 — which follows from: if s − ρ = 0 then s = ρ, then ζ(s) = ζ(ρ) = 0, contradicting zeta_ne_zero_horizontal ✓. So s−ρ ≠ 0 on the path ✓. Actually — hmm, more precisely: the capstone APPLICATION needs hζ : riemannZeta s ≠ 0 (given by zeta_ne_zero_horizontal) ✓. And the sum's continuity per term: (fun σ => (zeroMult ρ.1 : ℂ)/(↑σ + ↑T·I − ρ.1)) — continuous if the denominator never vanishes: (↑σ + ↑T·I) − ρ.1 ≠ 0 — as argued (else ζ zero) ✓. **H. The full mid-horizontal bound**: ‖Jlag‖ ≤ (e⁴+1)·‖logDeriv‖ pointwise; ∫‖Jlag‖ ≤ (e⁴+1)·∫‖logDeriv‖ (integral_mono_on with constant factor — actually: ∫(e⁴+1)·‖logDeriv‖ = (e⁴+1)·∫‖logDeriv‖ via intervalIntegral.integral_const_mul; pointwise mul_le_mul (kernel bound) ... (norm_nonneg) (e⁴+1 ≥ 0)). ∫‖logDeriv‖ ≤ ∫(‖R‖ + ‖Σ‖) = ∫‖R‖ + ∫‖Σ‖ — integral_mono_on (pointwise norm_le_insert') + intervalIntegral.integral_add (integrability both). Then ∫‖R‖ ≤ 5.5·K_R and ∫‖Σ‖ ≤ NI — total: ∫‖logDeriv‖ ≤ NI + 5.5·K_R — wait no: ∫‖logDeriv‖ ≤ ∫‖R‖ + ∫‖Σ‖ ≤ 5.5K_R + NI. Then ‖H(−5/2→3, T)‖ ≤ (e⁴+1)·(5.5·K_R + NI). And the kernel bound per σ: kFun_shift_norm_le_exp_four_add_one with s := ↑(σ−1) + ↑T·I: need kFun n (s+1) where s+1 = ↑σ + ↑T·I — cast: ↑(σ−1)+1 = ↑σ ✓ push_cast; ring. re s = σ−1 ≤ 3 ⟸ σ ≤ 4 ⟸ σ ≤ 3 ✓ (Icc). im s = T: n ≤ T² — from √n ≤ T: n = √n² ≤ T² (sq_le_sq... n = (√n)² ≤ T² via Real.sq_sqrt + sq_le_sq₀ or nlinarith [hTn, Real.sqrt_nonneg, sq_sqrt]). Hmm — kFun_shift_norm_le_exp_four_add_one (hσ : s.re ≤ 3) (hT : (n:ℝ) ≤ s.im ^ 2) : ‖kFun n (s + 1)‖ ≤ exp 4 + 1 — with s := ↑σ + ↑T·I − 1 (so s + 1 = ↑σ + ↑T·I): re s = σ−1, im s = T. Let me set s := (↑σ + ↑T*Complex.I) − 1 — then s + 1 = ↑σ + ↑T·I by ring (sub_add_cancel). re s = σ − 1: simp. im s = T: simp. **I. The bottom side (−T)**: conj trick cloning DL-5. Now — the ordering of lemmas in the file. This is a LOT. Let me also double-check the integrability of the per-ρ summand integrands for integral_finsetSum: (fun σ => (zeroMult ρ.1 : ℝ) * (1/√((σ−β)²+δ²))) — continuous (const · continuous) ✓ per-term integrable ✓. And for the ∫‖Σ‖ ≤ ∫Σ‖term‖ step: integral_mono_on needs the pointwise norm_sum_le ✓ and integrability of both sides: ‖Σ·‖ (norm of continuous ✓) and Σ‖·‖ (Finset.sum of continuous ✓). The equality chain ∫ Σ m/‖s−ρ‖ = Σ ∫ m/‖s−ρ‖: intervalIntegral.integral_finsetSum with per-term integrability ✓. And ∫ m·(1/√) = m·∫ 1/√: intervalIntegral.integral_const_mul ✓ (signature: integral_const_mul (c : ℝ) (f : ℝ → ℝ) : ∫ x in a..b, c * f x = c * ∫ x in a..b, f x ✓ exists). Hmm — actually, the near-sum's complex terms: the Σ in the capstone is over ℂ-valued terms (zeroMult ρ.1 : ℂ)/(s − ρ.1). The pointwise norm: ‖Σ_ρ term_ρ‖ ≤ Σ_ρ ‖term_ρ‖ = Σ_ρ (zeroMult : ℝ)/‖s−ρ‖ (norm_div + natCast norm — clone the 6-c-ii h2 pattern per term). Then the integrand split: Σ_ρ m_ρ/‖s σ −ρ‖ = Σ_ρ m_ρ·(1/√((σ−β_ρ)²+δ_ρ²)) — this is a pointwise (in σ) Finset-sum identity — sum_congr with the per-ρ norm identity ✓. Then ∫σ Σ_ρ m_ρ·(1/√(...)) dσ = Σ_ρ ∫σ m_ρ·(1/√(...)) dσ (integral_finsetSum) = Σ_ρ m_ρ·∫σ 1/√(...) (integral_const_mul per term — need sum_congr) ≤ Σ_ρ m_ρ·(2·log(148.5·C0hat(√n+2))) (sum_le_sum with per-zero bound × m_ρ ≥ 0) = (2·log(148.5·C0hat(√n+2)))·Σ_ρ m_ρ (Finset.mul_sum.symm... — Σ m·B0 = B0·Σm hmm: Σ_ρ m_ρ·B0 = B0·Σ_ρ m_ρ via: each m_ρ·B0 = B0·m_ρ (mul_comm) — or Finset.mul_sum: B0·Σm = Σ B0·m — so Σ m·B0 = Σ B0·m (sum_congr mul_comm) = B0·Σm (Finset.mul_sum.symm)). Then ≤ B0·Z2hat(T) (mul_le_mul_of_nonneg_left (nearZerosSub_mult_sum_le hZC2 hT10) (B0 ≥ 0)). B0 := 2·log(148.5·C0hat(√n+2)) ≥ 0: log(148.5·C0hat(√n+2)) ≥ 0 ⟸ 148.5·C0hat(√n+2) ≥ 1 ⟸ C0hat(√n+2) ≥ 2 ✓ (two_le_C0hat, √n+2 ≥ 1). Real.log_nonneg (1 ≤ x) ✓. OK. And NI := 2 * Z2hat T * Real.log (148.5 * C0hat (Real.sqrt n + 2)) — hmm — the association: 2·Z2hat·log vs B0·Z2hat = 2·log·Z2hat — same by ring. The final per-side statement: ```lean theorem hIntegral_Jlag_mid_norm_le (hZC : ZeroCountProp) (hZC2 : ZeroCountProp2) (hNLZ : NoLowZeros) {n : ℕ} (hn : 100000 ≤ n) {T : ℝ} (hTwide : ∀ ρ : ℂ, IsNontrivialZero ρ → Real.sqrt n - 1 < ρ.im → ρ.im < Real.sqrt n + 2 → 1 / (9 * C0hat (Real.sqrt n + 2)) ≤ |ρ.im - T|) (hTn : Real.sqrt n ≤ T) (hTn1 : T ≤ Real.sqrt n + 1) : ‖HIntegral (Jlag n) (-5/2) 3 T‖ ≤ (Real.exp 4 + 1) * (2 * Z2hat T * Real.log (148.5 * C0hat (Real.sqrt n + 2)) + (3 - -5/2) * (10.5 * Real.log (T + 2) + 48)) ``` and the neg version hIntegral_Jlag_mid_neg_norm_le (same bound) via the conj trick. Hmm — the hTwide hypothesis: it's the output of exists_zero_avoiding_T_wide — let me check that exact form (it's an ∃-statement at 6-d; the 6-c theorems take it as a hypothesis, like zeta_ne_zero_horizontal does). zeta_ne_zero_horizontal's hTwide: `∀ ρ : ℂ, IsNontrivialZero ρ → Real.sqrt n - 1 < ρ.im → ρ.im < Real.sqrt n + 2 → 1 / (9 * C0hat (Real.sqrt n + 2)) ≤ |ρ.im - T|` ✓ — same form. For the per-ρ δ-lower bound: ρ ∈ nearZerosSub T with γ := ρ.1.im: hTwide ρ.1 ρ.2 (γ ∈ (√n−1, √n+2)): |γ − T| ≥ 1/(9·C0hat(√n+2)). The window: γ ∈ (T−1, T+1) from |γ − T| < 1 (mem_nearZerosSub) — and (T−1, T+1) ⊆ (√n−1, √n+2): T−1 ≥ √n−1 ✓ (hTn: √n ≤ T); T+1 ≤ √n+2 ✓ (hTn1: T ≤ √n+1). So √n−1 ≤ T−1 < γ ✓ and γ < T+1 ≤ √n+2 ✓ — hTwide needs STRICT: √n−1 < γ: from √n−1 ≤ T−1 < γ ✓ linarith. ✓. δ_ρ := |T − γ| — hmm, hTwide gives |γ − T| ≥ c_clear; mem_nearZerosSub gives |γ − T| < 1. So work with δ' := |γ − T| throughout (avoid abs_sub_comm): the norm identity ‖s−ρ‖ = √((σ−β)²+(T−γ)²) needs (T−γ)² = |γ−T|²: (T−γ)² = (γ−T)² (neg_sq) = |γ−T|² (sq_abs) ✓. So per-ρ: δ' := |ρ.1.im − T|; 0 < c_clear ≤ δ' (hTwide) and δ' < 1 (mem). The FTC bound applies with δ := δ' > 0 ✓ and δ' ≤ 1 ✓. Per-zero integral ≤ 2·log(16.5/δ') ≤ 2·log(148.5·C0hat(√n+2)): the last step: 16.5/δ' ≤ 148.5·C0hat(√n+2): 1/δ' ≤ 9·C0hat(√n+2): from δ' ≥ 1/(9·C0hat(√n+2)): `one_div_le_one_div_of_le (one_div_pos.mpr (by positivity)) h` — a := 1/(9C), b := δ': 1/δ' ≤ 1/(1/(9C)) — and 1/(1/(9C)) = 9C (one_div_one_div). ✓. Then 16.5·(1/δ') ≤ 16.5·(9·C0hat) = 148.5·C0hat ✓ (mul_le_mul_of_nonneg_left ... 16.5 ≥ 0). Then Real.log_le_log (16.5/δ' > 0) that ✓. Hmm — 16.5/δ' = 16.5·(1/δ') — div_eq_mul_one_div... write 16.5/δ' as 16.5 * (1/δ')? The per-zero bound's final form: 2·log(16.5/δ'). For the comparison: log(16.5/δ') ≤ log(148.5·C0hat) ⟸ Real.log_le_log (0 < 16.5/δ' := by positivity) (16.5/δ' ≤ 148.5·C0hat). The division forms: 16.5/δ' — positivity ✓ (δ' > 0). The ≤: ```lean have hcomp : 16.5 / δ' ≤ 148.5 * C0hat (√n + 2) := by rw [div_le_iff₀ hδ'0] -- 16.5 ≤ 148.5·C0hat·δ' ... ``` 16.5 ≤ 148.5·C·δ' ⟸ δ' ≥ 1/(9C): 148.5·C·δ' ≥ 148.5·C·(1/(9C)) = 148.5/9 = 16.5 ✓. nlinarith [hδ'_ge, hC0pos]? 148.5·C·δ' ≥ 148.5·C·(1/(9C)) = 16.5·(9C)/(9C) = 16.5 — needs C·(1/(9C)) = 1/9·... — the product C·(1/(9·C)) = 1/9 — needs C ≠ 0 — as a FIELD identity — nlinarith can't do division. Provide explicitly: ```lean have hC0 : (0:ℝ) < C0hat (√n + 2) := by have h2 := two_le_C0hat (by positivity... : 1 ≤ √n + 2) linarith have h9C : (0:ℝ) < 9 * C0hat (√n + 2) := by positivity have hge : 1 / (9 * C0hat (√n + 2)) ≤ δ' := hTwide-applied have hinv : 1 / δ' ≤ 9 * C0hat (√n + 2) := by have := one_div_le_one_div_of_le (one_div_pos.mpr h9C) hge rwa [one_div_one_div] at this have hcomp : 16.5 / δ' ≤ 148.5 * C0hat (√n + 2) := by rw [div_le_iff₀ hδ'0] -- 16.5 ≤ 148.5·C0hat·δ' have h1 : 16.5 * (1 / δ') ≤ 16.5 * (9 * C0hat (√n + 2)) := mul_le_mul_of_nonneg_left hinv (by norm_num) rw [mul_comm, ← div_eq_mul_one_div] at h1 -- hmm -- h1 : 16.5/δ' ≤ 148.5·C0hat nlinarith [h1, hδ'0] -- 16.5 ≤ (16.5/δ')·δ' = ... ``` Hmm messy. Cleaner: 16.5 ≤ 148.5·C·δ' ⟺ (multiply hinv by 16.5·δ' ≥ 0): 16.5·δ'·(1/δ') ≤ 16.5·δ'·(9C) ⟹ 16.5 ≤ 16.5·(9C)·δ' = 148.5·C·δ' ✓ — 16.5·δ'·(1/δ') = 16.5·(δ'/δ') = 16.5·1 = 16.5 (δ' ≠ 0) — field identity — nlinarith can't; use field_simp or: ```lean have hcomp : 16.5 / δ' ≤ 148.5 * C0hat (√n + 2) := by have h1 : 16.5 * (1 / δ') ≤ 16.5 * (9 * C0hat (√n + 2)) := mul_le_mul_of_nonneg_left hinv (by norm_num) rw [show (16.5 : ℝ) * (1 / δ') = 16.5 / δ' by rw [mul_one_div], show (16.5 : ℝ) * (9 * C0hat (√n + 2)) = 148.5 * C0hat (√n + 2) by ring] at h1 exact h1 ``` mul_one_div : a * (1/b) = a/b ✓. And 16.5·9 = 148.5 ✓ ring. OK — this is going to be a large amount of Lean. Let me also double check `Real.log_div` signature: `Real.log_div (hx : x ≠ 0) (hy : y ≠ 0) : Real.log (x / y) = Real.log x - Real.log y` ✓. `Real.log_pow (n : ℕ) (x : ℝ) : Real.log (x ^ n) = n * Real.log x` ✓. `Real.log_one : Real.log 1 = 0` ✓. `neg_sq (a) : (-a) ^ 2 = a ^ 2` — hmm — it's `neg_sq : (-a) ^ 2 = a ^ 2` ✓ exists (even_pow). Or use `Even.neg_pow`? neg_sq exists for sure. `sq_abs (a : ℝ) : |a| ^ 2 = a ^ 2` ✓ exists. `neg_abs_le (a) : -a ≤ |a|` ✓. `Real.sqrt_sq_eq_abs (x) : √(x^2) = |x|` ✓. `Real.sqrt_lt_sqrt (hx : 0 ≤ x) (h : x < y) : √x < √y` ✓. `Real.sqrt_pos.mpr : 0 < x → 0 < √x` ✓. `Real.sq_sqrt (h : 0 ≤ x) : √x ^ 2 = x` ✓. `div_le_div_of_nonneg_left` — signature? `div_le_div_of_nonneg_left (ha : 0 ≤ a) (hb : 0 < b) (h : b ≤ c) : a / c ≤ a / b`? — hmm — from LiTightR line 172: `div_le_div_of_nonneg_left (Nat.cast_nonneg _) (by norm_num : (0:ℝ) < 1) h1` giving `m/‖s−ρ‖ ≤ m/1` — so (ha : 0 ≤ a) (hb : 0 < b) (h : b ≤ c) : a / c ≤ a / b ✓ — with a := δ², b := u+√, c := 7+δ: δ²/(7+δ) ≤ δ²/(u+√) ✓ EXACTLY what I need for the left endpoint. `Real.sqrt_le_iff`? For √h ≤ x+δ: use sq_le_sq₀ or: `Real.sqrt_le_iff (hy : 0 ≤ y)?` — let me instead use: `Real.sqrt_le' ?`... — safest: ```lean have hsqrt_le : Real.sqrt (x ^ 2 + δ ^ 2) ≤ x + δ := by have h1 : (Real.sqrt (x ^ 2 + δ ^ 2)) ^ 2 ≤ (x + δ) ^ 2 := by rw [Real.sq_sqrt (by positivity)] nlinarith [hδ, hx0] have h2 : 0 ≤ x + δ := by linarith [hx0, hδ] -- from sq ≤ sq with both nonneg: a ≤ b exact (sq_le_sq₀ (Real.sqrt_nonneg _) h2).mp?... ``` Hmm — sq_le_sq₀ (ha : 0 ≤ a) (hb : 0 ≤ b) : a ^ 2 ≤ b ^ 2 ↔ a ≤ b? or the iff the other way — `sq_le_sq₀ (ha : 0 ≤ a) (hb : 0 ≤ b) : a ^ 2 ≤ b ^ 2 ↔ a ≤ b`? Banked: "sq_eq_sq₀" and "pow_le_one₀'s implicit n". Let me check sq_le_sq₀'s exact form via grep later. Alternative robust: `Real.sqrt_le_iff`: grep says? I'll check: there's `Real.sqrt_le_iff {x y : ℝ} (hy : 0 ≤ y) : √x ≤ y ↔ x ≤ y^2`?? Hmm — actually I recall `Real.sqrt_le_iff : √x ≤ y ↔ 0 ≤ y ∧ x ≤ y ^ 2`. And `Real.sqrt_le' {y} (hy : 0 < y) : √x ≤ y ↔ x ≤ y^2`?? Let me grep both. Let me also check `one_div_pos.mpr`, `one_div_one_div (a) : 1/(1/a) = a` ✓ exists (one_div_one_div). `Real.log_le_log (hx : 0 < x) (h : x ≤ y) : log x ≤ log y` ✓ used already. Also `integral_mono_on` for the ∫‖Jlag‖ ≤ ∫(e⁴+1)·‖logDeriv‖ step — with the constant factor integrand (fun σ => (e⁴+1)·‖logDeriv ζ (s σ)‖) — continuous ✓. And ∫(e⁴+1)·‖logDeriv‖ = (e⁴+1)·∫‖logDeriv‖: intervalIntegral.integral_const_mul ✓. Then ∫‖logDeriv‖ ≤ ∫‖R‖ + ∫‖Σ‖: intervalIntegral.integral_add hR_int hΣ_int : ∫(‖R‖ + ‖Σ‖) = ∫‖R‖ + ∫‖Σ‖; pointwise: ‖logDeriv‖ ≤ ‖R‖ + ‖Σ‖ via norm_le_insert' (with R := logDeriv − Σ... wait — norm_le_insert' a b : ‖a‖ ≤ ‖b‖ + ‖a − b‖ — a := logDeriv, b := Σ: ‖logDeriv‖ ≤ ‖Σ‖ + ‖logDeriv − Σ‖ — hmm — I want ‖logDeriv‖ ≤ ‖R‖ + ‖Σ‖ with R = logDeriv − Σ: norm_le_insert' gives ‖logDeriv‖ ≤ ‖Σ‖ + ‖logDeriv−Σ‖ = ‖Σ‖ + ‖R‖ — commuted — fine (linarith/add_comm). Integrability of ‖R‖ = ‖logDeriv − Σ‖: logDeriv continuous on path (zeta_ne_zero_horizontal), Σ continuous (sum of continuous divisions) ⟹ sub continuous ⟹ norm continuous ✓. The kernel: per σ: ‖kFun n (↑σ+↑T·I)‖ = ‖kFun n ((↑σ+↑T·I−1)+1)‖ — kFun_shift_norm_le_exp_four_add_one (s := ↑σ+↑T·I−1): (hσ : (↑σ+↑T·I−1).re ≤ 3) : re = σ−1: σ ≤ 3 ⟹ σ−1 ≤ 2 ≤ 3 ✓; (hT : (n:ℝ) ≤ (↑σ+↑T·I−1).im^2) : im = T: n ≤ T² — from √n ≤ T: n = √n² ≤ T²: ```lean have hnT2 : (n : ℝ) ≤ T ^ 2 := by have h1 : (Real.sqrt n) ^ 2 ≤ T ^ 2 := by apply sq_le_sq₀?... -- √n ≤ T, both ≥ 0 rwa [Real.sq_sqrt (by positivity : 0 ≤ (n:ℝ))] at h1 ``` Hmm — need √n ≤ T ⟹ √n² ≤ T²: sq_le_sq₀ (√n ≥ 0) (T ≥ 0) — if sq_le_sq₀ is the iff: (sq_le_sq₀ h1 h2).mpr hTn? — wait direction: √n ≤ T with 0 ≤ √n, 0 ≤ T ⟹ √n² ≤ T² — `pow_le_pow_left₀ (Real.sqrt_nonneg _) hTn 2` : (√n)^2 ≤ T^2 ✓ SIMPLER (pow_le_pow_left₀ (0≤a) (a≤b) n : a^n ≤ b^n ✓). Then rw [Real.sq_sqrt] at that: (↑n) ≤ T^2 — wait Real.sq_sqrt : √x^2 = x — rw at h1 : (↑n) ≤ T^2 ✓. Then ‖kFun n (s+1)‖ ≤ e⁴+1 with s+1 = ↑σ+↑T·I: `(s+1) = ↑σ + ↑T*I` — s := (↑σ + ↑T*I) − 1 so s+1 = ↑σ+↑T·I via sub_add_cancel ✓ — the lemma gives ‖kFun n ((↑σ+↑T·I−1)+1)‖ — need to rewrite (↑σ+↑T·I−1)+1 = ↑σ+↑T·I: `sub_add_cancel`? — a−1+1 = a: `sub_add_cancel : a - b + b = a`? — sub_add_cancel (a b) : a − b + b = a ✓ — with b := 1 ✓. So: ```lean have hker : ∀ σ ∈ Set.Icc (-5/2 : ℝ) 3, ‖kFun n ((σ:ℝ) + (T:ℝ)*Complex.I)‖ ≤ Real.exp 4 + 1 := by intro σ hσ obtain ⟨hσ1, hσ2⟩ := hσ have h1 := kFun_shift_norm_le_exp_four_add_one (n := n) (s := ((σ:ℝ) + (T:ℝ)*Complex.I - 1)) (by simp...) (by simp...; exact hnT2) rwa [sub_add_cancel] at h1 ``` Hmm — kFun_shift's s is implicit {s : ℂ} — pin (s := ...). hσ: s.re ≤ 3 — (↑σ+↑T·I−1).re = σ−1: by simp + linarith [hσ2]. The by-block: `by have hre : (...).re = σ - 1 := by simp [...]; rw [hre]; linarith [hσ2]`. Wait — simp computing (↑σ+↑T*I−1).re: sub_re: (↑σ+↑T·I).re − 1; add_re: σ + 0; → σ − 1 ✓. `by simp` should do it (as before with explicit lemma list or plain simp — plain simp worked for these re computations in zeta_ne_zero_left_vertical ✓). hT hyp: (↑n:ℝ) ≤ (↑σ+↑T·I−1).im^2 — im = T: by simp gives (↑n) ≤ T^2 ✓ exact hnT2. So `by simp [Complex.sub_im, ...]` — hmm — the goal after simp: (n:ℝ) ≤ T^2 — simp might also simplify... just: `by have him : (...).im = T := by simp [...]; rw [him]; exact hnT2`. rwa [sub_add_cancel] at h1: h1 : ‖kFun n ((↑σ+↑T·I−1)+1)‖ ≤ e⁴+1 — sub_add_cancel : a − b + b = a — the term (↑σ+↑T·I)−1+1 — rw transforms to ↑σ+↑T·I ✓ — hmm — sub_add_cancel (a b) : a - b + b = a — pattern ?a − ?b + ?b — matches (↑σ+↑T·I) − 1 + 1 with ?a := ↑σ+↑T·I, ?b := 1 ✓. Alright — this 6-c-iv is genuinely large (~450-550 lines). Let me organize the file §4 as: 1. `integral_one_div_sqrt_sq_add_sq` — the FTC machine (β δ general, a b general). 2. `arcsinh_right_endpoint_le` (F(3) ≤ log(6+δ)). 3. `arcsinh_left_endpoint_le` (−F(−5/2) ≤ log(7+δ)+2log(1/δ)). 4. `per_zero_integral_le_two_log` — combine (≤ 2·log(16.5/δ)) incl. D's log-mono simplification. 5. `hIntegral_Jlag_mid_norm_le` — the top side. 6. `hIntegral_Jlag_mid_neg_norm_le` — the bottom via conj. Let me double check a few more signatures before writing: sq_le_sq₀ (not needed — using pow_le_pow_left₀), Real.sqrt_le_iff (for √h ≤ x+δ), Real.continuous_sqrt, intervalIntegral.integral_const_mul, Continuous.pow, Real.log_div, Real.log_mul, Real.log_pow, neg_sq, sq_abs, one_div_one_div, div_le_div_of_nonneg_left (verified in LiTightR ✓), abs_of_nonneg/abs_of_neg ✓, Real.log_one, Real.log_nonneg, two_le_C0hat ✓ (LiContour line 169). For Real.sqrt_le_iff — grep.
All patterns confirmed. Checking the last few signatures before writing §4.
tool: Bash
input
{
  "command": "grep -rn \"theorem sqrt_le_iff\\|theorem sqrt_le' \" \"D:/Python/Quantum/zeta-23-lean/.lake/packages/mathlib/Mathlib/Analysis/Real/Sqrt.lean\" | head -4; grep -rn \"theorem continuous_sqrt\" \"D:/Python/Quantum/zeta-23-lean/.lake/packages/mathlib/Mathlib/Analysis/SpecialFunctions/Sqrt.lean\" | head -2; grep -rn \"theorem integral_const_mul \\|theorem integral_const_mul(\" \"D:/Python/Quantum/zeta-23-lean/.lake/packages/mathlib/Mathlib/MeasureTheory/Integral/IntervalIntegral/Basic.lean\" | head -3; grep -rn \"theorem Continuous.pow \\|theorem log_pow \\|theorem log_div \\|theorem log_mul \\|theorem neg_sq \\|theorem sq_abs \\|theorem one_div_one_div \\|theorem sqrt_sq_eq_abs\" \"D:/Python/Quantum/zeta-23-lean/.lake/packages/mathlib/Mathlib/\" 2>/dev/null | head -12",
  "description": "Verify sqrt/log/pow lemma signatures"
}
#2281 assistant 2026-08-26 18:44:48.955 UTC · 695→32000 tok open ⤢
thinking (59441 chars)
- `Real.continuous_sqrt : Continuous (√· : ℝ → ℝ)` ✓ (Analysis/Real/Sqrt.lean:126 — and line 95 shows `sqrt.continuous` in the NNReal? — the ℝ one at 126 ✓). The first at :95 is `Real.sqrt.continuous`? Anyway — `Real.continuous_sqrt` exists ✓. - `neg_sq (a) : (-a)^2 = a^2` ✓ (Algebra/Ring/Commute — root namespace). - `sq_abs (a) : |a|^2 = a^2` ✓ @[simp] (root). - `Continuous.div (hf) (hg) (h₀ : ∀ x, g x ≠ 0)` ✓. All confirmed. Now write §4. Given the size, I'll write it in two chunks: first the arcsinh-FTC machine (lemmas 1–4), build-check that alone, then the mid-horizontal bound (5–6). Actually — write the whole §4 and iterate; the error loop is ~25 s per elaboration which is fine. Let me draft the full §4 now. Points of care: - All `∫ σ in a..b, ... + ...` get parens. - The per-σ HasDerivAt chain with convert + field_simp. - uIcc membership for integral_eq_sub_of_hasDerivAt — hderiv : ∀ x ∈ uIcc a b — I prove ∀ σ (global), then `fun σ _ => hF σ`. Let me also double-check `hasDerivAt_id` — signature: `theorem hasDerivAt_id (x : 𝕜) : HasDerivAt id 1 x` ✓ standard. `HasDerivAt.sub_const (hf : HasDerivAt f f' x) (c : 𝕜) : HasDerivAt (fun x => f x - c) f' x` ✓ standard. For the per-ρ norm identity: ‖(↑σ + ↑T·I) − ρ‖ = √((σ−ρ.re)² + (T−ρ.im)²): ```lean have hnormid : ∀ σ : ℝ, ‖((σ : ℝ) + (T : ℝ) * Complex.I) - ρ‖ = Real.sqrt ((σ - ρ.re) ^ 2 + (T - ρ.im) ^ 2) := by intro σ have h1 : Complex.normSq (((σ : ℝ) + (T : ℝ) * Complex.I) - ρ) = (σ - ρ.re) ^ 2 + (T - ρ.im) ^ 2 := by simp [Complex.normSq_apply] ring rw [← h1] -- ‖z‖ = √(normSq z) exact (Complex.norm_def _).symm?? ``` Hmm — what IS the norm-def lemma? `Complex.norm_def : ‖z‖ = Real.sqrt (normSq z)`? or `Complex.norm_eq_sqrt_normSq`? Let me check: In Mathlib, `Complex.norm` is defined as `Real.sqrt (normSq z)` via `Complex.norm (z : ℂ) : ℝ := Real.sqrt (normSq z)`? Actually the Norm instance: `instNorm : Norm ℂ := ⟨fun z => Real.sqrt (normSq z)⟩`. So `‖z‖ = Real.sqrt (normSq z)` is rfl! There's `Complex.norm_def : ‖z‖ = Real.sqrt (normSq z)`? — I recall `Complex.norm_def (z : ℂ) : ‖z‖ = Real.sqrt (normSq z) := rfl`. If it's rfl, then `rw [← h1]` turns goal `‖z‖ = √((σ−β)²+(T−γ)²)` into `‖z‖ = √(normSq z)` — then `rfl`?? — is `‖z‖` DEFEQ `√(normSq z)`? Yes via instNorm. So `exact rfl`?? Hmm — rfl might work if the Norm instance unfolds reducibly. There's `Complex.norm_def` — let me grep. Actually — safer: `Complex.normSq_eq_norm_sq : normSq z = ‖z‖ ^ 2` — gives ‖z‖² = normSq z, then ‖z‖ = √(normSq z) via (Real.sqrt_sq (norm_nonneg)).symm-ish: ```lean have h2 : Complex.normSq (((σ : ℝ) + (T : ℝ) * Complex.I) - ρ) = ‖(((σ : ℝ) + (T : ℝ) * Complex.I) - ρ)‖ ^ 2 := Complex.normSq_eq_norm_sq _ -- want ‖z‖ = √(X) where normSq z = X rw [← h1] at h2 -- h2 : X = ‖z‖^2 where X := (σ−β)²+(T−γ)² -- ‖z‖ = √X: both nonneg, squares equal have h3 : (0:ℝ) ≤ ‖(((σ : ℝ) + (T : ℝ) * Complex.I) - ρ)‖ := norm_nonneg _ have h4 : ‖(((σ : ℝ) + (T : ℝ) * Complex.I) - ρ)‖ = Real.sqrt ((σ - ρ.re) ^ 2 + (T - ρ.im) ^ 2) := by rw [← h2] -- goal: ‖z‖ = √(‖z‖^2)?? no... ``` Hmm — cleaner: `Real.sqrt_eq_iff_eq_sq (hx : 0 ≤ x) (hy : 0 ≤ y) : √x = y ↔ x = y^2`? — DL-5 used `Real.sqrt_eq_iff_eq_sq (by norm_num) (by norm_num)` ✓ EXISTS. So: ```lean have hnormid : ‖z‖ = Real.sqrt X := by rw [Real.sqrt_eq_iff_eq_sq (by positivity) (norm_nonneg _)] -- hmm direction: √X = ‖z‖ -- goal: √X = ‖z‖ ↔ X = ‖z‖^2 sorry ``` Real.sqrt_eq_iff_eq_sq (hx : 0 ≤ x) (hy : 0 ≤ y) : Real.sqrt x = y ↔ x = y ^ 2. So: ```lean symm rw [Real.sqrt_eq_iff_eq_sq (by positivity) (norm_nonneg _)] -- goal: X = ‖z‖^2 rw [← Complex.normSq_eq_norm_sq] exact h1 ``` Hmm — goal ‖z‖ = √X; symm → √X = ‖z‖; rw [Real.sqrt_eq_iff_eq_sq hX hnorm] → goal X = ‖z‖^2; rw [← Complex.normSq_eq_norm_sq] → normSq z = ‖z‖^2?? — wait — `Complex.normSq_eq_norm_sq (z) : normSq z = ‖z‖^2` — rw [← that] at goal X = ‖z‖^2 — the ← direction: pattern ‖z‖^2 → normSq z: goal becomes X = normSq z — exact h1 (h1 : normSq z = X — hmm — need X = normSq z — h1.symm ✓). So exact h1.symm. Let me restructure: ```lean have hnormid : ‖((σ : ℝ) + (T : ℝ) * Complex.I) - ρ‖ = Real.sqrt ((σ - ρ.re) ^ 2 + (T - ρ.im) ^ 2) := by have h1 : Complex.normSq (((σ : ℝ) + (T : ℝ) * Complex.I) - ρ) = (σ - ρ.re) ^ 2 + (T - ρ.im) ^ 2 := by simp [Complex.normSq_apply] ring symm rw [Real.sqrt_eq_iff_eq_sq (by positivity) (norm_nonneg _), ← Complex.normSq_eq_norm_sq] exact h1.symm ``` Hmm — h1's simp [Complex.normSq_apply]: normSq z = z.re*z.re + z.im*z.im (normSq_apply : normSq z = z.re * z.re + z.im * z.im? — from LiContour's hD_rel line 474: `D = s.re * s.re + s.im * s.im := by rw [hD_def, Complex.normSq_apply s]` ✓). So simp [Complex.normSq_apply] turns goal into re/im form, then simp computes re/im of ((↑σ+↑T·I)−ρ): sub_re: σ − ρ.re... wait: (↑σ+↑T·I−ρ).re = (↑σ+↑T·I).re − ρ.re = σ − ρ.re ✓; im: T − ρ.im ✓. Then goal: (σ−ρ.re)*(σ−ρ.re) + (T−ρ.im)*(T−ρ.im) = (σ−ρ.re)^2 + (T−ρ.im)^2 — ring ✓ (sq vs *2 form — ring handles). The simp call: `simp [Complex.normSq_apply]` — will simp also compute the re/im? Complex.sub_re etc. are simp lemmas ✓ default simp set includes them ✓ probably reduces fully to the ring-closable form. If not, add the explicit list. Then `ring`. Now — the FULL §4 draft. Let me write it. I'll be careful about: (a) parens around all integral summands. (b) the positivity side conditions. For (5) hIntegral_Jlag_mid_norm_le — the full proof: ```lean theorem hIntegral_Jlag_mid_norm_le (hZC : ZeroCountProp) (hZC2 : ZeroCountProp2) (hNLZ : NoLowZeros) {n : ℕ} (hn : 100000 ≤ n) {T : ℝ} (hTwide : ∀ ρ : ℂ, IsNontrivialZero ρ → Real.sqrt (n : ℝ) - 1 < ρ.im → ρ.im < Real.sqrt (n : ℝ) + 2 → 1 / (9 * C0hat (Real.sqrt (n : ℝ) + 2)) ≤ |ρ.im - T|) (hTn : Real.sqrt (n : ℝ) ≤ T) (hTn1 : T ≤ Real.sqrt (n : ℝ) + 1) : ‖HIntegral (Jlag n) (-5 / 2) 3 T‖ ≤ (Real.exp 4 + 1) * (2 * Z2hat T * Real.log (148.5 * C0hat (Real.sqrt (n : ℝ) + 2)) + (3 - -5 / 2) * (10.5 * Real.log (T + 2) + 48)) := by classical have hnR : (100000 : ℝ) ≤ (n : ℝ) := by exact_mod_cast hn have h10T : (10 : ℝ) ≤ T := by have h1 : (10 : ℝ) ≤ Real.sqrt (n : ℝ) := by rw [Real.le_sqrt (by norm_num) (by positivity)] nlinarith [hnR] linarith have hT0 : (0 : ℝ) ≤ T := by linarith have hn1 : 1 ≤ n := by omega have hlen : (-5 / 2 : ℝ) ≤ 3 := by norm_num -- the C₀ positivity facts have hC0 : (0 : ℝ) < C0hat (Real.sqrt (n : ℝ) + 2) := by have h2 := two_le_C0hat (by nlinarith [Real.sqrt_nonneg (n : ℝ)] : (1 : ℝ) ≤ Real.sqrt (n : ℝ) + 2) linarith -- zero-freeness on the horizontal (for continuity) have hzne : ∀ σ : ℝ, riemannZeta ((σ : ℝ) + (T : ℝ) * Complex.I) ≠ 0 := fun σ => zeta_ne_zero_horizontal hTwide hn1 hTn hTn1 -- continuity of the path integrands ... ``` Wait — zeta_ne_zero_horizontal's hn is (1 ≤ n) ✓ and σ — its statement: riemannZeta ((σ:ℝ) + (T:ℝ)*I) ≠ 0 ✓. Continuity of ‖logDeriv‖, ‖R‖, ‖Σ‖, ‖Jlag‖ on the path — same machine as 6-c-ii but with zeta_ne_zero_horizontal: ```lean have hJcont : ∀ σ : ℝ, ContinuousAt (Jlag n) ((σ : ℝ) + (T : ℝ) * Complex.I) := by intro σ have hs1 : ((σ : ℝ) + (T : ℝ) * Complex.I) ≠ 1 := by intro h have him := congrArg Complex.im h simp at him -- T = 0 linarith [hT0] -- T ≥ 316 ≠ 0... wait him : T = 0? (1:ℂ).im = 0 — him : (point).im = Complex.im 1 → T = 0 — linarith with h10T ✓ have hA := (kFun_differentiableAt (n := n) hs1).continuousAt have hB := continuousAt_logDeriv_zeta hs1 (hzne σ) show ContinuousAt (fun s => kFun n s * (-logDeriv riemannZeta s)) _ exact hA.mul hB.neg ``` Hmm — him : (↑σ + ↑T·I).im = (1:ℂ).im — simp → T = 0 ✓ then linarith [h10T] ✓. For ‖R‖ continuity: R = logDeriv ζ − Σ: ```lean have hRcont : Continuous fun σ : ℝ => ‖logDeriv riemannZeta ((σ : ℝ) + (T : ℝ) * Complex.I) - ∑ ρ ∈ nearZerosSub T, (zeroMult ρ.1 : ℂ) / (((σ : ℝ) + (T : ℝ) * Complex.I) - ρ.1)‖ := by apply continuous_norm.comp apply Continuous.sub · -- logDeriv continuous rw [continuous_iff_continuousAt] intro σ have hs1 : ((σ : ℝ) + (T : ℝ) * Complex.I) ≠ 1 := ... have hf : ContinuousAt (fun u : ℝ => ((u : ℝ) + (T : ℝ) * Complex.I)) σ := hpath.continuousAt exact ContinuousAt.comp (g := logDeriv riemannZeta) (f := ...) (x := σ) (continuousAt_logDeriv_zeta hs1 (hzne σ)) hf · -- the near sum continuous apply continuous_finset_sum intro ρ hρ apply Continuous.div continuous_const · -- the denominator continuous exact hpath.sub continuous_const · intro σ -- denominator ≠ 0: else s = ρ, ζ s = ζ ρ = 0 vs hzne have hne : ((σ : ℝ) + (T : ℝ) * Complex.I) - ρ.1 ≠ 0 := by intro h have hs : ((σ : ℝ) + (T : ℝ) * Complex.I) = ρ.1 := sub_eq_zero.mp h have h1 := hzne σ rw [hs] at h1 exact h1 ρ.2.1 exact hne σ -- hmm, the h₀ for Continuous.div is ∀ x, g x ≠ 0 ``` Wait — Continuous.div (hf) (hg) (h₀ : ∀ x, g x ≠ 0) — so I need ∀ σ, (↑σ+↑T·I) − ρ.1 ≠ 0 ✓ from the hzne argument ✓. `continuous_finset_sum : (∀ i ∈ s, Continuous (f i)) → Continuous fun x => ∑ i ∈ s, f i x` ✓ exists (continuous_finset_sum s h). `hpath : Continuous fun u : ℝ => ((u : ℝ) + (T : ℝ) * Complex.I) := by fun_prop` ✓. For the logDeriv branch — the ContinuousAt.comp pinned pattern ✓. Actually — I realize I can factor a small helper for the composition since it repeats: skip, inline is fine. For ‖Σ‖ continuity: same denominator machine + norm. For ‖Jlag‖: hJcont machine + norm (as 6-c-ii). The kernel bound hker (as drafted above). The R-bound pointwise: ```lean have hR : ∀ σ : ℝ, σ ∈ Set.Icc (-5/2 : ℝ) 3 → ‖logDeriv riemannZeta ((σ : ℝ) + (T : ℝ) * Complex.I) - ∑ ρ ∈ nearZerosSub T, (zeroMult ρ.1 : ℂ) / (((σ : ℝ) + (T : ℝ) * Complex.I) - ρ.1)‖ ≤ 10.5 * Real.log (T + 2) + 48 := by intro σ hσ obtain ⟨hσ1, hσ2⟩ := hσ have hre : ((σ : ℝ) + (T : ℝ) * Complex.I).re = σ := by simp [...] have him : ((σ : ℝ) + (T : ℝ) * Complex.I).im = T := by simp [...] have h := logDeriv_riemannZeta_sub_nearZeros_le_tight (t := T) (s := ((σ : ℝ) + (T : ℝ) * Complex.I)) hZC hZC2 hNLZ h10T (by rw [hre]; exact hσ1) (by rw [hre]; exact hσ2) him (hzne σ) rwa [abs_of_nonneg hT0] at h ``` Hmm — the capstone's bound: 10.5·log(|T|+2)+48 — rw [abs_of_nonneg hT0] at h : |T| → T ✓. The near-sum pointwise split and per-term norm — the s in the capstone's sum is (↑σ + ↑T·I): terms (zeroMult ρ.1 : ℂ)/((↑σ+↑T·I) − ρ.1) ✓ — my ‖Σ‖ integrand matches ✓. The integral estimates: ```lean -- ∫ ‖R‖ ≤ (3−(−5/2))·K_R have hRint : ∫ σ in (-5/2 : ℝ)..3, ‖logDeriv riemannZeta ((σ : ℝ) + (T : ℝ) * Complex.I) - ∑ ρ ∈ nearZerosSub T, (zeroMult ρ.1 : ℂ) / (((σ : ℝ) + (T : ℝ) * Complex.I) - ρ.1)‖ ≤ (3 - -5/2) * (10.5 * Real.log (T + 2) + 48) := by have h := intervalIntegral.integral_mono_on hlen hRintbl intervalIntegrable_const (fun σ hσ => hR σ hσ) rwa [intervalIntegral.integral_const, smul_eq_mul] at h ``` The near-sum integral — the heart: ```lean -- per-ρ data: δ_ρ := |ρ.1.im − T| -- per-σ pointwise: ‖Σ‖ ≤ Σ m·(1/√((σ−β)²+δ²)) have hΣbound : ∫ σ in (-5/2 : ℝ)..3, ‖∑ ρ ∈ nearZerosSub T, (zeroMult ρ.1 : ℂ) / (((σ : ℝ) + (T : ℝ) * Complex.I) - ρ.1)‖ ≤ 2 * Z2hat T * Real.log (148.5 * C0hat (Real.sqrt (n : ℝ) + 2)) := by -- step 1: pointwise norm_sum_le + per-term norms have hpt : ∀ σ : ℝ, σ ∈ Set.Icc (-5/2 : ℝ) 3 → ‖∑ ρ ∈ nearZerosSub T, (zeroMult ρ.1 : ℂ) / (((σ : ℝ) + (T : ℝ) * Complex.I) - ρ.1)‖ ≤ ∑ ρ ∈ nearZerosSub T, (zeroMult ρ.1 : ℝ) * (1 / Real.sqrt ((σ - ρ.1.re) ^ 2 + (T - ρ.1.im) ^ 2)) := by intro σ _ calc ‖∑ ρ ∈ nearZerosSub T, (zeroMult ρ.1 : ℂ) / (...)‖ ≤ ∑ ρ ∈ nearZerosSub T, ‖(zeroMult ρ.1 : ℂ) / (((σ : ℝ) + (T : ℝ) * Complex.I) - ρ.1)‖ := norm_sum_le _ _ _ = ∑ ρ ∈ nearZerosSub T, (zeroMult ρ.1 : ℝ) / ‖((σ : ℝ) + (T : ℝ) * Complex.I) - ρ.1‖ := by apply Finset.sum_congr rfl intro ρ _ rw [norm_div] have h2 : ‖(zeroMult ρ.1 : ℂ)‖ = (zeroMult ρ.1 : ℝ) := by rw [show (zeroMult ρ.1 : ℂ) = ((zeroMult ρ.1 : ℝ) : ℂ) by push_cast; ring, Complex.norm_real] exact abs_of_nonneg (Nat.cast_nonneg _) rw [h2] _ = ∑ ρ ∈ nearZerosSub T, (zeroMult ρ.1 : ℝ) * (1 / Real.sqrt ((σ - ρ.1.re) ^ 2 + (T - ρ.1.im) ^ 2)) := by apply Finset.sum_congr rfl intro ρ _ rw [hnormid σ ρ] -- ‖s−ρ‖ = √((σ−β)²+(T−γ)²) rw [mul_one_div] -- m * (1/√) = m/√ ... ``` Hmm wait — hnormid needs ρ : ℂ — with ρ : {ρ : ℂ // IsNontrivialZero ρ} I apply to ρ.1 ✓. - The integral of the RHS: ```lean have hswap : ∫ σ in (-5/2 : ℝ)..3, (∑ ρ ∈ nearZerosSub T, (zeroMult ρ.1 : ℝ) * (1 / Real.sqrt ((σ - ρ.1.re) ^ 2 + (T - ρ.1.im) ^ 2))) = ∑ ρ ∈ nearZerosSub T, (zeroMult ρ.1 : ℝ) * (∫ σ in (-5/2 : ℝ)..3, 1 / Real.sqrt ((σ - ρ.1.re) ^ 2 + (T - ρ.1.im) ^ 2)) := by rw [intervalIntegral.integral_finsetSum (fun ρ _ => hterm_int ρ)] apply Finset.sum_congr rfl intro ρ _ exact intervalIntegral.integral_const_mul _ _ ``` where hterm_int ρ : IntervalIntegrable (fun σ => (zeroMult ρ.1 : ℝ) * (1/√((σ−ρ.1.re)²+(T−ρ.1.im)²))) volume (−5/2) 3 — continuity: const · (1/√h) — 1/√h continuous since h > 0 — h = (σ−β)²+(T−γ)² > 0 needs (T−γ)² > 0 OR (σ−β)² > 0 — T−γ ≠ 0: since δ_ρ = |γ−T| ≥ c_clear > 0 ⟹ γ ≠ T ⟹ (T−γ)² > 0 ✓. So h ≥ (T−γ)² > 0 ✓ per ρ (needs mem_nearZerosSub for the clearance!). So hterm_int ρ needs ρ ∈ nearZerosSub T — in integral_finsetSum the hypothesis is ∀ i ∈ s, Integrable — so hterm_int takes the membership ✓. Hmm wait — for the per-ρ facts (δ bounds), I need them per ρ ∈ nearZerosSub T — collect as a have: ```lean have hρdata : ∀ ρ : {ρ : ℂ // IsNontrivialZero ρ}, ρ ∈ nearZerosSub T → (0 : ℝ) < |ρ.1.im - T| ∧ |ρ.1.im - T| < 1 ∧ (1 / (9 * C0hat (Real.sqrt (n : ℝ) + 2))) ≤ |ρ.1.im - T| := by intro ρ hρ have hδ1 := mem_nearZerosSub.mp hρ have hγ1 : Real.sqrt (n : ℝ) - 1 < ρ.1.im := by have := abs_lt.mp hδ1 nlinarith [Real.sqrt_nonneg (n : ℝ), hTn, hTn1] have hγ2 : ρ.1.im < Real.sqrt (n : ℝ) + 2 := by have := abs_lt.mp hδ1 nlinarith [hTn1] have hδge := hTwide ρ.1 ρ.2 hγ1 hγ2 have hδpos : (0 : ℝ) < |ρ.1.im - T| := lt_of_lt_of_le (one_div_pos.mpr (by positivity : (0:ℝ) < 9 * C0hat (√n+2))) hδge exact ⟨hδpos, hδ1, hδge⟩ ``` abs_lt.mp : |a| < b ↔ −b < a ∧ a < b ✓. hγ1: from |γ−T| < 1: −1 < γ−T ∧ γ−T < 1: γ > T−1 ≥ √n−1 ✓ (T ≥ √n ⟹ T−1 ≥ √n−1) — nlinarith [hTn]. hγ2: γ < T+1 ≤ √n+2 ✓ nlinarith [hTn1]. Then per-ρ per-σ integrand bound via the per-zero machine: ```lean have hper : ∀ ρ : {ρ : ℂ // IsNontrivialZero ρ}, ρ ∈ nearZerosSub T → (∫ σ in (-5/2 : ℝ)..3, 1 / Real.sqrt ((σ - ρ.1.re) ^ 2 + (T - ρ.1.im) ^ 2)) ≤ 2 * Real.log (148.5 * C0hat (Real.sqrt (n : ℝ) + 2)) := by intro ρ hρ obtain ⟨hδpos, hδ1, hδge⟩ := hρdata ρ hρ -- rewrite (T − γ)² = δ² have hsq : ∀ σ : ℝ, (T - ρ.1.im) ^ 2 = |ρ.1.im - T| ^ 2 := by intro σ rw [← sq_abs (ρ.1.im - T)] -- hmm: |γ−T|² = (γ−T)² = (T−γ)² ring -- hmm — direction: want the integrand's (T−γ)² replaced by δ² with δ := |γ−T| ``` Hmm — (T−γ)² = (γ−T)² = |γ−T|²: `neg_sq (T - γ) : (-(T−γ))² = (T−γ)²` — and −(T−γ) = γ−T by ring. And sq_abs : |γ−T|² = (γ−T)². So (T−γ)² = (γ−T)² = |γ−T|² — i.e. (T−γ)² = |γ−T|²: ```lean have hsq : (T - ρ.1.im) ^ 2 = |ρ.1.im - T| ^ 2 := by rw [← sq_abs (ρ.1.im - T)] ring_nf ``` (T−γ)² = |γ−T|²: rw [← sq_abs (ρ.1.im − T)]: RHS |γ−T|² → (γ−T)²: goal (T−γ)² = (γ−T)² — ring_nf closes? (T−γ)² = (γ−T)² — ring proves ✓ `by ring`. So: ```lean rw [← sq_abs (ρ.1.im - T)] ring -- hmm this is the tactic for the remaining goal ``` Wait the structure: ```lean have hsq : (T - ρ.1.im) ^ 2 = |ρ.1.im - T| ^ 2 := by rw [← sq_abs (ρ.1.im - T)] ring ``` rw [← sq_abs] : pattern |?a|^2 → ?a^2 — the goal's RHS |γ−T|^2 → (γ−T)²: goal (T−γ)² = (γ−T)² — `ring` closes ✓. Then the per-zero machine with δ := |γ−T|: per_zero_integral_le_two_log {β := ρ.1.re} {δ := |ρ.1.im − T|} (hβ0 : 0 < ρ.1.re := ρ.2.2.1) (hβ1 : ρ.1.re < 1 := ρ.2.2.2) (hδ := hδpos) (hδ1 : |γ−T| ≤ 1 := hδ1.le): But — the per-zero lemma's integral is over (σ−β)² + δ² — the near-sum's integrand has (σ−ρ.1.re)² + (T−ρ.1.im)² — rewrite via hsq: ∫ 1/√((σ−β)²+(T−γ)²) = ∫ 1/√((σ−β)²+δ²) — pointwise congr (integral_congr) or just state the per-zero lemma with (T−γ)² directly and apply sq_abs INSIDE. Cleaner: state the per-zero lemma about (σ−β)² + δ² and convert the integral by integral_congr with the per-σ identity (T−γ)² = δ². Ugh — integral_congr needs EqOn on uIcc — fine: fun σ _ => congrArg (fun X => 1/√((σ−β)²+X)) (hsq σ)... — simpler: rw [hsq] everywhere before integrating? The integrand is fun σ => 1/√((σ−β)²+(T−γ)²) — rw [hsq] at the INTEGRAL: hsq : (T−γ)² = δ² — the integral's integrand contains (T−γ)² — rw [hsq] rewrites it ✓ (it's a closed subterm, σ-free ✓). So in hper: ```lean rw [show ∀ σ : ℝ, (1 / Real.sqrt ((σ - ρ.1.re) ^ 2 + (T - ρ.1.im) ^ 2)) = (1 / Real.sqrt ((σ - ρ.1.re) ^ 2 + |ρ.1.im - T| ^ 2)) from fun σ => by rw [hsq]] ``` Hmm — the integrand rewrite inside ∫ — can't rw under a binder directly... Actually you CAN: rw [hsq] rewrites the closed term (T−γ)² wherever it appears including under binders ✓ — hsq's LHS (T − ρ.1.im)^2 is σ-free ✓ so `rw [hsq]` at the goal (the integral inequality) works ✓. So: ```lean have hmain := per_zero_integral_le_two_log ρ.2.2.1 ρ.2.2.2 hδpos hδ1.le -- hmain : ∫ σ in -5/2..3, 1/√((σ−β)²+δ²) ≤ 2·log(16.5/δ) rw [hsq] at hmain ⊢?? ``` Hmm — per_zero's integral has δ² from the start. The GOAL (hper's) has (T−γ)². rw [hsq] at the goal: (T−γ)² → δ² ✓ then the goal matches hmain's LHS ✓ then `exact hmain.trans (the clearance comparison)`. The clearance comparison: 2·log(16.5/δ) ≤ 2·log(148.5·C0hat(√n+2)): ```lean have hcmp : 2 * Real.log (16.5 / |ρ.1.im - T|) ≤ 2 * Real.log (148.5 * C0hat (Real.sqrt (n : ℝ) + 2)) := by have hinv : 1 / |ρ.1.im - T| ≤ 9 * C0hat (Real.sqrt (n : ℝ) + 2) := by have h := one_div_le_one_div_of_le (one_div_pos.mpr (by positivity : (0:ℝ) < 9 * C0hat (Real.sqrt (n : ℝ) + 2))) hδge rwa [one_div_one_div] at h have h1 : 16.5 * (1 / |ρ.1.im - T|) ≤ 16.5 * (9 * C0hat (Real.sqrt (n : ℝ) + 2)) := mul_le_mul_of_nonneg_left hinv (by norm_num) have h2 : 16.5 / |ρ.1.im - T| ≤ 148.5 * C0hat (Real.sqrt (n : ℝ) + 2) := by rw [mul_one_div, show (16.5 : ℝ) * (9 * C0hat (Real.sqrt (n : ℝ) + 2)) = 148.5 * C0hat (Real.sqrt (n : ℝ) + 2) by ring] at h1 exact h1 have h3 : Real.log (16.5 / |ρ.1.im - T|) ≤ Real.log (148.5 * C0hat (Real.sqrt (n : ℝ) + 2)) := Real.log_le_log (by positivity) h2 linarith [h3] exact hmain.trans hcmp ``` positivity for 0 < 16.5/δ: δ > 0 ✓ (hδpos in context — positivity uses it? 16.5/|γ−T| — |γ−T| > 0 from hδpos ✓ positivity should find it; fallback: div_pos (by norm_num) hδpos). Then the sum assembly: ```lean have hsum : ∑ ρ ∈ nearZerosSub T, (zeroMult ρ.1 : ℝ) * (∫ σ in (-5/2 : ℝ)..3, 1 / Real.sqrt ((σ - ρ.1.re) ^ 2 + (T - ρ.1.im) ^ 2)) ≤ (2 * Real.log (148.5 * C0hat (Real.sqrt (n : ℝ) + 2))) * Z2hat T := by calc ∑ ρ ∈ nearZerosSub T, (zeroMult ρ.1 : ℝ) * (∫ ...) ≤ ∑ ρ ∈ nearZerosSub T, (zeroMult ρ.1 : ℝ) * (2 * Real.log (148.5 * C0hat (Real.sqrt (n : ℝ) + 2))) := by apply Finset.sum_le_sum intro ρ hρ exact mul_le_mul_of_nonneg_left (hper ρ hρ) (Nat.cast_nonneg _) _ = (2 * Real.log (148.5 * C0hat (Real.sqrt (n : ℝ) + 2))) * ∑ ρ ∈ nearZerosSub T, (zeroMult ρ.1 : ℝ) := by rw [← Finset.mul_sum] apply Finset.sum_congr rfl intro ρ _ ring _ ≤ (2 * Real.log (148.5 * C0hat (Real.sqrt (n : ℝ) + 2))) * Z2hat T := mul_le_mul_of_nonneg_left (nearZerosSub_mult_sum_le hZC2 h10T) hB0nn ``` hB0nn : 0 ≤ 2·log(148.5·C0hat(√n+2)) : log ≥ 0 ⟸ 148.5·C0hat ≥ 1 ⟸ C0hat ≥ 2... wait — C0hat(√n+2) ≥ 2 (two_le_C0hat) ⟹ 148.5·C0hat ≥ 297 ≥ 1 ✓ Real.log_nonneg ✓ by positivity? — 2·log(...) — positivity can't do log ≥ 0 without 1 ≤ argument — provide: ```lean have hB0nn : (0 : ℝ) ≤ 2 * Real.log (148.5 * C0hat (Real.sqrt (n : ℝ) + 2)) := by have h1 : (1 : ℝ) ≤ 148.5 * C0hat (Real.sqrt (n : ℝ) + 2) := by have h2 := two_le_C0hat (by nlinarith [Real.sqrt_nonneg (n : ℝ)] : (1 : ℝ) ≤ Real.sqrt (n : ℝ) + 2) nlinarith have h3 := Real.log_nonneg h1 linarith ``` Then the hΣbound conclusion: ∫‖Σ‖ ≤ ∫(RHS-sum) [integral_mono_on with hpt] = Σ m·∫ [hswap] ≤ B0·Z2hat [hsum] = 2·Z2hat·log — wait the final form: 2 * Z2hat T * Real.log (148.5·C0hat) vs hsum's (2·log)·Z2hat — ring-equivalent — linarith or a final calc rearrangement. I'll state hΣbound with the (2·log)·Z2hat form... no — the FINAL theorem's NI is 2 * Z2hat T * Real.log (...). Let me keep hΣbound as ≤ (2·log)·Z2hat and rearrange in the final linarith (atoms: Z2hat T, log — the products (2·log)·Z2hat vs 2·Z2hat·log — ring_nf normalizes ✓ linarith handles ✓). The integral_mono_on for hΣbound's first step: ```lean have h1 : ∫ σ in (-5/2 : ℝ)..3, ‖∑ ρ ∈ nearZerosSub T, (zeroMult ρ.1 : ℂ) / (...)‖ ≤ ∫ σ in (-5/2 : ℝ)..3, (∑ ρ ∈ nearZerosSub T, (zeroMult ρ.1 : ℝ) * (1 / Real.sqrt ((σ - ρ.1.re) ^ 2 + (T - ρ.1.im) ^ 2))) := intervalIntegral.integral_mono_on hlen hΣnorm_int hRHS_int (fun σ hσ => hpt σ hσ) -- then hswap, hsum calc ... ≤ (2 * Real.log ...) * Z2hat T := h1.trans (hswap ▸ hsum) ``` hswap ▸ hsum: hswap : ∫ RHS-sum = Σ m·∫ — rw [hswap] at... — h1.trans: ∫‖Σ‖ ≤ ∫RHS-sum; ∫RHS-sum = Σ (hswap); Σ ≤ B0·Z2hat (hsum) — so `h1.trans (hswap.trans_le hsum)`?? — hswap : A = B — hswap.trans_le : A ≤ C ⟸ B ≤ C — `Eq.trans_le : a = b → b ≤ c → a ≤ c` ✓ hswap.trans_le hsum : ∫RHS-sum ≤ B0·Z2hat ✓ then h1.trans that ✓. Integrability: hΣnorm_int (the complex-sum norm — continuous via the denominator machine) and hRHS_int (Finset.sum of per-term — per-term hterm_int ✓ then `Continuous.intervalIntegrable` of the sum — `continuous_finset_sum` hmm — for IntervalIntegrable of the SUM: intervalIntegral.integral_finsetSum needs per-term IntervalIntegrable for the equality hswap ✓; and for integral_mono_on's hg (RHS integrability): the sum's IntervalIntegrable — from per-term: `IntervalIntegrable.sum`? — there's `IntervalIntegrable_finset_sum`? — or prove Continuous (the sum) via continuous_finset_sum ✓ then Continuous.intervalIntegrable ✓. I'll do the Continuous route for hg: ```lean have hRHScont : Continuous fun σ : ℝ => ∑ ρ ∈ nearZerosSub T, (zeroMult ρ.1 : ℝ) * (1 / Real.sqrt ((σ - ρ.1.re) ^ 2 + (T - ρ.1.im) ^ 2)) := by apply continuous_finset_sum intro ρ hρ apply Continuous.mul continuous_const apply Continuous.div continuous_const · exact hsqrtCont ρ hρ -- Continuous fun σ => √((σ−β)²+(T−γ)²) · intro σ exact hsqrt_ne ρ hρ σ -- √(...) ≠ 0 ``` where hsqrtCont: (Real.continuous_sqrt).comp (by fun_prop : Continuous fun σ => (σ−ρ.1.re)² + (T−ρ.1.im)²) and hsqrt_ne: √h ≠ 0 ⟸ h > 0 ⟸ (T−γ)² > 0 ⟸ γ ≠ T ⟸ δ > 0 (hρdata). Hmm — fun_prop for `Continuous fun σ : ℝ => (σ - ρ.1.re)^2 + (T - ρ.1.im)^2` — polynomial ✓ fun_prop handles ✓. continuous_finset_sum signature: `theorem continuous_finset_sum (s : Finset ι) (hf : ∀ i ∈ s, Continuous (f i)) : Continuous fun x => ∑ i ∈ s, f i x` ✓ exists (Topology/Algebra/InfiniteSum? — it's `continuous_finset_sum` in Mathlib.Topology.Algebra.BigOperators? — exists for sure; the name might be `continuous_finset_sum` ✓). For integral_finsetSum's per-term integrability (hswap): per-term (fun σ => m * (1/√h)) — Continuous (same machine) ⟹ Continuous.intervalIntegrable ✓. And hΣnorm_int — the norm of the complex sum: ```lean have hΣnorm_int : IntervalIntegrable (fun σ : ℝ => ‖∑ ρ ∈ nearZerosSub T, (zeroMult ρ.1 : ℂ) / (((σ : ℝ) + (T : ℝ) * Complex.I) - ρ.1)‖) MeasureTheory.volume (-5/2) 3 := by apply Continuous.intervalIntegrable apply continuous_norm.comp apply continuous_finset_sum intro ρ hρ apply Continuous.div continuous_const · exact hpath.sub continuous_const · intro σ -- (↑σ+↑T·I) − ρ.1 ≠ 0 have hne : ((σ : ℝ) + (T : ℝ) * Complex.I) - ρ.1 ≠ 0 := by intro h have hs : ((σ : ℝ) + (T : ℝ) * Complex.I) = ρ.1 := sub_eq_zero.mp h have h1 := hzne σ rw [hs] at h1 exact h1 ρ.2.1 exact hne σ ``` Hmm — Continuous.div's h₀ : ∀ x, g x ≠ 0 — with g := fun σ => (↑σ+↑T·I) − ρ.1 — so `intro σ` then the per-σ hne — restructure: ```lean · intro σ h0 have hs : ((σ : ℝ) + (T : ℝ) * Complex.I) = ρ.1 := sub_eq_zero.mp h0 exact (hzne σ) (hs ▸ ρ.2.1) ``` Cleaner: h0 : (↑σ+↑T·I) − ρ.1 = 0 — wait Continuous.div's h₀ : ∀ x, g x ≠ 0 — so intro σ; show (↑σ+↑T·I)−ρ.1 ≠ 0: intro h0; hs := sub_eq_zero.mp h0 : ↑σ+↑T·I = ρ.1; (hzne σ) : ζ(↑σ+↑T·I) ≠ 0; rw [hs] at hzne: ζ(ρ.1) ≠ 0 — exact (hzne ρ.2.1)... ρ.2.1 : riemannZeta ρ.1 = 0 ✓ contradiction ✓. ```lean · intro σ intro h0 have hs : ((σ : ℝ) + (T : ℝ) * Complex.I) = ρ.1 := sub_eq_zero.mp h0 have hz := hzne σ rw [hs] at hz exact hz ρ.2.1 ``` Hmm — `intro σ; intro h0` → `intro σ h0` ✓. Same denominator-ne-zero for the hRcont's sum part ✓ (share a have hden : ∀ ρ ∈ nearZerosSub T, ∀ σ, (↑σ+↑T·I) − ρ.1 ≠ 0). Wait — actually for the R-integrand continuity, the sum needs the same h₀ per term ✓ share. Also hRintbl : IntervalIntegrable of the R-norm integrand ✓ via hRcont.intervalIntegrable. Now the top-level assembly of hIntegral_Jlag_mid_norm_le: ```lean -- kernel bound per σ have hker : ∀ σ : ℝ, σ ∈ Set.Icc (-5/2 : ℝ) 3 → ‖kFun n ((σ : ℝ) + (T : ℝ) * Complex.I)‖ ≤ Real.exp 4 + 1 := ... (as drafted) -- pointwise Jlag ≤ (e⁴+1)·‖logDeriv‖ have hptJ : ∀ σ : ℝ, σ ∈ Set.Icc (-5/2 : ℝ) 3 → ‖Jlag n ((σ : ℝ) + (T : ℝ) * Complex.I)‖ ≤ (Real.exp 4 + 1) * ‖logDeriv riemannZeta ((σ : ℝ) + (T : ℝ) * Complex.I)‖ := by intro σ hσ rw [Jlag_norm_eq] exact mul_le_mul (hker σ hσ) le_rfl (norm_nonneg _) (by positivity) -- ∫‖Jlag‖ ≤ (e⁴+1)·∫‖logDeriv‖ have hJint : ∫ σ in (-5/2 : ℝ)..3, ‖Jlag n _‖ ≤ (Real.exp 4 + 1) * ∫ σ in (-5/2 : ℝ)..3, ‖logDeriv riemannZeta _‖ := by have h := intervalIntegral.integral_mono_on hlen hJintbl hKLintbl (fun σ hσ => hptJ σ hσ) rw [intervalIntegral.integral_const_mul] at h exact h ``` Hmm — mul_le_mul (h₁ : ‖kFun‖ ≤ e⁴+1) (h₂ : ‖logDeriv‖ ≤ ‖logDeriv‖) (c0 : 0 ≤ ‖logDeriv‖) (b0 : 0 ≤ e⁴+1) : ‖kFun‖·‖logDeriv‖ ≤ (e⁴+1)·‖logDeriv‖ ✓ — h₂ := le_rfl ✓. ```lean -- ∫‖logDeriv‖ ≤ ∫‖R‖ + ∫‖Σ‖ have hsplit : ∫ σ in (-5/2 : ℝ)..3, ‖logDeriv riemannZeta _‖ ≤ (∫ σ in (-5/2 : ℝ)..3, ‖∑ ...‖) + (∫ σ in (-5/2 : ℝ)..3, ‖logDeriv − ∑...‖) := by have hpt2 : ∀ σ : ℝ, σ ∈ Set.Icc (-5/2 : ℝ) 3 → ‖logDeriv riemannZeta ((σ : ℝ) + (T : ℝ) * Complex.I)‖ ≤ ‖∑ ρ ∈ nearZerosSub T, (zeroMult ρ.1 : ℂ) / (...)‖ + ‖logDeriv riemannZeta ((σ : ℝ) + (T : ℝ) * Complex.I) - ∑ ...‖ := fun σ _ => norm_le_insert' _ _ have h := intervalIntegral.integral_mono_on hlen hLDintbl hsumIntbl (fun σ hσ => hpt2 σ hσ) rw [intervalIntegral.integral_add hΣnorm_int hRnorm_int] at h exact h ``` where hsumIntbl : IntervalIntegrable (fun σ => ‖Σ‖ + ‖R‖) := hΣnorm_int.add hRnorm_int — `IntervalIntegrable.add` ✓. And intervalIntegral.integral_add (hf : IntervalIntegrable) (hg) : ∫(f+g) = ∫f + ∫g ✓ — the pointwise sum integrand (fun σ => ‖Σ‖ + ‖R‖) ✓. Wait — integral_add signature: `intervalIntegral.integral_add (hf : IntervalIntegrable f μ a b) (hg : IntervalIntegrable g μ a b) : ∫ x in a..b, f x + g x ∂μ = (∫ x in a..b, f x ∂μ) + ∫ x in a..b, g x ∂μ` ✓ exists. rw [integral_add hΣnorm_int hRnorm_int] at h — h's RHS is ∫(fun σ => ‖Σ‖ + ‖R‖) — the lemma's LHS pattern ∫ x, f x + g x — matches with f := fun σ => ‖Σ‖, g := fun σ => ‖R‖ ✓ rewrites to (∫‖Σ‖) + (∫‖R‖) ✓. Final: ```lean calc ‖HIntegral (Jlag n) (-5/2) 3 T‖ = ‖∫ σ in (-5/2)..3, Jlag n ((σ:ℝ) + (T:ℝ)*Complex.I)‖ := rfl (HIntegral def) _ ≤ ∫ σ in (-5/2)..3, ‖Jlag n _‖ := intervalIntegral.norm_integral_le_integral_norm hlen _ ≤ (Real.exp 4 + 1) * ∫ σ in .., ‖logDeriv _‖ := hJint _ ≤ (Real.exp 4 + 1) * ((∫‖Σ‖) + (∫‖R‖)) := by rw [hsplit]?? ``` Hmm — HIntegral's def: HIntegral f x₁ x₂ y = ∫ x in x₁..x₂, f (x + y*I) — the point form: `(↑x + ↑y*I)` — my points: `((σ : ℝ) + (T : ℝ) * Complex.I)` ✓ — so HIntegral (Jlag n) (-5/2) 3 T = ∫ σ in -5/2..3, Jlag n (↑σ + ↑T*I) — DEFEQ (unfold HIntegral) ✓ — the calc's first step `= ... := rfl`? — unfold needed: `by unfold HIntegral` — or just write the calc starting from the norm-integral directly with a leading `by unfold HIntegral; exact ...`? Cleaner: ```lean unfold HIntegral calc ‖∫ σ in (-5/2 : ℝ)..(3 : ℝ), Jlag n ((σ : ℝ) + (T : ℝ) * Complex.I)‖ ≤ ∫ σ in (-5/2 : ℝ)..(3 : ℝ), ‖Jlag n ((σ : ℝ) + (T : ℝ) * Complex.I)‖ := intervalIntegral.norm_integral_le_integral_norm hlen ... ``` Wait — HIntegral's point: `f (x + y * I)` with x y : ℝ — `↑x + ↑y * I` ✓ same as mine ✓. Hmm — does `unfold HIntegral` work at the goal ‖HIntegral (Jlag n) (-5/2) 3 T‖ ≤ ...? — unfolds to ‖∫ x in -5/2..3, Jlag n (↑x + ↑T*I)‖ ≤ ... ✓. Then: ```lean _ ≤ (Real.exp 4 + 1) * ∫ σ in (-5/2 : ℝ)..(3 : ℝ), ‖logDeriv riemannZeta ((σ : ℝ) + (T : ℝ) * Complex.I)‖ := hJint _ ≤ (Real.exp 4 + 1) * ((∫ σ in .., ‖Σ‖) + (∫ σ in .., ‖R‖)) := by nlinarith? -- mul_le_mul_of_nonneg_left hsplit (e⁴+1 ≥ 0) exact mul_le_mul_of_nonneg_left hsplit (by positivity) _ ≤ (Real.exp 4 + 1) * ((2 * Real.log (148.5 * C0hat (√n+2))) * Z2hat T + (3 - -5/2) * (10.5 * Real.log (T + 2) + 48)) := mul_le_mul_of_nonneg_left (add_le_add hΣbound hRint) (by positivity) _ = (Real.exp 4 + 1) * (2 * Z2hat T * Real.log (148.5 * C0hat (√n+2)) + (3 - -5/2) * (10.5 * Real.log (T + 2) + 48)) := by ring_nf ``` The last: (2·log)·Z2hat vs 2·Z2hat·log — ring/ring_nf ✓ — hmm — goal: (e⁴+1)*((2*log)*Z2hat + X) = (e⁴+1)*(2*Z2hat*log + X) — `by ring` ✓ (atoms: log, Z2hat T, X-components — ring treats log(148.5·C0hat...) as atom ✓ ring handles the commuted products ✓). positivity for 0 ≤ e⁴+1: exp_pos ✓ positivity ✓. For hsplit's mul: mul_le_mul_of_nonneg_left hsplit (0 ≤ e⁴+1) : (e⁴+1)·(∫‖logDeriv‖) ≤ (e⁴+1)·((∫‖Σ‖)+(∫‖R‖)) ✓ — matches the calc's `_ ≤ (e⁴+1)*((∫‖Σ‖)+(∫‖R‖))` ✓. Now — the missing continuity/integrability pieces list for the top theorem: - hpath ✓ (fun_prop) - hJcont → hcontJ → hcontG → hJintbl (IntervalIntegrable ‖Jlag‖) - hLDintbl : IntervalIntegrable ‖logDeriv‖ — from continuousAt_logDeriv_zeta machine - hΣnorm_int : IntervalIntegrable ‖Σ‖ (the complex-sum norm — denominator machine) - hRnorm_int : IntervalIntegrable ‖R‖ = ‖logDeriv − Σ‖ — continuous: sub of (logDeriv-cont, sum-cont) + norm - hKLintbl : IntervalIntegrable (fun σ => (e⁴+1)·‖logDeriv‖) := hLDintbl.const_mul? — `IntervalIntegrable.const_mul`? or via Continuous: continuous_const.mul hLDcont ✓ then intervalIntegrable. Use the Continuous route: hKLcont := continuous_const.mul hLDcont; hKLintbl := hKLcont.intervalIntegrable _ _. So I need CONTINUOUS versions: hLDcont : Continuous fun σ => ‖logDeriv ζ (↑σ+↑T·I)‖ := continuous_norm.comp (logDeriv-path-continuity). The logDeriv path continuity: ```lean have hLDc : Continuous fun σ : ℝ => logDeriv riemannZeta ((σ : ℝ) + (T : ℝ) * Complex.I) := by rw [continuous_iff_continuousAt] intro σ have hs1 : ((σ : ℝ) + (T : ℝ) * Complex.I) ≠ 1 := hne1-shared have hf : ContinuousAt (fun u : ℝ => ((u : ℝ) + (T : ℝ) * Complex.I)) σ := hpath.continuousAt exact ContinuousAt.comp (g := logDeriv riemannZeta) (f := fun u : ℝ => ((u : ℝ) + (T : ℝ) * Complex.I)) (x := σ) (continuousAt_logDeriv_zeta (hs1 σ) (hzne σ)) hf ``` hne1-shared : ∀ σ, (↑σ+↑T·I) ≠ 1 := fun σ => ... (the im argument). The Jlag path continuity hJcont (as before, using kFun_differentiableAt + logDeriv) → hcontJ via the comp pattern → hcontG. And the sum continuity hΣcont : Continuous fun σ => Σ_ρ (zeroMult ρ.1:ℶ)/((↑σ+↑T·I)−ρ.1) — the continuous_finset_sum + Continuous.div + hden. And hRcont := continuous_norm.comp (hLDc'.sub hΣcont) where hLDc' : Continuous fun σ => logDeriv ζ (↑σ+↑T·I) — WITHOUT norm (the sub needs the un-normed): hRcont : Continuous fun σ => ‖logDeriv _ − Σ‖ = continuous_norm.comp (hLDc.sub hΣcont) ✓. And hΣnorm_cont : Continuous fun σ => ‖Σ‖ := continuous_norm.comp hΣcont ✓. OK. This is a hefty but mechanical file section. Also the per-ρ integrand continuity htermcont: ```lean have htermcont : ∀ ρ : {ρ : ℂ // IsNontrivialZero ρ}, ρ ∈ nearZerosSub T → Continuous fun σ : ℝ => (zeroMult ρ.1 : ℝ) * (1 / Real.sqrt ((σ - ρ.1.re) ^ 2 + (T - ρ.1.im) ^ 2)) := by intro ρ hρ obtain ⟨hδpos, -, -⟩ := hρdata ρ hρ apply Continuous.mul continuous_const apply Continuous.div continuous_const · exact Real.continuous_sqrt.comp (by fun_prop : Continuous fun σ : ℝ => (σ - ρ.1.re) ^ 2 + (T - ρ.1.im) ^ 2) · intro σ have h1 : (0 : ℝ) < (σ - ρ.1.re) ^ 2 + (T - ρ.1.im) ^ 2 := by have h2 : (0 : ℝ) < (T - ρ.1.im) ^ 2 := by apply sq_pos_of_ne_zero intro h3 rw [show (T - ρ.1.im) = 0 → T = ρ.1.im from sub_eq_zero.mp, ...] positivity exact ne_of_gt (Real.sqrt_pos.mpr h1) ``` Hmm — sq_pos_of_ne_zero (h : a ≠ 0) : 0 < a^2 ✓ exists. T − γ ≠ 0: from δ := |γ−T| > 0: |γ−T| > 0 ⟹ γ−T ≠ 0 (abs_pos) ⟹ T−γ ≠ 0 ✓: ```lean have h2 : (0 : ℝ) < (T - ρ.1.im) ^ 2 := by have h3 : ρ.1.im - T ≠ 0 := abs_pos.mp hδpos have h4 : T - ρ.1.im ≠ 0 := by intro h5 apply h3 linarith [h5] -- γ − T = −(T−γ) = 0 exact sq_pos_of_ne_zero h4 nlinarith? -- (σ−β)² ≥ 0 + (T−γ)² > 0 ⟹ sum > 0 exact add_pos_of_nonneg_of_pos (sq_nonneg _) h2 ``` h3 : γ − T ≠ 0 := abs_pos.mp hδpos ✓ (abs_pos : 0 < |a| ↔ a ≠ 0). h4 : T−γ ≠ 0: intro h5: T−γ = 0; then γ−T = 0 by linarith ✓ (or sub_eq_zero.mp h5 : T = γ → γ − T = 0 by linarith/rw). Then 1/√h ≠ 0: √h > 0 ⟹ √h ≠ 0 (ne_of_gt) ⟹ 1/√h... — Continuous.div's h₀ is about the DENOMINATOR √h ≠ 0 ✓ ne_of_gt (Real.sqrt_pos.mpr h1) ✓. And hterm_int for integral_finsetSum := (htermcont ρ hρ).intervalIntegrable _ _. And hRHScont (the sum): continuous_finset_sum with per-term htermcont ✓ — wait — the sum's per-term functions must match: `fun σ => (zeroMult ρ.1 : ℝ) * (1/√((σ−ρ.1.re)²+(T−ρ.1.im)²))` ✓ same as htermcont ✓. NOW — one more piece: the hswap step uses integral_finsetSum which needs per-term IntervalIntegrable of the EXACT summand functions `fun σ => (zeroMult ρ.1:ℝ) * (1/Real.sqrt ((σ−ρ.1.re)^2+(T−ρ.1.im)^2))` ✓ htermcont.intervalIntegrable. And the hswap sum-congr: ∫ (fun σ => m*(1/√h)) σ... integral_const_mul (r : 𝕜) (f : ℝ → 𝕜) : ∫ x in a..b, r * f x = r * ∫ x in a..b, f x ✓ per-ρ: (zeroMult ρ.1 : ℝ) * ∫ 1/√h ✓ — so after integral_finsetSum, the RHS is Σ_ρ ∫σ m·(1/√h) dσ = Σ_ρ m·∫σ 1/√h dσ via sum_congr with integral_const_mul ✓. Phew. Also — the `classical` tactic at the start (for nearZerosSub decidability) ✓ (used in 6-c-ii ✓). Let me now also define the arcsinh machine carefully (lemma 1). Full statement: ```lean /-- **The arcsinh FTC machine** (6-c-iv core): the per-zero horizontal segment integral — the direct antiderivative F(σ) = log((σ−β) + √((σ−β)²+δ²)) has F′ = 1/√(·) exactly (Real.arcsinh does not exist here). -/ theorem integral_one_div_sqrt_sq_add_sq (β δ : ℝ) (hδ : 0 < δ) (a b : ℝ) : ∫ σ in a..b, 1 / Real.sqrt ((σ - β) ^ 2 + δ ^ 2) = Real.log ((b - β) + Real.sqrt ((b - β) ^ 2 + δ ^ 2)) - Real.log ((a - β) + Real.sqrt ((a - β) ^ 2 + δ ^ 2)) := by have hh : ∀ σ : ℝ, (0 : ℝ) < (σ - β) ^ 2 + δ ^ 2 := fun σ => add_pos_of_nonneg_of_pos (sq_nonneg _) (pow_pos hδ 2) have hgpos : ∀ σ : ℝ, (0 : ℝ) < (σ - β) + Real.sqrt ((σ - β) ^ 2 + δ ^ 2) := by intro σ have h1 : |(σ - β)| < Real.sqrt ((σ - β) ^ 2 + δ ^ 2) := by have h2 : (σ - β) ^ 2 < (σ - β) ^ 2 + δ ^ 2 := by nlinarith [hδ] have h3 := Real.sqrt_lt_sqrt (sq_nonneg _) h2 rwa [Real.sqrt_sq_eq_abs] at h3 have h4 : -(σ - β) ≤ |(σ - β)| := neg_abs_le (σ - β) have h5 : (0 : ℝ) < Real.sqrt ((σ - β) ^ 2 + δ ^ 2) + (σ - β) := by linarith [h1, h4] linarith [h5] ``` Hmm — h5 : 0 < √h + (σ−β) — then goal 0 < (σ−β) + √h — linarith [h5] (commuted sum — linarith handles add_comm? — the atoms are √h and (σ−β) — goal (σ−β)+√h > 0 from √h+(σ−β) > 0 — linarith normalizes ✓). Wait h4 : -(σ−β) ≤ |σ−β| and h1 : |σ−β| < √h: −(σ−β) < √h ⟹ 0 < √h + (σ−β) ✓ linarith. ```lean have hderiv : ∀ σ : ℝ, HasDerivAt (fun σ => Real.log ((σ - β) + Real.sqrt ((σ - β) ^ 2 + δ ^ 2))) (1 / Real.sqrt ((σ - β) ^ 2 + δ ^ 2)) σ := by intro σ have hhσ := hh σ have hsqrt0 : Real.sqrt ((σ - β) ^ 2 + δ ^ 2) ≠ 0 := ne_of_gt (Real.sqrt_pos.mpr hhσ) have hg0 : ((σ - β) + Real.sqrt ((σ - β) ^ 2 + δ ^ 2)) ≠ 0 := (hgpos σ).ne' have h1 : HasDerivAt (fun σ => σ - β) 1 σ := hasDerivAt_id.sub_const β have h2 : HasDerivAt (fun σ => (σ - β) ^ 2 + δ ^ 2) (2 * (σ - β)) σ := by have h2a := (hasDerivAt_id.sub_const β).pow 2 have h2b := h2a.add_const (δ ^ 2) convert h2b using 1 ring have h4 : HasDerivAt (fun σ => Real.sqrt ((σ - β) ^ 2 + δ ^ 2)) (2 * (σ - β) / (2 * Real.sqrt ((σ - β) ^ 2 + δ ^ 2))) σ := h2.sqrt hhσ.ne' have h5 : HasDerivAt (fun σ => (σ - β) + Real.sqrt ((σ - β) ^ 2 + δ ^ 2)) (1 + 2 * (σ - β) / (2 * Real.sqrt ((σ - β) ^ 2 + δ ^ 2))) σ := h1.add h4 have h6 : HasDerivAt (fun σ => (σ - β) + Real.sqrt ((σ - β) ^ 2 + δ ^ 2)) (((σ - β) + Real.sqrt ((σ - β) ^ 2 + δ ^ 2)) / Real.sqrt ((σ - β) ^ 2 + δ ^ 2)) σ := by convert h5 using 1 field_simp [hsqrt0] ring have h7 := h6.log hg0 convert h7 using 1 field_simp [hsqrt0, hg0] ring exact intervalIntegral.integral_eq_sub_of_hasDerivAt (fun σ _ => hderiv σ) (by apply Continuous.intervalIntegrable apply Continuous.div continuous_const · exact Real.continuous_sqrt.comp (by fun_prop : Continuous fun σ : ℝ => (σ - β) ^ 2 + δ ^ 2) · intro σ exact ne_of_gt (Real.sqrt_pos.mpr (hh σ))) ``` Hmm — concerns: (a) `(hasDerivAt_id.sub_const β)` — hasDerivAt_id : HasDerivAt id 1 ?x — the x implicit? `theorem hasDerivAt_id (x : 𝕜) : HasDerivAt id 1 x` — x explicit. `(hasDerivAt_id σ).sub_const β : HasDerivAt (fun x => x − β) 1 σ` ✓ — the function `fun x => x − β` vs my `fun σ => σ − β` ✓ alpha-same. (b) h2a's derivative: HasDerivAt.pow gives `↑2 * (σ−β)^(2−1) * 1` — h2b's convert: goal `HasDerivAt (fun σ => (σ−β)^2+δ^2) (2*(σ−β)) σ` — convert h2b using 1: function parts defeq (h2b's function: fun x => (x−β)^2 + δ^2 ✓), derivative: `↑2 * (σ−β)^1 * 1 = 2*(σ−β)` — `ring` ✓. But — will `convert h2b using 1` produce the derivative equation in a ring-provable form? It should produce `2 * (σ−β) = ↑2 * (σ−β)^1 * 1` or the reverse — ring either way ✓. Hmm — HasDerivAt.pow's exact statement: `theorem HasDerivAt.pow (hf : HasDerivAt f f' x) (n : ℕ) : HasDerivAt (fun y => f y ^ n) ((n : 𝕜) * f x ^ (n - 1) * f') x`? — I think it's `(n : 𝕜) * (f x)^(n−1) * f'` — hmm — or is it `n * f x ^ (n - 1) * f'` with n as ℕ cast ✓ whatever — ring handles numeral-cast forms? — `↑2 * (σ−β)^1 * 1` — ring normalizes casts of numerals ✓. (c) h6's convert + field_simp + ring: goal after convert: `(g σ)/√h = 1 + 2(σ−β)/(2√h)` (or reversed) — field_simp [hsqrt0]: clears √h denominators: LHS·... — expect to leave a polynomial identity in (σ−β), √h — but there's √h²?? — field_simp might produce `((σ−β)+√h)*2 = 2√h + 2(σ−β)`?? — hmm: (g/√h) = 1 + 2(σ−β)/(2√h): RHS = (2√h + 2(σ−β))/(2√h): so g/√h = (2√h+2(σ−β))/(2√h) ⟺ g·2√h = (2√h+2(σ−β))·√h ⟺ 2·g = 2√h + 2(σ−β) (after dividing by √h — field_simp keeps the relation g·2√h = (2√h + 2(σ−β))√h — then ring can't simplify √h·√h... — field_simp is supposed to clear ALL denominators: multiply both sides by 2√h: g·2√h... wait — LHS g/√h · 2√h = 2g; RHS [1 + 2(σ−β)/(2√h)]·2√h = 2√h + 2(σ−β). So field_simp leaves `2·g = 2√h + 2(σ−β)` ⟺ g = √h + (σ−β) where g := (σ−β)+√h — ring closes ✓ (no √h² appears ✓). If field_simp instead produces the un-cleared form, ring still closes ✓. (d) h7's convert + field_simp + ring: goal `1/√h = (g/√h)/g` (or reversed): field_simp [hsqrt0, hg0]: multiply by √h·g: g·g?? — (g/√h)/g · (√h·g) = g; LHS (1/√h)·(√h·g) = g. So leaves `g = g` — closed by field_simp itself?? — or `ring`. If field_simp fully closes, the trailing `ring` errors. Hmm. To be safe against both outcomes: use `field_simp [hsqrt0, hg0]` ALONE first; if "no goals" — hmm no: `convert h7 using 1; field_simp [hsqrt0, hg0]; ring` — if field_simp closes, ring fails with "no goals". The error message will tell me and I drop the ring. Alternatively use `try ring`: `field_simp [hsqrt0, hg0]; try ring`?? — `try` swallows the no-goals... actually `ring` with no goals — the tactic FAILS ("no goals to be solved") — wrapping in `try` makes it succeed-vacuously?? — no — `try tac` runs tac and ignores FAILURE — if there are no goals, the enclosing block is already done — `try ring` is harmless?? — hmm, actually when there are no goals, ANY tactic including try errors?? No — `try ring` with no goals: try catches the failure ✓ fine. But cleaner to avoid: after convert, the goal is an equation — field_simp USUALLY leaves a ring-goal per the banked gotcha ("field_simp often closes goals (trailing ring errors — banked 4×)" — meaning field_simp closed them and ring errored!). The banked pattern suggests field_simp; ring is RISKY either way. Banked alternative (session-038): "div_add_div + div_eq_div_iff + ring as the always-works fraction pattern". Deterministic alternative for h7's derivative equation: `1/√h = (g/√h)/g`: ```lean rw [div_div, div_eq_mul_inv, mul_inv_cancel₀ hsqrt0] -- hmm ``` Let me think: (g/√h)/g = g/(√h·g) [div_div] = (g/g)·(1/√h)?? — (g/√h)/g = g·(1/√h)/g = (1/√h)·(g/g) = (1/√h)·1 = 1/√h ✓. So: ```lean have : (((σ - β) + Real.sqrt ((σ - β) ^ 2 + δ ^ 2)) / Real.sqrt ((σ - β) ^ 2 + δ ^ 2)) / ((σ - β) + Real.sqrt ((σ - β) ^ 2 + δ ^ 2)) = 1 / Real.sqrt ((σ - β) ^ 2 + δ ^ 2) := by field_simp [hsqrt0, hg0] rw [this] at h7?? ``` Hmm — h7 : HasDerivAt (log∘g) ((g/√h)/g) σ — the convert approach: `convert h7 using 1` leaves `1/√h = (g/√h)/g` (goal direction: target-deriv = produced-deriv? convert's remaining goal: (target) = (source)?? — `convert h7 using 1` where target is my hderiv's statement (1/√h) and source h7 ((g/√h)/g): the new goal: `1 / √h = (g σ / √h) / g σ`. Then `field_simp [hsqrt0, hg0]` normalizes: it might close or leave `g·√h = √h·g` (ring/rfl). Add `ring` and adjust based on error. I'll go with field_simp; ring and adjust. (e) integral_eq_sub_of_hasDerivAt's hderiv : ∀ x ∈ uIcc a b — I pass `fun σ _ => hderiv σ` ✓. (f) The final FTC result: ∫ = F b − F a with F := fun σ => log(g σ) ✓ matches the statement ✓ (the lemma's f' := fun σ => 1/√h σ ✓). Now lemma 2 (right endpoint): ```lean theorem arcsinh_right_endpoint_le {β δ : ℝ} (hβ : β < 1) (hδ : 0 < δ) : Real.log ((3 - β) + Real.sqrt ((3 - β) ^ 2 + δ ^ 2)) ≤ Real.log (6 + δ) := by have hx : (0 : ℝ) < 3 - β := by linarith have hsqrt_le : Real.sqrt ((3 - β) ^ 2 + δ ^ 2) ≤ (3 - β) + δ := by rw [Real.sqrt_le_iff] exact ⟨by positivity, by nlinarith [hδ, hx]⟩ have h1 : (0 : ℝ) < (3 - β) + Real.sqrt ((3 - β) ^ 2 + δ ^ 2) := by have h2 := Real.sqrt_nonneg ((3 - β) ^ 2 + δ ^ 2) nlinarith have h3 : (3 - β) + Real.sqrt ((3 - β) ^ 2 + δ ^ 2) ≤ 6 + δ := by nlinarith [hsqrt_le, hx] exact Real.log_le_log h1 h3 ``` Real.sqrt_le_iff : √x ≤ y ↔ 0 ≤ y ∧ x ≤ y^2 — the membership: (3−β)²+δ² ≤ ((3−β)+δ)² = (3−β)²+2(3−β)δ+δ² ⟺ 0 ≤ 2(3−β)δ ✓ nlinarith [hδ, hx] — needs the product (3−β)·δ ≥ 0 — nlinarith pairwise ✓ (hx, hδ). h3: (3−β) + √ ≤ (3−β) + ((3−β)+δ) = 2(3−β)+δ ≤ 6+δ ⟺ 2(3−β) ≤ 6 ⟺ β ≥ 0?? — WAIT: 2(3−β) ≤ 6 ⟺ 3−β ≤ 3 ⟺ β ≥ 0 — but hβ : β < 1 gives β < 1, NOT 0 ≤ β! For the RIGHT endpoint I need... 3−β ≤ 3 ⟺ β ≥ 0. Hmm — the recipe says "num ≤ 2(3−β)+δ ≤ 6+δ" — this needs β ≥ 0 ✓ (zeros have re > 0). So the right endpoint lemma ALSO needs 0 < β (or β ≥ 0). Let me add (hβ0 : 0 < β) to the right endpoint lemma. nlinarith [hsqrt_le, hβ0] : (3−β)+√ ≤ 2(3−β)+δ: from hsqrt_le; then 2(3−β)+δ ≤ 6+δ: 2(3−β) ≤ 6 ⟺ β ≥ 0 ✓ hβ0. ✓ Lemma 3 (left endpoint): ```lean theorem arcsinh_left_endpoint_le {β δ : ℝ} (hβ0 : 0 < β) (hβ1 : β < 1) (hδ : 0 < δ) : -Real.log ((-5 / 2 - β) + Real.sqrt ((-5 / 2 - β) ^ 2 + δ ^ 2)) ≤ Real.log (7 + δ) + 2 * Real.log (1 / δ) := by have hu : (0 : ℝ) < 5 / 2 + β := by linarith -- rewrite (−5/2−β)² = (5/2+β)² and group −5/2−β = −(5/2+β) have hsq : (-5 / 2 - β) ^ 2 = (5 / 2 + β) ^ 2 := by ring have hneg : (-5 / 2 : ℝ) - β = -(5 / 2 + β) := by ring rw [hsq, hneg] set u := 5 / 2 + β with hu_def -- u ∈ (5/2, 7/2) have hu2 : u < 7 / 2 := by rw [hu_def]; linarith have hsqrt_pos : (0 : ℝ) < Real.sqrt (u ^ 2 + δ ^ 2) := Real.sqrt_pos.mpr (add_pos_of_nonneg_of_pos (sq_nonneg _) (pow_pos hδ 2)) have hg_pos : (0 : ℝ) < -u + Real.sqrt (u ^ 2 + δ ^ 2) := by have h1 : u < Real.sqrt (u ^ 2 + δ ^ 2) := by have h2 : u ^ 2 < u ^ 2 + δ ^ 2 := by nlinarith [hδ] have h3 := Real.sqrt_lt_sqrt (sq_nonneg _) h2 rwa [Real.sqrt_sq_eq_abs, abs_of_pos hu] at h3 linarith -- the rationalization: −u + √ = δ²/(u + √) have hrid : -u + Real.sqrt (u ^ 2 + δ ^ 2) = δ ^ 2 / (u + Real.sqrt (u ^ 2 + δ ^ 2)) := by rw [div_eq_iff (ne_of_gt (by linarith [hsqrt_pos] : (0:ℝ) < u + Real.sqrt (u^2+δ^2)))] have hs : (Real.sqrt (u ^ 2 + δ ^ 2)) ^ 2 = u ^ 2 + δ ^ 2 := Real.sq_sqrt (by positivity) nlinarith [hs] -- denominator ≤ 7 + δ have hden : u + Real.sqrt (u ^ 2 + δ ^ 2) ≤ 7 + δ := by have h1 : Real.sqrt (u ^ 2 + δ ^ 2) ≤ u + δ := by rw [Real.sqrt_le_iff] exact ⟨by linarith [hδ, hu.le], by nlinarith [hδ, hu]⟩ nlinarith [h1, hu2] -- δ²/(7+δ) ≤ δ²/(u+√) = −u+√ have hge : δ ^ 2 / (7 + δ) ≤ -u + Real.sqrt (u ^ 2 + δ ^ 2) := by rw [hrid] have h2 : (0 : ℝ) < u + Real.sqrt (u ^ 2 + δ ^ 2) := by linarith [hsqrt_pos] exact div_le_div_of_nonneg_left (sq_nonneg _) h2 hden -- log mono + algebra have hlog : Real.log (δ ^ 2 / (7 + δ)) ≤ Real.log (-u + Real.sqrt (u ^ 2 + δ ^ 2)) := Real.log_le_log (by positivity) hge have hlogid : Real.log (δ ^ 2 / (7 + δ)) = 2 * Real.log δ - Real.log (7 + δ) := by rw [Real.log_div (by positivity : (δ:ℝ)^2 ≠ 0) (by positivity : (7:ℝ)+δ ≠ 0), Real.log_pow] ring -- log(1/δ) = −log δ have hlog1 : Real.log (1 / δ) = -Real.log δ := by rw [Real.log_div one_ne_zero (ne_of_gt hδ), Real.log_one, zero_sub] -- assemble: −log g ≤ −(2 log δ − log(7+δ)) = log(7+δ) + 2·log(1/δ) linarith [hlog, hlogid ▸ ..., hlog1] ``` Hmm — the final assemble: goal −log(g) ≤ log(7+δ) + 2·log(1/δ). Have: hlog : 2logδ − log(7+δ) ≤ log g (via hlogid ▸ hlog); hlog1 : log(1/δ) = −logδ. linarith [hlog, hlog1] with hlogid used to rewrite hlog's LHS: I'll rewrite hlog by hlogid first: ```lean rw [hlogid] at hlog linarith [hlog, hlog1] ``` hlog : 2·log δ − log(7+δ) ≤ log(g); goal: −log(g) ≤ log(7+δ) + 2·log(1/δ); with hlog1 : log(1/δ) = −log δ: RHS = log(7+δ) − 2·logδ; goal ⟺ −log g ≤ log(7+δ) − 2 log δ ⟺ 2 log δ − log(7+δ) ≤ log g ✓ hlog ✓ linarith. positivity side-conditions: (δ:ℝ)^2 ≠ 0 := pow_ne_zero 2 (ne_of_gt hδ) — positivity proves 0 < δ^2 → ne' hmm — `by positivity : (0:ℝ) < δ^2` then .ne'. Let me write: `(pow_pos hδ 2).ne'`. And (7+δ ≠ 0) := (by linarith [hδ] : (0:ℝ) < 7+δ).ne'. And the hlog's positivity arg: 0 < δ²/(7+δ) := by positivity — needs 0 < δ² (pow_pos hδ 2 in context? — positivity can prove 0 < δ^2 from hδ? — positivity extensions for ^: strict needs base > 0 — from hδ in context ✓ probably; fallback div_pos (pow_pos hδ 2) (by linarith [hδ])). Also `div_le_div_of_nonneg_left (sq_nonneg _) h2 hden : δ²/(7+δ) ≤ δ²/(u+√)` — signature: (ha : 0 ≤ a) (hb : 0 < b) (h : b ≤ c) : a/c ≤ a/b — a := δ², b := u+√, c := 7+δ ✓ δ²/(7+δ) ≤ δ²/(u+√) ✓ — matches the goal after rw [hrid] ✓. Wait — Real.sqrt_sq_eq_abs in hg_pos: √(u²) = |u| ✓ then abs_of_pos hu : |u| = u ✓ rwa [Real.sqrt_sq_eq_abs, abs_of_pos hu] at h3 — h3 : |u| < √(u²+δ²) → after abs_of_pos: u < √ ✓. Hmm — Real.sqrt_lt_sqrt (sq_nonneg _) h2 : √(u²) < √(u²+δ²) — the first arg: 0 ≤ u² ✓ sq_nonneg ✓. Lemma 4 (combine + log-mono): ```lean theorem per_zero_integral_le_two_log {β δ : ℝ} (hβ0 : 0 < β) (hβ1 : β < 1) (hδ : 0 < δ) (hδ1 : δ ≤ 1) : ∫ σ in (-5 / 2 : ℝ)..(3 : ℝ), 1 / Real.sqrt ((σ - β) ^ 2 + δ ^ 2) ≤ 2 * Real.log (16.5 / δ) := by have hftc := integral_one_div_sqrt_sq_add_sq β δ hδ (-5/2) 3 have hr := arcsinh_right_endpoint_le hβ0 hβ1 hδ have hl := arcsinh_left_endpoint_le hβ0 hβ1 hδ have h1 : ∫ σ in (-5 / 2 : ℝ)..(3 : ℝ), 1 / Real.sqrt ((σ - β) ^ 2 + δ ^ 2) ≤ Real.log (6 + δ) + Real.log (7 + δ) + 2 * Real.log (1 / δ) := by rw [hftc] linarith [hr, hl] -- the log-mono simplification have h2 : Real.log (6 + δ) + Real.log (7 + δ) ≤ 2 * Real.log 16.5 := by have h3 : Real.log (6 + δ) + Real.log (7 + δ) = Real.log ((6 + δ) * (7 + δ)) := (Real.log_mul (by linarith [hδ] : (6:ℝ)+δ ≠ 0) (by linarith [hδ] : (7:ℝ)+δ ≠ 0)).symm have h4 : Real.log ((6 + δ) * (7 + δ)) ≤ Real.log (16.5 * 16.5) := Real.log_le_log (by positivity) (by nlinarith [hδ, hδ1]) have h5 : Real.log (16.5 * 16.5) = 2 * Real.log 16.5 := by rw [show (16.5 : ℝ) * 16.5 = 16.5 ^ 2 by ring, Real.log_pow] ring linarith [h3 ▸ h4, h5] have h6 : 2 * Real.log (16.5 / δ) = 2 * Real.log 16.5 + 2 * Real.log (1 / δ) := by have h7 : Real.log (1 / δ) = -Real.log δ := by rw [Real.log_div one_ne_zero (ne_of_gt hδ), Real.log_one, zero_sub] rw [Real.log_div (by norm_num) (ne_of_gt hδ), h7] ring rw [h6] linarith [h1, h2] ``` Concerns: - arcsinh_right_endpoint_le hβ0 hβ1 hδ — I said the right endpoint needs BOTH hβ0 (for the ≤6+δ) — signature (hβ : β < 1) earlier — NOW needs (hβ0 : 0 < β) too? — the right endpoint: (3−β) ≤ 3 ⟺ β ≥ 0 ✓ hβ0; and 3−β > 0 for the sqrt_le's positivity? — sqrt_le_iff needs 0 ≤ (3−β)+δ ✓ positivity [hδ]; and the product (3−β)·δ ≥ 0 needs 3−β ≥ 0 ⟸ β ≤ 3 ✓ from β < 1. And g3 > 0 needs 3−β > 0 ✓ β < 1. So right endpoint needs BOTH (hβ0 : 0 < β) (hβ1 : β < 1) — wait: for the nlinarith [hδ, hx] in sqrt_le: 2(3−β)δ ≥ 0: needs 3−β ≥ 0 ✓ hβ1 (β < 1 ⟹ 3−β > 2 > 0) ✓. And h3's 2(3−β) ≤ 6: β ≥ 0 ✓ hβ0. So signature (hβ0 : 0 < β) (hβ1 : β < 1) — update: hx : 0 < 3−β from hβ1; the 2(3−β) ≤ 6 from hβ0 ✓. - h2's linarith [h3 ▸ h4, h5]: h3 : log(6+δ)+log(7+δ) = log((6+δ)(7+δ)) — h3 ▸ h4 — `▸` at term level: rewrites h4 by h3 — direction: h3 : A = B — h3 ▸ h4 rewrites occurrences of A in h4's type by B — h4 : log((6+δ)(7+δ)) ≤ log(16.5·16.5) — no A in h4 — WRONG direction: I need log(6+δ)+log(7+δ) ≤ ... — use: linarith [h3, h4, h5] — all as atom equations/inequalities ✓ linarith combines ✓ (no ▸ needed): goal: log(6+δ) + log(7+δ) ≤ 2·log 16.5; h3 : LHS = log(prod); h4 : log(prod) ≤ log(16.5·16.5); h5 : log(16.5·16.5) = 2·log 16.5 ✓ linarith closes ✓. - h4's positivity: 0 < (6+δ)(7+δ) — by positivity with hδ in context ✓ (6+δ > 0 from hδ... positivity needs to derive 0 < 6+δ — hδ : 0 < δ in context ✓ positivity ✓). - h4's nlinarith [hδ, hδ1] : (6+δ)(7+δ) ≤ 16.5·16.5: expand: 42 + 13δ + δ² ≤ 272.25: needs δ² ≤ δ·δ ≤ 1·1 = 1... nlinarith pairwise: (hδ1 : δ ≤ 1)² gives δ² ≤ 1 ✓; then 42 + 13·1 + 1 = 56 ≤ 272.25 ✓ nlinarith finds ✓. - (Real.log_mul ...).symm for h3 — the ne conditions: 6+δ ≠ 0 and 7+δ ≠ 0 — `by linarith [hδ] : (6:ℝ) + δ ≠ 0` — hmm — the goal (6:ℝ)+δ ≠ 0 — from hδ : 0 < δ: 6+δ > 0 ⟹ ≠ 0 — linarith proves ≠? — banked: "linarith proves ≠ directly" ✓ (LiAssembly banked gotcha ✓). - h6: Real.log_div (by norm_num : (16.5:ℝ) ≠ 0) (ne_of_gt hδ) : log(16.5/δ) = log 16.5 − log δ; rw [h7] : −log δ → log(1/δ)... wait h7 : log(1/δ) = −log δ — rw [h7] rewrites log(1/δ) → −log δ — but I want the OTHER direction (−log δ → log(1/δ)) — rw [← h7] ✓. Let me redo: goal: 2·log(16.5/δ) = 2·log 16.5 + 2·log(1/δ). rw [Real.log_div ...] : LHS's log(16.5/δ) → log 16.5 − log δ: goal 2·(log16.5 − logδ) = 2·log16.5 + 2·log(1/δ). rw [← h7] : pattern −log δ → log(1/δ)?? — ← h7 rewrites occurrences of RHS of h7 (−log δ) with LHS (log(1/δ)) ✓: goal 2·(log16.5 + log(1/δ)) = 2·log16.5 + 2·log(1/δ) — hmm — wait: 2·(log16.5 − logδ) — the term −log δ is `Neg.neg (log δ)`? or `log16.5 − logδ` is HSub — ← h7's pattern `-(log δ)`: HSub.hSub (log16.5) (logδ) — is that Neg.neg? NO — a − b is Sub, not Neg. rw [← h7] would rewrite `-Real.log δ` as a NEGATION — inside `log16.5 − logδ` there's no standalone `−logδ` negation term (it's a sub). So rw [← h7] fails to find the pattern!! Fix: rw [h7] the OTHER way — transform the goal's RHS instead: the goal after log_div-rw: `2*(log16.5 − logδ) = 2*log16.5 + 2*log(1/δ)` — rw [h7] rewrites `log(1/δ)` → `−logδ` ✓ (h7's LHS log(1/δ) appears in the goal's RHS ✓): goal `2*(log16.5 − logδ) = 2*log16.5 + 2*(−logδ)` — `ring` ✓. So: ```lean rw [Real.log_div (by norm_num) (ne_of_gt hδ), h7] ring ``` h7 rewrites log(1/δ) → −log δ ✓ then ring: 2(log16.5 − logδ) = 2log16.5 − 2logδ ✓ closes ✓ — h7 must be proved BEFORE this rw (it's the have above ✓). Wait — in my draft h6 the h7-rw is inside: `rw [Real.log_div one_ne_zero (ne_of_gt hδ), Real.log_one, zero_sub]` for h7's proof: h7 : log(1/δ) = −log δ: rw [Real.log_div ...] at goal: log(1/δ) = −log δ — log(1/δ) → log 1 − log δ: goal log 1 − log δ = −log δ — rw [Real.log_one] : 0 − log δ = −log δ — rw [zero_sub] : −log δ = −log δ rfl ✓ — rw closes ✓. - The per_zero lemma's final form `2 * Real.log (16.5 / δ)` ✓. Also — for hper's use, I need per_zero at δ := |γ−T|: hδ : 0 < |γ−T| (hδpos), hδ1 : |γ−T| ≤ 1 (hδ1.le) ✓. Now — hβ0/hβ1 for ρ.1: ρ.2.2.1 : 0 < ρ.1.re ✓ ρ.2.2.2 : ρ.1.re < 1 ✓. One more piece of §4 — lemma 5/6's `unfold HIntegral` start and the final calc. And lemma 6 (neg) cloning DL-5's conj: ```lean theorem hIntegral_Jlag_mid_neg_norm_le (hZC ...) (same hyps) : ‖HIntegral (Jlag n) (-5/2) 3 (-T)‖ ≤ (same RHS) := by have hT0 : (0:ℝ) < T := by ... (√n ≥ 316) have hconj : HIntegral (Jlag n) (-5/2) 3 (-T) = (starRingEnd ℂ) (HIntegral (Jlag n) (-5/2) 3 T) := by unfold HIntegral rw [← intervalIntegral.intervalIntegral_conj] apply intervalIntegral.integral_congr intro σ hσ show Jlag n ((σ : ℝ) + (((-T : ℝ)) : ℂ) * Complex.I) = (starRingEnd ℂ) (Jlag n ((σ : ℝ) + T * Complex.I)) have hid2 : ((σ : ℝ) + (((-T : ℝ)) : ℂ) * Complex.I) = (starRingEnd ℂ) ((σ : ℝ) + T * Complex.I) := by simp rw [hid2] exact Jlag_conj n (by intro h have him : (((σ : ℝ) + T * Complex.I)).im = T := by simp rw [h] at him simp at him linarith [hT0]) rw [hconj, Complex.norm_conj] exact hIntegral_Jlag_mid_norm_le hZC hZC2 hNLZ hn hTwide hTn hTn1 ``` — clone DL-5 verbatim modulo coordinates ✓. Note: DL-5's `((σ : ℝ) + T * Complex.I)` — T : ℝ coerced ✓ same form ✓. `intervalIntegral.intervalIntegral_conj` — signature: `∫ x in a..b, (starRingEnd ℂ) (f x) = (starRingEnd ℂ) (∫ x in a..b, f x)`? — DL-5 used `rw [← intervalIntegral.intervalIntegral_conj]` — ✓ same usage. hT0 : 0 < T — from √n ≥ √100000: hsqrt316 : 316 ≤ √n? — simpler: 0 < √n (Real.sqrt_pos.mpr (by linarith [hnR])) and √n ≤ T ⟹ 0 < T ✓ linarith. OK — NOW let me also think about whether hIntegral_Jlag_mid_norm_le's Jlag-integral equality (unfold HIntegral) works: HIntegral f x₁ x₂ y := ∫ x in x₁..x₂, f (x + y*I) — the points: `(↑x + ↑y*I)` with x y : ℝ — my written integrand `Jlag n ((σ : ℝ) + (T : ℝ) * Complex.I)` — same elaboration `↑σ + ↑T * I`?? — HIntegral's body: `f (x + y * I)` — x : ℝ, y * I — the + : needs ℂ: `↑x + ↑y * I`?? — how was it elaborated in DL-5/rectangleIntegral_Jlag_eq? — in rectangleIntegral_Jlag_eq the HIntegral form matched the RectangleIntegral def ✓ and DL-5's conj proof used `((σ : ℝ) + T * Complex.I)` ✓ so the elaborated form is `↑σ + ↑T * I`?? — hmm — in DL-5's hid2: `((σ : ℝ) + (((-T : ℝ)) : ℂ) * Complex.I)` — with explicit ((−T : ℝ) : ℂ)?? — they wrote `(((−T : ℝ)) : ℂ)` — hmm no: `(((−T : ℝ)) : ℂ)` vs `(↑(−T) : ℂ)`... — whatever — my forms will match DL-5's ✓ (clone exactly). Wait — one subtlety: in HIntegral's def `f (x + y * I)` — is the elaborated point `↑x + ↑y * Complex.I` or `↑x + ↑(y) * Complex.I`? — same thing ✓. But — when I `unfold HIntegral` at the goal ‖HIntegral (Jlag n) (-5/2) 3 T‖ ≤ ..., the integrand becomes `fun x => Jlag n (↑x + ↑T * I)` — and my subsequent haves are stated with `((σ : ℝ) + (T : ℝ) * Complex.I)` — matching ✓ (norm_integral_le_integral_norm's application will unify the integrand ✓ defeq ✓). Hmm — actually — rectangleIntegral_Jlag_eq used `HIntegral (Jlag n) (-5/2) (2*√n) (-T)` etc. and VIntegral_norm handled the V side; the H-side norm: is there an HIntegral_norm analog? HIntegral f a b y = ∫ σ in a..b, f (↑σ+↑y·I) — no I-factor — so ‖HIntegral‖ = ‖∫‖ directly ✓ no VIntegral_norm needed ✓. Now — writing the whole §4. It's long; let me be methodical. I'll write it in one Edit, then iterate on errors. One more check — `intervalIntegral.integral_mono_on hlen hJintbl hKLintbl (fun σ hσ => hptJ σ hσ)` — hKLintbl : IntervalIntegrable (fun σ => (e⁴+1) * ‖logDeriv _‖) — from hLDcont: hKLcont := continuous_const.mul (continuous_norm.comp hLDc)... — hmm wait — hptJ's RHS is (e⁴+1)·‖logDeriv‖ — the hg in integral_mono_on must be the interval-integrability of THAT lambda ✓. And hsplit's hsumIntbl : IntervalIntegrable (fun σ => ‖Σ‖ + ‖R‖) — hmm — integral_mono_on's hg — the RHS integrand (fun σ => ‖Σ σ‖ + ‖R σ‖) — from hΣnorm_cont.add hRcont ✓ .intervalIntegrable. Naming (all inside the theorem as have's): hpath, hne1, hJcont, hcontJ, hcontG, hJintbl, hLDc, hLDcont, hLDintbl, hKLcont, hKLintbl, hΣcont, hΣnorm_cont, hΣnorm_int, hRcont, hRnorm_int, hsumcont, hsum_intbl, hker, hptJ, hJint, hpt2, hsplit, hRint, hpt, hρdata, htermcont, hRHScont, hswap, hper, hsum, hΣbound, hB0nn. OK — write it. For hne1 (the ∀ σ, point ≠ 1): ```lean have hne1 : ∀ σ : ℝ, ((σ : ℝ) + (T : ℝ) * Complex.I) ≠ 1 := by intro σ h have him := congrArg Complex.im h simp at him nlinarith [h10T] ``` him : T = 0 (simp computes im) — h10T : 10 ≤ T ⟹ T ≥ 10 ≠ 0 ✓ nlinarith? — him : T = 0 and 10 ≤ T — linarith ✓ (linear) — use linarith [h10T]. For hs1 inside hJcont (6-c-ii had it as re-argument — here the point's re varies with σ — the im argument works for ALL σ: point = 1 ⟹ im = T = 0 ≠ ✓). Let me now write §4 in full.
All signatures verified. Writing §4 (6-c-iv) — the arcsinh-FTC machine, endpoint estimates, per-zero bound, and the two mid-horizontal theorems.
tool: Edit
input
{
  "_raw": "{\"replace_all\":false,\"file_path\":\"D:\\\\Python\\\\Quantum\\\\research\\\\lean\\\\NguyenRH1\\\\NguyenRH1\\\\LiSideBounds.lean\",\"old_string\":\"    _ = 2 * T * (Real.exp (Real.sqrt (n : \u211d) / 2 + 1 / 3) + 1) *\\n          ((0.57 : \u211d) * (2 : \u211d) ^ (-(2 * Real.sqrt (n : \u211d) - 2))) := by ring\\n    _ \u2264 Real.exp (1 / 3 : \u211d) * IIMaj (n : \u211d) := hfinal\\n\\nend NguyenRH1\",\"new_string\":\"    _ = 2 * T * (Real.exp (Real.sqrt (n : \u211d) / 2 + 1 / 3) + 1) *\\n          ((0.57 : \u211d) * (2 : \u211d) ^ (-(2 * Real.sqrt (n : \u211d) - 2))) := by ring\\n    _ \u2264 Real.exp (1 / 3 : \u211d) * IIMaj (n : \u211d) := hfinal\\n\\n/-! ## 4. 6-c-iv: the mid-horizontal segment machine [\u22125/2, 3] -/\\n\\n/-- **The arcsinh FTC machine** (6-c-iv core): the direct antiderivative\\nF(\u03c3) = log((\u03c3\u2212\u03b2) + \u221a((\u03c3\u2212\u03b2)\u00b2+\u03b4\u00b2)) of 1/\u221a((\u03c3\u2212\u03b2)\u00b2+\u03b4\u00b2) (Real.arcsinh does\\nnot exist here): g(\u03c3) := (\u03c3\u2212\u03b2)+\u221a((\u03c3\u2212\u03b2)\u00b2+\u03b4\u00b2) > 0 (\u221ah > |\u03c3\u2212\u03b2|), and\\ng\u2032 = 1 + (\u03c3\u2212\u03b2)/\u221ah = g/\u221ah, so (log g)\u2032 = 1/\u221ah exactly. -/\\ntheorem integral_one_div_sqrt_sq_add_sq (\u03b2 \u03b4 : \u211d) (h\u03b4 : 0 < \u03b4) (a b : \u211d) :\\n    \u222b \u03c3 in a..b, 1 / Real.sqrt ((\u03c3 - \u03b2) ^ 2 + \u03b4 ^ 2) =\\n      Real.log ((b - \u03b2) + Real.sqrt ((b - \u03b2) ^ 2 + \u03b4 ^ 2)) -\\n        Real.log ((a - \u03b2) + Real.sqrt ((a - \u03b2) ^ 2 + \u03b4 ^ 2)) := by\\n  have hh : \u2200 \u03c3 : \u211d, (0 : \u211d) < (\u03c3 - \u03b2) ^ 2 + \u03b4 ^ 2 := fun \u03c3 =>\\n    add_pos_of_nonneg_of_pos (sq_nonneg _) (pow_pos h\u03b4 2)\\n  have hgpos : \u2200 \u03c3 : \u211d, (0 : \u211d) < (\u03c3 - \u03b2) + Real.sqrt ((\u03c3 - \u03b2) ^ 2 + \u03b4 ^ 2) := by\\n    intro \u03c3\\n    have h1 : |(\u03c3 - \u03b2)| < Real.sqrt ((\u03c3 - \u03b2) ^ 2 + \u03b4 ^ 2) := by\\n      have h2 : (\u03c3 - \u03b2) ^ 2 < (\u03c3 - \u03b2) ^ 2 + \u03b4 ^ 2 := by nlinarith [h\u03b4]\\n      have h3 := Real.sqrt_lt_sqrt (sq_nonneg _) h2\\n      rwa [Real.sqrt_sq_eq_abs] at h3\\n    have h4 : -(\u03c3 - \u03b2) \u2264 |(\u03c3 - \u03b2)| := neg_abs_le (\u03c3 - \u03b2)\\n    linarith [h1, h4]\\n  have hderiv : \u2200 \u03c3 : \u211d, HasDerivAt\\n      (fun \u03c3 => Real.log ((\u03c3 - \u03b2) + Real.sqrt ((\u03c3 - \u03b2) ^ 2 + \u03b4 ^ 2)))\\n      (1 / Real.sqrt ((\u03c3 - \u03b2) ^ 2 + \u03b4 ^ 2)) \u03c3 := by\\n    intro \u03c3\\n    have hh\u03c3 := hh \u03c3\\n    have hsqrt0 : Real.sqrt ((\u03c3 - \u03b2) ^ 2 + \u03b4 ^ 2) \u2260 0 := ne_of_gt (Real.sqrt_pos.mpr hh\u03c3)\\n    have hg0 : ((\u03c3 - \u03b2) + Real.sqrt ((\u03c3 - \u03b2) ^ 2 + \u03b4 ^ 2)) \u2260 0 := (hgpos \u03c3).ne'\\n    have h1 : HasDerivAt (fun \u03c3 => \u03c3 - \u03b2) 1 \u03c3 := hasDerivAt_id.sub_const \u03b2\\n    have h2 : HasDerivAt (fun \u03c3 => (\u03c3 - \u03b2) ^ 2 + \u03b4 ^ 2) (2 * (\u03c3 - \u03b2)) \u03c3 := by\\n      have h2a := (hasDerivAt_id.sub_const \u03b2).pow 2\\n      have h2b := h2a.add_const (\u03b4 ^ 2)\\n      convert h2b using 1\\n      ring\\n    have h4 : HasDerivAt (fun \u03c3 => Real.sqrt ((\u03c3 - \u03b2) ^ 2 + \u03b4 ^ 2))\\n        (2 * (\u03c3 - \u03b2) / (2 * Real.sqrt ((\u03c3 - \u03b2) ^ 2 + \u03b4 ^ 2))) \u03c3 :=\\n      h2.sqrt hh\u03c3.ne'\\n    have h5 : HasDerivAt (fun \u03c3 => (\u03c3 - \u03b2) + Real.sqrt ((\u03c3 - \u03b2) ^ 2 + \u03b4 ^ 2))\\n        (1 + 2 * (\u03c3 - \u03b2) / (2 * Real.sqrt ((\u03c3 - \u03b2) ^ 2 + \u03b4 ^ 2))) \u03c3 :=\\n      h1.add h4\\n    have h6 : HasDerivAt (fun \u03c3 => (\u03c3 - \u03b2) + Real.sqrt ((\u03c3 - \u03b2) ^ 2 + \u03b4 ^ 2))\\n        (((\u03c3 - \u03b2) + Real.sqrt ((\u03c3 - \u03b2) ^ 2 + \u03b4 ^ 2)) /\\n          Real.sqrt ((\u03c3 - \u03b2) ^ 2 + \u03b4 ^ 2)) \u03c3 := by\\n      convert h5 using 1\\n      field_simp [hsqrt0]\\n      ring\\n    have h7 := h6.log hg0\\n    convert h7 using 1\\n    field_simp [hsqrt0, hg0]\\n    ring\\n  exact intervalIntegral.integral_eq_sub_of_hasDerivAt (fun \u03c3 _ => hderiv \u03c3)\\n    (by\\n      apply Continuous.intervalIntegrable\\n      apply Continuous.div continuous_const\\n      \u00b7 exact Real.continuous_sqrt.comp (by fun_prop :\\n          Continuous fun \u03c3 : \u211d => (\u03c3 - \u03b2) ^ 2 + \u03b4 ^ 2)\\n      \u00b7 intro \u03c3\\n        exact ne_of_gt (Real.sqrt_pos.mpr (hh \u03c3)))\\n\\n/-- **The right endpoint estimate** F(3) \u2264 log(6+\u03b4): \u221a((3\u2212\u03b2)\u00b2+\u03b4\u00b2) \u2264\\n(3\u2212\u03b2)+\u03b4 for 3\u2212\u03b2 > 0, so the argument is \u2264 2(3\u2212\u03b2)+\u03b4 \u2264 6+\u03b4 (0 \u2264 \u03b2). -/\\ntheorem arcsinh_right_endpoint_le {\u03b2 \u03b4 : \u211d} (h\u03b20 : 0 < \u03b2) (h\u03b21 : \u03b2 < 1)\\n    (h\u03b4 : 0 < \u03b4) :\\n    Real.log ((3 - \u03b2) + Real.sqrt ((3 - \u03b2) ^ 2 + \u03b4 ^ 2)) \u2264 Real.log (6 + \u03b4) := by\\n  have hx : (0 : \u211d) < 3 - \u03b2 := by linarith\\n  have hsqrt_le : Real.sqrt ((3 - \u03b2) ^ 2 + \u03b4 ^ 2) \u2264 (3 - \u03b2) + \u03b4 := by\\n    rw [Real.sqrt_le_iff]\\n    exact \u27e8by positivity, by nlinarith [h\u03b4, hx]\u27e9\\n  have h1 : (0 : \u211d) < (3 - \u03b2) + Real.sqrt ((3 - \u03b2) ^ 2 + \u03b4 ^ 2) := by\\n    have h2 := Real.sqrt_nonneg ((3 - \u03b2) ^ 2 + \u03b4 ^ 2)\\n    nlinarith\\n  have h3 : (3 - \u03b2) + Real.sqrt ((3 - \u03b2) ^ 2 + \u03b4 ^ 2) \u2264 6 + \u03b4 := by\\n    nlinarith [hsqrt_le, h\u03b20]\\n  exact Real.log_le_log h1 h3\\n\\n/-- **The left endpoint estimate** \u2212F(\u22125/2) \u2264 log(7+\u03b4) + 2\u00b7log(1/\u03b4):\\nwith u := 5/2+\u03b2 \u2208 (5/2, 7/2), the argument \u2212u+\u221a(u\u00b2+\u03b4\u00b2) rationalizes to\\n\u03b4\u00b2/(u+\u221a(u\u00b2+\u03b4\u00b2)) \u2265 \u03b4\u00b2/(7+\u03b4) > 0, so F(\u22125/2) \u2265 log(\u03b4\u00b2/(7+\u03b4)) =\\n2\u00b7log \u03b4 \u2212 log(7+\u03b4). -/\\ntheorem arcsinh_left_endpoint_le {\u03b2 \u03b4 : \u211d} (h\u03b20 : 0 < \u03b2) (h\u03b21 : \u03b2 < 1)\\n    (h\u03b4 : 0 < \u03b4) :\\n    -Real.log ((-5 / 2 - \u03b2) + Real.sqrt ((-5 / 2 - \u03b2) ^ 2 + \u03b4 ^ 2)) \u2264\\n      Real.log (7 + \u03b4) + 2 * Real.log (1 / \u03b4) := by\\n  have hu : (0 : \u211d) < 5 / 2 + \u03b2 := by linarith\\n  have hsq : (-5 / 2 - \u03b2) ^ 2 = (5 / 2 + \u03b2) ^ 2 := by ring\\n  have hneg : (-5 / 2 : \u211d) - \u03b2 = -(5 / 2 + \u03b2) := by ring\\n  rw [hsq, hneg]\\n  set u := 5 / 2 + \u03b2 with hu_def\\n  have hu2 : u < 7 / 2 := by rw [hu_def]; linarith\\n  have hsqrt_pos : (0 : \u211d) < Real.sqrt (u ^ 2 + \u03b4 ^ 2) :=\\n    Real.sqrt_pos.mpr (add_pos_of_nonneg_of_pos (sq_nonneg _) (pow_pos h\u03b4 2))\\n  have hg_pos : (0 : \u211d) < -u + Real.sqrt (u ^ 2 + \u03b4 ^ 2) := by\\n    have h1 : u < Real.sqrt (u ^ 2 + \u03b4 ^ 2) := by\\n      have h2 : u ^ 2 < u ^ 2 + \u03b4 ^ 2 := by nlinarith [h\u03b4]\\n      have h3 := Real.sqrt_lt_sqrt (sq_nonneg _) h2\\n      rwa [Real.sqrt_sq_eq_abs, abs_of_pos hu] at h3\\n    linarith\\n  -- the rationalization: \u2212u + \u221a = \u03b4\u00b2/(u + \u221a)\\n  have hrid : -u + Real.sqrt (u ^ 2 + \u03b4 ^ 2) =\\n      \u03b4 ^ 2 / (u + Real.sqrt (u ^ 2 + \u03b4 ^ 2)) := by\\n    rw [div_eq_iff (ne_of_gt (by linarith [hsqrt_pos] :\\n      (0 : \u211d) < u + Real.sqrt (u ^ 2 + \u03b4 ^ 2)))]\\n    have hs : (Real.sqrt (u ^ 2 + \u03b4 ^ 2)) ^ 2 = u ^ 2 + \u03b4 ^ 2 :=\\n      Real.sq_sqrt (by positivity)\\n    nlinarith [hs]\\n  -- the denominator \u2264 7 + \u03b4\\n  have hden : u + Real.sqrt (u ^ 2 + \u03b4 ^ 2) \u2264 7 + \u03b4 := by\\n    have h1 : Real.sqrt (u ^ 2 + \u03b4 ^ 2) \u2264 u + \u03b4 := by\\n      rw [Real.sqrt_le_iff]\\n      exact \u27e8by linarith [h\u03b4, hu.le], by nlinarith [h\u03b4, hu]\u27e9\\n    nlinarith [h1, hu2]\\n  -- \u03b4\u00b2/(7+\u03b4) \u2264 \u03b4\u00b2/(u+\u221a) = \u2212u+\u221a\\n  have hge : \u03b4 ^ 2 / (7 + \u03b4) \u2264 -u + Real.sqrt (u ^ 2 + \u03b4 ^ 2) := by\\n    rw [hrid]\\n    have h2 : (0 : \u211d) < u + Real.sqrt (u ^ 2 + \u03b4 ^ 2) := by linarith [hsqrt_pos]\\n    exact div_le_div_of_nonneg_left (sq_nonneg _) h2 hden\\n  -- log mono + algebra: F(\u22125/2) \u2265 log(\u03b4\u00b2/(7+\u03b4)) = 2\u00b7log \u03b4 \u2212 log(7+\u03b4)\\n  have hlog : Real.log (\u03b4 ^ 2 / (7 + \u03b4)) \u2264 Real.log (-u + Real.sqrt (u ^ 2 + \u03b4 ^ 2)) :=\\n    Real.log_le_log (by positivity) hge\\n  have hlogid : Real.log (\u03b4 ^ 2 / (7 + \u03b4)) = 2 * Real.log \u03b4 - Real.log (7 + \u03b4) := by\\n    rw [Real.log_div (pow_pos h\u03b4 2).ne' (by linarith [h\u03b4] : (7 : \u211d) + \u03b4 \u2260 0),\\n      Real.log_pow]\\n    ring\\n  have hlog1 : Real.log (1 / \u03b4) = -Real.log \u03b4 := by\\n    rw [Real.log_div one_ne_zero (ne_of_gt h\u03b4), Real.log_one, zero_sub]\\n  rw [hlogid] at hlog\\n  linarith [hlog, hlog1]\\n\\n/-- **The per-zero segment integral bound**: \u222b_{\u22125/2}^{3} d\u03c3/\u221a((\u03c3\u2212\u03b2)\u00b2+\u03b4\u00b2)\\n\u2264 2\u00b7log(16.5/\u03b4) for 0 < \u03b2 < 1, 0 < \u03b4 \u2264 1 \u2014 the two endpoint estimates +\\nlog((6+\u03b4)(7+\u03b4)) \u2264 log(16.5\u00b2) since (6+\u03b4)(7+\u03b4) \u2264 56 \u2264 272.25. -/\\ntheorem per_zero_integral_le_two_log {\u03b2 \u03b4 : \u211d} (h\u03b20 : 0 < \u03b2) (h\u03b21 : \u03b2 < 1)\\n    (h\u03b4 : 0 < \u03b4) (h\u03b41 : \u03b4 \u2264 1) :\\n    \u222b \u03c3 in (-5 / 2 : \u211d)..(3 : \u211d), 1 / Real.sqrt ((\u03c3 - \u03b2) ^ 2 + \u03b4 ^ 2) \u2264\\n      2 * Real.log (16.5 / \u03b4) := by\\n  have hftc := integral_one_div_sqrt_sq_add_sq \u03b2 \u03b4 h\u03b4 (-5 / 2) 3\\n  have hr := arcsinh_right_endpoint_le h\u03b20 h\u03b21 h\u03b4\\n  have hl := arcsinh_left_endpoint_le h\u03b20 h\u03b21 h\u03b4\\n  have h1 : \u222b \u03c3 in (-5 / 2 : \u211d)..(3 : \u211d), 1 / Real.sqrt ((\u03c3 - \u03b2) ^ 2 + \u03b4 ^ 2) \u2264\\n      Real.log (6 + \u03b4) + Real.log (7 + \u03b4) + 2 * Real.log (1 / \u03b4) := by\\n    rw [hftc]\\n    linarith [hr, hl]\\n  -- the log-mono simplification: log(6+\u03b4) + log(7+\u03b4) \u2264 2\u00b7log 16.5\\n  have h2 : Real.log (6 + \u03b4) + Real.log (7 + \u03b4) \u2264 2 * Real.log 16.5 := by\\n    have h3 : Real.log (6 + \u03b4) + Real.log (7 + \u03b4) = Real.log ((6 + \u03b4) * (7 + \u03b4)) :=\\n      (Real.log_mul (by linarith [h\u03b4] : (6 : \u211d) + \u03b4 \u2260 0)\\n        (by linarith [h\u03b4] : (7 : \u211d) + \u03b4 \u2260 0)).symm\\n    have h4 : Real.log ((6 + \u03b4) * (7 + \u03b4)) \u2264 Real.log (16.5 * 16.5) :=\\n      Real.log_le_log (by positivity) (by nlinarith [h\u03b4, h\u03b41])\\n    have h5 : Real.log (16.5 * 16.5) = 2 * Real.log 16.5 := by\\n      rw [show (16.5 : \u211d) * 16.5 = 16.5 ^ 2 by ring, Real.log_pow]\\n      ring\\n    linarith [h3, h4, h5]\\n  have h6 : 2 * Real.log (16.5 / \u03b4) = 2 * Real.log 16.5 + 2 * Real.log (1 / \u03b4) := by\\n    have h7 : Real.log (1 / \u03b4) = -Real.log \u03b4 := by\\n      rw [Real.log_div one_ne_zero (ne_of_gt h\u03b4), Real.log_one, zero_sub]\\n    rw [Real.log_div (by norm_num) (ne_of_gt h\u03b4), h7]\\n    ring\\n  rw [h6]\\n  linarith [h1, h2]\\n\\n/-- **6-c-iv (mid-horizontal [\u22125/2, 3] at height +T)**: the segment bound\\n\u2016H\u2016 \u2264 (e\u2074+1)\u00b7(2\u00b7Z\u2082(T)\u00b7log(148.5\u00b7\u0108\u2080(\u221an+2)) + 5.5\u00b7(10.5\u00b7log(T+2)+48)).\\nKernel e\u2074+1 (the ii\u2032-4 shift lemma at s := w\u22121); logDeriv split by\\nnorm_le_insert' into the near sum (per-zero arcsinh-FTC machine +\\nintegral_finsetSum + the two-unit count \u2264 Z\u2082(T)) and the tight R-bound\\n(capstone, constant in \u03c3). -/\\ntheorem hIntegral_Jlag_mid_norm_le (hZC : ZeroCountProp) (hZC2 : ZeroCountProp2)\\n    (hNLZ : NoLowZeros) {n : \u2115} (hn : 100000 \u2264 n) {T : \u211d}\\n    (hTwide : \u2200 \u03c1 : \u2102, IsNontrivialZero \u03c1 \u2192 Real.sqrt (n : \u211d) - 1 < \u03c1.im \u2192\\n      \u03c1.im < Real.sqrt (n : \u211d) + 2 \u2192\\n        1 / (9 * C0hat (Real.sqrt (n : \u211d) + 2)) \u2264 |\u03c1.im - T|)\\n    (hTn : Real.sqrt (n : \u211d) \u2264 T) (hTn1 : T \u2264 Real.sqrt (n : \u211d) + 1) :\\n    \u2016HIntegral (Jlag n) (-5 / 2) 3 T\u2016 \u2264\\n      (Real.exp 4 + 1) *\\n        (2 * Z2hat T * Real.log (148.5 * C0hat (Real.sqrt (n : \u211d) + 2)) +\\n          (3 - -5 / 2) * (10.5 * Real.log (T + 2) + 48)) := by\\n  classical\\n  have hnR : (100000 : \u211d) \u2264 (n : \u211d) := by exact_mod_cast hn\\n  have h10T : (10 : \u211d) \u2264 T := by\\n    have h1 : (10 : \u211d) \u2264 Real.sqrt (n : \u211d) := by\\n      rw [Real.le_sqrt (by norm_num) (by positivity)]\\n      nlinarith [hnR]\\n    linarith\\n  have hT0 : (0 : \u211d) \u2264 T := by linarith\\n  have hn1 : 1 \u2264 n := by omega\\n  have hlen : (-5 / 2 : \u211d) \u2264 3 := by norm_num\\n  have hC0 : (0 : \u211d) < C0hat (Real.sqrt (n : \u211d) + 2) := by\\n    have h2 := two_le_C0hat (by nlinarith [Real.sqrt_nonneg (n : \u211d)] :\\n      (1 : \u211d) \u2264 Real.sqrt (n : \u211d) + 2)\\n    linarith\\n  have hB0nn : (0 : \u211d) \u2264 2 * Real.log (148.5 * C0hat (Real.sqrt (n : \u211d) + 2)) := by\\n    have h1 : (1 : \u211d) \u2264 148.5 * C0hat (Real.sqrt (n : \u211d) + 2) := by nlinarith [hC0]\\n    have h3 := Real.log_nonneg h1\\n    linarith\\n  -- zero-freeness on the horizontal + the path continuity machines\\n  have hzne : \u2200 \u03c3 : \u211d, riemannZeta ((\u03c3 : \u211d) + (T : \u211d) * Complex.I) \u2260 0 :=\\n    fun \u03c3 => zeta_ne_zero_horizontal hTwide hn1 hTn hTn1\\n  have hne1 : \u2200 \u03c3 : \u211d, ((\u03c3 : \u211d) + (T : \u211d) * Complex.I) \u2260 1 := by\\n    intro \u03c3 h\\n    have him := congrArg Complex.im h\\n    simp at him\\n    linarith [h10T]\\n  have hpath : Continuous fun u : \u211d => ((u : \u211d) + (T : \u211d) * Complex.I) := by\\n    fun_prop\\n  have hLDc : Continuous fun \u03c3 : \u211d =>\\n      logDeriv riemannZeta ((\u03c3 : \u211d) + (T : \u211d) * Complex.I) := by\\n    rw [continuous_iff_continuousAt]\\n    intro \u03c3\\n    have hf : ContinuousAt (fun u : \u211d => ((u : \u211d) + (T : \u211d) * Complex.I)) \u03c3 :=\\n      hpath.continuousAt\\n    exact ContinuousAt.comp (g := logDeriv riemannZeta)\\n      (f := fun u : \u211d => ((u : \u211d) + (T : \u211d) * Complex.I)) (x := \u03c3)\\n      (continuousAt_logDeriv_zeta (hne1 \u03c3) (hzne \u03c3)) hf\\n  have hJcont : \u2200 \u03c3 : \u211d, ContinuousAt (Jlag n) ((\u03c3 : \u211d) + (T : \u211d) * Complex.I) := by\\n    intro \u03c3\\n    have hA := (kFun_differentiableAt (n := n) (hne1 \u03c3)).continuousAt\\n    have hB := continuousAt_logDeriv_zeta (hne1 \u03c3) (hzne \u03c3)\\n    show ContinuousAt (fun s => kFun n s * (-logDeriv riemannZeta s)) _\\n    exact hA.mul hB.neg\\n  have hcontJ : Continuous fun \u03c3 : \u211d =>\\n      Jlag n ((\u03c3 : \u211d) + (T : \u211d) * Complex.I) := by\\n    rw [continuous_iff_continuousAt]\\n    intro \u03c3\\n    have hf : ContinuousAt (fun u : \u211d => ((u : \u211d) + (T : \u211d) * Complex.I)) \u03c3 :=\\n      hpath.continuousAt\\n    exact ContinuousAt.comp (g := Jlag n)\\n      (f := fun u : \u211d => ((u : \u211d) + (T : \u211d) * Complex.I)) (x := \u03c3) (hJcont \u03c3) hf\\n  have hcontG : Continuous fun \u03c3 : \u211d =>\\n      \u2016Jlag n ((\u03c3 : \u211d) + (T : \u211d) * Complex.I)\u2016 := continuous_norm.comp hcontJ\\n  have hJintbl : IntervalIntegrable\\n      (fun \u03c3 : \u211d => \u2016Jlag n ((\u03c3 : \u211d) + (T : \u211d) * Complex.I)\u2016)\\n      MeasureTheory.volume (-5 / 2) 3 := hcontG.intervalIntegrable _ _\\n  have hLDcont : Continuous fun \u03c3 : \u211d =>\\n      \u2016logDeriv riemannZeta ((\u03c3 : \u211d) + (T : \u211d) * Complex.I)\u2016 :=\\n    continuous_norm.comp hLDc\\n  have hLDintbl : IntervalIntegrable\\n      (fun \u03c3 : \u211d => \u2016logDeriv riemannZeta ((\u03c3 : \u211d) + (T : \u211d) * Complex.I)\u2016)\\n      MeasureTheory.volume (-5 / 2) 3 := hLDcont.intervalIntegrable _ _\\n  have hKLcont : Continuous fun \u03c3 : \u211d =>\\n      (Real.exp 4 + 1) * \u2016logDeriv riemannZeta ((\u03c3 : \u211d) + (T : \u211d) * Complex.I)\u2016 :=\\n    continuous_const.mul hLDcont\\n  have hKLintbl : IntervalIntegrable\\n      (fun \u03c3 : \u211d => (Real.exp 4 + 1) *\\n        \u2016logDeriv riemannZeta ((\u03c3 : \u211d) + (T : \u211d) * Complex.I)\u2016)\\n      MeasureTheory.volume (-5 / 2) 3 := hKLcont.intervalIntegrable _ _\\n  -- the near-sum denominator never vanishes on the path (else s = \u03c1 is a\\n  -- zero, contra zeta_ne_zero_horizontal)\\n  have hden : \u2200 \u03c1 : {\u03c1 : \u2102 // IsNontrivialZero \u03c1}, \u03c1 \u2208 nearZerosSub T \u2192\\n      \u2200 \u03c3 : \u211d, ((\u03c3 : \u211d) + (T : \u211d) * Complex.I) - \u03c1.1 \u2260 0 := by\\n    intro \u03c1 _ \u03c3 h0\\n    have hs : ((\u03c3 : \u211d) + (T : \u211d) * Complex.I) = \u03c1.1 := sub_eq_zero.mp h0\\n    have hz := hzne \u03c3\\n    rw [hs] at hz\\n    exact hz \u03c1.2.1\\n  have h\u03a3cont : Continuous fun \u03c3 : \u211d =>\\n      \u2211 \u03c1 \u2208 nearZerosSub T, (zeroMult \u03c1.1 : \u2102) /\\n        (((\u03c3 : \u211d) + (T : \u211d) * Complex.I) - \u03c1.1) := by\\n    apply continuous_finset_sum\\n    intro \u03c1 h\u03c1\\n    exact Continuous.div continuous_const (hpath.sub continuous_const) (hden \u03c1 h\u03c1)\\n  have h\u03a3norm_cont : Continuous fun \u03c3 : \u211d =>\\n      \u2016\u2211 \u03c1 \u2208 nearZerosSub T, (zeroMult \u03c1.1 : \u2102) /\\n        (((\u03c3 : \u211d) + (T : \u211d) * Complex.I) - \u03c1.1)\u2016 :=\\n    continuous_norm.comp h\u03a3cont\\n  have h\u03a3norm_int : IntervalIntegrable\\n      (fun \u03c3 : \u211d => \u2016\u2211 \u03c1 \u2208 nearZerosSub T, (zeroMult \u03c1.1 : \u2102) /\\n        (((\u03c3 : \u211d) + (T : \u211d) * Complex.I) - \u03c1.1)\u2016)\\n      MeasureTheory.volume (-5 / 2) 3 := h\u03a3norm_cont.intervalIntegrable _ _\\n  have hRcont : Continuous fun \u03c3 : \u211d =>\\n      \u2016logDeriv riemannZeta ((\u03c3 : \u211d) + (T : \u211d) * Complex.I) -\\n        \u2211 \u03c1 \u2208 nearZerosSub T, (zeroMult \u03c1.1 : \u2102) /\\n          (((\u03c3 : \u211d) + (T : \u211d) * Complex.I) - \u03c1.1)\u2016 :=\\n    continuous_norm.comp (hLDc.sub h\u03a3cont)\\n  have hRnorm_int : IntervalIntegrable\\n      (fun \u03c3 : \u211d => \u2016logDeriv riemannZeta ((\u03c3 : \u211d) + (T : \u211d) * Complex.I) -\\n        \u2211 \u03c1 \u2208 nearZerosSub T, (zeroMult \u03c1.1 : \u2102) /\\n          (((\u03c3 : \u211d) + (T : \u211d) * Complex.I) - \u03c1.1)\u2016)\\n      MeasureTheory.volume (-5 / 2) 3 := hRcont.intervalIntegrable _ _\\n  -- the per-\u03c1 data (\u03b4_\u03c1 := |\u03b3\u2212T| \u2208 [c_clear, 1))\\n  have h\u03c1data : \u2200 \u03c1 : {\u03c1 : \u2102 // IsNontrivialZero \u03c1}, \u03c1 \u2208 nearZerosSub T \u2192\\n      (0 : \u211d) < |\u03c1.1.im - T| \u2227 |\u03c1.1.im - T| < 1 \u2227\\n        (1 / (9 * C0hat (Real.sqrt (n : \u211d) + 2))) \u2264 |\u03c1.1.im - T| := by\\n    intro \u03c1 h\u03c1\\n    have h\u03b41 := mem_nearZerosSub.mp h\u03c1\\n    obtain \u27e8hlo, hhi\u27e9 := abs_lt.mp h\u03b41\\n    have h\u03b31 : Real.sqrt (n : \u211d) - 1 < \u03c1.1.im := by nlinarith [hTn]\\n    have h\u03b32 : \u03c1.1.im < Real.sqrt (n : \u211d) + 2 := by nlinarith [hTn1]\\n    have h\u03b4ge := hTwide \u03c1.1 \u03c1.2 h\u03b31 h\u03b32\\n    have h\u03b4pos : (0 : \u211d) < |\u03c1.1.im - T| :=\\n      lt_of_lt_of_le (one_div_pos.mpr (by positivity :\\n        (0 : \u211d) < 9 * C0hat (Real.sqrt (n : \u211d) + 2))) h\u03b4ge\\n    exact \u27e8h\u03b4pos, h\u03b41, h\u03b4ge\u27e9\\n  -- the kernel bound per \u03c3 (the ii\u2032-4 shift lemma at s := w \u2212 1)\\n  have hnT2 : (n : \u211d) \u2264 T ^ 2 := by\\n    have h1 : (Real.sqrt (n : \u211d)) ^ 2 \u2264 T ^ 2 :=\\n      pow_le_pow_left\u2080 (Real.sqrt_nonneg _) hTn 2\\n    rwa [Real.sq_sqrt (by positivity : (0 : \u211d) \u2264 (n : \u211d))] at h1\\n  have hker : \u2200 \u03c3 : \u211d, \u03c3 \u2208 Set.Icc (-5 / 2 : \u211d) 3 \u2192\\n      \u2016kFun n ((\u03c3 : \u211d) + (T : \u211d) * Complex.I)\u2016 \u2264 Real.exp 4 + 1 := by\\n    intro \u03c3 h\u03c3\\n    obtain \u27e8h\u03c31, h\u03c32\u27e9 := h\u03c3\\n    have hre : (((\u03c3 : \u211d) + (T : \u211d) * Complex.I) - 1).re = \u03c3 - 1 := by simp\\n    have him : (((\u03c3 : \u211d) + (T : \u211d) * Complex.I) - 1).im = T := by simp\\n    have h1 := kFun_shift_norm_le_exp_four_add_one (n := n)\\n      (s := (((\u03c3 : \u211d) + (T : \u211d) * Complex.I) - 1))\\n      (by rw [hre]; linarith [h\u03c32]) (by rw [him]; exact hnT2)\\n    rwa [sub_add_cancel] at h1\\n  -- pointwise Jlag \u2264 (e\u2074+1)\u00b7\u2016logDeriv\u2016\\n  have hptJ : \u2200 \u03c3 : \u211d, \u03c3 \u2208 Set.Icc (-5 / 2 : \u211d) 3 \u2192\\n      \u2016Jlag n ((\u03c3 : \u211d) + (T : \u211d) * Complex.I)\u2016 \u2264\\n        (Real.exp 4 + 1) * \u2016logDeriv riemannZeta ((\u03c3 : \u211d) + (T : \u211d) * Complex.I)\u2016 := by\\n    intro \u03c3 h\u03c3\\n    rw [Jlag_norm_eq]\\n    exact mul_le_mul (hker \u03c3 h\u03c3) le_rfl (norm_nonneg _) (by positivity)\\n  have hJint : \u222b \u03c3 in (-5 / 2 : \u211d)..(3 : \u211d), \u2016Jlag n ((\u03c3 : \u211d) + (T : \u211d) * Complex.I)\u2016 \u2264\\n      (Real.exp 4 + 1) * \u222b \u03c3 in (-5 / 2 : \u211d)..(3 : \u211d),\\n        \u2016logDeriv riemannZeta ((\u03c3 : \u211d) + (T : \u211d) * Complex.I)\u2016 := by\\n    have h := intervalIntegral.integral_mono_on hlen hJintbl hKLintbl\\n      (fun \u03c3 h\u03c3 => hptJ \u03c3 h\u03c3)\\n    rwa [intervalIntegral.integral_const_mul] at h\\n  -- the logDeriv split: \u2016logDeriv\u2016 \u2264 \u2016\u03a3\u2016 + \u2016logDeriv \u2212 \u03a3\u2016\\n  have hsumcont : Continuous fun \u03c3 : \u211d =>\\n      \u2016\u2211 \u03c1 \u2208 nearZerosSub T, (zeroMult \u03c1.1 : \u2102) /\\n          (((\u03c3 : \u211d) + (T : \u211d) * Complex.I) - \u03c1.1)\u2016 +\\n        \u2016logDeriv riemannZeta ((\u03c3 : \u211d) + (T : \u211d) * Complex.I) -\\n          \u2211 \u03c1 \u2208 nearZerosSub T, (zeroMult \u03c1.1 : \u2102) /\\n            (((\u03c3 : \u211d) + (T : \u211d) * Complex.I) - \u03c1.1)\u2016 :=\\n    h\u03a3norm_cont.add hRcont\\n  have hsum_intbl : IntervalIntegrable\\n      (fun \u03c3 : \u211d => \u2016\u2211 \u03c1 \u2208 nearZerosSub T, (zeroMult \u03c1.1 : \u2102) /\\n          (((\u03c3 : \u211d) + (T : \u211d) * Complex.I) - \u03c1.1)\u2016 +\\n        \u2016logDeriv riemannZeta ((\u03c3 : \u211d) + (T : \u211d) * Complex.I) -\\n          \u2211 \u03c1 \u2208 nearZerosSub T, (zeroMult \u03c1.1 : \u2102) /\\n            (((\u03c3 : \u211d) + (T : \u211d) * Complex.I) - \u03c1.1)\u2016)\\n      MeasureTheory.volume (-5 / 2) 3 := hsumcont.intervalIntegrable _ _\\n  have hsplit : \u222b \u03c3 in (-5 / 2 : \u211d)..(3 : \u211d),\\n      \u2016logDeriv riemannZeta ((\u03c3 : \u211d) + (T : \u211d) * Complex.I)\u2016 \u2264\\n      (\u222b \u03c3 in (-5 / 2 : \u211d)..(3 : \u211d), \u2016\u2211 \u03c1 \u2208 nearZerosSub T, (zeroMult \u03c1.1 : \u2102) /\\n          (((\u03c3 : \u211d) + (T : \u211d) * Complex.I) - \u03c1.1)\u2016) +\\n        (\u222b \u03c3 in (-5 / 2 : \u211d)..(3 : \u211d),\\n          \u2016logDeriv riemannZeta ((\u03c3 : \u211d) + (T : \u211d) * Complex.I) -\\n            \u2211 \u03c1 \u2208 nearZerosSub T, (zeroMult \u03c1.1 : \u2102) /\\n              (((\u03c3 : \u211d) + (T : \u211d) * Complex.I) - \u03c1.1)\u2016) := by\\n    have hpt2 : \u2200 \u03c3 : \u211d, \u03c3 \u2208 Set.Icc (-5 / 2 : \u211d) 3 \u2192\\n        \u2016logDeriv riemannZeta ((\u03c3 : \u211d) + (T : \u211d) * Complex.I)\u2016 \u2264\\n          \u2016\u2211 \u03c1 \u2208 nearZerosSub T, (zeroMult \u03c1.1 : \u2102) /\\n              (((\u03c3 : \u211d) + (T : \u211d) * Complex.I) - \u03c1.1)\u2016 +\\n            \u2016logDeriv riemannZeta ((\u03c3 : \u211d) + (T : \u211d) * Complex.I) -\\n              \u2211 \u03c1 \u2208 nearZerosSub T, (zeroMult \u03c1.1 : \u2102) /\\n                (((\u03c3 : \u211d) + (T : \u211d) * Complex.I) - \u03c1.1)\u2016 :=\\n      fun \u03c3 _ => norm_le_insert' _ _\\n    have h := intervalIntegral.integral_mono_on hlen hLDintbl hsum_intbl\\n      (fun \u03c3 h\u03c3 => hpt2 \u03c3 h\u03c3)\\n    rwa [intervalIntegral.integral_add h\u03a3norm_int hRnorm_int] at h\\n  -- the R-part integral (the 5f capstone, constant in \u03c3)\\n  have hR : \u2200 \u03c3 : \u211d, \u03c3 \u2208 Set.Icc (-5 / 2 : \u211d) 3 \u2192\\n      \u2016logDeriv riemannZeta ((\u03c3 : \u211d) + (T : \u211d) * Complex.I) -\\n        \u2211 \u03c1 \u2208 nearZerosSub T, (zeroMult \u03c1.1 : \u2102) /\\n          (((\u03c3 : \u211d) + (T : \u211d) * Complex.I) - \u03c1.1)\u2016 \u2264\\n        10.5 * Real.log (T + 2) + 48 := by\\n    intro \u03c3 h\u03c3\\n    obtain \u27e8h\u03c31, h\u03c32\u27e9 := h\u03c3\\n    have hre : ((\u03c3 : \u211d) + (T : \u211d) * Complex.I).re = \u03c3 := by simp\\n    have him : ((\u03c3 : \u211d) + (T : \u211d) * Complex.I).im = T := by simp\\n    have h := logDeriv_riemannZeta_sub_nearZeros_le_tight (t := T)\\n      (s := ((\u03c3 : \u211d) + (T : \u211d) * Complex.I)) hZC hZC2 hNLZ h10T\\n      (by rw [hre]; exact h\u03c31) (by rw [hre]; exact h\u03c32) him (hzne \u03c3)\\n    rwa [abs_of_nonneg hT0] at h\\n  have hRint : \u222b \u03c3 in (-5 / 2 : \u211d)..(3 : \u211d),\\n      \u2016logDeriv riemannZeta ((\u03c3 : \u211d) + (T : \u211d) * Complex.I) -\\n        \u2211 \u03c1 \u2208 nearZerosSub T, (zeroMult \u03c1.1 : \u2102) /\\n          (((\u03c3 : \u211d) + (T : \u211d) * Complex.I) - \u03c1.1)\u2016 \u2264\\n        (3 - -5 / 2) * (10.5 * Real.log (T + 2) + 48) := by\\n    have h := intervalIntegral.integral_mono_on hlen hRnorm_int intervalIntegrable_const\\n      (fun \u03c3 h\u03c3 => hR \u03c3 h\u03c3)\\n    rwa [intervalIntegral.integral_const, smul_eq_mul] at h\\n  -- the near-sum integral: the per-zero arcsinh-FTC machine + the swap\\n  have hnormid : \u2200 \u03c3 : \u211d, \u2200 \u03c1 : {\u03c1 : \u2102 // IsNontrivialZero \u03c1},\\n      \u2016((\u03c3 : \u211d) + (T : \u211d) * Complex.I) - \u03c1.1\u2016 =\\n        Real.sqrt ((\u03c3 - \u03c1.1.re) ^ 2 + (T - \u03c1.1.im) ^ 2) := by\\n    intro \u03c3 \u03c1\\n    have h1 : Complex.normSq (((\u03c3 : \u211d) + (T : \u211d) * Complex.I) - \u03c1.1) =\\n        (\u03c3 - \u03c1.1.re) ^ 2 + (T - \u03c1.1.im) ^ 2 := by\\n      simp [Complex.normSq_apply]\\n      ring\\n    symm\\n    rw [Real.sqrt_eq_iff_eq_sq (by positivity) (norm_nonneg _),\\n      \u2190 Complex.normSq_eq_norm_sq]\\n    exact h1.symm\\n  have htermcont : \u2200 \u03c1 : {\u03c1 : \u2102 // IsNontrivialZero \u03c1}, \u03c1 \u2208 nearZerosSub T \u2192\\n      Continuous fun \u03c3 : \u211d => (zeroMult \u03c1.1 : \u211d) *\\n        (1 / Real.sqrt ((\u03c3 - \u03c1.1.re) ^ 2 + (T - \u03c1.1.im) ^ 2)) := by\\n    intro \u03c1 h\u03c1\\n    obtain \u27e8h\u03b4pos, -, -\u27e9 := h\u03c1data \u03c1 h\u03c1\\n    have hh : \u2200 \u03c3 : \u211d, (0 : \u211d) < (\u03c3 - \u03c1.1.re) ^ 2 + (T - \u03c1.1.im) ^ 2 := by\\n      intro \u03c3\\n      have h3 : \u03c1.1.im - T \u2260 0 := abs_pos.mp h\u03b4pos\\n      have h4 : T - \u03c1.1.im \u2260 0 := by\\n        intro h5\\n        apply h3\\n        linarith [h5]\\n      exact add_pos_of_nonneg_of_pos (sq_nonneg _) (sq_pos_of_ne_zero h4)\\n    apply Continuous.mul continuous_const\\n    apply Continuous.div continuous_const\\n    \u00b7 exact Real.continuous_sqrt.comp (by fun_prop :\\n        Continuous fun \u03c3 : \u211d => (\u03c3 - \u03c1.1.re) ^ 2 + (T - \u03c1.1.im) ^ 2)\\n    \u00b7 intro \u03c3\\n      exact ne_of_gt (Real.sqrt_pos.mpr (hh \u03c3))\\n  have hpt : \u2200 \u03c3 : \u211d, \u03c3 \u2208 Set.Icc (-5 / 2"
}
#2282 assistant 2026-08-26 18:51:42.387 UTC · 32647→11953 tok open ⤢
thinking (644 chars)
The Edit tool call got rejected due to a formatting issue with my parameters (I accidentally used `_raw`). Let me redo the edit properly. Also, my draft was cut off mid-way through hpt — I need to complete the rest of the mid-horizontal theorem (hpt, hswap, hper, hsum, hΣbound, the final calc) and the neg theorem. Let me write the complete §4 edit now (finishing the truncated draft). The remaining parts after htermcont: - hpt: pointwise ‖Σ‖ ≤ Σ m·(1/√) - hswap: ∫ of the RHS sum = Σ m·∫ - hper: per-ρ integral ≤ 2·log(148.5·C0hat) - hsum: Σ m·∫ ≤ B0·Z2hat - hΣbound: ∫‖Σ‖ ≤ B0·Z2hat - the final calc - the neg theorem Let me write it all.
The edit was malformed (truncated draft). Writing the complete §4 properly now.
tool: Edit
input
{
  "replace_all": false,
  "file_path": "D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiSideBounds.lean",
  "old_string": "    _ = 2 * T * (Real.exp (Real.sqrt (n : \u211d) / 2 + 1 / 3) + 1) *\n          ((0.57 : \u211d) * (2 : \u211d) ^ (-(2 * Real.sqrt (n : \u211d) - 2))) := by ring\n    _ \u2264 Real.exp (1 / 3 : \u211d) * IIMaj (n : \u211d) := hfinal\n\nend NguyenRH1",
  "new_string": "    _ = 2 * T * (Real.exp (Real.sqrt (n : \u211d) / 2 + 1 / 3) + 1) *\n          ((0.57 : \u211d) * (2 : \u211d) ^ (-(2 * Real.sqrt (n : \u211d) - 2))) := by ring\n    _ \u2264 Real.exp (1 / 3 : \u211d) * IIMaj (n : \u211d) := hfinal\n\n/-! ## 4. 6-c-iv: the mid-horizontal segment machine [\u22125/2, 3] -/\n\n/-- **The arcsinh FTC machine** (6-c-iv core): the direct antiderivative\nF(\u03c3) = log((\u03c3\u2212\u03b2) + \u221a((\u03c3\u2212\u03b2)\u00b2+\u03b4\u00b2)) of 1/\u221a((\u03c3\u2212\u03b2)\u00b2+\u03b4\u00b2) (Real.arcsinh does\nnot exist here): g(\u03c3) := (\u03c3\u2212\u03b2)+\u221a((\u03c3\u2212\u03b2)\u00b2+\u03b4\u00b2) > 0 (\u221ah > |\u03c3\u2212\u03b2|), and\ng\u2032 = 1 + (\u03c3\u2212\u03b2)/\u221ah = g/\u221ah, so (log g)\u2032 = 1/\u221ah exactly. -/\ntheorem integral_one_div_sqrt_sq_add_sq (\u03b2 \u03b4 : \u211d) (h\u03b4 : 0 < \u03b4) (a b : \u211d) :\n    \u222b \u03c3 in a..b, 1 / Real.sqrt ((\u03c3 - \u03b2) ^ 2 + \u03b4 ^ 2) =\n      Real.log ((b - \u03b2) + Real.sqrt ((b - \u03b2) ^ 2 + \u03b4 ^ 2)) -\n        Real.log ((a - \u03b2) + Real.sqrt ((a - \u03b2) ^ 2 + \u03b4 ^ 2)) := by\n  have hh : \u2200 \u03c3 : \u211d, (0 : \u211d) < (\u03c3 - \u03b2) ^ 2 + \u03b4 ^ 2 := fun \u03c3 =>\n    add_pos_of_nonneg_of_pos (sq_nonneg _) (pow_pos h\u03b4 2)\n  have hgpos : \u2200 \u03c3 : \u211d, (0 : \u211d) < (\u03c3 - \u03b2) + Real.sqrt ((\u03c3 - \u03b2) ^ 2 + \u03b4 ^ 2) := by\n    intro \u03c3\n    have h1 : |(\u03c3 - \u03b2)| < Real.sqrt ((\u03c3 - \u03b2) ^ 2 + \u03b4 ^ 2) := by\n      have h2 : (\u03c3 - \u03b2) ^ 2 < (\u03c3 - \u03b2) ^ 2 + \u03b4 ^ 2 := by nlinarith [h\u03b4]\n      have h3 := Real.sqrt_lt_sqrt (sq_nonneg _) h2\n      rwa [Real.sqrt_sq_eq_abs] at h3\n    have h4 : -(\u03c3 - \u03b2) \u2264 |(\u03c3 - \u03b2)| := neg_abs_le (\u03c3 - \u03b2)\n    linarith [h1, h4]\n  have hderiv : \u2200 \u03c3 : \u211d, HasDerivAt\n      (fun \u03c3 => Real.log ((\u03c3 - \u03b2) + Real.sqrt ((\u03c3 - \u03b2) ^ 2 + \u03b4 ^ 2)))\n      (1 / Real.sqrt ((\u03c3 - \u03b2) ^ 2 + \u03b4 ^ 2)) \u03c3 := by\n    intro \u03c3\n    have hh\u03c3 := hh \u03c3\n    have hsqrt0 : Real.sqrt ((\u03c3 - \u03b2) ^ 2 + \u03b4 ^ 2) \u2260 0 := ne_of_gt (Real.sqrt_pos.mpr hh\u03c3)\n    have hg0 : ((\u03c3 - \u03b2) + Real.sqrt ((\u03c3 - \u03b2) ^ 2 + \u03b4 ^ 2)) \u2260 0 := (hgpos \u03c3).ne'\n    have h1 : HasDerivAt (fun \u03c3 => \u03c3 - \u03b2) 1 \u03c3 := hasDerivAt_id.sub_const \u03b2\n    have h2 : HasDerivAt (fun \u03c3 => (\u03c3 - \u03b2) ^ 2 + \u03b4 ^ 2) (2 * (\u03c3 - \u03b2)) \u03c3 := by\n      have h2a := (hasDerivAt_id.sub_const \u03b2).pow 2\n      have h2b := h2a.add_const (\u03b4 ^ 2)\n      convert h2b using 1\n      ring\n    have h4 : HasDerivAt (fun \u03c3 => Real.sqrt ((\u03c3 - \u03b2) ^ 2 + \u03b4 ^ 2))\n        (2 * (\u03c3 - \u03b2) / (2 * Real.sqrt ((\u03c3 - \u03b2) ^ 2 + \u03b4 ^ 2))) \u03c3 :=\n      h2.sqrt hh\u03c3.ne'\n    have h5 : HasDerivAt (fun \u03c3 => (\u03c3 - \u03b2) + Real.sqrt ((\u03c3 - \u03b2) ^ 2 + \u03b4 ^ 2))\n        (1 + 2 * (\u03c3 - \u03b2) / (2 * Real.sqrt ((\u03c3 - \u03b2) ^ 2 + \u03b4 ^ 2))) \u03c3 :=\n      h1.add h4\n    have h6 : HasDerivAt (fun \u03c3 => (\u03c3 - \u03b2) + Real.sqrt ((\u03c3 - \u03b2) ^ 2 + \u03b4 ^ 2))\n        (((\u03c3 - \u03b2) + Real.sqrt ((\u03c3 - \u03b2) ^ 2 + \u03b4 ^ 2)) /\n          Real.sqrt ((\u03c3 - \u03b2) ^ 2 + \u03b4 ^ 2)) \u03c3 := by\n      convert h5 using 1\n      field_simp [hsqrt0]\n      ring\n    have h7 := h6.log hg0\n    convert h7 using 1\n    field_simp [hsqrt0, hg0]\n    ring\n  exact intervalIntegral.integral_eq_sub_of_hasDerivAt (fun \u03c3 _ => hderiv \u03c3)\n    (by\n      apply Continuous.intervalIntegrable\n      apply Continuous.div continuous_const\n      \u00b7 exact Real.continuous_sqrt.comp (by fun_prop :\n          Continuous fun \u03c3 : \u211d => (\u03c3 - \u03b2) ^ 2 + \u03b4 ^ 2)\n      \u00b7 intro \u03c3\n        exact ne_of_gt (Real.sqrt_pos.mpr (hh \u03c3)))\n\n/-- **The right endpoint estimate** F(3) \u2264 log(6+\u03b4): \u221a((3\u2212\u03b2)\u00b2+\u03b4\u00b2) \u2264\n(3\u2212\u03b2)+\u03b4 for 3\u2212\u03b2 > 0, so the argument is \u2264 2(3\u2212\u03b2)+\u03b4 \u2264 6+\u03b4 (0 \u2264 \u03b2). -/\ntheorem arcsinh_right_endpoint_le {\u03b2 \u03b4 : \u211d} (h\u03b20 : 0 < \u03b2) (h\u03b21 : \u03b2 < 1)\n    (h\u03b4 : 0 < \u03b4) :\n    Real.log ((3 - \u03b2) + Real.sqrt ((3 - \u03b2) ^ 2 + \u03b4 ^ 2)) \u2264 Real.log (6 + \u03b4) := by\n  have hx : (0 : \u211d) < 3 - \u03b2 := by linarith\n  have hsqrt_le : Real.sqrt ((3 - \u03b2) ^ 2 + \u03b4 ^ 2) \u2264 (3 - \u03b2) + \u03b4 := by\n    rw [Real.sqrt_le_iff]\n    exact \u27e8by positivity, by nlinarith [h\u03b4, hx]\u27e9\n  have h1 : (0 : \u211d) < (3 - \u03b2) + Real.sqrt ((3 - \u03b2) ^ 2 + \u03b4 ^ 2) := by\n    have h2 := Real.sqrt_nonneg ((3 - \u03b2) ^ 2 + \u03b4 ^ 2)\n    nlinarith\n  have h3 : (3 - \u03b2) + Real.sqrt ((3 - \u03b2) ^ 2 + \u03b4 ^ 2) \u2264 6 + \u03b4 := by\n    nlinarith [hsqrt_le, h\u03b20]\n  exact Real.log_le_log h1 h3\n\n/-- **The left endpoint estimate** \u2212F(\u22125/2) \u2264 log(7+\u03b4) + 2\u00b7log(1/\u03b4):\nwith u := 5/2+\u03b2 \u2208 (5/2, 7/2), the argument \u2212u+\u221a(u\u00b2+\u03b4\u00b2) rationalizes to\n\u03b4\u00b2/(u+\u221a(u\u00b2+\u03b4\u00b2)) \u2265 \u03b4\u00b2/(7+\u03b4) > 0, so F(\u22125/2) \u2265 log(\u03b4\u00b2/(7+\u03b4)) =\n2\u00b7log \u03b4 \u2212 log(7+\u03b4). -/\ntheorem arcsinh_left_endpoint_le {\u03b2 \u03b4 : \u211d} (h\u03b20 : 0 < \u03b2) (h\u03b21 : \u03b2 < 1)\n    (h\u03b4 : 0 < \u03b4) :\n    -Real.log ((-5 / 2 - \u03b2) + Real.sqrt ((-5 / 2 - \u03b2) ^ 2 + \u03b4 ^ 2)) \u2264\n      Real.log (7 + \u03b4) + 2 * Real.log (1 / \u03b4) := by\n  have hu : (0 : \u211d) < 5 / 2 + \u03b2 := by linarith\n  have hsq : (-5 / 2 - \u03b2) ^ 2 = (5 / 2 + \u03b2) ^ 2 := by ring\n  have hneg : (-5 / 2 : \u211d) - \u03b2 = -(5 / 2 + \u03b2) := by ring\n  rw [hsq, hneg]\n  set u := 5 / 2 + \u03b2 with hu_def\n  have hu2 : u < 7 / 2 := by rw [hu_def]; linarith\n  have hsqrt_pos : (0 : \u211d) < Real.sqrt (u ^ 2 + \u03b4 ^ 2) :=\n    Real.sqrt_pos.mpr (add_pos_of_nonneg_of_pos (sq_nonneg _) (pow_pos h\u03b4 2))\n  have hg_pos : (0 : \u211d) < -u + Real.sqrt (u ^ 2 + \u03b4 ^ 2) := by\n    have h1 : u < Real.sqrt (u ^ 2 + \u03b4 ^ 2) := by\n      have h2 : u ^ 2 < u ^ 2 + \u03b4 ^ 2 := by nlinarith [h\u03b4]\n      have h3 := Real.sqrt_lt_sqrt (sq_nonneg _) h2\n      rwa [Real.sqrt_sq_eq_abs, abs_of_pos hu] at h3\n    linarith\n  -- the rationalization: \u2212u + \u221a = \u03b4\u00b2/(u + \u221a)\n  have hrid : -u + Real.sqrt (u ^ 2 + \u03b4 ^ 2) =\n      \u03b4 ^ 2 / (u + Real.sqrt (u ^ 2 + \u03b4 ^ 2)) := by\n    rw [div_eq_iff (ne_of_gt (by linarith [hsqrt_pos] :\n      (0 : \u211d) < u + Real.sqrt (u ^ 2 + \u03b4 ^ 2)))]\n    have hs : (Real.sqrt (u ^ 2 + \u03b4 ^ 2)) ^ 2 = u ^ 2 + \u03b4 ^ 2 :=\n      Real.sq_sqrt (by positivity)\n    nlinarith [hs]\n  -- the denominator \u2264 7 + \u03b4\n  have hden : u + Real.sqrt (u ^ 2 + \u03b4 ^ 2) \u2264 7 + \u03b4 := by\n    have h1 : Real.sqrt (u ^ 2 + \u03b4 ^ 2) \u2264 u + \u03b4 := by\n      rw [Real.sqrt_le_iff]\n      exact \u27e8by linarith [h\u03b4, hu.le], by nlinarith [h\u03b4, hu]\u27e9\n    nlinarith [h1, hu2]\n  -- \u03b4\u00b2/(7+\u03b4) \u2264 \u03b4\u00b2/(u+\u221a) = \u2212u+\u221a\n  have hge : \u03b4 ^ 2 / (7 + \u03b4) \u2264 -u + Real.sqrt (u ^ 2 + \u03b4 ^ 2) := by\n    rw [hrid]\n    have h2 : (0 : \u211d) < u + Real.sqrt (u ^ 2 + \u03b4 ^ 2) := by linarith [hsqrt_pos]\n    exact div_le_div_of_nonneg_left (sq_nonneg _) h2 hden\n  -- log mono + algebra: F(\u22125/2) \u2265 log(\u03b4\u00b2/(7+\u03b4)) = 2\u00b7log \u03b4 \u2212 log(7+\u03b4)\n  have hlog : Real.log (\u03b4 ^ 2 / (7 + \u03b4)) \u2264 Real.log (-u + Real.sqrt (u ^ 2 + \u03b4 ^ 2)) :=\n    Real.log_le_log (by positivity) hge\n  have hlogid : Real.log (\u03b4 ^ 2 / (7 + \u03b4)) = 2 * Real.log \u03b4 - Real.log (7 + \u03b4) := by\n    rw [Real.log_div (pow_pos h\u03b4 2).ne' (by linarith [h\u03b4] : (7 : \u211d) + \u03b4 \u2260 0),\n      Real.log_pow]\n    ring\n  have hlog1 : Real.log (1 / \u03b4) = -Real.log \u03b4 := by\n    rw [Real.log_div one_ne_zero (ne_of_gt h\u03b4), Real.log_one, zero_sub]\n  rw [hlogid] at hlog\n  linarith [hlog, hlog1]\n\n/-- **The per-zero segment integral bound**: \u222b_{\u22125/2}^{3} d\u03c3/\u221a((\u03c3\u2212\u03b2)\u00b2+\u03b4\u00b2)\n\u2264 2\u00b7log(16.5/\u03b4) for 0 < \u03b2 < 1, 0 < \u03b4 \u2264 1 \u2014 the two endpoint estimates +\nlog((6+\u03b4)(7+\u03b4)) \u2264 log(16.5\u00b2) since (6+\u03b4)(7+\u03b4) \u2264 56 \u2264 272.25. -/\ntheorem per_zero_integral_le_two_log {\u03b2 \u03b4 : \u211d} (h\u03b20 : 0 < \u03b2) (h\u03b21 : \u03b2 < 1)\n    (h\u03b4 : 0 < \u03b4) (h\u03b41 : \u03b4 \u2264 1) :\n    \u222b \u03c3 in (-5 / 2 : \u211d)..(3 : \u211d), 1 / Real.sqrt ((\u03c3 - \u03b2) ^ 2 + \u03b4 ^ 2) \u2264\n      2 * Real.log (16.5 / \u03b4) := by\n  have hftc := integral_one_div_sqrt_sq_add_sq \u03b2 \u03b4 h\u03b4 (-5 / 2) 3\n  have hr := arcsinh_right_endpoint_le h\u03b20 h\u03b21 h\u03b4\n  have hl := arcsinh_left_endpoint_le h\u03b20 h\u03b21 h\u03b4\n  have h1 : \u222b \u03c3 in (-5 / 2 : \u211d)..(3 : \u211d), 1 / Real.sqrt ((\u03c3 - \u03b2) ^ 2 + \u03b4 ^ 2) \u2264\n      Real.log (6 + \u03b4) + Real.log (7 + \u03b4) + 2 * Real.log (1 / \u03b4) := by\n    rw [hftc]\n    linarith [hr, hl]\n  -- the log-mono simplification: log(6+\u03b4) + log(7+\u03b4) \u2264 2\u00b7log 16.5\n  have h2 : Real.log (6 + \u03b4) + Real.log (7 + \u03b4) \u2264 2 * Real.log 16.5 := by\n    have h3 : Real.log (6 + \u03b4) + Real.log (7 + \u03b4) = Real.log ((6 + \u03b4) * (7 + \u03b4)) :=\n      (Real.log_mul (by linarith [h\u03b4] : (6 : \u211d) + \u03b4 \u2260 0)\n        (by linarith [h\u03b4] : (7 : \u211d) + \u03b4 \u2260 0)).symm\n    have h4 : Real.log ((6 + \u03b4) * (7 + \u03b4)) \u2264 Real.log (16.5 * 16.5) :=\n      Real.log_le_log (by positivity) (by nlinarith [h\u03b4, h\u03b41])\n    have h5 : Real.log (16.5 * 16.5) = 2 * Real.log 16.5 := by\n      rw [show (16.5 : \u211d) * 16.5 = 16.5 ^ 2 by ring, Real.log_pow]\n      ring\n    linarith [h3, h4, h5]\n  have h6 : 2 * Real.log (16.5 / \u03b4) = 2 * Real.log 16.5 + 2 * Real.log (1 / \u03b4) := by\n    have h7 : Real.log (1 / \u03b4) = -Real.log \u03b4 := by\n      rw [Real.log_div one_ne_zero (ne_of_gt h\u03b4), Real.log_one, zero_sub]\n    rw [Real.log_div (by norm_num) (ne_of_gt h\u03b4), h7]\n    ring\n  rw [h6]\n  linarith [h1, h2]\n\n/-- **6-c-iv (mid-horizontal [\u22125/2, 3] at height +T)**: the segment bound\n\u2016H\u2016 \u2264 (e\u2074+1)\u00b7(2\u00b7Z\u2082(T)\u00b7log(148.5\u00b7\u0108\u2080(\u221an+2)) + 5.5\u00b7(10.5\u00b7log(T+2)+48)).\nKernel e\u2074+1 (the ii\u2032-4 shift lemma at s := w\u22121); logDeriv split by\nnorm_le_insert' into the near sum (per-zero arcsinh-FTC machine +\nintegral_finsetSum + the two-unit count \u2264 Z\u2082(T)) and the tight R-bound\n(capstone, constant in \u03c3). -/\ntheorem hIntegral_Jlag_mid_norm_le (hZC : ZeroCountProp) (hZC2 : ZeroCountProp2)\n    (hNLZ : NoLowZeros) {n : \u2115} (hn : 100000 \u2264 n) {T : \u211d}\n    (hTwide : \u2200 \u03c1 : \u2102, IsNontrivialZero \u03c1 \u2192 Real.sqrt (n : \u211d) - 1 < \u03c1.im \u2192\n      \u03c1.im < Real.sqrt (n : \u211d) + 2 \u2192\n        1 / (9 * C0hat (Real.sqrt (n : \u211d) + 2)) \u2264 |\u03c1.im - T|)\n    (hTn : Real.sqrt (n : \u211d) \u2264 T) (hTn1 : T \u2264 Real.sqrt (n : \u211d) + 1) :\n    \u2016HIntegral (Jlag n) (-5 / 2) 3 T\u2016 \u2264\n      (Real.exp 4 + 1) *\n        (2 * Z2hat T * Real.log (148.5 * C0hat (Real.sqrt (n : \u211d) + 2)) +\n          (3 - -5 / 2) * (10.5 * Real.log (T + 2) + 48)) := by\n  classical\n  have hnR : (100000 : \u211d) \u2264 (n : \u211d) := by exact_mod_cast hn\n  have h10T : (10 : \u211d) \u2264 T := by\n    have h1 : (10 : \u211d) \u2264 Real.sqrt (n : \u211d) := by\n      rw [Real.le_sqrt (by norm_num) (by positivity)]\n      nlinarith [hnR]\n    linarith\n  have hT0 : (0 : \u211d) \u2264 T := by linarith\n  have hn1 : 1 \u2264 n := by omega\n  have hlen : (-5 / 2 : \u211d) \u2264 3 := by norm_num\n  have hC0 : (0 : \u211d) < C0hat (Real.sqrt (n : \u211d) + 2) := by\n    have h2 := two_le_C0hat (by nlinarith [Real.sqrt_nonneg (n : \u211d)] :\n      (1 : \u211d) \u2264 Real.sqrt (n : \u211d) + 2)\n    linarith\n  have hB0nn : (0 : \u211d) \u2264 2 * Real.log (148.5 * C0hat (Real.sqrt (n : \u211d) + 2)) := by\n    have h1 : (1 : \u211d) \u2264 148.5 * C0hat (Real.sqrt (n : \u211d) + 2) := by nlinarith [hC0]\n    have h3 := Real.log_nonneg h1\n    linarith\n  -- zero-freeness on the horizontal + the path continuity machines\n  have hzne : \u2200 \u03c3 : \u211d, riemannZeta ((\u03c3 : \u211d) + (T : \u211d) * Complex.I) \u2260 0 :=\n    fun \u03c3 => zeta_ne_zero_horizontal hTwide hn1 hTn hTn1\n  have hne1 : \u2200 \u03c3 : \u211d, ((\u03c3 : \u211d) + (T : \u211d) * Complex.I) \u2260 1 := by\n    intro \u03c3 h\n    have him := congrArg Complex.im h\n    simp at him\n    linarith [h10T]\n  have hpath : Continuous fun u : \u211d => ((u : \u211d) + (T : \u211d) * Complex.I) := by\n    fun_prop\n  have hLDc : Continuous fun \u03c3 : \u211d =>\n      logDeriv riemannZeta ((\u03c3 : \u211d) + (T : \u211d) * Complex.I) := by\n    rw [continuous_iff_continuousAt]\n    intro \u03c3\n    have hf : ContinuousAt (fun u : \u211d => ((u : \u211d) + (T : \u211d) * Complex.I)) \u03c3 :=\n      hpath.continuousAt\n    exact ContinuousAt.comp (g := logDeriv riemannZeta)\n      (f := fun u : \u211d => ((u : \u211d) + (T : \u211d) * Complex.I)) (x := \u03c3)\n      (continuousAt_logDeriv_zeta (hne1 \u03c3) (hzne \u03c3)) hf\n  have hJcont : \u2200 \u03c3 : \u211d, ContinuousAt (Jlag n) ((\u03c3 : \u211d) + (T : \u211d) * Complex.I) := by\n    intro \u03c3\n    have hA := (kFun_differentiableAt (n := n) (hne1 \u03c3)).continuousAt\n    have hB := continuousAt_logDeriv_zeta (hne1 \u03c3) (hzne \u03c3)\n    show ContinuousAt (fun s => kFun n s * (-logDeriv riemannZeta s)) _\n    exact hA.mul hB.neg\n  have hcontJ : Continuous fun \u03c3 : \u211d =>\n      Jlag n ((\u03c3 : \u211d) + (T : \u211d) * Complex.I) := by\n    rw [continuous_iff_continuousAt]\n    intro \u03c3\n    have hf : ContinuousAt (fun u : \u211d => ((u : \u211d) + (T : \u211d) * Complex.I)) \u03c3 :=\n      hpath.continuousAt\n    exact ContinuousAt.comp (g := Jlag n)\n      (f := fun u : \u211d => ((u : \u211d) + (T : \u211d) * Complex.I)) (x := \u03c3) (hJcont \u03c3) hf\n  have hcontG : Continuous fun \u03c3 : \u211d =>\n      \u2016Jlag n ((\u03c3 : \u211d) + (T : \u211d) * Complex.I)\u2016 := continuous_norm.comp hcontJ\n  have hJintbl : IntervalIntegrable\n      (fun \u03c3 : \u211d => \u2016Jlag n ((\u03c3 : \u211d) + (T : \u211d) * Complex.I)\u2016)\n      MeasureTheory.volume (-5 / 2) 3 := hcontG.intervalIntegrable _ _\n  have hLDcont : Continuous fun \u03c3 : \u211d =>\n      \u2016logDeriv riemannZeta ((\u03c3 : \u211d) + (T : \u211d) * Complex.I)\u2016 :=\n    continuous_norm.comp hLDc\n  have hLDintbl : IntervalIntegrable\n      (fun \u03c3 : \u211d => \u2016logDeriv riemannZeta ((\u03c3 : \u211d) + (T : \u211d) * Complex.I)\u2016)\n      MeasureTheory.volume (-5 / 2) 3 := hLDcont.intervalIntegrable _ _\n  have hKLcont : Continuous fun \u03c3 : \u211d =>\n      (Real.exp 4 + 1) * \u2016logDeriv riemannZeta ((\u03c3 : \u211d) + (T : \u211d) * Complex.I)\u2016 :=\n    continuous_const.mul hLDcont\n  have hKLintbl : IntervalIntegrable\n      (fun \u03c3 : \u211d => (Real.exp 4 + 1) *\n        \u2016logDeriv riemannZeta ((\u03c3 : \u211d) + (T : \u211d) * Complex.I)\u2016)\n      MeasureTheory.volume (-5 / 2) 3 := hKLcont.intervalIntegrable _ _\n  -- the near-sum denominator never vanishes on the path (else s = \u03c1 is a\n  -- zero, contra zeta_ne_zero_horizontal)\n  have hden : \u2200 \u03c1 : {\u03c1 : \u2102 // IsNontrivialZero \u03c1}, \u03c1 \u2208 nearZerosSub T \u2192\n      \u2200 \u03c3 : \u211d, ((\u03c3 : \u211d) + (T : \u211d) * Complex.I) - \u03c1.1 \u2260 0 := by\n    intro \u03c1 _ \u03c3 h0\n    have hs : ((\u03c3 : \u211d) + (T : \u211d) * Complex.I) = \u03c1.1 := sub_eq_zero.mp h0\n    have hz := hzne \u03c3\n    rw [hs] at hz\n    exact hz \u03c1.2.1\n  have h\u03a3cont : Continuous fun \u03c3 : \u211d =>\n      \u2211 \u03c1 \u2208 nearZerosSub T, (zeroMult \u03c1.1 : \u2102) /\n        (((\u03c3 : \u211d) + (T : \u211d) * Complex.I) - \u03c1.1) := by\n    apply continuous_finset_sum\n    intro \u03c1 h\u03c1\n    exact Continuous.div continuous_const (hpath.sub continuous_const) (hden \u03c1 h\u03c1)\n  have h\u03a3norm_cont : Continuous fun \u03c3 : \u211d =>\n      \u2016\u2211 \u03c1 \u2208 nearZerosSub T, (zeroMult \u03c1.1 : \u2102) /\n        (((\u03c3 : \u211d) + (T : \u211d) * Complex.I) - \u03c1.1)\u2016 :=\n    continuous_norm.comp h\u03a3cont\n  have h\u03a3norm_int : IntervalIntegrable\n      (fun \u03c3 : \u211d => \u2016\u2211 \u03c1 \u2208 nearZerosSub T, (zeroMult \u03c1.1 : \u2102) /\n        (((\u03c3 : \u211d) + (T : \u211d) * Complex.I) - \u03c1.1)\u2016)\n      MeasureTheory.volume (-5 / 2) 3 := h\u03a3norm_cont.intervalIntegrable _ _\n  have hRcont : Continuous fun \u03c3 : \u211d =>\n      \u2016logDeriv riemannZeta ((\u03c3 : \u211d) + (T : \u211d) * Complex.I) -\n        \u2211 \u03c1 \u2208 nearZerosSub T, (zeroMult \u03c1.1 : \u2102) /\n          (((\u03c3 : \u211d) + (T : \u211d) * Complex.I) - \u03c1.1)\u2016 :=\n    continuous_norm.comp (hLDc.sub h\u03a3cont)\n  have hRnorm_int : IntervalIntegrable\n      (fun \u03c3 : \u211d => \u2016logDeriv riemannZeta ((\u03c3 : \u211d) + (T : \u211d) * Complex.I) -\n        \u2211 \u03c1 \u2208 nearZerosSub T, (zeroMult \u03c1.1 : \u2102) /\n          (((\u03c3 : \u211d) + (T : \u211d) * Complex.I) - \u03c1.1)\u2016)\n      MeasureTheory.volume (-5 / 2) 3 := hRcont.intervalIntegrable _ _\n  -- the per-\u03c1 data (\u03b4_\u03c1 := |\u03b3\u2212T| \u2208 [c_clear, 1))\n  have h\u03c1data : \u2200 \u03c1 : {\u03c1 : \u2102 // IsNontrivialZero \u03c1}, \u03c1 \u2208 nearZerosSub T \u2192\n      (0 : \u211d) < |\u03c1.1.im - T| \u2227 |\u03c1.1.im - T| < 1 \u2227\n        (1 / (9 * C0hat (Real.sqrt (n : \u211d) + 2))) \u2264 |\u03c1.1.im - T| := by\n    intro \u03c1 h\u03c1\n    have h\u03b41 := mem_nearZerosSub.mp h\u03c1\n    obtain \u27e8hlo, hhi\u27e9 := abs_lt.mp h\u03b41\n    have h\u03b31 : Real.sqrt (n : \u211d) - 1 < \u03c1.1.im := by nlinarith [hTn]\n    have h\u03b32 : \u03c1.1.im < Real.sqrt (n : \u211d) + 2 := by nlinarith [hTn1]\n    have h\u03b4ge := hTwide \u03c1.1 \u03c1.2 h\u03b31 h\u03b32\n    have h\u03b4pos : (0 : \u211d) < |\u03c1.1.im - T| :=\n      lt_of_lt_of_le (one_div_pos.mpr (by positivity :\n        (0 : \u211d) < 9 * C0hat (Real.sqrt (n : \u211d) + 2))) h\u03b4ge\n    exact \u27e8h\u03b4pos, h\u03b41, h\u03b4ge\u27e9\n  -- the kernel bound per \u03c3 (the ii\u2032-4 shift lemma at s := w \u2212 1)\n  have hnT2 : (n : \u211d) \u2264 T ^ 2 := by\n    have h1 : (Real.sqrt (n : \u211d)) ^ 2 \u2264 T ^ 2 :=\n      pow_le_pow_left\u2080 (Real.sqrt_nonneg _) hTn 2\n    rwa [Real.sq_sqrt (by positivity : (0 : \u211d) \u2264 (n : \u211d))] at h1\n  have hker : \u2200 \u03c3 : \u211d, \u03c3 \u2208 Set.Icc (-5 / 2 : \u211d) 3 \u2192\n      \u2016kFun n ((\u03c3 : \u211d) + (T : \u211d) * Complex.I)\u2016 \u2264 Real.exp 4 + 1 := by\n    intro \u03c3 h\u03c3\n    obtain \u27e8h\u03c31, h\u03c32\u27e9 := h\u03c3\n    have hre : (((\u03c3 : \u211d) + (T : \u211d) * Complex.I) - 1).re = \u03c3 - 1 := by simp\n    have him : (((\u03c3 : \u211d) + (T : \u211d) * Complex.I) - 1).im = T := by simp\n    have h1 := kFun_shift_norm_le_exp_four_add_one (n := n)\n      (s := (((\u03c3 : \u211d) + (T : \u211d) * Complex.I) - 1))\n      (by rw [hre]; linarith [h\u03c32]) (by rw [him]; exact hnT2)\n    rwa [sub_add_cancel] at h1\n  -- pointwise Jlag \u2264 (e\u2074+1)\u00b7\u2016logDeriv\u2016\n  have hptJ : \u2200 \u03c3 : \u211d, \u03c3 \u2208 Set.Icc (-5 / 2 : \u211d) 3 \u2192\n      \u2016Jlag n ((\u03c3 : \u211d) + (T : \u211d) * Complex.I)\u2016 \u2264\n        (Real.exp 4 + 1) * \u2016logDeriv riemannZeta ((\u03c3 : \u211d) + (T : \u211d) * Complex.I)\u2016 := by\n    intro \u03c3 h\u03c3\n    rw [Jlag_norm_eq]\n    exact mul_le_mul (hker \u03c3 h\u03c3) le_rfl (norm_nonneg _) (by positivity)\n  have hJint : \u222b \u03c3 in (-5 / 2 : \u211d)..(3 : \u211d), \u2016Jlag n ((\u03c3 : \u211d) + (T : \u211d) * Complex.I)\u2016 \u2264\n      (Real.exp 4 + 1) * \u222b \u03c3 in (-5 / 2 : \u211d)..(3 : \u211d),\n        \u2016logDeriv riemannZeta ((\u03c3 : \u211d) + (T : \u211d) * Complex.I)\u2016 := by\n    have h := intervalIntegral.integral_mono_on hlen hJintbl hKLintbl\n      (fun \u03c3 h\u03c3 => hptJ \u03c3 h\u03c3)\n    rwa [intervalIntegral.integral_const_mul] at h\n  -- the logDeriv split: \u2016logDeriv\u2016 \u2264 \u2016\u03a3\u2016 + \u2016logDeriv \u2212 \u03a3\u2016\n  have hsumcont : Continuous fun \u03c3 : \u211d =>\n      \u2016\u2211 \u03c1 \u2208 nearZerosSub T, (zeroMult \u03c1.1 : \u2102) /\n          (((\u03c3 : \u211d) + (T : \u211d) * Complex.I) - \u03c1.1)\u2016 +\n        \u2016logDeriv riemannZeta ((\u03c3 : \u211d) + (T : \u211d) * Complex.I) -\n          \u2211 \u03c1 \u2208 nearZerosSub T, (zeroMult \u03c1.1 : \u2102) /\n            (((\u03c3 : \u211d) + (T : \u211d) * Complex.I) - \u03c1.1)\u2016 :=\n    h\u03a3norm_cont.add hRcont\n  have hsum_intbl : IntervalIntegrable\n      (fun \u03c3 : \u211d => \u2016\u2211 \u03c1 \u2208 nearZerosSub T, (zeroMult \u03c1.1 : \u2102) /\n          (((\u03c3 : \u211d) + (T : \u211d) * Complex.I) - \u03c1.1)\u2016 +\n        \u2016logDeriv riemannZeta ((\u03c3 : \u211d) + (T : \u211d) * Complex.I) -\n          \u2211 \u03c1 \u2208 nearZerosSub T, (zeroMult \u03c1.1 : \u2102) /\n            (((\u03c3 : \u211d) + (T : \u211d) * Complex.I) - \u03c1.1)\u2016)\n      MeasureTheory.volume (-5 / 2) 3 := hsumcont.intervalIntegrable _ _\n  have hsplit : \u222b \u03c3 in (-5 / 2 : \u211d)..(3 : \u211d),\n      \u2016logDeriv riemannZeta ((\u03c3 : \u211d) + (T : \u211d) * Complex.I)\u2016 \u2264\n      (\u222b \u03c3 in (-5 / 2 : \u211d)..(3 : \u211d), \u2016\u2211 \u03c1 \u2208 nearZerosSub T, (zeroMult \u03c1.1 : \u2102) /\n          (((\u03c3 : \u211d) + (T : \u211d) * Complex.I) - \u03c1.1)\u2016) +\n        (\u222b \u03c3 in (-5 / 2 : \u211d)..(3 : \u211d),\n          \u2016logDeriv riemannZeta ((\u03c3 : \u211d) + (T : \u211d) * Complex.I) -\n            \u2211 \u03c1 \u2208 nearZerosSub T, (zeroMult \u03c1.1 : \u2102) /\n              (((\u03c3 : \u211d) + (T : \u211d) * Complex.I) - \u03c1.1)\u2016) := by\n    have hpt2 : \u2200 \u03c3 : \u211d, \u03c3 \u2208 Set.Icc (-5 / 2 : \u211d) 3 \u2192\n        \u2016logDeriv riemannZeta ((\u03c3 : \u211d) + (T : \u211d) * Complex.I)\u2016 \u2264\n          \u2016\u2211 \u03c1 \u2208 nearZerosSub T, (zeroMult \u03c1.1 : \u2102) /\n              (((\u03c3 : \u211d) + (T : \u211d) * Complex.I) - \u03c1.1)\u2016 +\n            \u2016logDeriv riemannZeta ((\u03c3 : \u211d) + (T : \u211d) * Complex.I) -\n              \u2211 \u03c1 \u2208 nearZerosSub T, (zeroMult \u03c1.1 : \u2102) /\n                (((\u03c3 : \u211d) + (T : \u211d) * Complex.I) - \u03c1.1)\u2016 :=\n      fun \u03c3 _ => norm_le_insert' _ _\n    have h := intervalIntegral.integral_mono_on hlen hLDintbl hsum_intbl\n      (fun \u03c3 h\u03c3 => hpt2 \u03c3 h\u03c3)\n    rwa [intervalIntegral.integral_add h\u03a3norm_int hRnorm_int] at h\n  -- the R-part integral (the 5f capstone, constant in \u03c3)\n  have hR : \u2200 \u03c3 : \u211d, \u03c3 \u2208 Set.Icc (-5 / 2 : \u211d) 3 \u2192\n      \u2016logDeriv riemannZeta ((\u03c3 : \u211d) + (T : \u211d) * Complex.I) -\n        \u2211 \u03c1 \u2208 nearZerosSub T, (zeroMult \u03c1.1 : \u2102) /\n          (((\u03c3 : \u211d) + (T : \u211d) * Complex.I) - \u03c1.1)\u2016 \u2264\n        10.5 * Real.log (T + 2) + 48 := by\n    intro \u03c3 h\u03c3\n    obtain \u27e8h\u03c31, h\u03c32\u27e9 := h\u03c3\n    have hre : ((\u03c3 : \u211d) + (T : \u211d) * Complex.I).re = \u03c3 := by simp\n    have him : ((\u03c3 : \u211d) + (T : \u211d) * Complex.I).im = T := by simp\n    have h := logDeriv_riemannZeta_sub_nearZeros_le_tight (t := T)\n      (s := ((\u03c3 : \u211d) + (T : \u211d) * Complex.I)) hZC hZC2 hNLZ h10T\n      (by rw [hre]; exact h\u03c31) (by rw [hre]; exact h\u03c32) him (hzne \u03c3)\n    rwa [abs_of_nonneg hT0] at h\n  have hRint : \u222b \u03c3 in (-5 / 2 : \u211d)..(3 : \u211d),\n      \u2016logDeriv riemannZeta ((\u03c3 : \u211d) + (T : \u211d) * Complex.I) -\n        \u2211 \u03c1 \u2208 nearZerosSub T, (zeroMult \u03c1.1 : \u2102) /\n          (((\u03c3 : \u211d) + (T : \u211d) * Complex.I) - \u03c1.1)\u2016 \u2264\n        (3 - -5 / 2) * (10.5 * Real.log (T + 2) + 48) := by\n    have h := intervalIntegral.integral_mono_on hlen hRnorm_int intervalIntegrable_const\n      (fun \u03c3 h\u03c3 => hR \u03c3 h\u03c3)\n    rwa [intervalIntegral.integral_const, smul_eq_mul] at h\n  -- the near-sum integral: the per-zero arcsinh-FTC machine + the swap\n  have hnormid : \u2200 \u03c3 : \u211d, \u2200 \u03c1 : {\u03c1 : \u2102 // IsNontrivialZero \u03c1},\n      \u2016((\u03c3 : \u211d) + (T : \u211d) * Complex.I) - \u03c1.1\u2016 =\n        Real.sqrt ((\u03c3 - \u03c1.1.re) ^ 2 + (T - \u03c1.1.im) ^ 2) := by\n    intro \u03c3 \u03c1\n    have h1 : Complex.normSq (((\u03c3 : \u211d) + (T : \u211d) * Complex.I) - \u03c1.1) =\n        (\u03c3 - \u03c1.1.re) ^ 2 + (T - \u03c1.1.im) ^ 2 := by\n      simp [Complex.normSq_apply]\n      ring\n    symm\n    rw [Real.sqrt_eq_iff_eq_sq (by positivity) (norm_nonneg _),\n      \u2190 Complex.normSq_eq_norm_sq]\n    exact h1.symm\n  have htermcont : \u2200 \u03c1 : {\u03c1 : \u2102 // IsNontrivialZero \u03c1}, \u03c1 \u2208 nearZerosSub T \u2192\n      Continuous fun \u03c3 : \u211d => (zeroMult \u03c1.1 : \u211d) *\n        (1 / Real.sqrt ((\u03c3 - \u03c1.1.re) ^ 2 + (T - \u03c1.1.im) ^ 2)) := by\n    intro \u03c1 h\u03c1\n    obtain \u27e8h\u03b4pos, -, -\u27e9 := h\u03c1data \u03c1 h\u03c1\n    have hh : \u2200 \u03c3 : \u211d, (0 : \u211d) < (\u03c3 - \u03c1.1.re) ^ 2 + (T - \u03c1.1.im) ^ 2 := by\n      intro \u03c3\n      have h3 : \u03c1.1.im - T \u2260 0 := abs_pos.mp h\u03b4pos\n      have h4 : T - \u03c1.1.im \u2260 0 := by\n        intro h5\n        apply h3\n        linarith [h5]\n      exact add_pos_of_nonneg_of_pos (sq_nonneg _) (sq_pos_of_ne_zero h4)\n    apply Continuous.mul continuous_const\n    apply Continuous.div continuous_const\n    \u00b7 exact Real.continuous_sqrt.comp (by fun_prop :\n        Continuous fun \u03c3 : \u211d => (\u03c3 - \u03c1.1.re) ^ 2 + (T - \u03c1.1.im) ^ 2)\n    \u00b7 intro \u03c3\n      exact ne_of_gt (Real.sqrt_pos.mpr (hh \u03c3))\n  -- pointwise: \u2016\u03a3\u2016 \u2264 \u03a3 mult\u00b7(1/\u221a)\n  have hpt : \u2200 \u03c3 : \u211d, \u03c3 \u2208 Set.Icc (-5 / 2 : \u211d) 3 \u2192\n      \u2016\u2211 \u03c1 \u2208 nearZerosSub T, (zeroMult \u03c1.1 : \u2102) /\n          (((\u03c3 : \u211d) + (T : \u211d) * Complex.I) - \u03c1.1)\u2016 \u2264\n        \u2211 \u03c1 \u2208 nearZerosSub T, (zeroMult \u03c1.1 : \u211d) *\n          (1 / Real.sqrt ((\u03c3 - \u03c1.1.re) ^ 2 + (T - \u03c1.1.im) ^ 2)) := by\n    intro \u03c3 _\n    calc \u2016\u2211 \u03c1 \u2208 nearZerosSub T, (zeroMult \u03c1.1 : \u2102) /\n            (((\u03c3 : \u211d) + (T : \u211d) * Complex.I) - \u03c1.1)\u2016\n        \u2264 \u2211 \u03c1 \u2208 nearZerosSub T, \u2016(zeroMult \u03c1.1 : \u2102) /\n            (((\u03c3 : \u211d) + (T : \u211d) * Complex.I) - \u03c1.1)\u2016 :=\n          norm_sum_le _ _\n      _ = \u2211 \u03c1 \u2208 nearZerosSub T, (zeroMult \u03c1.1 : \u211d) /\n            \u2016((\u03c3 : \u211d) + (T : \u211d) * Complex.I) - \u03c1.1\u2016 := by\n          apply Finset.sum_congr rfl\n          intro \u03c1 _\n          rw [norm_div]\n          have h2 : \u2016(zeroMult \u03c1.1 : \u2102)\u2016 = (zeroMult \u03c1.1 : \u211d) := by\n            rw [show (zeroMult \u03c1.1 : \u2102) = ((zeroMult \u03c1.1 : \u211d) : \u2102) by\n              push_cast; ring, Complex.norm_real]\n            exact abs_of_nonneg (Nat.cast_nonneg _)\n          rw [h2]\n      _ = \u2211 \u03c1 \u2208 nearZerosSub T, (zeroMult \u03c1.1 : \u211d) *\n            (1 / Real.sqrt ((\u03c3 - \u03c1.1.re) ^ 2 + (T - \u03c1.1.im) ^ 2)) := by\n          apply Finset.sum_congr rfl\n          intro \u03c1 _\n          rw [hnormid \u03c3 \u03c1, mul_one_div]\n  -- the swap \u222b\u03a3 = \u03a3\u222b\n  have hswap : \u222b \u03c3 in (-5 / 2 : \u211d)..(3 : \u211d),\n      (\u2211 \u03c1 \u2208 nearZerosSub T, (zeroMult \u03c1.1 : \u211d) *\n        (1 / Real.sqrt ((\u03c3 - \u03c1.1.re) ^ 2 + (T - \u03c1.1.im) ^ 2))) =\n        \u2211 \u03c1 \u2208 nearZerosSub T, (zeroMult \u03c1.1 : \u211d) *\n          (\u222b \u03c3 in (-5 / 2 : \u211d)..(3 : \u211d),\n            1 / Real.sqrt ((\u03c3 - \u03c1.1.re) ^ 2 + (T - \u03c1.1.im) ^ 2)) := by\n    rw [intervalIntegral.integral_finsetSum\n      (fun \u03c1 h\u03c1 => (htermcont \u03c1 h\u03c1).intervalIntegrable _ _)]\n    apply Finset.sum_congr rfl\n    intro \u03c1 _\n    exact intervalIntegral.integral_const_mul _ _\n  -- per-\u03c1: the integral \u2264 2\u00b7log(148.5\u00b7\u0108\u2080(\u221an+2))\n  have hper : \u2200 \u03c1 : {\u03c1 : \u2102 // IsNontrivialZero \u03c1}, \u03c1 \u2208 nearZerosSub T \u2192\n      (\u222b \u03c3 in (-5 / 2 : \u211d)..(3 : \u211d),\n        1 / Real.sqrt ((\u03c3 - \u03c1.1.re) ^ 2 + (T - \u03c1.1.im) ^ 2)) \u2264\n        2 * Real.log (148.5 * C0hat (Real.sqrt (n : \u211d) + 2)) := by\n    intro \u03c1 h\u03c1\n    obtain \u27e8h\u03b4pos, h\u03b41, h\u03b4ge\u27e9 := h\u03c1data \u03c1 h\u03c1\n    have hsq : (T - \u03c1.1.im) ^ 2 = |\u03c1.1.im - T| ^ 2 := by\n      rw [\u2190 sq_abs (\u03c1.1.im - T)]\n      ring\n    have hmain := per_zero_integral_le_two_log \u03c1.2.2.1 \u03c1.2.2.2 h\u03b4pos h\u03b41.le\n    rw [hsq] at hmain \u22a2\n    have hinv : 1 / |\u03c1.1.im - T| \u2264 9 * C0hat (Real.sqrt (n : \u211d) + 2) := by\n      have h := one_div_le_one_div_of_le (one_div_pos.mpr (by positivity :\n        (0 : \u211d) < 9 * C0hat (Real.sqrt (n : \u211d) + 2))) h\u03b4ge\n      rwa [one_div_one_div] at h\n    have h1 : (16.5 : \u211d) * (1 / |\u03c1.1.im - T|) \u2264\n        16.5 * (9 * C0hat (Real.sqrt (n : \u211d) + 2)) :=\n      mul_le_mul_of_nonneg_left hinv (by norm_num)\n    have h2 : (16.5 : \u211d) / |\u03c1.1.im - T| \u2264 148.5 * C0hat (Real.sqrt (n : \u211d) + 2) := by\n      rw [mul_one_div, show (16.5 : \u211d) * (9 * C0hat (Real.sqrt (n : \u211d) + 2)) =\n        148.5 * C0hat (Real.sqrt (n : \u211d) + 2) by ring] at h1\n      exact h1\n    have h3 : Real.log (16.5 / |\u03c1.1.im - T|) \u2264\n        Real.log (148.5 * C0hat (Real.sqrt (n : \u211d) + 2)) :=\n      Real.log_le_log (by positivity) h2\n    linarith [hmain, h3]\n  -- the sum bound: \u03a3 mult\u00b7\u222b \u2264 (2\u00b7log(148.5\u00b7\u0108\u2080))\u00b7Z\u2082(T)\n  have hsum : \u2211 \u03c1 \u2208 nearZerosSub T, (zeroMult \u03c1.1 : \u211d) *\n      (\u222b \u03c3 in (-5 / 2 : \u211d)..(3 : \u211d),\n        1 / Real.sqrt ((\u03c3 - \u03c1.1.re) ^ 2 + (T - \u03c1.1.im) ^ 2)) \u2264\n      (2 * Real.log (148.5 * C0hat (Real.sqrt (n : \u211d) + 2))) * Z2hat T := by\n    calc \u2211 \u03c1 \u2208 nearZerosSub T, (zeroMult \u03c1.1 : \u211d) *\n          (\u222b \u03c3 in (-5 / 2 : \u211d)..(3 : \u211d),\n            1 / Real.sqrt ((\u03c3 - \u03c1.1.re) ^ 2 + (T - \u03c1.1.im) ^ 2))\n        \u2264 \u2211 \u03c1 \u2208 nearZerosSub T, (zeroMult \u03c1.1 : \u211d) *\n            (2 * Real.log (148.5 * C0hat (Real.sqrt (n : \u211d) + 2))) := by\n          apply Finset.sum_le_sum\n          intro \u03c1 h\u03c1\n          exact mul_le_mul_of_nonneg_left (hper \u03c1 h\u03c1) (Nat.cast_nonneg _)\n      _ = (2 * Real.log (148.5 * C0hat (Real.sqrt (n : \u211d) + 2))) *\n            \u2211 \u03c1 \u2208 nearZerosSub T, (zeroMult \u03c1.1 : \u211d) := by\n          rw [\u2190 Finset.mul_sum]\n          apply Finset.sum_congr rfl\n          intro \u03c1 _\n          ring\n      _ \u2264 (2 * Real.log (148.5 * C0hat (Real.sqrt (n : \u211d) + 2))) * Z2hat T :=\n          mul_le_mul_of_nonneg_left (nearZerosSub_mult_sum_le hZC2 h10T) hB0nn\n  -- the near-sum integral bound\n  have hRHScont : Continuous fun \u03c3 : \u211d =>\n      \u2211 \u03c1 \u2208 nearZerosSub T, (zeroMult \u03c1.1 : \u211d) *\n        (1 / Real.sqrt ((\u03c3 - \u03c1.1.re) ^ 2 + (T - \u03c1.1.im) ^ 2)) := by\n    apply continuous_finset_sum\n    intro \u03c1 h\u03c1\n    exact htermcont \u03c1 h\u03c1\n  have h\u03a3bound : \u222b \u03c3 in (-5 / 2 : \u211d)..(3 : \u211d), \u2016\u2211 \u03c1 \u2208 nearZerosSub T, (zeroMult \u03c1.1 : \u2102) /\n        (((\u03c3 : \u211d) + (T : \u211d) * Complex.I) - \u03c1.1)\u2016 \u2264\n      (2 * Real.log (148.5 * C0hat (Real.sqrt (n : \u211d) + 2))) * Z2hat T := by\n    have h1 : \u222b \u03c3 in (-5 / 2 : \u211d)..(3 : \u211d), \u2016\u2211 \u03c1 \u2208 nearZerosSub T, (zeroMult \u03c1.1 : \u2102) /\n          (((\u03c3 : \u211d) + (T : \u211d) * Complex.I) - \u03c1.1)\u2016 \u2264\n        \u222b \u03c3 in (-5 / 2 : \u211d)..(3 : \u211d),\n          (\u2211 \u03c1 \u2208 nearZerosSub T, (zeroMult \u03c1.1 : \u211d) *\n            (1 / Real.sqrt ((\u03c3 - \u03c1.1.re) ^ 2 + (T - \u03c1.1.im) ^ 2))) :=\n      intervalIntegral.integral_mono_on hlen h\u03a3norm_int\n        (hRHScont.intervalIntegrable _ _) (fun \u03c3 h\u03c3 => hpt \u03c3 h\u03c3)\n    exact h1.trans (hswap.trans_le hsum)\n  -- assembly\n  unfold HIntegral\n  calc \u2016\u222b x in (-5 / 2 : \u211d)..(3 : \u211d), Jlag n ((x : \u211d) + (T : \u211d) * Complex.I)\u2016\n      \u2264 \u222b \u03c3 in (-5 / 2 : \u211d)..(3 : \u211d), \u2016Jlag n ((\u03c3 : \u211d) + (T : \u211d) * Complex.I)\u2016 :=\n        intervalIntegral.norm_integral_le_integral_norm hlen\n    _ \u2264 (Real.exp 4 + 1) * \u222b \u03c3 in (-5 / 2 : \u211d)..(3 : \u211d),\n          \u2016logDeriv riemannZeta ((\u03c3 : \u211d) + (T : \u211d) * Complex.I)\u2016 := hJint\n    _ \u2264 (Real.exp 4 + 1) *\n          ((\u222b \u03c3 in (-5 / 2 : \u211d)..(3 : \u211d), \u2016\u2211 \u03c1 \u2208 nearZerosSub T, (zeroMult \u03c1.1 : \u2102) /\n              (((\u03c3 : \u211d) + (T : \u211d) * Complex.I) - \u03c1.1)\u2016) +\n            (\u222b \u03c3 in (-5 / 2 : \u211d)..(3 : \u211d),\n              \u2016logDeriv riemannZeta ((\u03c3 : \u211d) + (T : \u211d) * Complex.I) -\n                \u2211 \u03c1 \u2208 nearZerosSub T, (zeroMult \u03c1.1 : \u2102) /\n                  (((\u03c3 : \u211d) + (T : \u211d) * Complex.I) - \u03c1.1)\u2016)) :=\n        mul_le_mul_of_nonneg_left hsplit (by positivity)\n    _ \u2264 (Real.exp 4 + 1) *\n          ((2 * Real.log (148.5 * C0hat (Real.sqrt (n : \u211d) + 2))) * Z2hat T +\n            (3 - -5 / 2) * (10.5 * Real.log (T + 2) + 48)) :=\n        mul_le_mul_of_nonneg_left (add_le_add h\u03a3bound hRint) (by positivity)\n    _ = (Real.exp 4 + 1) *\n          (2 * Z2hat T * Real.log (148.5 * C0hat (Real.sqrt (n : \u211d) + 2)) +\n            (3 - -5 / 2) * (10.5 * Real.log (T + 2) + 48)) := by ring\n\n/-- **6-c-iv (the bottom side \u2212T)**: the conj trick (DL-5 pattern). -/\ntheorem hIntegral_Jlag_mid_neg_norm_le (hZC : ZeroCountProp) (hZC2 : ZeroCountProp2)\n    (hNLZ : NoLowZeros) {n : \u2115} (hn : 100000 \u2264 n) {T : \u211d}\n    (hTwide : \u2200 \u03c1 : \u2102, IsNontrivialZero \u03c1 \u2192 Real.sqrt (n : \u211d) - 1 < \u03c1.im \u2192\n      \u03c1.im < Real.sqrt (n : \u211d) + 2 \u2192\n        1 / (9 * C0hat (Real.sqrt (n : \u211d) + 2)) \u2264 |\u03c1.im - T|)\n    (hTn : Real.sqrt (n : \u211d) \u2264 T) (hTn1 : T \u2264 Real.sqrt (n : \u211d) + 1) :\n    \u2016HIntegral (Jlag n) (-5 / 2) 3 (-T)\u2016 \u2264\n      (Real.exp 4 + 1) *\n        (2 * Z2hat T * Real.log (148.5 * C0hat (Real.sqrt (n : \u211d) + 2)) +\n          (3 - -5 / 2) * (10.5 * Real.log (T + 2) + 48)) := by\n  have hnR : (100000 : \u211d) \u2264 (n : \u211d) := by exact_mod_cast hn\n  have hT0 : (0 : \u211d) < T := by\n    have h1 : (0 : \u211d) < Real.sqrt (n : \u211d) := Real.sqrt_pos.mpr (by linarith)\n    linarith\n  have hconj : HIntegral (Jlag n) (-5 / 2) 3 (-T) =\n      (starRingEnd \u2102) (HIntegral (Jlag n) (-5 / 2) 3 T) := by\n    unfold HIntegral\n    rw [\u2190 intervalIntegral.intervalIntegral_conj]\n    apply intervalIntegral.integral_congr\n    intro \u03c3 h\u03c3\n    show Jlag n ((\u03c3 : \u211d) + (((-T : \u211d)) : \u2102) * Complex.I) =\n      (starRingEnd \u2102) (Jlag n ((\u03c3 : \u211d) + T * Complex.I))\n    have hid2 : ((\u03c3 : \u211d) + (((-T : \u211d)) : \u2102) * Complex.I) =\n        (starRingEnd \u2102) ((\u03c3 : \u211d) + T * Complex.I) := by simp\n    rw [hid2]\n    exact Jlag_conj n (by\n      intro h\n      have him : (((\u03c3 : \u211d) + T * Complex.I)).im = T := by simp\n      rw [h] at him\n      simp at him\n      linarith [hT0])\n  rw [hconj, Complex.norm_conj]\n  exact hIntegral_Jlag_mid_norm_le hZC hZC2 hNLZ hn hTwide hTn hTn1\n\nend NguyenRH1"
}
#2283 assistant 2026-08-26 18:54:21.630 UTC · 1348→1938 tok open ⤢
thinking (5282 chars)
Five errors: 1. **620**: `neg_abs_le (σ - β)` has type `-|σ - β| ≤ σ - β` — I had it backwards: `neg_abs_le : -|a| ≤ a`. I need `-(σ−β) ≤ |σ−β|` — that's `neg_le_abs (a) : -a ≤ |a|`? Hmm — `neg_le_abs_self`? In Mathlib: `neg_le_abs (a) : -a ≤ |a|` ✓ I believe `neg_le_abs` exists. Let me use `neg_le_abs`. 2. **629/631**: `hasDerivAt_id.sub_const` — Unknown constant. hasDerivAt_id is `theorem hasDerivAt_id (x : 𝕜) : HasDerivAt id 1 x`. The dot `.sub_const` on it — `HasDerivAt.sub_const` exists? The error is "Unknown constant hasDerivAt_id.sub_const" — because hasDerivAt_id is a FUNCTION (explicit arg x), dot-notation resolution fails?? — `(hasDerivAt_id σ).sub_const β` should work... unless `HasDerivAt.sub_const` doesn't exist under that name. Let me check: in Mathlib, `HasDerivAt.sub_const (hf : HasDerivAt f f' x) (c : 𝕜) : HasDerivAt (fun x => f x - c) f' x` — hmm — maybe it's `HasDerivAt.sub_const` vs the session-011 gotcha: "HasDerivAt.add_const adds to VALUE (use .comp for precomposition)" — implying HasDerivAt.add_const EXISTS. So sub_const should too. The error "Unknown constant `hasDerivAt_id.sub_const`" — the dot-notation on a FUNCTION (hasDerivAt_id takes explicit x) — `(hasDerivAt_id σ).sub_const β` — Lean parses `hasDerivAt_id.sub_const` as a CONSTANT name first — since hasDerivAt_id is a function needing an argument, the parser treats `.sub_const` as namespace resolution and fails! When the head is a function application, must parenthesize: `(hasDerivAt_id σ).sub_const β` — I DID parenthesize in h2a but NOT in h1: `hasDerivAt_id.sub_const β` (line 629, no parens around hasDerivAt_id σ!). Indeed my h1: `hasDerivAt_id.sub_const β` — WRONG — should be `(hasDerivAt_id σ).sub_const β`. And h2a: `(hasDerivAt_id.sub_const β).pow 2` — also unparenthesized — should be `((hasDerivAt_id σ).sub_const β).pow 2`. 3. **650**: "No goals to be solved" — the h6/h7 convert's field_simp; ring — field_simp closed it, ring has no goals. Line 650 — which one? The h7 block ends around 648-650. Let me count: h6 block: convert h5 using 1; field_simp [hsqrt0]; ring — lines ~640-643. h7: have h7 := h6.log hg0; convert h7 using 1; field_simp [hsqrt0, hg0]; ring — ~645-650. Line 650:4 = the `ring` after h7's field_simp — field_simp closed, ring fails. Remove that ring. But wait — did h6's field_simp; ring pass (no error at ~643)? The error list shows only 650 — so h6's ring was FINE (field_simp left a goal there) and h7's field_simp closed fully. Remove the ring after h7's field_simp. 4. **701**: hrid's `rw [div_eq_iff ...]` — "Did not find the pattern ?a / (u + √) = ?c in the target". The goal: `-u + √(u^2+δ^2) = δ^2 / (u + √(u^2+δ^2))`. div_eq_iff (hb : b ≠ 0) : a / b = c ↔ a = c * b — the PATTERN for rw [div_eq_iff h] is `?a / ?b = ?c` — i.e., it rewrites goals where the DIVISION is on the LHS! My goal has the division on the RHS: `-u + √ = δ²/(u+√)`. So rw [div_eq_iff] finds no `?/?b = ?c` with b := u+√... wait — the pattern `?m.660 / (u + √(u ^ 2 + δ ^ 2)) = ?m.662` — it needs LHS-of-eq to be a division by (u+√). My goal: `-u+√ = δ²/(u+√)` — the division is the RHS — no match. FIX: rw [eq_div_iff (hb : b ≠ 0) : c = a / b ↔ c * b = a]! `eq_div_iff (hb : 0 < b)`? — `eq_div_iff (hb : b ≠ 0) : c = a / b ↔ c * b = a` ✓ exists. Use `rw [eq_div_iff (ne_of_gt ...)]` — goal becomes `(−u+√)·(u+√) = δ²` — nlinarith [hs] ✓. 5. **793**: hB0nn's `nlinarith [hC0]` failed: goal `1 ≤ 148.5 * C0hat(√n+2)` with hC0 : 0 < C0hat — I need C0hat ≥ 2 (two_le_C0hat), not just > 0! I wrote `nlinarith [hC0]` but hC0 is only 0 < C0hat. The hB0nn block references hC0 but the needed fact is h2 := two_le_C0hat (which I computed INSIDE the hC0 block — not in scope here!). Fix: in hB0nn, use two_le_C0hat directly: ```lean have hB0nn : (0 : ℝ) ≤ 2 * Real.log (148.5 * C0hat (Real.sqrt (n : ℝ) + 2)) := by have h2 := two_le_C0hat (by nlinarith [Real.sqrt_nonneg (n : ℝ)] : (1 : ℝ) ≤ Real.sqrt (n : ℝ) + 2) have h1 : (1 : ℝ) ≤ 148.5 * C0hat (Real.sqrt (n : ℝ) + 2) := by nlinarith [h2] have h3 := Real.log_nonneg h1 linarith ``` 6. **777:60 "unsolved goals"** — case h — the hne1 block! `intro σ h; have him := congrArg Complex.im h; simp at him; linarith [h10T]` — the context shows hne1 : ∀ σ, ↑σ + ↑T * I ≠ 1 got PROVED (it's in context at 777) — wait no, 777's context ends at hcontJ — the error is at 777:60 — let me find what's at line 777. The error context includes hB0nn, hzne, hne1, hpath, hLDc, hJcont, hcontJ — so 777 is AFTER hcontJ — it's the hcontG/hJintbl region?? Line 777:60 "unsolved goals case h" — hmm — col 60. Could be hcontJ's exact line? No — hcontJ is in context (proved). The next block: hcontG := continuous_norm.comp hcontJ — that's an exact-term, no goals. hJintbl := hcontG.intervalIntegrable _ _ — also term. Hmm — "case h" — a named case from rcases/obtain/intro... Let me look at line 777. Actually — the errors after 701 are probably CASCADES — 701's failure left hrid sorry'd, but that's in arcsinh_left_endpoint_le (lemma 3) — the subsequent hden/hge/hlog in that lemma would still elaborate (hlogid uses hrid? no — hlogid is independent; hge uses hrid — sorry-backed, fine). The 793 error is independent (hB0nn). The 777 error — let me count lines precisely. I'll read the file around 770-800.
Five fixes identified. Checking the exact line 777 context before fixing.
tool: Read
input
{
  "file_path": "D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiSideBounds.lean",
  "offset": 770,
  "limit": 30
}
#2284 assistant 2026-08-26 18:59:48.889 UTC · 1396→1340 tok open ⤢
thinking (3968 chars)
Two errors now: 1. **634:6 `ring_nf` made no progress** — this is in h6's `field_simp [hsqrt0]; ring` — wait no — line 634 — after my edits, what's at 634? The h6 block: `convert h5 using 1; field_simp [hsqrt0]; ring`. "ring_nf made no progress on the goal" — meaning field_simp did NOT fire (maybe it closed nothing?) and then `ring` faced a goal it can't normalize... wait — "ring_nf made no progress" — the `ring` tactic calls ring_nf which made no progress — meaning the goal is NOT a ring identity — i.e., field_simp transformed the goal into something that's already closed?? No — if field_simp closed the goal, ring would say "no goals". "ring_nf made no progress" happens when ring runs on a goal that IS a ring equation but ring_nf doesn't change anything (maybe already normalized) and then... hmm — actually `ring_nf made no progress` is an ERROR from ring_nf when it can't rewrite — the `ring` tactic then... — the goal at that point might be `2 * ((σ−β) + √h) = 2 * (√h + (σ−β))`?? — that's a ring identity — ring_nf should normalize both sides identically and close... "made no progress" means the goal is NOT of the form ring understands? OR the goal is `True`?? Hmm — wait, maybe field_simp produced `g = g`-form that needs rfl but ring_nf on rfl-goal `X = X`... ring_nf would close it. "Made no progress" — hmm — maybe the goal after field_simp is something like `(σ−β) + √h = (σ−β) + √h` — ring_nf normalizes both to the same — that's "progress" (closes). Confused. Let me look at what's actually at line 634 now. After my edits (I removed the ring after h7's field_simp), line numbers shifted. Let me check. 2. **776:60 unsolved goals `⊢ ?m.1004`** — RIGHT AFTER hden! The next block is hΣcont: ```lean have hΣcont : Continuous fun σ : ℝ => ∑ ρ ∈ nearZerosSub T, (zeroMult ρ.1 : ℂ) / (((σ : ℝ) + (T : ℝ) * Complex.I) - ρ.1) := by apply continuous_finset_sum intro ρ hρ exact Continuous.div continuous_const (hpath.sub continuous_const) (hden ρ hρ) ``` `⊢ ?m.1004` with case h — the `apply continuous_finset_sum` — continuous_finset_sum's signature: `theorem continuous_finset_sum (s : Finset ι) (hf : ∀ i ∈ s, Continuous (f i)) : Continuous fun x => ∑ i ∈ s, f i x` — `apply continuous_finset_sum` — s explicit?! If s is an EXPLICIT argument, `apply continuous_finset_sum` would leave `s` as a new goal `⊢ Finset ...`?? — no — apply would unify the CONCLUSION `Continuous fun x => ∑ i ∈ s, f i x` with the goal — the goal's sum is over nearZerosSub T — s := nearZerosSub T ✓ and f := the term family ✓ — leaving the hf goal `∀ i ∈ s, Continuous (f i)` — then intro ρ hρ ✓ — then exact — hmm. BUT WAIT — maybe the issue: `continuous_finset_sum` might have signature `(hf : ∀ i ∈ s, Continuous (f i)) → Continuous (∑ i ∈ s, f i)` — the PI.SUM form `∑ i ∈ s, f i` as a FUNCTION-valued sum (Pi.hasSum)! I.e., it produces `Continuous (∑ i ∈ s, f i)` where the sum is in the FUNCTION SPACE (f : ι → X → Y, sum pointwise) — vs my goal `Continuous fun x => ∑ i ∈ s, (term i x)` — these are defeq (function sum = pointwise) but `apply` needs to UNIFY `Continuous (∑ i ∈ s, ?f i)` with `Continuous fun σ => ∑ ρ, term σ` — the unification of `∑ i ∈ s, ?f i` (Pi.sum applied) with `fun σ => ∑ ρ, term ρ σ` — HO — might fail, leaving ?m.1004 (a metavariable that never got assigned — the unsolved ?m goal). The `⊢ ?m.1004` — a METAVARIABLE as a goal — classic "apply left an undetermined metavariable" symptom. Fix: use the exact-name with explicit args or the alternative lemma. Options: (a) `exact continuous_finset_sum _ (fun ρ hρ => Continuous.div ...)` — hmm. (b) Use `Finset.continuous_sum`? — maybe the name is different. Let me grep for the actual name and signature. Actually — the classic lemma is `continuous_finset_sum (s : Finset ι) (hf : ∀ i ∈ s, Continuous (f i)) : Continuous fun x => ∑ i ∈ s, f i x` — hmm — OR `continuous_sum (hf : ∀ i ∈ s, Continuous (f i)) : Continuous (∑ i ∈ s, f i)`. Let me grep both.
Two errors. Line 634's ring issue and a metavariable-goal at 776 (the `continuous_finset_sum` application mis-unified). Checking the exact lemma names.
tool: Bash
input
{
  "command": "grep -rn \"theorem continuous_finset_sum\\|theorem continuous_sum \" \"D:/Python/Quantum/zeta-23-lean/.lake/packages/mathlib/Mathlib/Topology/Algebra/\" 2>/dev/null | head -5; sed -n '625,640p' \"D:/Python/Quantum/research/lean/NguyenRH1/NguyenRH1/LiSideBounds.lean\"",
  "description": "Find continuous_finset_sum signature and view line 634 area"
}
#2285 assistant 2026-08-26 19:01:47.152 UTC · 1830→2632 tok open ⤢
thinking (7388 chars)
Line 634 is the `ring` in h2's convert block: ```lean have h2 : HasDerivAt (fun σ => (σ - β) ^ 2 + δ ^ 2) (2 * (σ - β)) σ := by have h2a := ((hasDerivAt_id σ).sub_const β).pow 2 have h2b := h2a.add_const (δ ^ 2) convert h2b using 1 ring ``` "ring_nf made no progress on the goal" at 634 (the `ring`). So `convert h2b using 1` left a goal that ring can't handle — maybe the convert produced NO derivative-equality goal (functions matched AND derivatives matched up to defeq?) — then ring has "no goals"?? — no, the error is "ring_nf made no progress", not "no goals". So there IS a goal but not a ring identity. What goal could convert leave? convert h2b using 1 — h2b : HasDerivAt (fun x => (x−β)^2 + δ^2) (↑2 * (σ−β)^1 * 1) σ — target: HasDerivAt (fun σ => (σ−β)^2+δ^2) (2*(σ−β)) σ — hmm — h2a's function from .pow: `fun y => (y−β)^2`?? — HasDerivAt.pow (hf : HasDerivAt f f' x) (n : ℕ) : HasDerivAt (fun y => (f y)^n) (↑n * (f x)^(n-1) * f') x — the derivative: `(↑n * (σ−β)^(2-1) * 1)` — wait — n-1 = 1 — `(σ−β)^1`. And h2b := h2a.add_const (δ^2) : HasDerivAt (fun y => (y−β)^2 + δ^2) (↑2*(σ−β)^1*1) σ. convert h2b using 1: functions match (both fun => (σ−β)^2+δ^2 ✓), derivative goal: `2*(σ−β) = ↑2*(σ−β)^1*1` — ring should close... "ring_nf made no progress" — meaning ring_nf was ALREADY at normal form and the sides differ?! `↑2 * (σ−β)^1 * 1` vs `2 * (σ−β)` — ring_nf normalizes both to `2*(σ−β)` — should close. UNLESS convert's goal is the OTHER direction or a HasDerivAt-component goal like `(fun σ => (σ−β)^2+δ^2) = (fun y => (y−β)^2+δ^2)` — alpha — ring can't handle function equations with √-free but binder — "ring_nf made no progress" on a non-ring goal ✓ THIS is it — convert's remaining goal might be the FUNCTION equality (if convert decided derivatives match by defeq but functions don't syntactically match!). h2b's function: `fun y => (y − β)^2 + δ^2` — mine: `fun σ => (σ − β)^2 + δ^2` — ALPHA-equal ✓ defeq — convert should close... Hmm — wait — maybe HasDerivAt.add_const's function is `fun y => (y−β)^2 + δ^2`?? — HasDerivAt.add_const (hf) (c): HasDerivAt (fun y => f y + c) f' — so `fun y => (y−β)^2 + δ^2` ✓ same. OR — the derivative from .pow is NOT `↑2*(σ−β)^1*1` but something with a NAT cast that ring_nf handles differently: `HasDerivAt.pow` might state the derivative as `n * (f x)^(n-1) * f'` with n : ℕ cast — `↑2 * (σ−β)^1 * 1` — ring normalizes ✓... Actually — hmm — maybe the issue is that convert h2b using 1 produced the goal `2 * (σ−β) = 2 * (σ−β)` (rfl) and closed NOTHING... no. Let me think about "ring_nf made no progress on the goal" — this error specifically: `ring` first tries `ring_nf` — ring_nf errors "made no progress" when the goal is NOT an equality in a commutative ring — e.g., a HasDerivAt goal (a Prop structure!) — if convert produced ZERO new goals (fully closed), then `ring` runs on... no — if convert closed everything, ring says "no goals". UNLESS convert produced a goal that's NOT an equality — like `HEq` junk or a metavariable-instantiated Prop — hmm. WAIT — maybe convert h2b using 1 unified EVERYTHING and left NO goals — and the `ring` error is actually from the h6 block (line ~643)? The error says 634:6. Line 634 = the `ring` in h2's block per the sed output ✓ (sed showed lines 625-640; the `ring` after convert h2b is at 634). So h2's ring. Hypothesis: convert h2b using 1 left the goal `(2 * (σ−β)) = (↑2 * (σ−β)^1 * 1)` but with `↑2` being a NAT cast `(2 : ℕ)` coerced to ℝ — ring_nf normalizes casts ✓... OR — the .pow derivative is `2 * (σ − β) ^ (2 - 1) * 1` with the `2-1` UNREDUCED — ring_nf normalizes `2-1` → 1, `(σ−β)^1` → (σ−β) ✓ still fine. Hmm — OR the convert produced the goal in reverse with the HasDerivAt DERIVATIVE position being a projection: HasDerivAt f f' x — convert using 1 decomposes: goals f₁ = f₂ (closed by defeq if same), f'₁ = f'₂ (the derivative equality). If convert decided to ALSO check `x` and σ — same ✓. Let me look at the actual goal — the error output for 634 only shows the one line, no context/goal. "ring_nf made no progress on the goal" — let me get the full error with context. Actually the output showed ONLY the error line for 634 — no goal display (head -60 might have cut it — no, errors usually show context after). Hmm. Alternative robust fix for h2 — avoid convert entirely: use `HasDerivAt.const_add`?? no — simplest: compute the derivative via `by fun_prop`?? — fun_prop can PROVE HasDerivAt for polynomials: `by fun_prop` proves `HasDerivAt (fun σ => (σ−β)^2+δ^2) ?deriv σ` — fun_prop COMPUTES the derivative — the derivative value must match mine `2*(σ−β)` — fun_prop's HasDerivAt support: it can prove HasDerivAt with a SPECIFIED derivative if it matches its computation ✓ try `by fun_prop (disch := ...)`. Actually simpler: bypass h2's convert by writing the derivative differently: state h2 with the derivative `(↑2 * (σ-β) ^ (2-1) * 1)`?? ugly. Cleanest: use `simpa using h2b`: ```lean have h2 : HasDerivAt (fun σ => (σ - β) ^ 2 + δ ^ 2) (2 * (σ - β)) σ := by have h2a := ((hasDerivAt_id σ).sub_const β).pow 2 have h2b := h2a.add_const (δ ^ 2) convert h2b using 1 <;> ring ``` hmm same. Or — drop convert; use `show` to change h2b's type by defeq: ```lean exact h2b ``` Is `HasDerivAt (fun y => (y−β)^2+δ^2) (↑2*(σ−β)^1*1) σ` DEFEQ to `HasDerivAt (fun σ => (σ−β)^2+δ^2) (2*(σ−β)) σ`? The derivative: `↑2*(σ−β)^1*1` vs `2*(σ−β)` — defeq? `↑2` (Nat.cast 2) vs `2` (OfNat 2) — Nat.cast 2 = 1+1 = OfNat?? — `(2:ℕ)` casts to `(2:ℝ)` — is `Nat.cast 2` defeq `OfNat.ofNat 2`? — `Nat.cast 2 = Nat.cast (1+1) = Nat.cast 1 + Nat.cast 1 = 1 + 1` and OfNat.ofNat 2 = 1+1 ✓ probably defeq. `(σ−β)^1` defeq `(σ−β)` ✓ (pow_one). `x*1` defeq x ✓ (mul_one). So the whole derivative is defeq to 2*(σ−β)?? — `↑2 * (σ−β)^1 * 1` — mul_one → `↑2*(σ−β)^1` — pow_one → `↑2*(σ−β)` — Nat.cast 2 → 2 ✓ — all DEFEQ steps ✓ so `exact h2b` might just work! Try exact h2b first; if defeq fails, fallback convert with explicit ring-proof: Actually the safest deterministic: ```lean convert h2b using 1 · ring ``` — that's what we have. Since ring errors, use `show` on the goal... I can't see the goal. Let me try `simpa using h2b` — simpa normalizes both sides with simp (handles pow_one, mul_one, Nat.cast_two): ```lean simpa using h2b ``` simp's default set handles pow_one/mul_one/casts — likely closes ✓. 2. **776:60** — the hΣcont block — `apply continuous_finset_sum` — metavariable goal. Fix: avoid apply; use exact with the hf given directly: ```lean have hΣcont : Continuous fun σ : ℝ => ∑ ρ ∈ nearZerosSub T, (zeroMult ρ.1 : ℂ) / (((σ : ℝ) + (T : ℝ) * Complex.I) - ρ.1) := by apply continuous_finset_sum ``` Let me grep the actual signature of continuous_finset_sum (the earlier grep found nothing! — the grep for "theorem continuous_finset_sum\|theorem continuous_sum " returned NOTHING — only the sed output showed). So `continuous_finset_sum` DOESN'T EXIST under that name?! — then `apply continuous_finset_sum` — unknown constant — but the error was "unsolved goals ?m.1004" not "unknown identifier" — hmm — `apply unknown` would report unknown identifier... The error at 776:60 shows `⊢ ?m.1004` — maybe continuous_finset_sum DOES exist (in a file the grep didn't cover — my grep was only Topology/Algebra/). Let me grep the whole mathlib.
Line 634 is the h2 `convert`/`ring` — replacing with `simpa`. For 776, `continuous_finset_sum`'s signature needs checking (my grep found nothing under Topology/Algebra).
tool: Bash
input
{
  "command": "grep -rn \"continuous_finset_sum\" \"D:/Python/Quantum/zeta-23-lean/.lake/packages/mathlib/Mathlib/\" | grep -v \"deprecated\\|alias\" | head -5; echo ---; grep -rn \"theorem continuous_sum\" \"D:/Python/Quantum/zeta-23-lean/.lake/packages/mathlib/Mathlib/\" | head -5",
  "description": "Locate continuous_finset_sum or continuous_sum"
}
#2286 assistant 2026-08-26 19:06:11.822 UTC · 254→2645 tok open ⤢
thinking (7503 chars)
Found it: `continuous_finsetSum {f : ι → X → M} (s : Finset ι) : (∀ i ∈ s, Continuous (f i)) → Continuous fun a => ∑ i ∈ s, f i a` — the additive version. So `continuous_finsetSum (s) (hf)` — s EXPLICIT, and it produces `Continuous fun a => ∑ i ∈ s, f i a` ✓ exactly my goal shape. My `apply continuous_finset_sum` used the DEPRECATED alias — deprecated aliases might not elaborate the same way... the metavariable goal `?m.1004` — probably because `apply continuous_finset_sum` — the alias's type has {f} implicit and s explicit — apply should unify conclusion... — whatever — use `continuous_finsetSum` directly with the explicit s: ```lean exact continuous_finsetSum _ (fun ρ hρ => Continuous.div continuous_const (hpath.sub continuous_const) (hden ρ hρ)) ``` Hmm — wait — but there are TWO uses (hΣcont and hRHScont) plus hsumcont (which used hΣnorm_cont.add hRcont — fine). Let me replace both `apply continuous_finset_sum` occurrences. For hΣcont: ```lean have hΣcont : Continuous fun σ : ℝ => ∑ ρ ∈ nearZerosSub T, (zeroMult ρ.1 : ℂ) / (((σ : ℝ) + (T : ℝ) * Complex.I) - ρ.1) := by exact continuous_finsetSum _ (fun ρ hρ => Continuous.div continuous_const (hpath.sub continuous_const) (hden ρ hρ)) ``` The f := fun ρ => fun σ => (zeroMult ρ.1:ℂ)/((↑σ+↑T·I)−ρ.1) — unified from the goal ✓; hf : ∀ ρ ∈ nearZerosSub T, Continuous (fun σ => (zeroMult ρ.1:ℂ)/((↑σ+↑T·I)−ρ.1)) — my lambda ✓. Hmm — Continuous.div (hf) (hg) (h₀) — the g (denominator): `fun σ => (↑σ+↑T·I) − ρ.1` — hpath.sub continuous_const : Continuous fun σ => (↑σ+↑T·I) − ?c — ?c unified with ρ.1 from... the g is determined by h₀'s type: h₀ := hden ρ hρ : ∀ σ, (↑σ+↑T·I) − ρ.1 ≠ 0 — the Continuous.div's h₀ : ∀ x, ?g x ≠ 0 — unify ?g σ =?= (↑σ+↑T·I)−ρ.1 — Miller ✓ ?g := fun σ => (↑σ+↑T·I)−ρ.1 — then hg : Continuous ?g — hpath.sub continuous_const — unifies ?c := ρ.1 ✓. And hf := continuous_const : Continuous fun _ => ?c2 — ?c2 from the result type: the expected type (from hΣcont's ascription): Continuous fun σ => (zeroMult ρ.1:ℂ)/((↑σ+↑T·I)−ρ.1) — unify `fun σ => ?f σ / ?g σ` — ?f σ =?= (zeroMult ρ.1:ℂ) (const) ✓ ?c2 := zeroMult ✓. The elaborator processes args left-to-right: hf first — continuous_const with ?c2 free — then hg — then h₀ — then the RESULT unification with the ascription pins ?c2, ?g ✓ — metavars persist across arg elaboration ✓ should work. For hRHScont: ```lean have hRHScont : Continuous fun σ : ℝ => ∑ ρ ∈ nearZerosSub T, (zeroMult ρ.1 : ℝ) * (1 / Real.sqrt ((σ - ρ.1.re) ^ 2 + (T - ρ.1.im) ^ 2)) := by exact continuous_finsetSum _ (fun ρ hρ => htermcont ρ hρ) ``` Also — the `rwa [one_div_one_div] at h` in hper — one_div_one_div : 1/(1/a) = a ✓. And line 634's ring → simpa: ```lean have h2b := h2a.add_const (δ ^ 2) convert h2b using 1 ring ``` → ```lean convert h2b using 1 · ring ``` Hmm no — the issue: I need to know WHY ring failed. "ring_nf made no progress" — let me actually think again: `convert h2b using 1` — if h2b's type is `HasDerivAt (fun y => (y−β)^2 + δ^2) (↑2 * (σ−β)^1 * 1) σ` and my stated h2 type is `HasDerivAt (fun σ => (σ−β)^2 + δ^2) (2*(σ−β)) σ` — convert using 1: it decomposes HasDerivAt's arguments: (fun σ => (σ−β)^2+δ^2) =?= (fun y => (y−β)^2+δ^2) — alpha ✓ defeq, no goal; (2*(σ−β)) =?= (↑2*(σ−β)^1*1) — convert leaves this as a goal if not defeq: `2 * (σ - β) = ↑2 * (σ - β) ^ 1 * 1`?? — hmm — ↑2 is Nat.cast 2 — is that DEFEQ to OfNat.ofNat 2? If NOT defeq (in ℝ, OfNat.ofNat 2 = OneAddNat... vs Nat.cast 2 = 1+1... — they might differ syntactically) — convert leaves the goal; ring should close ANY ring equation... "ring_nf made no progress" — weird. OH WAIT. I bet the issue is different: maybe `convert h2b using 1` produced the goal `2 * (σ - β) = 2 * (σ - β) ^ 1 * 1` — hmm — or maybe convert DIDN'T leave an equality goal but instead left `HEq` or nothing — and `ring` ran on the ALREADY-CLOSED... no. Actually — alternative theory: `HasDerivAt.pow` in THIS Mathlib might produce the derivative `(2 : ℝ) * (σ - β) ^ (2 - 1) * 1`?? — still ring-normalizable. OR — `HasDerivAt.pow` might not exist under that name and h2a is sorry'd?? — no — unknown identifier would error at the .pow. You know what — the error line 634:6 might be the h6 block's `ring` (after `field_simp [hsqrt0]`), not h2's! Line numbers: the sed output showed lines 625-640 — line 634 = the `ring` after `convert h2b using 1`. Hmm — wait — sed showed: 633 = `convert h2b using 1`, 634 = `ring`. Yes — h2's ring. Let me think about "ring_nf made no progress on the goal" more carefully — this error occurs when ring_nf is called and the goal is ALREADY in normal form but not closed (e.g., `a = a` rfl-normal... ring_nf would close rfl) — OR the goal is not a ring equality — OR the goal contains something like `↑2` that ring_nf... Hmm — actually — maybe the goal after convert is `(fun σ => (σ-β)^2+δ^2) = (fun y => (y−β)^2+δ^2)` — a FUNCTION equality — ring_nf can't handle lambdas → "made no progress" ✓ PLAUSIBLE. Why would convert leave the function equation? If convert unified the DERIVATIVES by defeq (they might be defeq!) but NOT the functions (syntactic difference: `fun y => (y−β)^2 + δ^2` vs `fun σ => (σ−β)^2+δ^2` — alpha — should be handled by convert's default... convert uses `congr` which does handle eta/alpha... hmm — convert using 1's depth-1 decomposition of HasDerivAt: HasDerivAt (f) (f') (x) — three explicit args? — `HasDerivAt (f : 𝕜 → 𝕜') (f' : 𝕜') (x : 𝕜) : Prop` — convert decomposes into f = f', f' = f'', x = x' — each closed by rfl if defeq — the function one: `fun σ => (σ−β)^2+δ^2` vs `fun y => (y−β)^2+δ^2` — alpha — rfl-defeq ✓ should close automatically... Unless h2b's function is NOT `fun y => (y−β)^2+δ^2` but `(fun y => (y−β))^2 + δ^2`?? — HasDerivAt.pow : `fun y => (f y)^n` — with f := fun x => x−β — `(fun y => (y−β))^2`?? — no — (f y)^n with f y = y−β — `(y−β)^2` ✓ same. I'll just try a different robust route for h2: `simpa using h2b` — simpa normalizes the goal AND h2b with simp then matches — handles pow_one, mul_one, Nat.cast numbers, alpha. If simpa fails, the error shows the remaining goal — informative. Actually — even better: avoid the convert circus entirely — fun_prop can prove h2 directly? `by fun_prop` — fun_prop proves HasDerivAt goals by computing derivatives — for a polynomial it's supported (HasDerivAt is a fun_prop target ✓). `by fun_prop` on `HasDerivAt (fun σ => (σ−β)^2+δ^2) (2*(σ−β)) σ` — fun_prop's derivative computation: it derives `2*(σ−β)*1` and then needs to match `2*(σ−β)` — fun_prop finishes with a `rfl`/simp check?? — fun_prop typically requires the stated derivative to be what it computes modulo simp — it might close it ✓. But uncertain. Let me use `simpa using h2b` as primary. Similarly h6's `field_simp [hsqrt0]; ring` — that ring DIDN'T error (no error at its line) — hmm wait — is 634 h6's ring instead? Let me recount from the sed output: ``` 625: intro σ 626: have hhσ := hh σ 627: have hsqrt0 : ... 628: have hg0 : ... 629: have h1 : HasDerivAt ... := (hasDerivAt_id σ).sub_const β 630: have h2 : ... := by 631: have h2a := ((hasDerivAt_id σ).sub_const β).pow 2 632: have h2b := h2a.add_const (δ ^ 2) 633: convert h2b using 1 634: ring ``` Yes — 634 = h2's ring. Replace with... let me think about what goal convert leaves: `2 * (σ - β) = ↑(2 : ℕ) * (σ - β) ^ 1 * 1`?? Hmm — HasDerivAt.pow's statement in this Mathlib — let me actually check it.
The lemma is `continuous_finsetSum` (new name). Also checking `HasDerivAt.pow`'s exact derivative form to understand line 634.
tool: Bash
input
{
  "command": "grep -rn -A3 \"theorem HasDerivAt.pow \" \"D:/Python/Quantum/zeta-23-lean/.lake/packages/mathlib/Mathlib/Analysis/Calculus/Deriv/\" | head -8",
  "description": "Show HasDerivAt.pow signature"
}
#2287 assistant 2026-08-26 19:07:29.855 UTC · 2860→1817 tok open ⤢
thinking (2684 chars)
`HasDerivAt.pow (h) (n) : HasDerivAt (f ^ n) (n * f x ^ (n - 1) * f') x` — the function is `(f ^ n)` — PI.POWER — `f ^ 2` where f := fun x => x − β — the function is `(fun x => x−β)^2` (Pi.pow!) — NOT `fun y => (y−β)^2`! And h2b := h2a.add_const (δ^2) : HasDerivAt ((fun x => x−β)^2 + fun _ => δ^2)?? — HasDerivAt.add_const (hf) (c) : HasDerivAt (fun y => f y + c) f' — applied to `f ^ n` — gives `fun y => (f^2) y + δ^2` — hmm — `fun y => ((fun x => x−β)^2) y + δ^2` — with Pi.pow applied — beta-reducible to (y−β)^2 ✓ defeq. The derivative: `n * f x ^ (n - 1) * f'` with n := 2, f x := σ−β, f' := 1: `(2:ℕ... ` — wait `n * f x ^ (n - 1) * f'` — n : ℕ — the multiplication `n * ...` — n coerced to ℝ? — `↑2 * (σ−β)^(2−1) * 1`. So convert's goal could be: - function: `fun σ => (σ−β)^2+δ^2` vs `fun y => ((fun x => x−β)^2) y + δ^2` — the Pi.pow beta-redex — convert using 1's defeq-check should reduce it ✓... - OR convert leaves the function goal AND the derivative goal — with "case h" being one of them... Ugh — I've been speculating too long. The robust fix for the whole h2: use `by fun_prop`? — fun_prop with the Pi.pow form — it handles `fun σ => (σ−β)^2+δ^2` — computes derivative — the stated derivative 2*(σ−β) — fun_prop's HasDerivAt support: `HasDerivAt (fun x => (x-c)^2) (2*(x-c)) x` — fun_prop should prove via its rules (HasDerivAt.pow is a fun_prop lemma? or it uses deriv computation). Let me check: fun_prop does support HasDerivAt for elementary functions ✓ (it's the standard way: `by fun_prop`). But the EXACT derivative match: fun_prop computes the derivative via its own simp-normal form and then compares with the stated one using `rfl`/simp — 2*(σ−β) vs its computed 2*(σ−β)*1 or similar — usually closes ✓. Alternative safest: sidestep h2 entirely — prove the sqrt-chain derivative via the OUTER composition: HasDerivAt (fun σ => √((σ−β)^2+δ^2)) — I can ALSO get it from `(hasDerivAt_id σ).sub_const β |>.pow 2 |>.add_const _ |>.sqrt _` with simpa wrappers. Let me write: ```lean have h4 : HasDerivAt (fun σ => Real.sqrt ((σ - β) ^ 2 + δ ^ 2)) (2 * (σ - β) / (2 * Real.sqrt ((σ - β) ^ 2 + δ ^ 2))) σ := by have h2 : HasDerivAt (fun σ => (σ - β) ^ 2 + δ ^ 2) (2 * (σ - β)) σ := by have h2a := ((hasDerivAt_id σ).sub_const β).pow 2 have h2b := h2a.add_const (δ ^ 2) simpa using h2b exact h2.sqrt hhσ.ne' ``` `simpa using h2b` — simp normalizes: Pi.pow → applied pow (via Pi.pow_apply?), pow_one, mul_one, Nat.cast_ofNat (or Nat.cast_two → 2), — simp should turn h2b's statement into exactly the goal ✓ likely. Let me apply: h2's convert;ring → simpa using h2b. And hΣcont/hRHScont → continuous_finsetSum.
`HasDerivAt.pow` produces a `Pi.pow` form — that's what tripped the convert/ring. Fixing with `simpa` and the `continuous_finsetSum` name.
tool: Edit
input
{
  "replace_all": false,
  "file_path": "D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiSideBounds.lean",
  "old_string": "    have h2 : HasDerivAt (fun \u03c3 => (\u03c3 - \u03b2) ^ 2 + \u03b4 ^ 2) (2 * (\u03c3 - \u03b2)) \u03c3 := by\n      have h2a := ((hasDerivAt_id \u03c3).sub_const \u03b2).pow 2\n      have h2b := h2a.add_const (\u03b4 ^ 2)\n      convert h2b using 1\n      ring",
  "new_string": "    have h2 : HasDerivAt (fun \u03c3 => (\u03c3 - \u03b2) ^ 2 + \u03b4 ^ 2) (2 * (\u03c3 - \u03b2)) \u03c3 := by\n      have h2a := ((hasDerivAt_id \u03c3).sub_const \u03b2).pow 2\n      have h2b := h2a.add_const (\u03b4 ^ 2)\n      simpa using h2b"
}
tool: Edit
input
{
  "replace_all": false,
  "file_path": "D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiSideBounds.lean",
  "old_string": "  have h\u03a3cont : Continuous fun \u03c3 : \u211d =>\n      \u2211 \u03c1 \u2208 nearZerosSub T, (zeroMult \u03c1.1 : \u2102) /\n        (((\u03c3 : \u211d) + (T : \u211d) * Complex.I) - \u03c1.1) := by\n    apply continuous_finset_sum\n    intro \u03c1 h\u03c1\n    exact Continuous.div continuous_const (hpath.sub continuous_const) (hden \u03c1 h\u03c1)",
  "new_string": "  have h\u03a3cont : Continuous fun \u03c3 : \u211d =>\n      \u2211 \u03c1 \u2208 nearZerosSub T, (zeroMult \u03c1.1 : \u2102) /\n        (((\u03c3 : \u211d) + (T : \u211d) * Complex.I) - \u03c1.1) := by\n    exact continuous_finsetSum _ (fun \u03c1 h\u03c1 =>\n      Continuous.div continuous_const (hpath.sub continuous_const) (hden \u03c1 h\u03c1))"
}
tool: Edit
input
{
  "replace_all": false,
  "file_path": "D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiSideBounds.lean",
  "old_string": "  have hRHScont : Continuous fun \u03c3 : \u211d =>\n      \u2211 \u03c1 \u2208 nearZerosSub T, (zeroMult \u03c1.1 : \u211d) *\n        (1 / Real.sqrt ((\u03c3 - \u03c1.1.re) ^ 2 + (T - \u03c1.1.im) ^ 2)) := by\n    apply continuous_finset_sum\n    intro \u03c1 h\u03c1\n    exact htermcont \u03c1 h\u03c1",
  "new_string": "  have hRHScont : Continuous fun \u03c3 : \u211d =>\n      \u2211 \u03c1 \u2208 nearZerosSub T, (zeroMult \u03c1.1 : \u211d) *\n        (1 / Real.sqrt ((\u03c3 - \u03c1.1.re) ^ 2 + (T - \u03c1.1.im) ^ 2)) := by\n    exact continuous_finsetSum _ (fun \u03c1 h\u03c1 => htermcont \u03c1 h\u03c1)"
}
#2288 assistant 2026-08-26 19:11:38.472 UTC · 1507→4361 tok open ⤢
thinking (13653 chars)
Still `⊢ ?m.1004` right after hden — the hΣcont block STILL fails with a metavariable goal. The context shows it stops at hden — so the failing block is hΣcont: ```lean have hΣcont : Continuous fun σ : ℝ => ∑ ρ ∈ nearZerosSub T, (zeroMult ρ.1 : ℂ) / (((σ : ℝ) + (T : ℝ) * Complex.I) - ρ.1) := by exact continuous_finsetSum _ (fun ρ hρ => Continuous.div continuous_const (hpath.sub continuous_const) (hden ρ hρ)) ``` `⊢ ?m.1004` — the unsolved metavariable. The issue: `Continuous.div continuous_const (hpath.sub continuous_const) (hden ρ hρ)` — Continuous.div's THREE explicit args (hf) (hg) (h₀) — hmm wait — Continuous.div (hf : Continuous f) (hg : Continuous g) (h₀ : ∀ x, g x ≠ 0) : Continuous fun x => f x / g x — the f and g are implicit. When elaborating `Continuous.div continuous_const (hpath.sub continuous_const) (hden ρ hρ)`: - hf := continuous_const : Continuous fun x => ?c₁ — ?c₁ free. - hg := hpath.sub continuous_const : Continuous fun σ => (↑σ+↑T·I) − ?c₂ — ?c₂ free. - h₀ := hden ρ hρ : ∀ σ, (↑σ+↑T·I) − ρ.1 ≠ 0 — this should unify ?g := fun σ => (↑σ+↑T·I) − ρ.1 (pinning ?c₂ := ρ.1)... but wait — h₀'s type needs to match `∀ x, ?g x ≠ 0` where ?g is determined by hg's type: hg : Continuous (fun σ => (↑σ+↑T·I) − ?c₂) — so ?g is ALREADY the lambda `fun σ => (↑σ+↑T·I) − ?c₂` — then h₀'s expected type: ∀ x, (↑x+↑T·I) − ?c₂ ≠ 0 — hden ρ hρ : ∀ σ, (↑σ+↑T·I) − ρ.1 ≠ 0 — unify ?c₂ := ρ.1 ✓. - Then the RESULT: Continuous fun x => ?c₁ / ((↑x+↑T·I) − ρ.1) — unify with the expected type (from continuous_finsetSum's hf for this ρ): `Continuous fun σ => (zeroMult ρ.1:ℂ) / ((↑σ+↑T·I)−ρ.1)` — ?c₁ := (zeroMult ρ.1:ℂ) ✓. Hmm — that should work... UNLESS the expected-type propagation through `continuous_finsetSum _ (fun ρ hρ => ...)` doesn't give the lambda its expected type per-ρ. `continuous_finsetSum _ (fun ρ hρ => Continuous.div ...)` — the SECOND arg's expected type: ∀ i ∈ ?s, Continuous (?f i) — with ?f implicit — the elaborator must determine ?f from the goal (the hΣcont ascription): `Continuous fun σ => ∑ ρ ∈ nearZerosSub T, (zeroMult ρ.1:ℂ)/((↑σ+↑T·I)−ρ.1)` vs `Continuous fun a => ∑ i ∈ ?s, ?f i a` — unify `∑ ρ ∈ nearZerosSub T, TERM` with `∑ i ∈ ?s, ?f i a` — ?s := nearZerosSub T ✓ and `?f ρ a =?= (zeroMult ρ.1:ℂ)/((↑a+↑T·I)−ρ.1)` — HO — Miller pattern (ρ, a distinct bound vars) ✓ ?f := fun ρ a => (zeroMult ρ.1:ℂ)/((↑a+↑T·I)−ρ.1) ✓. Then the hf arg's expected type: ∀ ρ ∈ nearZerosSub T, Continuous (fun a => (zeroMult ρ.1:ℂ)/((↑a+↑T·I)−ρ.1)) — the provided lambda fun ρ hρ => ... gets this type ✓ then Continuous.div's elaboration as above ✓. So why `?m.1004`?? — maybe the `_` for s: `continuous_finsetSum _` — s explicit — `_` creates a metavariable to be unified with the goal — with the expected hΣcont type ✓ fine... Hmm — OR the elaboration of `continuous_finsetSum` goes through its TO_ADDITIVE generated form and the `open` namespaces cause a resolution issue — maybe it's picking the SEMINORM continuous_finsetSum (Analysis/Seminorm.lean:1165)?? — that one is `theorem continuous_finsetSum [TopologicalSpace E] ...` — seminorm-related — if the root-namespace resolution finds the Seminorm one FIRST (it's not in a namespace?), the unification fails producing ?m garbage. Hmm — the Seminorm one at :1165 — let me check its full signature/namespace. If it's `theorem continuous_finsetSum [TopologicalSpace E] {𝕜 E : Type*} ... (s : Finset ι) (hφ : ∀ i ∈ s, Continuous (φ i)) : Continuous (φ s)`?? — for seminorms: `Continuous (s.sum φ)`? — different conclusion — if Lean resolved `continuous_finsetSum` to THIS one, the `exact` would fail with a mismatch (not ?m). The error display: "unsolved goals ... ⊢ ?m.1004" — a METAVARIABLE goal remains. This happens when the elaborator defers a goal whose TYPE is an unassigned metavariable — e.g., a typeclass instance goal for a type that was never determined! Like `Continuous ?f` needing `TopologicalSpace ?X` with ?X free, or the `_` for s generating an mvar that leaked. OR — the `Continuous.div`'s f: `Continuous fun x => ?c₁ / ?g x` — needs the DIVISION structure — fine for ℂ. Hmm — wait — maybe the problem: `continuous_finsetSum` (the to_additive from continuous_finsetProd) — the to_additive-generated name — does `exact continuous_finsetSum ...` resolve? If the additive version wasn't generated properly (e.g., needs `open`?) — "unknown constant" would error instead. The ?m error suggests the constant RESOLVED but elaboration left a dangling mvar. Let me try a different approach for the sum-continuity that's fully explicit and avoids the problematic elaboration path — via `ContinuousOn`/`Finset.sum` induction? NO — simpler: use `by fun_prop`! The fun_prop attribute was on continuous_finsetProd (to_additive attr := continuity, fun_prop) — so fun_prop KNOWS how to prove finset-sum continuity: ```lean have hΣcont : Continuous fun σ : ℝ => ∑ ρ ∈ nearZerosSub T, (zeroMult ρ.1 : ℂ) / (((σ : ℝ) + (T : ℝ) * Complex.I) - ρ.1) := by fun_prop (disch := ...) ``` Hmm — but fun_prop needs to prove each term continuous: (zeroMult ρ.1:ℂ)/((↑σ+↑T·I)−ρ.1) — fun_prop's div rule requires the denominator-nonzero side condition — discharged via the discharger — fun_prop's default discharger can't prove hden. fun_prop (disch := ...) with a custom side-goal handler: `fun_prop (disch := intro σ h0; ...)`?? — the disch gets side goals like `(↑σ+↑T·I) − ρ.1 ≠ 0` — hmm, actually fun_prop for `Continuous fun σ => f σ / g σ` requires `∀ x, g x ≠ 0` as a side goal passed to the discharger. Writing a custom disch tactic block: ```lean fun_prop (disch := first | (intro h0; exact hden _ ‹_› _ h0) | assumption) ``` — messy and fragile. ALTERNATIVE — the cleanest: prove the sum-continuity by FINSET INDUCTION manually? No — simplest robust: use `ContinuousOn`-free direct: `continuous_finsetSum` with EVERYTHING pinned by ascription of the hf argument. The problem might be that `exact continuous_finsetSum _ (fun ρ hρ => ...)` elaborates the lambda BEFORE the head's type is fully determined. Give the full term with ascribed types: ```lean have hΣcont : Continuous fun σ : ℝ => ∑ ρ ∈ nearZerosSub T, (zeroMult ρ.1 : ℂ) / (((σ : ℝ) + (T : ℝ) * Complex.I) - ρ.1) := by have hterm : ∀ ρ ∈ nearZerosSub T, Continuous fun σ : ℝ => (zeroMult ρ.1 : ℂ) / (((σ : ℝ) + (T : ℝ) * Complex.I) - ρ.1) := by intro ρ hρ exact Continuous.div continuous_const (hpath.sub continuous_const) (hden ρ hρ) exact continuous_finsetSum _ hterm ``` With hterm's type ascribed per-ρ, the Continuous.div elaboration has a fixed expected type ✓ pins ?c₁ := zeroMult, ?c₂ := ρ.1 ✓. Then continuous_finsetSum _ hterm — s := nearZerosSub T unified from the goal ✓. Hmm — but would the ?m problem persist in hterm's exact? `Continuous.div continuous_const (hpath.sub continuous_const) (hden ρ hρ)` against the ascribed `Continuous fun σ => (zeroMult ρ.1:ℂ)/((↑σ+↑T·I)−ρ.1)` — the elaborator: Continuous.div's result `fun x => ?f x / ?g x` — unify with ascription — ?f := fun _ => zeroMult? — hmm — ?f x =?= (zeroMult ρ.1:ℂ) — Miller (const) ✓; ?g x =?= (↑x+↑T·I)−ρ.1 ✓ — then hf := continuous_const : Continuous ?f — ?f = fun x => zeroMult ✓; hg := hpath.sub continuous_const : Continuous fun σ => (↑σ+↑T·I) − ?c₂ — unify with ?g = fun σ => (↑σ+↑T·I)−ρ.1 — ?c₂ := ρ.1 ✓; h₀ : ∀ x, ?g x ≠ 0 ✓ hden ✓. Should work with the ascription. But WAIT — why did the metavariable appear in the FIRST place? `⊢ ?m.1004` as a GOAL — this looks like a `case h` goal — "case h" — the hΣcont block after `exact ...` should produce NO goals if exact succeeded; if exact FAILED, we'd see a type mismatch. A metavariable-typed goal `?m.1004` — I recall this happens when `elabTerm` defers a goal for an mvar that turns out to be a synthetic-Opaque goal — hmm — OR it's the `have h := ...` from `Continuous.div`'s TYPECLASS instances: `Continuous fun x => ?f x / ?g x` needs `Div ?Y`, `TopologicalSpace ?Y` — ?Y := ℂ ✓ pinned by ascription... Whatever — the ascribed-hterm route is more robust. Actually — let me try an even MORE robust formulation avoiding Continuous.div's implicit-matching entirely: prove hterm via the compositional route: the term is const · (denom)⁻¹ — `Continuous.mul continuous_const (Continuous.inv₀ (hpath.sub continuous_const) hden')`: `Continuous.inv₀ (hf : Continuous f) (h0 : ∀ x, f x ≠ 0) : Continuous fun x => (f x)⁻¹` — hmm — then m·(1/g) vs m/g: 1/g = g⁻¹ (one_div_eq_inv)... rewriting needed. OR use `Continuous.div` with named args: ```lean exact Continuous.div (f := fun _ : ℝ => (zeroMult ρ.1 : ℂ)) (g := fun σ : ℝ => ((σ : ℝ) + (T : ℝ) * Complex.I) - ρ.1) continuous_const (hpath.sub continuous_const) (hden ρ hρ) ``` With f, g pinned explicitly, no HO unification needed: hf := continuous_const : Continuous fun _ => zeroMult ✓ (defeq with f ✓); hg := hpath.sub continuous_const : Continuous fun σ => (↑σ+↑T·I) − ?c — vs g := fun σ => (↑σ+↑T·I)−ρ.1 — ?c₂ := ρ.1 ✓; h₀ := hden ρ hρ ✓. Result: Continuous fun x => (zeroMult)/((↑x+↑T·I)−ρ.1) — matches hterm's ascription ✓. Hmm — but wait, are f and g NAMED implicit in Continuous.div? From the grep: `theorem Continuous.div (hf : Continuous f) (hg : Continuous g) (h₀ : ∀ x, g x ≠ 0)` — f g come from section variables — likely `variable {f g : X → M}`. Names f and g probably. Risk: names differ. The ascribed-hterm route avoids name-dependence for the div... no it doesn't — same elaboration. Hmm. Let me just try the ascribed-hterm first — if the elaboration still leaves ?m, pin named args. Actually — hold on. Let me reconsider the ?m.1004: it appears at "case h" — maybe it's the SIDE GOAL of Continuous.div for a DIFFERENT reason: Continuous.div in GroupWithZero.lean — `theorem Continuous.div (hf : Continuous f) (hg : Continuous g) (h₀ : ∀ x, g x ≠ 0) : Continuous fun x => f x / g x` — with [TopologicalSpace X] [TopologicalSpace M] [GroupWithZero M] [ContinuousMul M]?? — instances — for ℂ fine. Hmm — the `case h` prefix — in the PREVIOUS run (before continuous_finsetSum fix), the error was identical at the same spot — so the ?m.1004 is NOT from continuous_finsetSum — it's from Continuous.div (shared by both versions)! Wait — but the context display ends at hden — the failing block is hΣcont (the first Continuous.div use)... — OR — is it possible the error is actually inside hden itself?? hden's context: `hden : ∀ ρ ∈ nearZerosSub T, ∀ (σ : ℝ), ↑σ + ↑T * I - ↑ρ ≠ 0` — hden is IN the context (proved ✓). So the failure is after hden — hΣcont ✓. Hmm — `⊢ ?m.1004` — maybe the problem is that `Continuous.div continuous_const ...` — `continuous_const` — signature: `theorem continuous_const {c : Y} : Continuous fun _ => c`?? — or `continuous_const (c : Y)`? — if continuous_const takes c EXPLICITLY, then `continuous_const` bare has type `Continuous fun _ => ?c` ✓ still fine. You know what — `?m.1004` with "case h" — I recall: when `exact` fails to unify, the error is a mismatch — but when the TERM contains a metavariable that blocks unification, Lean reports "unsolved goals ⊢ ?m.NNN" — the ?m is the TYPE of some subterm that couldn't be determined. In `Continuous.div continuous_const (hpath.sub continuous_const) (hden ρ hρ)` — the hf's function: `fun _ => ?c₁` — and the RESULT unification with hΣcont's ascribed type — WAIT — in the CURRENT version (after my edit), the hΣcont's ascription exists ✓ — but the error persists IDENTICALLY — meaning the elaboration never even reached the ascription unification?? — OR the ?m.1004 is generated inside Continuous.div's elaboration BEFORE result unification: Lean elaborates `Continuous.div hf hg h₀` by FIRST computing the head's type with metavars, then elaborating args AGAINST the domain types, THEN unifying the result with the expected type. The args: hf := continuous_const — its type `Continuous fun x => ?c` — hmm — `continuous_const`'s type: `Continuous (fun (x : ?X) => ?c)` — the DOMAIN TYPE ?X is ALSO a metavariable!! Unifying hf's type with `Continuous ?f` (the expected domain from Continuous.div's hf-slot) — Continuous.div (hf : Continuous ?f) — ?f := fun x => ?c₁ — hf : Continuous fun x => ?c — unify `fun (x : ?X) => ?c` with `fun (x : ℝ) => ?c₁`?? — domain ?X := ℝ ✓ ?c := ?c₁ ✓ fine. `hpath.sub continuous_const` — Continuous.sub (hf) (hg) — hpath : Continuous fun u : ℝ => (↑u+↑T·I) ✓; continuous_const : Continuous fun x => ?c₂ ✓ — result: Continuous fun x => (↑x+↑T·I) − ?c₂ ✓. h₀ := hden ρ hρ : ∀ σ : ℝ, (↑σ+↑T·I)−ρ.1 ≠ 0 — expected: ∀ x, ?g x ≠ 0 where ?g from hg := fun x => (↑x+↑T·I)−?c₂ — expected ∀ x, (↑x+↑T·I)−?c₂ ≠ 0 — unify with hden's: ?c₂ := ρ.1 ✓. Result: `Continuous fun x => ?c₁ / ((↑x+↑T·I)−ρ.1)` — unify with ascription `Continuous fun σ => (zeroMult ρ.1:ℂ)/((↑σ+↑T·I)−ρ.1)` — ?c₁ := zeroMult ✓ ALL DETERMINED. I genuinely don't see the mvar source. UNLESS — the `Continuous.div` I found is NOT the one being resolved! Maybe there's a NAMESPACED `Continuous.div` (e.g., `Continuous.div` in another file with different arg structure — like `Continuous.div (hf) (hg) : Continuous (f / g)` — the PI-DIVISION form `f / g` (function-space division) — NOT `fun x => f x / g x`!!). If THIS Mathlib's Continuous.div produces `Continuous (f / g)` (Pi.div), then unifying `Continuous (?f / ?g)` with `Continuous fun σ => (zeroMult)/((↑σ+↑T·I)−ρ.1)` — `(?f / ?g)` vs `fun σ => ?f σ / ?g σ` — Pi.div defeq — unify `?f / ?g =?= fun σ => ...` — HO problem — might leave ?m!!! The GroupWithZero.lean grep showed: `theorem Continuous.div (hf : Continuous f) (hg : Continuous g) (h₀ : ∀ x, g x ≠ 0) :` — the CONCLUSION line was cut! Let me check what the conclusion actually is.
The `?m.1004` persists at the same spot — I need to see `Continuous.div`'s actual conclusion (my grep truncated it).
tool: Bash
input
{
  "command": "sed -n '228,236p' \"D:/Python/Quantum/zeta-23-lean/.lake/packages/mathlib/Mathlib/Topology/Algebra/GroupWithZero.lean\"",
  "description": "Show Continuous.div full statement"
}
#2289 assistant 2026-08-26 19:16:36.469 UTC · 1441→566 tok open ⤢
thinking (1282 chars)
STILL `⊢ ?m.1004` at the same place (line 775:60, right after hden) — and ?m.1004 is the SAME metavariable number across runs?! That's suspicious — ?m.1004 identical across three different code versions. That means the failing point is NOT in hΣcont — it's a metavariable created EARLIER that never got assigned. Line 775:60 — let me look at what's EXACTLY at line 775 col 60 now. Line 775 in the current file — the theorem statement's `:= by` at the end of the statement!! Let me check: earlier read showed line 774-777: `‖HIntegral (Jlag n) (-5/2) 3 T‖ ≤ (Real.exp 4 + 1) * (2 * Z2hat T * Real.log (...) + (3 - -5/2) * (10.5 * Real.log (T + 2) + 48)) := by`. So 775:60 is INSIDE the statement — around `(3 - -5/2) * (10.5 * Real.log (T + 2) + 48))` — hmm col 60. WAIT — "unsolved goals ... ⊢ ?m.1004" AT THE THEOREM STATEMENT LINE — the elaboration of the STATEMENT creates a metavariable?! The statement: `‖HIntegral (Jlag n) (-5 / 2) 3 T‖ ≤ ...` — HIntegral (Jlag n) (-5/2) 3 T — HIntegral's args: f : ℂ → E, x₁ x₂ y : ℝ — E := ℂ ✓ — hmm. ?m.1004 — a metavariable from the STATEMENT elaboration that persists into the proof and is reported at the end?? — OR — the `Real.exp 4 + 1` — no. Hmm wait — let me count lines precisely in the CURRENT file. Let me read around 770-780.
The `?m.1004` is identical across versions — it's created earlier than I thought. Let me see exactly what's at line 775 col 60 now.
tool: Read
input
{
  "file_path": "D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiSideBounds.lean",
  "offset": 762,
  "limit": 45
}
#2290 assistant 2026-08-26 19:18:28.810 UTC · 1738→3098 tok open ⤢
thinking (8984 chars)
Line 775:60 — `(3 - -5 / 2) * (10.5 * Real.log (T + 2) + 48)) := by` — col 60 is right around the `))` before `:= by` — i.e., the error position is at the END of the theorem STATEMENT — the "unsolved goals ⊢ ?m.1004" is reported at the statement's end position. This means the elaboration of the STATEMENT itself creates an unassigned metavariable ?m.1004! Look at the statement: `‖HIntegral (Jlag n) (-5 / 2) 3 T‖ ≤ (Real.exp 4 + 1) * (2 * Z2hat T * Real.log (148.5 * C0hat (Real.sqrt (n : ℝ) + 2)) + (3 - -5 / 2) * (10.5 * Real.log (T + 2) + 48))`. Hmm — `Real.exp 4 + 1` — the `4` here — `Real.exp 4` — the numeral 4 : ℝ ✓. `Z2hat T` ✓. `C0hat (Real.sqrt (n:ℝ) + 2)` ✓. `148.5` ✓. `HIntegral (Jlag n) (-5/2) 3 T` — HIntegral (f : ℂ → E) (x₁ x₂ y : ℝ) : E — E := ℂ from Jlag n : ℂ → ℂ ✓. Wait — `(3 - -5 / 2)` — hmm — `3 - -5 / 2` parses as `3 - (-5/2)` ✓ (unary minus binds tighter) ✓ fine. What creates ?m.1004 in the STATEMENT? — OH WAIT. I bet it's `HIntegral`!! Look at HIntegral's def: `noncomputable def HIntegral (f : ℂ → E) (x₁ x₂ y : ℝ) : E := ∫ x in x₁..x₂, f (x + y * I)` — E is a TYPE VARIABLE — `HIntegral (Jlag n) (-5/2) 3 T` — E should unify to ℂ ✓... — but what if E is an implicit that DOESN'T get pinned?! Jlag n : ℂ → ℂ — so f : ℂ → E with E := ℂ ✓ pinned. Hmm — but wait, the error context lists all the haves UP TO hden — and the goal `⊢ ?m.1004`. The position 775:60 (statement end) — Lean reports "unsolved goals" at the END of the declaration when the proof finishes with goals OR when the TERM contains unassigned mvars. The ?m.1004 in the CONTEXT is AFTER hden — so it's created by the hΣcont block — and reported at the DECLARATION's end (775:60 = the `:= by`... hmm no — col 60 of line 775 — the line is ` (3 - -5 / 2) * (10.5 * Real.log (T + 2) + 48)) := by` — col 60 ≈ the `))` just before `:= by`. Hmm — actually the error's LINE number is where elaboration gave up — for "unsolved goals" it's typically the start of the proof or the declaration name... Hmm wait — maybe I should count differently: maybe the error is at hΣcont and the reported POSITION (775:60) is where the hΣcont block STARTS?? Line 775 in the CURRENT file is the statement's last line. hΣcont is around line 820+. So 775:60 is the statement. The ?m.1004 lives in the STATEMENT?! Look VERY carefully at the statement for something producing a metavariable: `‖HIntegral (Jlag n) (-5 / 2) 3 T‖` — hmm — `(-5 / 2)` for x₁ : ℝ ✓. `3` ✓. `T` ✓. `(Real.exp 4 + 1)` ✓. `(2 * Z2hat T * Real.log (148.5 * C0hat (Real.sqrt (n : ℝ) + 2)) + (3 - -5 / 2) * (10.5 * Real.log (T + 2) + 48))` ✓ all ℝ. Hmm — nothing metavariable-ish. UNLESS — the `:= by` proof contains ?m.1004 from hΣcont (unassigned ?c₁ from Continuous.div — my show-bridge SHOULD have fixed it — but wait, did my show-bridge actually make it into the file? The error ?m.1004 is IDENTICAL across three versions — including BEFORE the show-bridge. If the show-bridge were in effect, the mvar NUMBER would change (elaboration order changes) — SAME number ?m.1004 ⟹ the failing point is BEFORE hΣcont — something in the blocks UP TO hden creates ?m.1004, and hΣcont/htermcont etc. never even elaborate?! WAIT — the context display shows hden PROVED (in context). So blocks up to hden elaborated. The mvar ?m.1004 — created by hden's block?? hden: ```lean have hden : ∀ ρ : {ρ : ℂ // IsNontrivialZero ρ}, ρ ∈ nearZerosSub T → ∀ σ : ℝ, ((σ : ℝ) + (T : ℝ) * Complex.I) - ρ.1 ≠ 0 := by intro ρ _ σ h0 have hs : ((σ : ℝ) + (T : ℝ) * Complex.I) = ρ.1 := sub_eq_zero.mp h0 have hz := hzne σ rw [hs] at hz exact hz ρ.2.1 ``` Hmm — `intro ρ _ σ h0` — the statement has ∀ ρ, ρ ∈ nearZerosSub T → ∀ σ, ... — intro ρ _ σ h0 ✓. sub_eq_zero.mp h0 : (↑σ+↑T·I) = ρ.1 — wait — sub_eq_zero : a − b = 0 ↔ a = b ✓. rw [hs] at hz — hz : ζ (↑σ+↑T·I) ≠ 0 — rw [hs] rewrites (↑σ+↑T·I) → ρ.1 — hz : ζ ρ.1 ≠ 0 — exact hz ρ.2.1 : ρ.2.1 : ζ ρ.1 = 0 — hz ρ.2.1 : False ✓. Fine — no mvar. Hmm — then the NEXT block is hΣcont — ?m.1004 SHOULD be from hΣcont. But identical number across the pre-show-bridge and post-show-bridge versions... — actually mvar numbering depends on the WHOLE elaboration history — my show-bridge edit changed hΣcont's internals but NOT the blocks before it — if the failing block were hΣcont, the ?m number would likely CHANGE (the blocks' internal elaboration changed). SAME number ⟹ the failure is BEFORE hΣcont — in a block whose elaboration is IDENTICAL across versions ⟹ hden or earlier. And the "case h" — hmm. WAIT — maybe it's `hpath.sub continuous_const` INSIDE hΣcont's hterm — no... Hold on — let me look at the error message ONE more time: "775:60: error: unsolved goals ... ⊢ ?m.1004" followed by TWO context dumps. The FIRST context dump ends at hden; the second (truncated) at hLDintbl. TWO dumps = TWO unsolved goals? No — one error, two dumps shown sequentially (the second is the full context, truncated by head -60). Hmm — actually the two dumps differ: first ends at hden, second at hLDintbl (truncated by head). So the error has ONE goal ?m.1004 with context ending at hden. OK — which block comes RIGHT after hden? hΣcont. And ?m.1004 is its metavariable. The number being identical — maybe ?m.1004 is created at a FIXED point early in hΣcont's elaboration, and my show-bridge edit didn't actually change the elaboration path that creates it... My current hΣcont: ```lean have hΣcont : Continuous fun σ : ℝ => ∑ ρ ∈ nearZerosSub T, (zeroMult ρ.1 : ℂ) / (((σ : ℝ) + (T : ℝ) * Complex.I) - ρ.1) := by have hterm : ∀ ρ ∈ nearZerosSub T, Continuous fun σ : ℝ => (zeroMult ρ.1 : ℂ) / (((σ : ℝ) + (T : ℝ) * Complex.I) - ρ.1) := by intro ρ hρ show Continuous ((fun _ : ℝ => (zeroMult ρ.1 : ℂ)) / (fun σ : ℝ => ((σ : ℝ) + (T : ℝ) * Complex.I) - ρ.1)) exact Continuous.div continuous_const (hpath.sub continuous_const) (hden ρ hρ) exact continuous_finsetSum _ hterm ``` If the show-bridge works, hterm elaborates cleanly. Then `exact continuous_finsetSum _ hterm` — continuous_finsetSum (s) (hf) : Continuous fun a => ∑ i ∈ s, f i a — unify with hΣcont's ascription — ?s := nearZerosSub T, ?f := fun ρ σ => (zeroMult ρ.1:ℂ)/((↑σ+↑T·I)−ρ.1) — then hf's expected type: ∀ i ∈ ?s, Continuous (?f i) — hterm : ∀ ρ ∈ nearZerosSub T, Continuous fun σ => (zeroMult ρ.1:ℂ)/((↑σ+↑T·I)−ρ.1) — unify `Continuous (?f ρ)` with `Continuous fun σ => TERM` — ?f := fun ρ σ => TERM ✓. Hmm — `∀ ρ ∈ nearZerosSub T, ...` vs `∀ i ∈ ?s, ...` — the ∈-form: `∀ i, i ∈ ?s → ...` — matches ✓. STILL — ?m.1004 identical. So the failure is NOT in hΣcont! What about... — WAIT. Maybe the failing block is hLDc's ContinuousAt.comp!! Look at hLDc: ```lean have hLDc : Continuous fun σ : ℝ => logDeriv riemannZeta ((σ : ℝ) + (T : ℝ) * Complex.I) := by rw [continuous_iff_continuousAt] intro σ have hf : ContinuousAt (fun u : ℝ => ((u : ℝ) + (T : ℝ) * Complex.I)) σ := hpath.continuousAt exact ContinuousAt.comp (g := logDeriv riemannZeta) (f := fun u : ℝ => ((u : ℝ) + (T : ℝ) * Complex.I)) (x := σ) (continuousAt_logDeriv_zeta (hne1 σ) (hzne σ)) hf ``` hLDc is IN the context (proved ✓). hJcont, hcontJ, hcontG, hJintbl, hLDcont, hLDintbl, hKLcont, hKLintbl, hden — ALL in context ✓. So the failing block is DEFINITELY hΣcont (first block after hden)... — unless the error's context is from an EARLIER elaboration state and ?m.1004 was created by a block that APPEARS in context (its TYPE was proved but with a sorry due to the mvar — Lean inserts sorry for failed proofs and CONTINUES — so hΣcont could be "in context" as a sorry-backed hypothesis... — but hΣcont is NOT in the context dump! The dump ends at hden ✓ — so the failure is hΣcont ✓✓ (hΣcont would appear if it had elaborated, even as sorry). Hmm — wait — actually when a `have` fails, does Lean continue past it? For `have h : T := by tac` failing — Lean reports the error, assigns h := sorry, and CONTINUES elaborating the rest ✓ — so subsequent blocks WOULD be elaborated and appear in later contexts — the second dump (truncated at hLDintbl) is from a DIFFERENT part?? — no — one error = one goal = one context. The second dump is the SAME context displayed fuller (the first dump was abbreviated for the goal display, the second is "case h" full?). Actually the format "unsolved goals\ncase h\n<context>\n⊢ ?m.1004\n\n<context2>" — hmm — Lean 4's "unsolved goals" message: "unsolved goals\ncase h\n<local context>\n⊢ GOAL" — ONE context. Then ANOTHER context dump follows — maybe a SECOND unsolved goal collapsed into one error line by head? — the `head -60` cut whatever follows. There might be more errors below! Let me get the FULL error list (no head limit) — maybe there are MULTIPLE errors and I'm only seeing the first. The ?m.1004 might be from htermcont (which comes later) and hΣcont actually FAILED differently (its error might be below the cut). Let me re-run and capture ALL errors with grep for "error:".
The identical `?m.1004` across versions means I'm only seeing part of the picture. Getting the complete error list.
tool: Bash
input
{
  "command": "export PATH=\"$HOME/.elan/bin:$PATH\" && cd \"D:/Python/Quantum/research/lean/NguyenRH1\" && lake env lean NguyenRH1/LiSideBounds.lean 2>&1 | grep -E \"^NguyenRH1.*error\" | head -20",
  "description": "List all error lines in LiSideBounds",
  "timeout": 600000
}
#2291 assistant 2026-08-26 19:25:11.535 UTC · 1288→2581 tok open ⤢
thinking (5133 chars)
Error 1070: `rw [hsq] at hmain ⊢` — I wrote `rw [hsq] at hmain ⊢` — TWO targets — hsq : (T − γ)² = |γ−T|² — hmain's statement has `(T−γ)²` inside the integrand ✓ should rewrite there... wait no — the error says it didn't find `(T - (↑ρ).im)^2` in the TARGET (⊢). The target: `∫ σ in -5/2..3, 1/√((σ−β)² + |γ−T|²) ≤ 2·log(16.5/|γ−T|)` — WAIT — the target ALREADY has |γ−T|²! — that's hmain's form after... no — hold on. The hper goal: `∫ σ, 1/√((σ−ρ.1.re)² + (T−ρ.1.im)²) ≤ 2·log(148.5·C0hat)`. And hmain := per_zero_integral_le_two_log ... : `∫ σ, 1/√((σ−β)² + δ²) ≤ 2·log(16.5/δ)` with δ := |γ−T| — hmain's integrand has |γ−T|² (from δ²). I wrote `rw [hsq] at hmain ⊢` — hsq : (T−γ)² = |γ−T|²: - at hmain: hmain has |γ−T|² (not (T−γ)²) — rw [hsq] looks for (T−γ)² — NOT FOUND in hmain either!! — hmm — wait, actually — hmain := per_zero_integral_le_two_log ρ.2.2.1 ρ.2.2.2 hδpos hδ1.le — per_zero's δ is implicit {δ} — determined by hδpos : 0 < |γ−T| — so δ := |γ−T| — hmain : ∫ σ, 1/√((σ−ρ.1.re)² + |γ−T|²) ≤ 2·log(16.5/|γ−T|) — contains |γ−T|² not (T−γ)² — so rw [hsq] at hmain fails (but the error shown is about ⊢). And at ⊢: the goal has (T−γ)² ✓ rw [hsq] at ⊢ should find it... — the error message shows the ⊢ target WITH |γ−T|² already — meaning hsq got rewritten somewhere?? — no — the error says the PATTERN (T−γ)² was not found in the displayed target — which ALREADY shows |γ−T|². OH — I see — `rw [hsq] at hmain ⊢` — Lean processes the locations left-to-right: `at hmain` FIRST — fails (no (T−γ)² in hmain)... — hmm, but then the error would be about hmain. The displayed target is the GOAL ⊢ — with |γ−T|² — so the rw at ⊢ also ran and... — wait — if `at hmain` fails, the whole rw fails with the error about hmain. The error shows the pattern not found "in the target expression" — "target" = the GOAL ⊢ — and the displayed goal already has |γ−T|². Hmm — but the goal should have (T−γ)² (from hper's statement)... WAIT — look at the displayed target: `∫ σ in -5/2..3, 1/√((σ−β)² + |γ−T|²) ≤ 2·log(16.5/|γ−T|)` — the RHS is `2·log(16.5/|γ−T|)` — that's hmain's STATEMENT, not hper's goal (which has `2·log(148.5·C0hat)`)! So this "target" IS hmain (the rw failed at hmain first). The error message's "target expression" is hmain's type. So: `rw [hsq] at hmain ⊢` — hsq's LHS (T−γ)² not in hmain (hmain has |γ−T|² — hsq's LHS is (T−γ)² — these are DIFFERENT subterms!) — I had the direction BACKWARDS again. What I actually want: hper's goal has (T−γ)²; hmain has δ² = |γ−T|². To connect: rewrite the GOAL's (T−γ)² into |γ−T|² — that's hsq : (T−γ)² = |γ−T|² applied FORWARD at ⊢: `rw [hsq]` — but hsq's own proof needs the direction I just analyzed: `(T−γ)² = |γ−T|²` — prove by: rw [sq_abs] turning |γ−T|² → (γ−T)², then ring (since (T−γ)² = (γ−T)²). So: ```lean have hsq : (T - ρ.1.im) ^ 2 = |ρ.1.im - T| ^ 2 := by rw [sq_abs (ρ.1.im - T)] ring have hmain := per_zero_integral_le_two_log ρ.2.2.1 ρ.2.2.2 hδpos hδ1.le rw [hsq] -- only at the goal, not at hmain ``` Then goal: ∫ 1/√((σ−β)²+|γ−T|²) ≤ 2·log(148.5·C0hat) — matches hmain's LHS ✓ then linarith [hmain, h3] as before ✓. Error 1101: hsum's Finset.mul_sum step: ```lean _ = (2 * Real.log (...)) * ∑ ρ ∈ nearZerosSub T, (zeroMult ρ.1 : ℝ) := by rw [← Finset.mul_sum] apply Finset.sum_congr rfl intro ρ _ ring ``` The error: `rw [← Finset.mul_sum]` — pattern `∑ i ∈ ?s, ?a * ?f i` not found in target `∑ ρ, ↑(zeroMult ↑ρ) * (2*log(...)) = (2*log(...)) * ∑ ρ, ↑(zeroMult ↑ρ)`. Hmm — Finset.mul_sum (s) (f) (a) : a * ∑ i ∈ s, f i = ∑ i ∈ s, a * f i — ← direction: pattern `∑ i ∈ ?s, ?a * ?f i` — my LHS terms: `↑(zeroMult ρ) * (2·log(...))` — the constant factor is on the RIGHT (m * B0), not LEFT (a * f i)! The pattern needs `?a * ?f i` — mine is `?f i * ?a` — no match (rw is syntactic). Fix: first rewrite each term's product to constant-left, or use Finset.sum_mul (the constant-RIGHT version): `Finset.sum_mul (s) (f) (a) : (∑ i ∈ s, f i) * a = ∑ i ∈ s, f i * a` — ← Finset.sum_mul : pattern `∑ i ∈ s, ?f i * ?a` ✓ matches my LHS → `(∑ ρ, ↑(zeroMult ρ)) * (2·log)` — then the goal's RHS is `(2·log) * ∑` — mul_comm. So: ```lean _ = (2 * Real.log (148.5 * C0hat (Real.sqrt (n : ℝ) + 2))) * ∑ ρ ∈ nearZerosSub T, (zeroMult ρ.1 : ℝ) := by rw [← Finset.sum_mul, mul_comm] ``` rw [← Finset.sum_mul]: `∑ ρ, ↑(zeroMult ρ) * B0` → `(∑ ρ, ↑(zeroMult ρ)) * B0` — goal becomes `(∑ m) * B0 = B0 * (∑ m)` — rw [mul_comm] — rewrites the FIRST product found — `(∑ m) * B0` → `B0 * (∑ m)` — goal `B0 * (∑ m) = B0 * (∑ m)` ✓ rfl-closed ✓. Hmm — but mul_comm might rewrite the WRONG product — the goal after ← Finset.sum_mul: `(∑ ρ ∈ nearZerosSub T, ↑(zeroMult ↑ρ)) * (2·log(...)) = (2·log(...)) * ∑ ρ, ↑(zeroMult ↑ρ)` — products: the outer two — mul_comm rewrites the first found (LHS outer) ✓ then rfl ✓. Safe. Let me check Finset.sum_mul exists: in Mathlib/Algebra/BigOperators/Ring/Finset.lean next to mul_sum ✓ (`lemma sum_mul (s : Finset ι) (f : ι → R) (a : R) : (∑ i ∈ s, f i) * a = ∑ i ∈ s, f i * a` — inside namespace Finset ✓). Apply all three fixes.
Three direction fixes: `sq_abs` forward (not reverse), drop the `at hmain ⊢` double-target, and `Finset.sum_mul` (constant-right) instead of `Finset.mul_sum`.
tool: Edit
input
{
  "replace_all": false,
  "file_path": "D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiSideBounds.lean",
  "old_string": "    have hsq : (T - \u03c1.1.im) ^ 2 = |\u03c1.1.im - T| ^ 2 := by\n      rw [\u2190 sq_abs (\u03c1.1.im - T)]\n      ring\n    have hmain := per_zero_integral_le_two_log \u03c1.2.2.1 \u03c1.2.2.2 h\u03b4pos h\u03b41.le\n    rw [hsq] at hmain \u22a2",
  "new_string": "    have hsq : (T - \u03c1.1.im) ^ 2 = |\u03c1.1.im - T| ^ 2 := by\n      rw [sq_abs (\u03c1.1.im - T)]\n      ring\n    have hmain := per_zero_integral_le_two_log \u03c1.2.2.1 \u03c1.2.2.2 h\u03b4pos h\u03b41.le\n    rw [hsq]"
}
tool: Edit
input
{
  "replace_all": false,
  "file_path": "D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiSideBounds.lean",
  "old_string": "      _ = (2 * Real.log (148.5 * C0hat (Real.sqrt (n : \u211d) + 2))) *\n            \u2211 \u03c1 \u2208 nearZerosSub T, (zeroMult \u03c1.1 : \u211d) := by\n          rw [\u2190 Finset.mul_sum]\n          apply Finset.sum_congr rfl\n          intro \u03c1 _\n          ring",
  "new_string": "      _ = (2 * Real.log (148.5 * C0hat (Real.sqrt (n : \u211d) + 2))) *\n            \u2211 \u03c1 \u2208 nearZerosSub T, (zeroMult \u03c1.1 : \u211d) := by\n          rw [\u2190 Finset.sum_mul, mul_comm]"
}
#2292 assistant 2026-08-26 19:25:22.637 UTC · 2702→152 tok open ⤢
thinking (158 chars)
The file was modified since read (my sed rename command modified it outside the Edit tool's tracking). I need to re-read the relevant sections before editing.
The sed rename invalidated the read cache. Re-reading the relevant regions.
tool: Read
input
{
  "file_path": "D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiSideBounds.lean",
  "offset": 1055,
  "limit": 55
}
#2293 assistant 2026-08-26 19:27:10.315 UTC · 387→264 tok open ⤢
thinking (121 chars)
CLEAN — zero errors/warnings. §4 (6-c-iv) elaborates green. Now: mark §4 DONE in the header, full build, audit, commit.
§4 clean. Header marker, full build, audit, commit.
tool: Edit
input
{
  "replace_all": false,
  "file_path": "D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiSideBounds.lean",
  "old_string": "\u00a74 (6-c-iv): the mid-horizontal segment machine [\u22125/2, 3].\n\u00a75 (6-c-v): the full horizontal assembly (mid + DL-5 tails, both sides).",
  "new_string": "\u00a74 (6-c-iv) DONE: the mid-horizontal segment machine [\u22125/2, 3].\n\u00a75 (6-c-v): the full horizontal assembly (mid + DL-5 tails, both sides)."
}
#2294 assistant 2026-08-26 19:31:58.371 UTC · 1060→2265 tok open ⤢
thinking (4589 chars)
6-c-iv committed (1fed1df). Now 6-c-v — the horizontal assembly. This is much simpler: **6-c-v RECIPE:** HIntegral_split at 3 (integrability via zeta_ne_zero_horizontal for the mid, re ≥ 3 zero-free for the tail) ⟹ ‖H(−5/2→2√n, ±T)‖ ≤ mid + DogLegRight (dogLegUpper_two_sides, hIntegral_Jlag_low(_neg)_norm_le_dogLegUpper ✓ PROVED); both sides: ≤ 2(e⁴+1)(NI + 5.5(...)) + 2·DogLegRight. So: - ‖H(−5/2→2√n, T)‖ = ‖H(−5/2→3, T) + H(3→2√n, T)‖ ≤ ‖H(−5/2→3, T)‖ + ‖H(3→2√n, T)‖ ≤ mid-bound + DL-tail-bound. The DL-5 tail bounds: - `hIntegral_Jlag_low_le_dogLegUpper {n} {T} (hn : 16 ≤ n) (hTn) (hTn1) : ‖HIntegral (Jlag n) 3 (2√n) T‖ ≤ ((√n+1)/2)(e^{5/2}+1)/2 + 8.38 + IIMaj/4` - `hIntegral_Jlag_low_neg_norm_le_dogLegUpper` — same for (−T). And DogLegRight (n : ℝ) — from LiSkeleton line 762: `def DogLegRight (n : ℝ) : ℝ := ...` — let me check its exact def (it should be ((√n+1)/2)(e^{5/2}+1)/2 + 8.38 + IIMaj/4 — the per-side charge). And `dogLegUpper_two_sides` (DL-5 capstone): ‖H(3→2√n, T)‖ + ‖H(3→2√n, −T)‖ ≤ 2·DogLegRight... wait — dogLegUpper_two_sides's statement (lines 1404-1408): ‖HIntegral (Jlag n) 3 (2√n) T‖ + ‖HIntegral (Jlag n) 3 (2√n) (−T)‖ ≤ 2·((√n+1)/2·(e^{5/2}+1)/2 + 8.38 + IIMaj/4) — hmm — the RHS is written as 2·(...) — and DogLegRight is presumably exactly that (...). Let me check DogLegRight's def. The 6-c-v deliverable (per the 6-d recipe): the horizontal charge is `2(e⁴+1)(NI + 5.5(...)) + 2·DogLegRight` where NI = 2·Z2hat(T)·log(148.5·C0hat(√n+2)). Hmm — actually for the FOLD (6-d), what form is most useful? The fold: |Sf n| ≤ (1/2π)(I_III + Horiz + e^{1/3}IIMaj) + (4F+1) where Horiz = ‖H(−5/2→2√n, T)‖ + ‖H(−5/2→2√n, −T)‖... — wait — actually the RectangleIntegral decomposes as H(−T) − H(T) + V(right) − V(left) — so ‖RectInt‖ ≤ ‖H(−T)‖ + ‖H(T)‖ + ‖V(right)‖ + ‖V(left)‖ — the "sides" sum. The horizontal assembly's deliverable: ‖H(−5/2→2√n, T)‖ + ‖H(−5/2→2√n, −T)‖ ≤ 2·mid-bound + 2·DogLegRight — i.e., a two-sides lemma like dogLegUpper_two_sides. Per 6-c-v recipe: "both sides: ≤ 2(e⁴+1)(NI + 5.5(...)) + 2·DogLegRight". So: ```lean theorem hIntegral_Jlag_full_two_sides (hZC) (hZC2) (hNLZ) {n} (hn : 100000 ≤ n) {T} (hTwide) (hTn) (hTn1) : ‖HIntegral (Jlag n) (-5/2) (2 * Real.sqrt n) T‖ + ‖HIntegral (Jlag n) (-5/2) (2 * Real.sqrt n) (-T)‖ ≤ 2 * ((Real.exp 4 + 1) * (2 * Z2hat T * Real.log (148.5 * C0hat (√n+2)) + (3 - -5/2) * (10.5 * Real.log (T + 2) + 48))) + 2 * DogLegRight (n : ℝ) ``` Steps: 1. HIntegral_split at 3 for each side: H(−5/2→2√n, ±T) = H(−5/2→3, ±T) + H(3→2√n, ±T) — needs integrability on [−5/2,3] and [3,2√n] of the Jlag path at ±T. - On [−5/2,3]: continuity via zeta_ne_zero_horizontal (my hcontJ from 6-c-iv — but that's inside the mid theorem's proof — I need it again here or factor it... hmm — hcontJ was a local have in hIntegral_Jlag_mid_norm_le. For 6-c-v I need the integrability again. Factor a global continuity lemma? — actually the HIntegral_split (from §1 scaffolding) takes (h1 : IntervalIntegrable ... a b) (h2 : ... b c). For [−5/2,3] — rebuild the hcontJ machine (or factor). For [3,2√n] — re ≥ 3 zero-free — the DL continuity machine: `holomorphicOn_Jlag_dogleg` or `Jlag_differentiableAt_dogleg` (DL-1) — gives holomorphicity on Re ≥ 3! So IntervalIntegrable from the DL lemmas? Let me check what DL-5 used for integrability when it split... — DL-5's hIntegral_Jlag_low_le_dogLegUpper is about H(3→2√n) — its own proof handled the integrability internally. For MY split at 3, I need IntervalIntegrable on both [−5/2,3] and [3,2√n] of (fun σ => Jlag n (↑σ + ↑T·I)). - [−5/2,3]: hcontJ from 6-c-iv (rebuild — it's ~15 lines). - [3,2√n]: Jlag continuous on re ≥ 3 path — from DL-1's Jlag_differentiableAt_dogleg / holomorphicOn_Jlag_dogleg — those are about the Jlag on the dog-leg region. Let me check their exact statements — if they give DifferentiableAt Jlag at each point with re ≥ 3, then continuousAt ⟹ intervalIntegrable via Continuous.intervalIntegrable with the comp pattern (fun_prop path + ContinuousAt.comp pinned) — or via holomorphicOn → continuousOn → intervalIntegrable: `ContinuousOn.intervalIntegrable (hab : a ≤ b) (h : ContinuousOn f (uIcc a b))`. The path's image {σ + iT : σ ∈ [3, 2√n]} ⊆ {re ≥ 3} ✓. 2. Then ‖H(−5/2→2√n, ±T)‖ ≤ ‖H(−5/2→3, ±T)‖ + ‖H(3→2√n, ±T)‖ (norm_add_le) ≤ mid + DL-tail (the 6-c-iv mid bound + DL-5's hIntegral_Jlag_low(_neg)_norm_le_dogLegUpper). 3. Sum the two sides + final arithmetic to the stated form. Let me check the exact DL lemma names/statements + DogLegRight's def + holomorphicOn_Jlag_dogleg.
6-c-iv committed (`1fed1df`). Now 6-c-v (horizontal assembly) — the lightest of the four. Checking the DL-5 split-integrability machinery and `DogLegRight`'s definition.
tool: TodoWrite ok
input
{
  "todos": [
    {
      "content": "6-c-ii: I_III left-vertical bound vIntegral_Jlag_left_norm_le",
      "status": "completed",
      "activeForm": "Proving 6-c-ii I_III left-vertical bound"
    },
    {
      "content": "6-c-iii: I_I right-vertical bound (F-E form)",
      "status": "completed",
      "activeForm": "Proving 6-c-iii I_I bound"
    },
    {
      "content": "6-c-iv: mid-horizontal segment machine",
      "status": "completed",
      "activeForm": "Proving 6-c-iv mid-horizontal machine"
    },
    {
      "content": "6-c-v: horizontal assembly (mid + DL-5 tails)",
      "status": "in_progress",
      "activeForm": "Assembling 6-c-v horizontal bound"
    },
    {
      "content": "6-d: fold \u27f9 sfBoundProp (then commit, update archive)",
      "status": "pending",
      "activeForm": "Proving 6-d fold to sfBoundProp"
    }
  ]
}
result 2026-08-26 19:31:58Z → 2026-08-26 19:31:58Z
{
  "oldTodos": [
    "content='6-c-ii: I_III left-vertical bound vIntegral_Jlag_left_norm_le' status='completed' active_form='Proving 6-c-ii I_III left-vertical bound'",
    "content='6-c-iii: I_I right-vertical bound (F-E form)' status='completed' active_form='Proving 6-c-iii I_I bound'",
    "content='6-c-iv: mid-horizontal segment machine' status='in_progress' active_form='Proving 6-c-iv mid-horizontal machine'",
    "content='6-c-v: horizontal assembly (mid + DL-5 tails)' status='pending' active_form='Assembling 6-c-v horizontal bound'",
    "content='6-d: fold \u27f9 sfBoundProp (then commit, update archive)' status='pending' active_form='Proving 6-d fold to sfBoundProp'"
  ],
  "newTodos": [
    "content='6-c-ii: I_III left-vertical bound vIntegral_Jlag_left_norm_le' status='completed' active_form='Proving 6-c-ii I_III left-vertical bound'",
    "content='6-c-iii: I_I right-vertical bound (F-E form)' status='completed' active_form='Proving 6-c-iii I_I bound'",
    "content='6-c-iv: mid-horizontal segment machine' status='completed' active_form='Proving 6-c-iv mid-horizontal machine'",
    "content='6-c-v: horizontal assembly (mid + DL-5 tails)' status='in_progress' active_form='Assembling 6-c-v horizontal bound'",
    "content='6-d: fold \u27f9 sfBoundProp (then commit, update archive)' status='pending' active_form='Proving 6-d fold to sfBoundProp'"
  ],
  "cleared": false
}
tool: Grep
input
{
  "pattern": "def DogLegRight|theorem holomorphicOn_Jlag_dogleg|theorem Jlag_differentiableAt_dogleg|theorem ContinuousOn\\.intervalIntegrable|theorem continuousOn_intervalIntegrable",
  "path": "D:\\Python\\Quantum",
  "output_mode": "content",
  "-A": 8,
  "glob": "*.lean",
  "head_limit": 60
}
#2295 assistant 2026-08-26 19:36:08.686 UTC · 3654→5496 tok open ⤢
thinking (15540 chars)
Key facts: - `DogLegRight (n : ℝ) := ((√n + 1)/2)·(e^{5/2}+1)/2 + 8.38 + IIMaj n / 4` — exactly the DL-5 per-side bound ✓. - `Jlag_differentiableAt_dogleg {n} {s} (hre : 3 ≤ s.re) (him : 0 < s.im) : DifferentiableAt ℂ (Jlag n) s` — for the [3, 2√n] tail at height +T: points σ+iT have re = σ ≥ 3 ✓ im = T > 0 ✓. For the BOTTOM side (−T): im = −T < 0 — him : 0 < s.im FAILS! — hmm — the dog-leg differentiability needs 0 < im. For the bottom side, ζ is zero-free on re ≥ 1 regardless of im sign — `riemannZeta_ne_zero_of_one_lt_re` works for any im ✓ — so for the bottom I can build the differentiability via `Jlag_differentiableAt` directly (like DL-1's proof but without the him condition). Let me check `Jlag_differentiableAt`'s signature — line 50: `apply Jlag_differentiableAt` with two side goals: s ≠ 1 (rintro rfl; simp at him) and ζ s ≠ 0 (riemannZeta_ne_zero_of_one_lt_re). So `Jlag_differentiableAt (h1 : s ≠ 1) (hζ : riemannZeta s ≠ 0) : DifferentiableAt ℂ (Jlag n) s` presumably. For the bottom side's tail integrability, I need Jlag continuous on the path σ+i(−T), σ ∈ [3, 2√n] — via Jlag_differentiableAt (s ≠ 1 since re ≥ 3 > 1... wait s ≠ 1: re = 3 ≤ s.re... s with re ≥ 3: if s = 1 then re = 1 ≠ ≥ 3 ✓) (ζ s ≠ 0 via riemannZeta_ne_zero_of_one_lt_re (1 < re since 3 ≤ re ⟹ 1 < re ✓)). So a small helper: for both signs, Jlag differentiableAt at points with re ≥ 3 and im = ±T — actually im doesn't matter (ζ zero-free on re > 1). So the tail integrability helper (both signs of T): ```lean have hJcontTail : ∀ σ : ℝ, (3 : ℝ) ≤ σ → ∀ y : ℝ, ContinuousAt (Jlag n) ((σ : ℝ) + (y : ℝ) * Complex.I) := by intro σ hσ y have hre : (((σ : ℝ) + (y : ℝ) * Complex.I)).re = σ := by simp have hs1 : ((σ : ℝ) + (y : ℝ) * Complex.I) ≠ 1 := by intro h rw [h] at hre simp at hre linarith [hσ] have hζ : riemannZeta ((σ : ℝ) + (y : ℝ) * Complex.I) ≠ 0 := by apply riemannZeta_ne_zero_of_one_lt_re rw [hre] linarith [hσ] exact Jlag_differentiableAt hs1 hζ |>.continuousAt ``` Hmm — wait — `Jlag_differentiableAt` — its signature: from DL-1's usage: `apply Jlag_differentiableAt` produced goals `s ≠ 1`? (rintro rfl — hmm — rintro rfl introduces s = 1?? — the goal after apply was `(s = 1) → False`?? — no — `rintro rfl` on goal `s ≠ 1` = `(s = 1) → False` — rintro rfl intro's s := 1, then `simp at him` — him : 0 < (1:ℂ).im = 0 — contradiction ✓). And second goal `riemannZeta s ≠ 0` (via riemannZeta_ne_zero_of_one_lt_re) ✓. So Jlag_differentiableAt (h1 : s ≠ 1) (hζ : ζ s ≠ 0) : DifferentiableAt ℂ (Jlag n) s ✓ (let me verify its exact signature — grep). Then the tail IntervalIntegrable: ```lean have hTailInt : ∀ y : ℝ, IntervalIntegrable (fun σ : ℝ => Jlag n ((σ : ℝ) + (y : ℝ) * Complex.I)) MeasureTheory.volume 3 (2 * Real.sqrt (n : ℝ)) := by intro y apply Continuous.intervalIntegrable rw [continuous_iff_continuousAt] intro σ -- need 3 ≤ σ?? — the path covers ALL σ (not just [3, 2√n]) — Continuous.intervalIntegrable needs GLOBAL continuity ``` Hmm — Continuous.intervalIntegrable needs GLOBAL continuity of the integrand — but Jlag is only continuous where ζ ≠ 0. The [3, 2√n] path is zero-free, but the GLOBAL function t ↦ Jlag (↑σ+↑y·I) isn't continuous everywhere (it hits zeros for σ < 1!). So I need `ContinuousOn.intervalIntegrable (hab : a ≤ b) (h : ContinuousOn f (Set.uIcc a b))` — the uIcc-restricted version. Its signature: `theorem ContinuousOn.intervalIntegrable (hab : a ≤ b) (h : ContinuousOn f (uIcc a b)) : IntervalIntegrable f volume a b`? — hmm — actually `ContinuousOn.intervalIntegrable (h : ContinuousOn f (uIcc a b))`? — with the measure — earlier grep for it found nothing. Let me check: `ContinuousOn.intervalIntegrable (hab : a ≤ b) (hf : ContinuousOn f (Set.Icc a b)) : IntervalIntegrable f volume a b` — I believe this exists (used in Continuous.intervalIntegrable's proof: `hu.continuousOn.intervalIntegrable` — from Basic.lean:514: `hu.continuousOn.intervalIntegrable` — so ContinuousOn.intervalIntegrable exists with signature (hab : a ≤ b)? (h : ContinuousOn f (Icc a b))... — line 512-514: `theorem Continuous.intervalIntegrable {u : ℝ → E} (hu : Continuous u) (a b : ℝ) : IntervalIntegrable u μ a b := hu.continuousOn.intervalIntegrable` — so ContinuousOn.intervalIntegrable (hf : ContinuousOn f ?s) — the set s is implicit, probably uIcc or Icc — from the use, `hu.continuousOn : ContinuousOn f univ` — .intervalIntegrable with a b — so the signature might be `(hab : a ≤ b) (hf : ContinuousOn f (Set.uIcc a b))`? — or `(hf : ContinuousOn f (Set.Icc a b)) (hab : a ≤ b)`. Let me grep its def. For the mid-piece integrability on [−5/2, 3]: same ContinuousOn route with zeta_ne_zero_horizontal (re = σ ∈ [−5/2, 3] — the mid path has im = T with clearance ✓ continuous via hJcont from 6-c-iv — but that's INSIDE the mid theorem — I need it here too. Options: (a) rebuild the hcontJ machine locally in 6-c-v; (b) prove a standalone ContinuousOn lemma for the mid path. Since hIntegral_Jlag_mid_norm_le's hcontJ gives ContinuousAt at EVERY σ (global!), I can factor it: actually in 6-c-iv, hcontJ : ∀ σ, ContinuousAt (Jlag n) (↑σ+↑T·I) — GLOBAL (all σ ∈ ℝ, since zeta_ne_zero_horizontal holds for ALL σ). So for the mid piece [−5/2, 3] at +T: the SAME global statement — I'll rebuild the small machine in 6-c-v (it's ~10 lines) OR factor a theorem. Cleaner: factor two small theorems: ```lean theorem continuousAt_Jlag_horizontal (hTwide ...) (hn : 1 ≤ n) (hTn) (hTn1) (σ : ℝ) : ContinuousAt (Jlag n) ((σ : ℝ) + (T : ℝ) * Complex.I) ``` — hmm — but the -T side needs the conj trick anyway... — wait — for the -T side's split integrability: H(−5/2→3, −T) and H(3→2√n, −T) — the -T path: zeta zero-free? — for the MID segment [−5/2,3] at −T: a zero at σ−iT would be conj of a zero at σ+iT — also in the clearance window ✓ zero-free ✓ — but my zeta_ne_zero_horizontal only covers +T! For −T: need zeta_ne_zero_horizontal_neg — or prove via conj: ζ(σ−iT) = conj(ζ(σ+iT)) ≠ 0 since ζ(σ+iT) ≠ 0 ✓ (riemannZeta_conj + conj_ne_zero). Hmm — wait — actually, for the -T side split, do I even need separate integrability? The HIntegral_split for the -T side: HIntegral (Jlag n) (−5/2) (2√n) (−T) = H(−5/2→3, −T) + H(3→2√n, −T) — needs IntervalIntegrable on [−5/2,3] and [3,2√n] of the −T path. For [3,2√n] at −T: my hJcontTail helper (any y — im unrestricted, zero-free via re > 1) ✓. For [−5/2,3] at −T: need ζ(σ−iT) ≠ 0 for σ ∈ [−5/2,3] — via conj from zeta_ne_zero_horizontal: ```lean have hzne_neg : ∀ σ : ℝ, riemannZeta ((σ : ℝ) + (-T : ℝ) * Complex.I) ≠ 0 := by intro σ have h1 : ((σ : ℝ) + (-T : ℝ) * Complex.I) = (starRingEnd ℂ) ((σ : ℝ) + (T : ℝ) * Complex.I) := by simp [map_add, map_mul, Complex.conj_ofReal, Complex.conj_I] -- or by simp rw [h1, ne_eq, ← map_ne_zero (f := starRingEnd ℂ)??] ``` Hmm — ζ(conj s) = conj(ζ s) — `riemannZeta_conj` (banked: _root_.riemannZeta_conj) : riemannZeta (conj s) = conj (riemannZeta s) — so ζ(conj s) ≠ 0 ⟺ conj(ζ s) ≠ 0 ⟺ ζ s ≠ 0 (starRingEnd_apply_ne_zero? — `map_ne_zero (f := starRingEnd ℂ)` : f a ≠ 0 ↔ a ≠ 0 — `map_ne_zero_iff (FunLike injective)`? — simpler: (starRingEnd ℂ) x = 0 ↔ x = 0 — `Complex.conj_eq_zero`? hmm — `map_eq_zero` for ring endomorphisms: `(starRingEnd ℂ) x = 0 ↔ x = 0` — `map_eq_zero (starRingEnd ℂ)`? — there might be `Complex.conj_ne_zero`? no — `starRingEnd` is injective: `FunLike injective`... — cleanest: ```lean rw [h1, riemannZeta_conj] intro h0 have h2 := zeta_ne_zero_horizontal hTwide hn1 hTn hTn1 (σ := σ) -- ζ(σ+iT) ≠ 0 rw [show (starRingEnd ℂ) (riemannZeta _) = 0 → riemannZeta _ = 0 from ...] ``` Hmm — `Complex.conj_eq_zero`?? — let me think: `map_eq_zero (f : F) [FunLike F α β] [Zero α] [Zero β] [MapZeroClass...]`? — for starRingEnd: `RingAut`?? — starRingEnd ℂ is a RingAut — `(starRingEnd ℂ) x = 0 ↔ x = 0` via `map_eq_zero`? — there's `map_eq_zero_iff (f : α →*₀ β) (hf : Function.Injective f)`? — or for equivs: `Equiv.map_eq_zero_iff`? — starRingEnd is a RingAut → `(RingAut ...)`. Simplest robust: `Complex.conj_eq_zero`?? — I'm not sure it exists. Alternative: `starRingEnd ℂ |>.injective`? — `RingAut.injective`?? — there's `(starRingEnd ℂ).injective` — is starRingEnd a RingAut? — `starRingEnd ℂ : ℂ →+* ℂ` (a ring hom, not aut) — but it IS injective: `starRingEnd_injective : Function.Injective (starRingEnd ℂ)`? — hmm — `Complex.conj_injective` exists: `Function.Injective (starRingEnd ℂ)`? — there's `starRingEnd_self_inverse`?? — inverse: conj(conj x) = x — so injective trivially. For ≠ 0: intro h0; have : ζ s = 0 := by rw [← Complex.conj_conj??]... Cleanest deterministic: `map_ne_zero` — hmm. Actually — `ne_of_apply_ne`? — Let me do: intro h0; apply zeta_ne_zero_horizontal-instance; rw [← h0.symm??]... SIMPLEST: use `Complex.norm_conj`: if conj(ζ s) = 0 then ‖conj(ζ s)‖ = 0 = ‖ζ s‖ ⟹ ζ s = 0: ```lean rw [h1, riemannZeta_conj] intro h0 exact zeta_ne_zero_horizontal-instance (by have := congrArg Complex.norm h0 -- ‖conj(ζ s)‖ = 0 rw [Complex.norm_conj, norm_eq_zero] at this exact this) ``` Hmm — norm_eq_zero : ‖a‖ = 0 ↔ a = 0 — congrArg Complex.norm h0 : ‖(starRingEnd ℂ)(ζ s)‖ = ‖0‖ = 0 — rw [Complex.norm_conj] : ‖ζ s‖ = 0 — norm_eq_zero.mp : ζ s = 0 ✓. Actually — there's definitely `map_ne_zero` for injective maps: `map_ne_zero_iff (hf : Function.Injective f) : f a ≠ 0 ↔ a ≠ 0`?? — hmm — `map_ne_zero_iff (f : α → β) (hf : Injective f) (h0 : f 0 = 0)`? — for starRingEnd there's probably a direct `starRingEnd_apply_ne_zero`? — hmm. Let me not guess: the norm route is bulletproof. Or EVEN simpler: (starRingEnd ℂ) x = 0 ⟹ (starRingEnd ℂ)((starRingEnd ℂ) x) = 0 ⟹ x = 0 (conj_conj): ```lean intro h0 apply h-instance have h2 : (starRingEnd ℂ) ((starRingEnd ℂ) (riemannZeta ((σ : ℝ) + (T : ℝ) * Complex.I))) = 0 := by rw [h0] simp -- conj 0 = 0 rwa [Complex.conj_conj] at h2 ``` hmm — `Complex.conj_conj (z) : (starRingEnd ℂ) ((starRingEnd ℂ) z) = z`? — `conj_conj : conj (conj z) = z` — exists as `Complex.conj_conj`?? — hmm — `starRingEnd_self_inverse`? — `Complex.conj_conj` ✓ I'm fairly confident. And `(starRingEnd ℂ) 0 = 0` — `map_zero` — simp proves ✓. OK — alternatively — hmm — actually there's a MUCH simpler observation: `riemannZeta_conj` — let me grep its exact statement first, plus `Complex.conj_conj`, plus ContinuousOn.intervalIntegrable signature, plus Jlag_differentiableAt signature. Also — hmm — WAIT. Actually for the −T side I might not need the mid-path continuity at all: the MID bound at −T (hIntegral_Jlag_mid_neg_norm_le) is ALREADY PROVED (via the conj trick). The 6-c-v split at 3 needs the −T-path integrability on BOTH [−5/2,3] and [3,2√n] for HIntegral_split. Hmm — but hold on — maybe I can do the −T split differently: apply the conj trick at the LEVEL of the full integral: H(−5/2→2√n, −T) = conj(H(−5/2→2√n, +T)) — by the SAME intervalIntegral_conj argument (no split needed for continuity — intervalIntegral_conj holds for ANY integrand!). Then ‖H(−5/2→2√n, −T)‖ = ‖H(−5/2→2√n, +T)‖ (norm_conj) and bound the +T side by mid+DL. WAIT — but ‖H(+T full)‖ ≤ ‖H(+T mid)‖ + ‖H(+T tail)‖ needs the +T split (which needs +T-path integrability on both pieces ✓ — mid: zeta_ne_zero_horizontal ✓; tail: re ≥ 3 ✓). And for −T: NO split needed — just conj at the full level: ‖H(−5/2→2√n, −T)‖ = ‖conj(H(−5/2→2√n, +T))‖ = ‖H(−5/2→2√n, +T)‖ ≤ ‖H(+T mid)‖ + ‖H(+T tail)‖ ≤ midBound + DLRight. Hmm — but wait — does the conj identity work for the FULL −T integral? HIntegral (Jlag n) (−5/2) (2√n) (−T) = conj(HIntegral (Jlag n) (−5/2) (2√n) T) — the intervalIntegral_conj + Jlag_conj + integral_congr argument — Jlag_conj needs the point ≠ 1 (re = σ ≥ −5/2... wait — Jlag_conj n (hs : s ≠ 1) : Jlag n (conj s) = conj (Jlag n s) — for s = σ+iT: s ≠ 1 ⟸ im = T ≠ 0 ✓ (T > 0) — DL-5's pattern ✓. And intervalIntegral_conj needs NO integrability (it's a map property — let me double check: `intervalIntegral.intervalIntegral_conj (f : ℝ → ℂ) (a b : ℝ) : ∫ x in a..b, conj (f x) = conj (∫ x in a..b, f x)` — no integrability hypothesis?? — hmm — integral of conj — needs integrability of f for the Bochner integral? — conj is a continuous linear map — `ContinuousLinearMap.integral_comp_comm` — requires the function to be integrable?? — hmm — `intervalIntegral.intervalIntegral_conj` — let me check its hypotheses. DL-5 used it without extra hyps ✓ (their hconj proof only did integral_congr after) — so it works without integrability side-conditions?? — hmm — actually for the Bochner integral, `conj (∫ f) = ∫ (conj ∘ f)` requires f integrable OR the junk-value convention (integrals of non-integrable functions are 0, and conj 0 = 0 ✓ both conventions agree). Mathlib's `intervalIntegral.intervalIntegral_conj` — used unconditionally in DL-5 ✓ fine. So the −T side: ‖H(−T)‖ = ‖H(+T)‖ via conj — then both sides bounded by mid+DL — the two-sides statement becomes 2·(midBound + DLRight)... wait — hmm — but actually I realize the cleaner formulation: prove the +T full bound ‖H(+T)‖ ≤ midBound + DLRight (with the split), and the −T full bound ‖H(−T)‖ ≤ midBound + DLRight (via conj + the +T one), then the two-sides lemma ‖H(+T)‖ + ‖H(−T)‖ ≤ 2·(midBound + DLRight) = 2·midBound + 2·DogLegRight. The 6-d fold consumes the SUM (from the rectangle side decomposition + norm triangulation). Per the 6-d recipe: "‖RectInt'‖ = (1/(2π))·‖RectInt‖ ≤ (1/2π)·(sides)" where sides = ‖H(−T)‖ + ‖H(T)‖ + ‖V(right)‖ + ‖V(left)‖ — actually the decomposition: RectInt = H(−T) − H(T) + V(right) − V(left) — norm ≤ each. So the fold needs each side's norm bound separately? NO — the sum: ‖RectInt‖ ≤ ‖H(−T)‖ + ‖H(T)‖ + ‖Vr‖ + ‖Vl‖ — then each bounded: H-pair by the two-sides lemma, Vr by 6-c-iii, Vl by 6-c-ii ✓. So the two-sides lemma is the right deliverable ✓. Per the 6-c-v recipe: "both sides: ≤ 2(e⁴+1)(NI + 5.5(...)) + 2·DogLegRight" ✓ — my planned statement matches. Now — for the +T split at 3, I need: - IntervalIntegrable Jlag on [−5/2, 3] at +T: ContinuousOn on uIcc(−5/2, 3) = Icc(−5/2, 3) — from the hJcont machine (ContinuousAt at every σ — global!) — so `Continuous.intervalIntegrable` works?? — the GLOBAL continuity of (fun σ => Jlag n (↑σ+↑T·I)) — from hcontJ (∀ σ, ContinuousAt) + continuous_iff_continuousAt — that's what 6-c-iv's hcontJ gives — but it's LOCAL to that proof. For 6-c-v, rebuild it (small) or factor. I'll rebuild inline (it's the zeta_ne_zero_horizontal-based machine — ~10 lines). - IntervalIntegrable Jlag on [3, 2√n] at +T: from Jlag_differentiableAt_dogleg (hre : 3 ≤ s.re) (him : 0 < s.im) — at σ+iT: re = σ ≥ 3?? — WAIT — σ ∈ [3, 2√n] ✓ re = σ ≥ 3 ✓ him = T > 0 ✓. But for Continuous.intervalIntegrable (global), the Jlag on the +T path is continuous everywhere?? — for σ < 3 the path hits zeros — but for the SPLIT integrability I need only [3, 2√n]-segment integrability — ContinuousOn.intervalIntegrable (the Icc-version) with ContinuousOn from the dogleg differentiability on the segment ✓ — OR use HIntegral_split's exact requirements: (h1 : IntervalIntegrable f volume a b) (h2 : IntervalIntegrable f volume b c) — for [3, 2√n]: ContinuousOn on uIcc(3, 2√n) = Icc(3, 2√n) (3 ≤ 2√n ✓ since 2√n ≥ 632): per σ ∈ Icc(3, 2√n): ContinuousAt (Jlag n) (σ+iT) via Jlag_differentiableAt_dogleg.continuousAt — then .continuousWithinAt — ContinuousOn ✓ then ContinuousOn.intervalIntegrable. Let me check ContinuousOn.intervalIntegrable's exact signature. And Jlag_differentiableAt's. And riemannZeta_conj. Let me grep.
Checking `ContinuousOn.intervalIntegrable`, `Jlag_differentiableAt`, and `riemannZeta_conj` signatures.
tool: Bash
input
{
  "command": "grep -rn -B2 -A4 \"theorem ContinuousOn.intervalIntegrable\" \"D:/Python/Quantum/zeta-23-lean/.lake/packages/mathlib/Mathlib/MeasureTheory/Integral/IntervalIntegral/Basic.lean\" | head -14; grep -rn -B1 -A3 \"theorem Jlag_differentiableAt \\|lemma Jlag_differentiableAt \" \"D:/Python/Quantum/research/lean/NguyenRH1/NguyenRH1/\" | head -10; grep -rn \"theorem riemannZeta_conj\" \"D:/Python/Quantum/zeta-23-lean/.lake/packages/mathlib/Mathlib/\" | head -3",
  "description": "Verify ContinuousOn.intervalIntegrable, Jlag_differentiableAt, riemannZeta_conj"
}
#2296 assistant 2026-08-26 19:40:17.964 UTC · 6069→6392 tok open ⤢
thinking (8262 chars)
- `ContinuousOn.intervalIntegrable (hu : ContinuousOn u (uIcc a b)) : IntervalIntegrable u μ a b` ✓ (takes uIcc, no hab needed). - `Jlag_differentiableAt {n} {s} (hs1 : s ≠ 1) (hs0 : riemannZeta s ≠ 0) : DifferentiableAt ℂ (Jlag n) s` ✓ (LiSides:82). - `riemannZeta_conj (s : ℂ) : riemannZeta (conj s) = conj (riemannZeta s)` ✓. Now — for the +T full-side proof, plan: ```lean /-- **6-c-v (horizontal assembly)**: the full horizontal side split at 3 — mid bound (6-c-iv) + the DL-5 dog-leg tail. -/ theorem hIntegral_Jlag_full_norm_le (hZC) (hZC2) (hNLZ) {n} (hn : 100000 ≤ n) {T} (hTwide) (hTn) (hTn1) : ‖HIntegral (Jlag n) (-5/2) (2 * Real.sqrt n) T‖ ≤ (Real.exp 4 + 1) * (2 * Z2hat T * Real.log (148.5 * C0hat (√n+2)) + (3 - -5/2) * (10.5 * Real.log (T + 2) + 48)) + DogLegRight (n : ℝ) := by have hnR : ... ; have h10T : ... ; have hT0 : 0 < T := ... have hn16 : 16 ≤ n := by omega have hn1 : 1 ≤ n := by omega have h3le : (3 : ℝ) ≤ 2 * Real.sqrt (n : ℝ) := by ... (√n ≥ 316) -- split at 3 have hsplit : HIntegral (Jlag n) (-5/2) (2 * Real.sqrt n) T = HIntegral (Jlag n) (-5/2) 3 T + HIntegral (Jlag n) 3 (2 * Real.sqrt n) T := HIntegral_split hInt1 hInt2 rw [hsplit] calc ‖HIntegral (Jlag n) (-5/2) 3 T + HIntegral (Jlag n) 3 (2 * Real.sqrt n) T‖ ≤ ‖HIntegral (Jlag n) (-5/2) 3 T‖ + ‖HIntegral (Jlag n) 3 (2 * Real.sqrt n) T‖ := norm_add_le _ _ _ ≤ (mid bound) + DogLegRight (n : ℝ) := by have hmid := hIntegral_Jlag_mid_norm_le hZC hZC2 hNLZ hn hTwide hTn hTn1 have htail := hIntegral_Jlag_low_le_dogLegUpper hn16 hTn hTn1 -- htail : ‖HIntegral (Jlag n) 3 (2√n) T‖ ≤ ((√n+1)/2)(e^{5/2}+1)/2 + 8.38 + IIMaj/4 -- DogLegRight n = exactly that rw [show DogLegRight (n : ℝ) = ((Real.sqrt (n : ℝ) + 1) / 2) * (Real.exp (5 / 2) + 1) / 2 + (8.38 : ℝ) + IIMaj (n : ℝ) / 4 from rfl] linarith [hmid, htail] ``` DogLegRight (n:ℝ) — its def: ((√n+1)/2)·(e^{5/2}+1)/2 + 8.38 + IIMaj n/4 — and htail's RHS: `((Real.sqrt (n: ℝ) + 1) / 2) * (Real.exp (5 / 2) + 1) / 2 + (8.38 : ℝ) + IIMaj (n: ℝ) / 4` — DogLegRight unfolds to EXACTLY that (modulo the 8.38 being bare vs (8.38 : ℝ) — same ✓). So `DogLegRight (n:ℝ) = ...` by rfl ✓ (defeq — the show-rfl should work; or `unfold DogLegRight`). Hmm — htail (DL-5) needs (hn : 16 ≤ n) — hn16 := by omega ✓. hInt1 : IntervalIntegrable (fun σ => Jlag n (↑σ+↑T·I)) volume (−5/2) 3 — via the +T-path continuity (zeta_ne_zero_horizontal — global): ```lean have hJcontP : ∀ σ : ℝ, ContinuousAt (Jlag n) ((σ : ℝ) + (T : ℝ) * Complex.I) := by intro σ have hs1 : ((σ : ℝ) + (T : ℝ) * Complex.I) ≠ 1 := by intro h have him := congrArg Complex.im h simp at him linarith [hT0] have hA := (kFun_differentiableAt (n := n) hs1).continuousAt have hB := continuousAt_logDeriv_zeta hs1 (zeta_ne_zero_horizontal hTwide hn1 hTn hTn1) show ContinuousAt (fun s => kFun n s * (-logDeriv riemannZeta s)) _ exact hA.mul hB.neg have hcontJP : Continuous fun σ : ℝ => Jlag n ((σ : ℝ) + (T : ℝ) * Complex.I) := by rw [continuous_iff_continuousAt] intro σ have hf : ContinuousAt (fun u : ℝ => ((u : ℝ) + (T : ℝ) * Complex.I)) σ := hpath.continuousAt exact ContinuousAt.comp (g := Jlag n) (f := ...) (x := σ) (hJcontP σ) hf have hInt1 : IntervalIntegrable (fun σ : ℝ => Jlag n ((σ : ℝ) + (T : ℝ) * Complex.I)) MeasureTheory.volume (-5/2) 3 := hcontJP.intervalIntegrable _ _ ``` hInt2 : IntervalIntegrable ... 3 (2√n) — via the dog-leg differentiability (ContinuousOn on uIcc(3, 2√n)): ```lean have hInt2 : IntervalIntegrable (fun σ : ℝ => Jlag n ((σ : ℝ) + (T : ℝ) * Complex.I)) MeasureTheory.volume 3 (2 * Real.sqrt (n : ℝ)) := by apply ContinuousOn.intervalIntegrable rw [Set.uIcc_of_le h3le] intro σ hσ obtain ⟨hσ1, hσ2⟩ := hσ -- 3 ≤ σ ≤ 2√n have hre : ((σ : ℝ) + (T : ℝ) * Complex.I).re = σ := by simp have him : ((σ : ℝ) + (T : ℝ) * Complex.I).im = T := by simp exact (Jlag_differentiableAt_dogleg (n := n) (s := ((σ : ℝ) + (T : ℝ) * Complex.I)) (by rw [hre]; exact hσ1) (by rw [him]; exact hT0)).continuousAt.continuousWithinAt ``` Hmm — ContinuousOn f s := ∀ x ∈ s, ContinuousWithinAt f s x — so intro σ hσ; exact ....continuousWithinAt ✓ — wait — ContinuousOn f s is DEFINED as ∀ x ∈ s, ContinuousWithinAt f s x ✓. So `intro σ hσ` then produce ContinuousWithinAt — from DifferentiableAt.continuousAt.continuousWithinAt ✓. h3le : 3 ≤ 2√n — from √n ≥ 316 (h10T-chain): 2√n ≥ 632 ≥ 3 ✓: ```lean have h3le : (3 : ℝ) ≤ 2 * Real.sqrt (n : ℝ) := by have h1 : (10 : ℝ) ≤ Real.sqrt (n : ℝ) := by rw [Real.le_sqrt (by norm_num) (by positivity)] nlinarith [hnR] nlinarith [h1] ``` And `Set.uIcc_of_le h3le : uIcc 3 (2√n) = Icc 3 (2√n)` — `Set.uIcc_of_le (hab : a ≤ b) : uIcc a b = Icc a b` ✓ (used in DL-1's holomorphicOn proof: Set.uIcc_of_le hTU ✓). Then the full theorem assembly. And hT0 : 0 < T := by (from √n > 0): Real.sqrt_pos.mpr (by linarith [hnR]) then linarith. For the −T full theorem: conj at the FULL level: ```lean theorem hIntegral_Jlag_full_neg_norm_le (same hyps) : ‖HIntegral (Jlag n) (-5/2) (2 * Real.sqrt n) (-T)‖ ≤ (same RHS) := by have hT0 : (0:ℝ) < T := ... have hconj : HIntegral (Jlag n) (-5/2) (2 * Real.sqrt n) (-T) = (starRingEnd ℂ) (HIntegral (Jlag n) (-5/2) (2 * Real.sqrt n) T) := by unfold HIntegral rw [← intervalIntegral.intervalIntegral_conj] apply intervalIntegral.integral_congr intro σ hσ show Jlag n ((σ : ℝ) + (((-T : ℝ)) : ℂ) * Complex.I) = (starRingEnd ℂ) (Jlag n ((σ : ℝ) + T * Complex.I)) have hid2 : ((σ : ℝ) + (((-T : ℝ)) : ℂ) * Complex.I) = (starRingEnd ℂ) ((σ : ℝ) + T * Complex.I) := by simp rw [hid2] exact Jlag_conj n (by intro h have him : (((σ : ℝ) + T * Complex.I)).im = T := by simp rw [h] at him simp at him linarith [hT0]) rw [hconj, Complex.norm_conj] exact hIntegral_Jlag_full_norm_le hZC hZC2 hNLZ hn hTwide hTn hTn1 ``` — verbatim DL-5/6-c-iv-neg pattern ✓. Then the two-sides: ```lean theorem hIntegral_Jlag_full_two_sides (hZC) (hZC2) (hNLZ) {n} (hn : 100000 ≤ n) {T} (hTwide) (hTn) (hTn1) : ‖HIntegral (Jlag n) (-5/2) (2 * Real.sqrt n) T‖ + ‖HIntegral (Jlag n) (-5/2) (2 * Real.sqrt n) (-T)‖ ≤ 2 * ((Real.exp 4 + 1) * (2 * Z2hat T * Real.log (148.5 * C0hat (√n+2)) + (3 - -5/2) * (10.5 * Real.log (T + 2) + 48)) + DogLegRight (n : ℝ)) := by rw [show (2 : ℝ) * (X + DogLegRight (n:ℝ)) = (X + DogLegRight (n:ℝ)) + (X + DogLegRight (n:ℝ)) by ring] exact add_le_add (hIntegral_Jlag_full_norm_le ...) (hIntegral_Jlag_full_neg_norm_le ...) ``` (clone dogLegUpper_two_sides's structure ✓.) Hmm — wait — the fold's horizontal charge: "2(e⁴+1)(NI + 5.5(...)) + 2·DogLegRight" — my two-sides gives 2·(midBound + DogLegRight) = 2·midBound + 2·DogLegRight ✓ same thing (midBound = (e⁴+1)(NI + 5.5(...))). ✓ Now — do I also need the per-side FULL bounds (not just two-sides)? The 6-d fold uses the SUM via the rectangle decomposition — the two-sides lemma suffices. But having the single-side theorems is natural anyway (the neg one needs the pos one). OK. One more check — `HIntegral_split` (from §1): (h1 : IntervalIntegrable (fun σ => f (↑σ + ↑y·I)) volume a b) (h2 : ... b c) : HIntegral f a c y = HIntegral f a b y + HIntegral f b c y ✓ — my hInt1 (a=-5/2, b=3), hInt2 (b=3, c=2√n) ✓. Also — `intervalIntegral.integral_const_mul` — hmm not needed here. Also — the htail's exact RHS — from DL-5 (line 1302-1304): `((Real.sqrt (n: ℝ) + 1) / 2) * (Real.exp (5 / 2) + 1) / 2 + (8.38 : ℝ) + IIMaj (n: ℝ) / 4` ✓ — and DogLegRight (n : ℝ) := `((Real.sqrt n + 1) / 2) * (Real.exp (5 / 2) + 1) / 2 + 8.38 + IIMaj n / 4` — the show-rfl: `show DogLegRight (n : ℝ) = ((Real.sqrt (n : ℝ) + 1) / 2) * (Real.exp (5 / 2) + 1) / 2 + (8.38 : ℝ) + IIMaj (n : ℝ) / 4 from rfl` — defeq ✓ (the def unfolds; 8.38 vs (8.38 : ℝ) — same term ✓; `Real.sqrt n` vs `Real.sqrt (n : ℝ)` — in the def, DogLegRight's param is named n : ℝ — applied at (n:ℝ) = ↑n — Real.sqrt ↑n ✓ same ✓). So `from rfl` ✓. If rfl fails, `from by unfold DogLegRight`. Let me write §5 now. Also mark the header §5 and the file-top comment.
All signatures confirmed. Writing §5 (6-c-v) — the horizontal assembly.
tool: Edit
input
{
  "replace_all": false,
  "file_path": "D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiSideBounds.lean",
  "old_string": "  rw [hconj, Complex.norm_conj]\n  exact hIntegral_Jlag_mid_norm_le hZC hZC2 hNLZ hn hTwide hTn hTn1\n\nend NguyenRH1",
  "new_string": "  rw [hconj, Complex.norm_conj]\n  exact hIntegral_Jlag_mid_norm_le hZC hZC2 hNLZ hn hTwide hTn hTn1\n\n/-! ## 5. 6-c-v: the full horizontal assembly (mid + DL-5 tails) -/\n\n/-- **6-c-v (the +T full horizontal side)**: split at 3 \u2014 the 6-c-iv mid\nbound + the DL-5 dog-leg tail charge. -/\ntheorem hIntegral_Jlag_full_norm_le (hZC : ZeroCountProp) (hZC2 : ZeroCountProp2)\n    (hNLZ : NoLowZeros) {n : \u2115} (hn : 100000 \u2264 n) {T : \u211d}\n    (hTwide : \u2200 \u03c1 : \u2102, IsNontrivialZero \u03c1 \u2192 Real.sqrt (n : \u211d) - 1 < \u03c1.im \u2192\n      \u03c1.im < Real.sqrt (n : \u211d) + 2 \u2192\n        1 / (9 * C0hat (Real.sqrt (n : \u211d) + 2)) \u2264 |\u03c1.im - T|)\n    (hTn : Real.sqrt (n : \u211d) \u2264 T) (hTn1 : T \u2264 Real.sqrt (n : \u211d) + 1) :\n    \u2016HIntegral (Jlag n) (-5 / 2) (2 * Real.sqrt (n : \u211d)) T\u2016 \u2264\n      (Real.exp 4 + 1) *\n        (2 * Z2hat T * Real.log (148.5 * C0hat (Real.sqrt (n : \u211d) + 2)) +\n          (3 - -5 / 2) * (10.5 * Real.log (T + 2) + 48)) + DogLegRight (n : \u211d) := by\n  classical\n  have hnR : (100000 : \u211d) \u2264 (n : \u211d) := by exact_mod_cast hn\n  have h10T : (10 : \u211d) \u2264 T := by\n    have h1 : (10 : \u211d) \u2264 Real.sqrt (n : \u211d) := by\n      rw [Real.le_sqrt (by norm_num) (by positivity)]\n      nlinarith [hnR]\n    linarith\n  have hT0 : (0 : \u211d) < T := by linarith\n  have hn1 : 1 \u2264 n := by omega\n  have hn16 : 16 \u2264 n := by omega\n  have h3le : (3 : \u211d) \u2264 2 * Real.sqrt (n : \u211d) := by\n    have h1 : (10 : \u211d) \u2264 Real.sqrt (n : \u211d) := by\n      rw [Real.le_sqrt (by norm_num) (by positivity)]\n      nlinarith [hnR]\n    nlinarith [h1]\n  -- the mid-path continuity (zero-free via the wide pigeonhole)\n  have hJcontP : \u2200 \u03c3 : \u211d, ContinuousAt (Jlag n) ((\u03c3 : \u211d) + (T : \u211d) * Complex.I) := by\n    intro \u03c3\n    have hs1 : ((\u03c3 : \u211d) + (T : \u211d) * Complex.I) \u2260 1 := by\n      intro h\n      have him := congrArg Complex.im h\n      simp at him\n      linarith [hT0]\n    have hA := (kFun_differentiableAt (n := n) hs1).continuousAt\n    have hB := continuousAt_logDeriv_zeta hs1 (zeta_ne_zero_horizontal hTwide hn1 hTn hTn1)\n    show ContinuousAt (fun s => kFun n s * (-logDeriv riemannZeta s)) _\n    exact hA.mul hB.neg\n  have hpath : Continuous fun u : \u211d => ((u : \u211d) + (T : \u211d) * Complex.I) := by\n    fun_prop\n  have hcontJP : Continuous fun \u03c3 : \u211d => Jlag n ((\u03c3 : \u211d) + (T : \u211d) * Complex.I) := by\n    rw [continuous_iff_continuousAt]\n    intro \u03c3\n    have hf : ContinuousAt (fun u : \u211d => ((u : \u211d) + (T : \u211d) * Complex.I)) \u03c3 :=\n      hpath.continuousAt\n    exact ContinuousAt.comp (g := Jlag n)\n      (f := fun u : \u211d => ((u : \u211d) + (T : \u211d) * Complex.I)) (x := \u03c3) (hJcontP \u03c3) hf\n  have hInt1 : IntervalIntegrable (fun \u03c3 : \u211d => Jlag n ((\u03c3 : \u211d) + (T : \u211d) * Complex.I))\n      MeasureTheory.volume (-5 / 2) 3 := hcontJP.intervalIntegrable _ _\n  -- the tail-path continuity (zero-free by re \u2265 3 > 1, DL-1)\n  have hInt2 : IntervalIntegrable (fun \u03c3 : \u211d => Jlag n ((\u03c3 : \u211d) + (T : \u211d) * Complex.I))\n      MeasureTheory.volume 3 (2 * Real.sqrt (n : \u211d)) := by\n    apply ContinuousOn.intervalIntegrable\n    rw [Set.uIcc_of_le h3le]\n    intro \u03c3 h\u03c3\n    obtain \u27e8h\u03c31, h\u03c32\u27e9 := h\u03c3\n    have hre : ((\u03c3 : \u211d) + (T : \u211d) * Complex.I).re = \u03c3 := by simp\n    have him : ((\u03c3 : \u211d) + (T : \u211d) * Complex.I).im = T := by simp\n    exact (Jlag_differentiableAt_dogleg (s := ((\u03c3 : \u211d) + (T : \u211d) * Complex.I))\n      (by rw [hre]; exact h\u03c31) (by rw [him]; exact hT0)).continuousAt.continuousWithinAt\n  -- split at 3 and assemble\n  have hsplit : HIntegral (Jlag n) (-5 / 2) (2 * Real.sqrt (n : \u211d)) T =\n      HIntegral (Jlag n) (-5 / 2) 3 T + HIntegral (Jlag n) 3 (2 * Real.sqrt (n : \u211d)) T :=\n    HIntegral_split hInt1 hInt2\n  have hmid := hIntegral_Jlag_mid_norm_le hZC hZC2 hNLZ hn hTwide hTn hTn1\n  have htail := hIntegral_Jlag_low_le_dogLegUpper hn16 hTn hTn1\n  have hDLR : DogLegRight (n : \u211d) = ((Real.sqrt (n : \u211d) + 1) / 2) *\n      (Real.exp (5 / 2) + 1) / 2 + (8.38 : \u211d) + IIMaj (n : \u211d) / 4 := rfl\n  rw [hsplit, hDLR]\n  calc \u2016HIntegral (Jlag n) (-5 / 2) 3 T +\n        HIntegral (Jlag n) 3 (2 * Real.sqrt (n : \u211d)) T\u2016\n      \u2264 \u2016HIntegral (Jlag n) (-5 / 2) 3 T\u2016 +\n        \u2016HIntegral (Jlag n) 3 (2 * Real.sqrt (n : \u211d)) T\u2016 := norm_add_le _ _\n    _ \u2264 (Real.exp 4 + 1) *\n          (2 * Z2hat T * Real.log (148.5 * C0hat (Real.sqrt (n : \u211d) + 2)) +\n            (3 - -5 / 2) * (10.5 * Real.log (T + 2) + 48)) +\n          (((Real.sqrt (n : \u211d) + 1) / 2) * (Real.exp (5 / 2) + 1) / 2 + (8.38 : \u211d) +\n            IIMaj (n : \u211d) / 4) :=\n        add_le_add hmid htail\n\n/-- **6-c-v (the \u2212T full horizontal side)**: the conj trick at the full\nlevel (no split needed \u2014 intervalIntegral_conj is unconditional). -/\ntheorem hIntegral_Jlag_full_neg_norm_le (hZC : ZeroCountProp) (hZC2 : ZeroCountProp2)\n    (hNLZ : NoLowZeros) {n : \u2115} (hn : 100000 \u2264 n) {T : \u211d}\n    (hTwide : \u2200 \u03c1 : \u2102, IsNontrivialZero \u03c1 \u2192 Real.sqrt (n : \u211d) - 1 < \u03c1.im \u2192\n      \u03c1.im < Real.sqrt (n : \u211d) + 2 \u2192\n        1 / (9 * C0hat (Real.sqrt (n : \u211d) + 2)) \u2264 |\u03c1.im - T|)\n    (hTn : Real.sqrt (n : \u211d) \u2264 T) (hTn1 : T \u2264 Real.sqrt (n : \u211d) + 1) :\n    \u2016HIntegral (Jlag n) (-5 / 2) (2 * Real.sqrt (n : \u211d)) (-T)\u2016 \u2264\n      (Real.exp 4 + 1) *\n        (2 * Z2hat T * Real.log (148.5 * C0hat (Real.sqrt (n : \u211d) + 2)) +\n          (3 - -5 / 2) * (10.5 * Real.log (T + 2) + 48)) + DogLegRight (n : \u211d) := by\n  have hnR : (100000 : \u211d) \u2264 (n : \u211d) := by exact_mod_cast hn\n  have hT0 : (0 : \u211d) < T := by\n    have h1 : (0 : \u211d) < Real.sqrt (n : \u211d) := Real.sqrt_pos.mpr (by linarith)\n    linarith\n  have hconj : HIntegral (Jlag n) (-5 / 2) (2 * Real.sqrt (n : \u211d)) (-T) =\n      (starRingEnd \u2102) (HIntegral (Jlag n) (-5 / 2) (2 * Real.sqrt (n : \u211d)) T) := by\n    unfold HIntegral\n    rw [\u2190 intervalIntegral.intervalIntegral_conj]\n    apply intervalIntegral.integral_congr\n    intro \u03c3 h\u03c3\n    show Jlag n ((\u03c3 : \u211d) + (((-T : \u211d)) : \u2102) * Complex.I) =\n      (starRingEnd \u2102) (Jlag n ((\u03c3 : \u211d) + T * Complex.I))\n    have hid2 : ((\u03c3 : \u211d) + (((-T : \u211d)) : \u2102) * Complex.I) =\n        (starRingEnd \u2102) ((\u03c3 : \u211d) + T * Complex.I) := by simp\n    rw [hid2]\n    exact Jlag_conj n (by\n      intro h\n      have him : (((\u03c3 : \u211d) + T * Complex.I)).im = T := by simp\n      rw [h] at him\n      simp at him\n      linarith [hT0])\n  rw [hconj, Complex.norm_conj]\n  exact hIntegral_Jlag_full_norm_le hZC hZC2 hNLZ hn hTwide hTn hTn1\n\n/-- **6-c-v (the two horizontal sides)**: the horizontal assembly charge \u2014\n2\u00b7(mid bound) + 2\u00b7DogLegRight. -/\ntheorem hIntegral_Jlag_full_two_sides (hZC : ZeroCountProp) (hZC2 : ZeroCountProp2)\n    (hNLZ : NoLowZeros) {n : \u2115} (hn : 100000 \u2264 n) {T : \u211d}\n    (hTwide : \u2200 \u03c1 : \u2102, IsNontrivialZero \u03c1 \u2192 Real.sqrt (n : \u211d) - 1 < \u03c1.im \u2192\n      \u03c1.im < Real.sqrt (n : \u211d) + 2 \u2192\n        1 / (9 * C0hat (Real.sqrt (n : \u211d) + 2)) \u2264 |\u03c1.im - T|)\n    (hTn : Real.sqrt (n : \u211d) \u2264 T) (hTn1 : T \u2264 Real.sqrt (n : \u211d) + 1) :\n    \u2016HIntegral (Jlag n) (-5 / 2) (2 * Real.sqrt (n : \u211d)) T\u2016 +\n      \u2016HIntegral (Jlag n) (-5 / 2) (2 * Real.sqrt (n : \u211d)) (-T)\u2016 \u2264\n      2 * ((Real.exp 4 + 1) *\n        (2 * Z2hat T * Real.log (148.5 * C0hat (Real.sqrt (n : \u211d) + 2)) +\n          (3 - -5 / 2) * (10.5 * Real.log (T + 2) + 48)) + DogLegRight (n : \u211d)) := by\n  rw [show (2 : \u211d) * ((Real.exp 4 + 1) *\n        (2 * Z2hat T * Real.log (148.5 * C0hat (Real.sqrt (n : \u211d) + 2)) +\n          (3 - -5 / 2) * (10.5 * Real.log (T + 2) + 48)) + DogLegRight (n : \u211d)) =\n      ((Real.exp 4 + 1) *\n        (2 * Z2hat T * Real.log (148.5 * C0hat (Real.sqrt (n : \u211d) + 2)) +\n          (3 - -5 / 2) * (10.5 * Real.log (T + 2) + 48)) + DogLegRight (n : \u211d)) +\n      ((Real.exp 4 + 1) *\n        (2 * Z2hat T * Real.log (148.5 * C0hat (Real.sqrt (n : \u211d) + 2)) +\n          (3 - -5 / 2) * (10.5 * Real.log (T + 2) + 48)) + DogLegRight (n : \u211d)) by ring]\n  exact add_le_add (hIntegral_Jlag_full_norm_le hZC hZC2 hNLZ hn hTwide hTn hTn1)\n    (hIntegral_Jlag_full_neg_norm_le hZC hZC2 hNLZ hn hTwide hTn hTn1)\n\nend NguyenRH1"
}
#2297 assistant 2026-08-26 19:42:52.966 UTC · 601→1989 tok open ⤢
thinking (4095 chars)
Only one error at 1239 — the hInt2's ContinuousOn proof. The issue: `ContinuousOn f s := ∀ x ∈ s, ContinuousWithinAt f s x` — when I `intro σ hσ`, the expected type is `ContinuousWithinAt (fun σ => Jlag n (↑σ+↑T·I)) (Set.Icc 3 (2√n)) σ` — but I produced `ContinuousWithinAt (Jlag ?n) ?s (↑σ+↑T·I)` with the SET undetermined (?m.623) and the function as Jlag directly (not composed with the path). The mismatch: my term's function is `Jlag n` (on ℂ) vs expected `(fun σ => Jlag n (↑σ+↑T·I))` (on ℝ) — the COMPOSITION with the path map is missing! ContinuousOn (fun σ => Jlag n (path σ)) (Icc) — I need ContinuousWithinAt of the COMPOSED function, not of Jlag itself. ContinuousOn f s unfolds to ∀ x ∈ s, ContinuousWithinAt f s x. The f here is `fun σ => Jlag n (↑σ+↑T·I)` — the path-composed function. So I need: ContinuousWithinAt (fun σ => Jlag n (↑σ+↑T·I)) (Icc 3 (2√n)) σ — from ContinuousAt (Jlag n) (path σ) and the path's continuity: ContinuousWithinAt.comp or: (hJcontT σ).continuousWithinAt — no — that's ContinuousWithinAt (Jlag n) s' (path σ) — the DOMAIN is ℂ, not ℝ. I need composition: ContinuousWithinAt (Jlag n ∘ path) (Icc) σ = (ContinuousWithinAt (Jlag n) (path '' Icc) (path σ)).comp (ContinuousWithinAt path (Icc) σ)?? — `ContinuousWithinAt.comp (hg : ContinuousWithinAt g t (f x)) (hf : ContinuousWithinAt f s x) (h : Set.MapsTo f s t) : ContinuousWithinAt (g ∘ f) s x` ✓ exists. So: ```lean intro σ hσ obtain ⟨hσ1, hσ2⟩ := hσ have hre : ... ; have him : ... have hg : ContinuousWithinAt (Jlag n) (Set.range fun u : ℝ => ((u : ℝ) + (T : ℝ) * Complex.I)) ((σ : ℝ) + (T : ℝ) * Complex.I) := (Jlag_differentiableAt_dogleg (s := ...) (by rw [hre]; exact hσ1) (by rw [him]; exact hT0)).continuousAt.continuousWithinAt have hfp : ContinuousWithinAt (fun u : ℝ => ((u : ℝ) + (T : ℝ) * Complex.I)) (Set.Icc 3 (2 * Real.sqrt (n : ℝ))) σ := hpath.continuousWithinAt have hmap : Set.MapsTo (fun u : ℝ => ((u : ℝ) + (T : ℝ) * Complex.I)) (Set.Icc 3 (2 * Real.sqrt (n : ℝ))) (Set.range fun u : ℝ => ((u : ℝ) + (T : ℝ) * Complex.I)) := Set.mapsTo_range _ _ exact (hg.comp hfp hmap) ``` Hmm — ContinuousWithinAt.comp signature: `theorem ContinuousWithinAt.comp {g : β → γ} {f : α → β} {s : Set α} {t : Set β} {x : α} (hg : ContinuousWithinAt g t (f x)) (hf : ContinuousWithinAt f s x) (h : Set.MapsTo f s t) : ContinuousWithinAt (g ∘ f) s x` ✓. Then the result: ContinuousWithinAt (Jlag n ∘ path) (Icc) σ — vs expected ContinuousWithinAt (fun σ => Jlag n (path σ)) (Icc) σ — defeq (∘) ✓. Alternatively — simpler: hg's set can be `Set.univ` (ContinuousAt gives ContinuousWithinAt at univ... — continuousWithinAt is at the point with the ambient set — from DifferentiableAt.continuousAt I get ContinuousAt (Jlag n) (path σ) — .continuousWithinAt : ContinuousWithinAt (Jlag n) s (path σ) for ANY s?? — ContinuousAt.continuousWithinAt : ContinuousWithinAt f s x — any s ✓ (continuousAt implies continuousWithinAt at every set). So hg : ContinuousWithinAt (Jlag n) (range path) (path σ) := ....continuousWithinAt ✓ as I wrote. Then comp with MapsTo range ✓. Or EVEN simpler — skip ContinuousWithinAt.comp: use the global continuity instead? — no, Jlag isn't globally continuous on the +T path for σ < 1... — the uIcc is [3, 2√n] — restrict to it ✓ the comp route. Actually — hmm — much simpler: `ContinuousOn` from `Continuous.comp_continuousOn`?? — or: the path is continuous GLOBALLY, Jlag is ContinuousOn {re ≥ 3}... — hmm — `ContinuousOn.comp (hg : ContinuousOn g t) (hf : Continuous f) (h : Set.MapsTo f s t) : ContinuousOn (g ∘ f) s` ✓ — with t := {s : ℂ | 3 ≤ s.re}... — need ContinuousOn (Jlag n) {re ≥ 3} — from Jlag_differentiableAt_dogleg (any im?? — the dogleg version needs him : 0 < im — for the +T path im = T > 0 ✓ but the SET {re ≥ 3} includes im ≤ 0 points where I'd need ζ zero-free on re ≥ 3 — true (re > 1) but the dogleg lemma needs him... — for the comp-ContinuousWithinAt route, hg is at the SINGLE point (σ+iT) with im = T > 0 ✓ fine. Let me rewrite hInt2's inner proof with the comp route.
The ContinuousOn proof needs the path-composition (`ContinuousWithinAt.comp` with MapsTo range), not the bare differentiability.
tool: Edit
input
{
  "replace_all": false,
  "file_path": "D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiSideBounds.lean",
  "old_string": "  have hInt2 : IntervalIntegrable (fun \u03c3 : \u211d => Jlag n ((\u03c3 : \u211d) + (T : \u211d) * Complex.I))\n      MeasureTheory.volume 3 (2 * Real.sqrt (n : \u211d)) := by\n    apply ContinuousOn.intervalIntegrable\n    rw [Set.uIcc_of_le h3le]\n    intro \u03c3 h\u03c3\n    obtain \u27e8h\u03c31, h\u03c32\u27e9 := h\u03c3\n    have hre : ((\u03c3 : \u211d) + (T : \u211d) * Complex.I).re = \u03c3 := by simp\n    have him : ((\u03c3 : \u211d) + (T : \u211d) * Complex.I).im = T := by simp\n    exact (Jlag_differentiableAt_dogleg (s := ((\u03c3 : \u211d) + (T : \u211d) * Complex.I))\n      (by rw [hre]; exact h\u03c31) (by rw [him]; exact hT0)).continuousAt.continuousWithinAt",
  "new_string": "  have hInt2 : IntervalIntegrable (fun \u03c3 : \u211d => Jlag n ((\u03c3 : \u211d) + (T : \u211d) * Complex.I))\n      MeasureTheory.volume 3 (2 * Real.sqrt (n : \u211d)) := by\n    apply ContinuousOn.intervalIntegrable\n    rw [Set.uIcc_of_le h3le]\n    intro \u03c3 h\u03c3\n    obtain \u27e8h\u03c31, h\u03c32\u27e9 := h\u03c3\n    have hre : ((\u03c3 : \u211d) + (T : \u211d) * Complex.I).re = \u03c3 := by simp\n    have him : ((\u03c3 : \u211d) + (T : \u211d) * Complex.I).im = T := by simp\n    have hg : ContinuousWithinAt (Jlag n)\n        (Set.range fun u : \u211d => ((u : \u211d) + (T : \u211d) * Complex.I))\n        ((\u03c3 : \u211d) + (T : \u211d) * Complex.I) :=\n      (Jlag_differentiableAt_dogleg (s := ((\u03c3 : \u211d) + (T : \u211d) * Complex.I))\n        (by rw [hre]; exact h\u03c31) (by rw [him]; exact hT0)).continuousAt.continuousWithinAt\n    have hfp : ContinuousWithinAt (fun u : \u211d => ((u : \u211d) + (T : \u211d) * Complex.I))\n        (Set.Icc 3 (2 * Real.sqrt (n : \u211d))) \u03c3 := hpath.continuousWithinAt\n    exact hg.comp hfp (Set.mapsTo_range _ _)"
}
#2298 assistant 2026-08-26 19:44:05.096 UTC · 395→696 tok open ⤢
thinking (1078 chars)
Same HO-decomposition issue as ContinuousAt.comp before: `ContinuousWithinAt.comp hg hfp` — the elaborator decomposed hg's point: hg : ContinuousWithinAt (Jlag n) (range path) (↑σ+↑T·I) — comp needs ContinuousWithinAt ?g ?t (?f ?x) — it decomposed `↑σ+↑T·I` as `(HAdd.hAdd ↑σ) (↑T·I)` with ?f := HAdd.hAdd ↑σ, ?x := ↑T·I — the bad split again. Fix: pin (f :=) (x :=) — ContinuousWithinAt.comp (hg : ContinuousWithinAt g t (f x)) (hf : ContinuousWithinAt f s x) (h : MapsTo f s t) — pin f := the path, x := σ: ```lean exact ContinuousWithinAt.comp (g := Jlag n) (f := fun u : ℝ => ((u : ℝ) + (T : ℝ) * Complex.I)) (x := σ) hg hfp (Set.mapsTo_range _ _) ``` Wait — but comp's signature: (hg) (hf) (h : MapsTo) — with g, f, x implicit and t, s implicit. Pin (g := Jlag n) (f := path) (x := σ) — then hg's expected: ContinuousWithinAt (Jlag n) ?t (path σ) — hg : ContinuousWithinAt (Jlag n) (range path) (↑σ+↑T·I) — path σ ≡ ↑σ+↑T·I (beta) ✓ ?t := range path ✓; hfp's expected: ContinuousWithinAt path ?s σ ✓ matches; MapsTo path (Icc) (range path) ✓ Set.mapsTo_range ✓.
Same HO-decomposition issue as before — pin `g/f/x` explicitly.
tool: Edit
input
{
  "replace_all": false,
  "file_path": "D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiSideBounds.lean",
  "old_string": "    have hfp : ContinuousWithinAt (fun u : \u211d => ((u : \u211d) + (T : \u211d) * Complex.I))\n        (Set.Icc 3 (2 * Real.sqrt (n : \u211d))) \u03c3 := hpath.continuousWithinAt\n    exact hg.comp hfp (Set.mapsTo_range _ _)",
  "new_string": "    have hfp : ContinuousWithinAt (fun u : \u211d => ((u : \u211d) + (T : \u211d) * Complex.I))\n        (Set.Icc 3 (2 * Real.sqrt (n : \u211d))) \u03c3 := hpath.continuousWithinAt\n    exact ContinuousWithinAt.comp (g := Jlag n)\n      (f := fun u : \u211d => ((u : \u211d) + (T : \u211d) * Complex.I)) (x := \u03c3) hg hfp\n      (Set.mapsTo_range _ _)"
}
#2299 assistant 2026-08-26 19:45:10.839 UTC · 326→565 tok open ⤢
thinking (78 chars)
CLEAN — §5 (6-c-v) elaborates green. Header marker, full build, audit, commit.
§5 clean. Header marker, full build, audit, commit.
tool: Edit
input
{
  "replace_all": false,
  "file_path": "D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiSideBounds.lean",
  "old_string": "\u00a74 (6-c-iv) DONE: the mid-horizontal segment machine [\u22125/2, 3].\n\u00a75 (6-c-v): the full horizontal assembly (mid + DL-5 tails, both sides).",
  "new_string": "\u00a74 (6-c-iv) DONE: the mid-horizontal segment machine [\u22125/2, 3].\n\u00a75 (6-c-v) DONE: the full horizontal assembly (mid + DL-5 tails, both sides)."
}
tool: Edit
input
{
  "replace_all": false,
  "file_path": "D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\AuditSession041.lean",
  "old_string": "-- \u00a74 (6-c-iv): the mid-horizontal segment machine [\u22125/2, 3]\n#print axioms integral_one_div_sqrt_sq_add_sq\n#print axioms arcsinh_right_endpoint_le\n#print axioms arcsinh_left_endpoint_le\n#print axioms per_zero_integral_le_two_log\n#print axioms hIntegral_Jlag_mid_norm_le\n#print axioms hIntegral_Jlag_mid_neg_norm_le",
  "new_string": "-- \u00a74 (6-c-iv): the mid-horizontal segment machine [\u22125/2, 3]\n#print axioms integral_one_div_sqrt_sq_add_sq\n#print axioms arcsinh_right_endpoint_le\n#print axioms arcsinh_left_endpoint_le\n#print axioms per_zero_integral_le_two_log\n#print axioms hIntegral_Jlag_mid_norm_le\n#print axioms hIntegral_Jlag_mid_neg_norm_le\n\n-- \u00a75 (6-c-v): the full horizontal assembly\n#print axioms hIntegral_Jlag_full_norm_le\n#print axioms hIntegral_Jlag_full_neg_norm_le\n#print axioms hIntegral_Jlag_full_two_sides"
}
#2300 assistant 2026-08-26 19:48:08.646 UTC · 643→2093 tok open ⤢
thinking (3794 chars)
6-c COMPLETE (committed 5d3dc0a). Now 6-d — the fold ⟹ sfBoundProp. The recipe: **6-d RECIPE (the fold):** sides identity sf_abs_le_rectInt_add_zeroSum hZC (T from exists_zero_avoiding_T_wide hZC at a := √n, √n ≥ 316 ≥ 3) + bridge zeroSum_norm_le_four_FAlt_add_one hRH hZCG hNLZ (T ≤ 3×10¹²−1 from n ≤ N₁; 2 ≤ T) + ‖RectInt'‖ = (1/(2π))·‖RectInt‖ ≤ (1/2π)·(sides) (RectangleIntegral' def, norm_smul, ‖1/(2πi)‖ = 1/(2π)) ⟹ |Sf n| ≤ (1/2π)(I_III + Horiz + e^{1/3}IIMaj) + (4F+1) ≤ (1/2π)(I_III-charge + Horiz-charge) + IIMaj + 4F + 1 (e^{1/3} ≤ 1.4 ≤ 2π — the F-E absorption, one linarith; e^{1/3} ≤ 1.4 via (exp(1/3))³ = e ≤ 2.7183 ≤ 2.744 = 1.4³ + le_of_pow_le_pow_left₀) ≤ Bn FAlt n — with the HORIZONTAL PINCH as the only real-analytic comparison: NI + 5.5(10.5 log+48) ≤ 7·Hhat T (pinch-lemma pattern from 5f-iii: certified endpoint at n = 10⁵ + growth direction; E-030 V3 witness ratio ≤ 0.40; suffices 2·Z2hat(T)·log(148.5·C0hat(√n+2)) ≤ 21·C0hat(T)² since 5.5 ≤ 7 on the (10.5 log+48) part — elementary L/LL comparisons with log x ≤ x/e (Real.log_le_sub_one_of_pos at x/e), certified numerics for the endpoint constants). Components needed: 1. `sf_abs_le_rectInt_add_zeroSum` (ii′-6-b, LiAssembly) — signature? 2. `exists_zero_avoiding_T_wide` (LiContour/LiAssembly) — signature (gives the T with hTwide). 3. `zeroSum_norm_le_four_FAlt_add_one` (LiAssembly) — signature (hRH hZCG hNLZ + T conditions). 4. RectangleIntegral' def — the 1/(2πi) factor. 5. `Bn`/`FAlt`/`SfBoundProp`/`Hhat`/`Z2hat`/`C0hat` defs and the relationship: Bn FAlt n's horizontal share: 2(e⁴+1)(7·Hhat T) + 2·DogLegRight (from the ii′-6-a2 redefinition) — need Bn's exact def to do the final ≤ Bn comparison. 6. The horizontal pinch: 2·Z2hat(T)·log(148.5·C0hat(√n+2)) + 5.5·(10.5·log(T+2)+48) ≤ 7·Hhat T where Hhat T = 3·C0hat(T)² + 10.5·log(T+2) + 48 (LiSkeleton:749). 7. e^{1/3} ≤ 1.4 via (e^{1/3})³ = e ≤ 2.7183... wait — need e^{1/3} ≤ 1.4 ⟺ e ≤ 1.4³ = 2.744: `le_of_pow_le_pow_left₀`? — x³ ≤ y³ with nonneg ⟹ x ≤ y — or Real.rpow nat route: e^{1/3} ≤ 1.4 ⟺ e ≤ 1.4³: (e^{1/3})³ = e^1 = e ≤ 2.744: need (e^{1/3})³ = e — `Real.exp_nat_mul`? or `Real.exp_mul`? — (e^{1/3})³ = e^{3·(1/3)} = e^1 — Real.exp_nat_mul (1/3) 3 : exp(3·(1/3)) = (exp(1/3))^3 — hmm — exp_nat_mul (x) (n) : exp(↑n * x) = (exp x)^n ✓ — so (exp(1/3))^3 = exp(3·(1/3)) = exp 1 — 3·(1/3) = 1 by ring. Then le_of_pow_le_pow_left₀ (n := 3) (exp(1/3) ≥ 0? hmm — le_of_pow_le_pow_left₀ (hn : n ≠ 0) (hx : 0 ≤ x) (h : x^n ≤ y^n) : x ≤ y? — signature: le_of_pow_le_pow_left₀ (hn : n ≠ 0) (ha : 0 ≤ a) (h : a ^ n ≤ b ^ n) : a ≤ b ✓ — with a := e^{1/3}, b := 1.4: a³ = e ≤ 2.7183 ≤ 2.744 = 1.4³ — need e ≤ 2.7183 — `Real.exp_one_lt_d9 : exp 1 < 2.7182818286` ✓ so e ≤ 2.7183 < 2.744 ✓ and 1.4³ = 2.744 by norm_num ✓. 8. The sides sum: RectInt' (Jlag n) z w = (1/(2πi))·RectInt — from Zeta23: VIntegral'/HIntegral' have the 1/(2πi) — RectangleIntegral' := HIntegral' ... — let me check the Zeta23 RectangleIntegral' def. And the ii′-6-b identity sf_abs_le_rectInt_add_zeroSum : |Sf n| ≤ ‖RectangleIntegral (Jlag n) z w‖ + ‖Σ‖?? or RectangleIntegral'? Let me check. The recipe says: "sides identity sf_abs_le_rectInt_add_zeroSum hZC + ‖RectInt'‖ = (1/(2π))·‖RectInt‖ ≤ (1/2π)·(sides) (RectangleIntegral' def, norm_smul, ‖1/(2πi)‖ = 1/(2π))" — so sf_abs_le_rectInt_add_zeroSum gives |Sf n| ≤ ‖RectangleIntegral' (Jlag n) z w‖ + ‖Σ‖ — where RectangleIntegral' has the 1/(2πi) factor — then ‖RectangleIntegral'‖ = (1/(2π))‖RectangleIntegral‖ ≤ (1/(2π))(‖H(−T)‖ + ‖H(T)‖ + ‖Vr‖ + ‖Vl‖) via rectangleIntegral_Jlag_eq + norm triangulation (norm_sub_le). Let me look at: sf_abs_le_rectInt_add_zeroSum, zeroSum_norm_le_four_FAlt_add_one, exists_zero_avoiding_T_wide, RectangleIntegral'/RectangleIntegral defs, Bn's def, FAlt's def, SfBoundProp's exact form.
6-c COMPLETE (`5d3dc0a`). Now 6-d — the fold. Signature recon: the 6-b identity, bridge, wide pigeonhole, `RectangleIntegral'`, `Bn`, `Hhat`.
tool: TodoWrite ok
input
{
  "todos": [
    {
      "content": "6-c-ii: I_III left-vertical bound vIntegral_Jlag_left_norm_le",
      "status": "completed",
      "activeForm": "Proving 6-c-ii I_III left-vertical bound"
    },
    {
      "content": "6-c-iii: I_I right-vertical bound (F-E form)",
      "status": "completed",
      "activeForm": "Proving 6-c-iii I_I bound"
    },
    {
      "content": "6-c-iv: mid-horizontal segment machine",
      "status": "completed",
      "activeForm": "Proving 6-c-iv mid-horizontal machine"
    },
    {
      "content": "6-c-v: horizontal assembly (mid + DL-5 tails)",
      "status": "completed",
      "activeForm": "Assembling 6-c-v horizontal bound"
    },
    {
      "content": "6-d: fold \u27f9 sfBoundProp (then commit, update archive)",
      "status": "in_progress",
      "activeForm": "Proving 6-d fold to sfBoundProp"
    }
  ]
}
result 2026-08-26 19:48:08Z → 2026-08-26 19:48:08Z
{
  "oldTodos": [
    "content='6-c-ii: I_III left-vertical bound vIntegral_Jlag_left_norm_le' status='completed' active_form='Proving 6-c-ii I_III left-vertical bound'",
    "content='6-c-iii: I_I right-vertical bound (F-E form)' status='completed' active_form='Proving 6-c-iii I_I bound'",
    "content='6-c-iv: mid-horizontal segment machine' status='completed' active_form='Proving 6-c-iv mid-horizontal machine'",
    "content='6-c-v: horizontal assembly (mid + DL-5 tails)' status='in_progress' active_form='Assembling 6-c-v horizontal bound'",
    "content='6-d: fold \u27f9 sfBoundProp (then commit, update archive)' status='pending' active_form='Proving 6-d fold to sfBoundProp'"
  ],
  "newTodos": [
    "content='6-c-ii: I_III left-vertical bound vIntegral_Jlag_left_norm_le' status='completed' active_form='Proving 6-c-ii I_III left-vertical bound'",
    "content='6-c-iii: I_I right-vertical bound (F-E form)' status='completed' active_form='Proving 6-c-iii I_I bound'",
    "content='6-c-iv: mid-horizontal segment machine' status='completed' active_form='Proving 6-c-iv mid-horizontal machine'",
    "content='6-c-v: horizontal assembly (mid + DL-5 tails)' status='completed' active_form='Assembling 6-c-v horizontal bound'",
    "content='6-d: fold \u27f9 sfBoundProp (then commit, update archive)' status='in_progress' active_form='Proving 6-d fold to sfBoundProp'"
  ],
  "cleared": false
}
tool: Grep
input
{
  "pattern": "theorem sf_abs_le_rectInt_add_zeroSum|theorem zeroSum_norm_le_four_FAlt_add_one|theorem exists_zero_avoiding_T_wide|def Bn\\b|def FAlt|noncomputable def RectangleIntegral|def SfBoundProp|def Sf\\b",
  "path": "D:\\Python\\Quantum",
  "output_mode": "content",
  "-A": 12,
  "glob": "*.lean",
  "head_limit": 120
}
#2301 assistant 2026-08-26 19:55:50.612 UTC · 6274→10954 tok open ⤢
thinking (20312 chars)
All the pieces: 1. `sf_abs_le_rectInt_add_zeroSum (hZC) {n} (hn : 1 ≤ n) {T} (hT1 : 1 ≤ T) (hTn : √n ≤ T) (hTn1 : T ≤ √n+1) (hTwide) : |Sf n| ≤ ‖RectangleIntegral' (Jlag n) (rectZ T) (rectW n T)‖ + ‖∑ ρ ∈ (zeroBothWindow_finite T).toFinset ∪ {(-2:ℂ)}, (zeroMult ρ : ℂ) * (1 - (ρ/(ρ-1))^n)‖` 2. `zeroSum_norm_le_four_FAlt_add_one (hRH : RHUpTo (3*10^12)) (hZCG : ZeroCountGlobalProp) (hNLZ : NoLowZeros) {n} {T} (hTle : T ≤ 3*10^12−1) (hT2 : 2 ≤ T) : ‖Σ‖ ≤ 4·FAlt T + 1` 3. `exists_zero_avoiding_T_wide (hZC) {a} (ha : 3 ≤ a) : ∃ T, a ≤ T ∧ T ≤ a+1 ∧ ∀ ρ, IsNontrivialZero ρ → a−1 < ρ.im → ρ.im < a+2 → 1/(9·C0hat(a+2)) ≤ |ρ.im − T|` 4. `Bn (F : ℝ → ℝ) (n : ℝ) := let T := √n + 1; 4·F T + 4·T·((2/3)·Z2hat T + 10.5·log(T+2) + 48) + 40·1.3352 + 2·(e⁴+1)·(7·Hhat T) + 2·DogLegRight n + IIMaj n + 1` 5. `SfBoundProp := ∀ n : ℕ, 100000 ≤ n → (n:ℝ) ≤ (3·10^12 − 1)^2 → |Sf n| ≤ Bn FAlt n` 6. `Hhat T := 3·C0hat T² + 10.5·log(T+2) + 48`. 7. `FAlt T := (1/(2π))·(T·log T) + 0.137·log T + 0.443·log(log T) + (7/8 + 1.588)`. Also need: `RectangleIntegral'` def — the 1/(2πi) factor. And `rectangleIntegral_Jlag_eq` (§1): RectangleIntegral (Jlag n) (rectZ T) (rectW n T) = H(−T) − H(T) + V(right) − V(left). Let me check RectangleIntegral' quickly. From Zeta23's ResidueCalcOnRectangles: HIntegral'/VIntegral' have (1/(2πi))• factors — RectangleIntegral' presumably := HIntegral' ... combination. Let me check. The final goal structure per the recipe: |Sf n| ≤ ‖RectInt'‖ + ‖Σ‖ [6-b identity] ≤ (1/2π)·‖RectInt‖ + (4·FAlt T + 1) [RectInt' def + norm_smul + bridge] ‖RectInt‖ ≤ ‖H(−T)‖ + ‖H(T)‖ + ‖Vr‖ + ‖Vl‖ [rectangleIntegral_Jlag_eq + norm_sub_le] ≤ [two-sides horizontal] + e^{1/3}·IIMaj + [I_III charge] So |Sf n| ≤ (1/2π)·(Horiz2 + e^{1/3}·IIMaj + I_IIIcharge) + 4F + 1 ≤ (1/2π)·(2(e⁴+1)(NI + 5.5R) + 2·DogLegRight + 2π·IIMaj + I_IIIcharge) + 4F + 1 [e^{1/3} ≤ 2π] ≤ (1/2π)·(2(e⁴+1)(7·Hhat T) + 2·DogLegRight + I_IIIcharge) + IIMaj + 4F + 1 [the horizontal pinch NI + 5.5R ≤ 7·Hhat; e^{1/3}·IIMaj/(2π) ≤ IIMaj] wait — let me redo the arithmetic carefully. Bn's horizontal share: 2·(e⁴+1)·(7·Hhat T) + 2·DogLegRight. My 6-c-v two-sides bound: 2·((e⁴+1)·(NI + 5.5·R(T)) + DogLegRight) where NI = 2·Z2hat(T)·log(148.5·C0hat(√n+2)) and R(T) = 10.5·log(T+2)+48. The pinch: NI + 5.5·R(T) ≤ 7·Hhat T ⟺ 2·Z2hat(T)·log(148.5·C0hat(√n+2)) + 5.5·(10.5 log(T+2)+48) ≤ 7·(3·C0hat(T)² + 10.5 log(T+2) + 48). Per the recipe: "suffices 2·Z2hat(T)·log(148.5·C0hat(√n+2)) ≤ 21·C0hat(T)² since 5.5 ≤ 7 on the (10.5 log+48) part". So the pinch splits: - 5.5·(10.5 log(T+2)+48) ≤ 7·(10.5 log(T+2)+48) ✓ (5.5 ≤ 7, the bracket ≥ 0). - 2·Z2hat(T)·log(148.5·C0hat(√n+2)) ≤ 21·C0hat(T)² — THE real-analytic comparison. Hmm — with C0hat(T)² — C0hat t = 0.433·log(t+2) + 0.886·loglog(t+2) + 2.9 and Z2hat t = 0.6·log(t+2) + 0.886·loglog(t+2) + 2.7. The argument of log on the LHS: 148.5·C0hat(√n+2) vs the RHS's 21·C0hat(T)². E-030's V3 witness says the ratio ≤ 0.40 — the honest pinch. The recipe: "elementary L/LL comparisons with log x ≤ x/e (Real.log_le_sub_one_of_pos at x/e), certified numerics for the endpoint constants". Let me think about the honest proof of 2·Z2hat(T)·log(148.5·C0hat(√n+2)) ≤ 21·C0hat(T)² for T ∈ [√10⁵, 3·10¹²]: Ingredients: - C0hat(T) ≥ 2 (two_le_C0hat, T ≥ 1) — so C0hat(T)² ≥ 2·C0hat(T). - log(148.5·C0hat(√n+2)): with T ≤ √n+1 ⟹ √n+2 ≤ T+1... wait — T ∈ [√n, √n+1] so √n+2 ≤ T+2?? — √n ≤ T ⟹ √n+2 ≤ T+2 ✓. So C0hat(√n+2) ≤ C0hat(T+2) (C0hat_mono, 1 ≤ √n+2 ✓... wait — need √n+2 ≤ T+2 ✓ and C0hat_mono (1 ≤ a) (a ≤ b) ✓ with a := √n+2 ≥ 1 ✓). So log(148.5·C0hat(√n+2)) ≤ log(148.5·C0hat(T+2)). - C0hat(T+2) vs C0hat(T): C0hat(T+2) = 0.433·log(T+4) + 0.886·loglog(T+4) + 2.9. log(T+4) vs log(T+2): log(T+4) ≤ log(T+2) + log((T+4)/(T+2)) ≤ log(T+2) + 2/(T+2) (log(1+x) ≤ x)... For T ≥ 316: 2/(T+2) ≤ 0.0063. So C0hat(T+2) ≤ C0hat(T) + 0.433·0.0063 + 0.886·(loglog(T+4) − loglog(T+2)) + 0 ≈ C0hat(T) + 0.003. This is getting into certified-numerics territory. Alternative simpler route: 148.5·C0hat(√n+2) vs something cleaner. Note log(148.5·C0hat(√n+2)) = log 148.5 + log(C0hat(√n+2)). log 148.5 ≤ 5.001 (e⁵ ≈ 148.41 < 148.5 — hmm — log 148.5: e^5 = 148.413, so log 148.5 ≈ 5.0006 — certify log 148.5 ≤ 5.001 via 148.5 ≤ e^5.001?? — e^5.001 = e^5·e^0.001 ≥ 148.413·1.001 = 148.56 ≥ 148.5 ✓ — messy certified numerics). Hmm — alternatively: 148.5·C0hat(√n+2) ≤ (T+2)²?? — for T ≥ 316: C0hat(√n+2) ≤ C0hat(T+2) ≤ ~10 (C0hat at 3·10¹²+2 ≈ 0.433·29.5 + 0.886·3.4 + 2.9 ≈ 18.6 — hmm — C0hat(T+2) ≤ 19 on the whole range). 148.5·19 ≈ 2822 ≤ (318)² = 101124 ✓ — so log(148.5·C0hat(√n+2)) ≤ 2·log(T+2) — WAIT is that right? 148.5·C0hat(√n+2) ≤ (T+2)² ⟸ C0hat(√n+2) ≤ (T+2)²/148.5 — for T ≥ 316: (T+2)²/148.5 ≥ 318²/148.5 ≈ 681 ≥ 19 ≥ C0hat(√n+2) ✓ — need C0hat(√n+2) ≤ 681 — C0hat(√n+2) ≤ C0hat(T+2) and C0hat(T+2) ≤ 681?? — C0hat is increasing — on T ≤ 3·10¹²: C0hat(3·10¹²+2) ≈ 18.6 ≤ 681 ✓ — but formally bounding C0hat(T+2) by 681 needs a bound over the range... — C0hat(t) ≤ t for t ≥ ?? — C0hat(t) = 0.433·log(t+2) + 0.886·loglog(t+2) + 2.9 ≤ t for t ≥ 4-ish (log(t+2) ≤ t+2−1, loglog(t+2) ≤ log(t+2) ≤ t+2) ⟹ C0hat(t) ≤ 0.433(t+1) + 0.886(t+1) + 2.9 = 1.32(t+1) + 2.9 ≤ t for t ≥ 5ish. So C0hat(√n+2) ≤ √n+2 for √n+2 ≥ 6 (n ≥ 16) — then 148.5·(√n+2) ≤ (√n+2)² ⟺ 148.5 ≤ √n+2 ⟺ √n ≥ 146.5 ⟺ n ≥ 21463 ✓ (n ≥ 10⁵). THEN: log(148.5·C0hat(√n+2)) ≤ log((√n+2)²) = 2·log(√n+2) ≤ 2·log(T+2) (√n ≤ T ⟹ √n+2 ≤ T+2 ✓). So log(148.5·C0hat(√n+2)) ≤ 2·log(T+2)!! Clean. Then: 2·Z2hat(T)·log(148.5·C0hat(√n+2)) ≤ 2·Z2hat(T)·2·log(T+2) = 4·Z2hat(T)·log(T+2). And Z2hat(T) = 0.6·log(T+2) + 0.886·loglog(T+2) + 2.7. Hmm — Z2hat(T) vs C0hat(T) = 0.433·log(T+2) + 0.886·loglog(T+2) + 2.9: Z2hat(T) ≤ 2·C0hat(T)?? — 0.6L + 0.886LL + 2.7 ≤ 2·(0.433L + 0.886LL + 2.9) = 0.866L + 1.772LL + 5.8 ⟺ 0 ≤ 0.266L + 0.886LL + 3.1 ✓ TRUE (L, LL ≥ 0 for T ≥ 1). So Z2hat(T) ≤ 2·C0hat(T) (elementary, no numerics). Then 4·Z2hat(T)·log(T+2) ≤ 8·C0hat(T)·log(T+2). And we want ≤ 21·C0hat(T)² ⟸ 8·log(T+2) ≤ 21·C0hat(T) ⟺ C0hat(T) ≥ (8/21)·log(T+2). C0hat(T) = 0.433·log(T+2) + 0.886·loglog(T+2) + 2.9 ≥ 0.433·log(T+2) ≥ (8/21)·log(T+2) ⟺ 0.433 ≥ 8/21 ≈ 0.381 ✓ TRUE. So: 2·Z2hat(T)·log(148.5·C0hat(√n+2)) ≤ 4·Z2hat(T)·log(T+2) ≤ 8·C0hat(T)·log(T+2) ≤ 8·(C0hat(T)/0.433)·C0hat(T)?? — hmm — cleaner: C0hat(T) ≥ 0.433·log(T+2) ⟹ log(T+2) ≤ C0hat(T)/0.433. So 8·C0hat(T)·log(T+2) ≤ 8·C0hat(T)²/0.433 = 18.48·C0hat(T)² ≤ 21·C0hat(T)² ✓ (18.48 ≤ 21 — with 8/0.433 = 18.4757... ≤ 21 ✓). Hmm — 8/0.433: 0.433·21 = 9.093 > 8 ✓ so 8·C0hat·log(T+2) ≤ 8·C0hat²/0.433 ≤ 21·C0hat² ⟺ 8/0.433 ≤ 21 ⟺ 8 ≤ 9.093 ✓. So the chain (all elementary, no certified numerics!): 1. C0hat(√n+2) ≤ √n+2 [if C0hat(t) ≤ t for t ≥ some bound — need to prove: C0hat(t) = 0.433·log(t+2) + 0.886·loglog(t+2) + 2.9 ≤ t for t ≥ ?? — let me verify at what t: at t = 148: 0.433·log(150) ≈ 0.433·5.01 ≈ 2.17, 0.886·log(5.01) ≈ 0.886·1.61 ≈ 1.43, +2.9 ≈ 6.5 ≤ 148 ✓. The bound C0hat(t) ≤ t holds WAY earlier. Formally: log(t+2) ≤ (t+2)/e·e?? — log x ≤ x/e (Real.log_le_sub_one_of_pos gives log x ≤ x−1 — log(t+2) ≤ t+1; loglog(t+2) ≤ log(t+2) ≤ t+1. So C0hat(t) ≤ 0.433(t+1) + 0.886(t+1) + 2.9 = 1.319·(t+1) + 2.9 ≤ t ⟺ 2.9 + 1.319 ≤ t − 1.319t + ... hmm — 1.319(t+1)+2.9 ≤ t ⟺ 1.319t + 1.319 + 2.9 ≤ t ⟺ 4.219 ≤ −0.319t — FALSE. So C0hat(t) ≤ t does NOT hold for small t via this crude route — at t = 20: 1.319·21 + 2.9 = 30.6 > 20 ✗. At t = 100: 1.319·101+2.9 = 136 > 100 ✗. At t = 200: 1.319·201+2.9 = 268 > 200 ✗. Hmm — the crude log x ≤ x−1 is too lossy. With log x ≤ 2√x (softer): log(t+2) ≤ 2√(t+2), loglog(t+2) ≤ √(t+2)... 0.433·2√ + 0.886·√ + 2.9 = 1.752√(t+2) + 2.9 ≤ t ⟺ ... at t = 148: 1.752·12.2 + 2.9 = 24.3 ≤ 148 ✓ — at t = 20: 1.752·4.7+2.9 = 11.1 ≤ 20 ✓ — at t = 10: 1.752·3.46+2.9 = 8.96 ≤ 10 ✓ — at t = 8: 1.752·3.16+2.9 = 8.44 > 8 ✗ — so C0hat(t) ≤ t for t ≥ ~10 via log x ≤ 2√x. Proving log x ≤ 2√x in Lean: Real.log_le_sub_one_of_pos at √x: log(√x) ≤ √x − 1 — log(√x) = (1/2)log x ⟹ log x ≤ 2(√x−1) ≤ 2√x ✓ — via Real.log_sqrt (log √x = (log x)/2) + Real.log_le_sub_one_of_pos (√x > 0). So: C0hat(t) ≤ t for t ≥ 10: C0hat(t) = 0.433L + 0.886LL + 2.9 with L := log(t+2), LL := log L. L ≤ 2√(t+2) (via the above); LL = log L ≤ L/e·e?? — LL ≤ L − 1 ≤ L ≤ 2√(t+2) hmm — or LL ≤ √(t+2)?? — log L ≤ L − 1 ≤ L ≤ 2√(t+2) — use LL ≤ 2√(t+2) too. So C0hat(t) ≤ (0.433+0.886)·2√(t+2) + 2.9 = 2.638·√(t+2) + 2.9 ≤ t for t ≥ 10: at t = 10: 2.638·3.46 + 2.9 = 12.0 > 10 ✗. Hmm — at t = 15: 2.638·4.12+2.9 = 13.8 ≤ 15 ✓. at t = 13: 2.638·3.87+2.9 = 13.1 > 13 ✗. at t = 14: 2.638·4+2.9 = 13.45 ≤ 14 ✓. So C0hat(t) ≤ t for t ≥ 14. Formally: 2.638·√(t+2) + 2.9 ≤ t ⟺ 2.638·√(t+2) ≤ t − 2.9 ⟺ (2.638)²(t+2) ≤ (t−2.9)² for t ≥ 14: 6.96(t+2) ≤ (t−2.9)² = t² − 5.8t + 8.41 ⟺ 0 ≤ t² − 12.76t − 5.5 — at t = 14: 196 − 178.6 − 5.5 = 11.9 > 0 ✓ and increasing for t ≥ 14 (derivative 2t − 12.76 > 0 for t > 6.4) ✓. So C0hat(t) ≤ t for t ≥ 14, provable by: square both sides (nonneg), quadratic comparison, elementary. Hmm — this is honest but fiddly. Since √n ≥ 316 (n ≥ 10⁵), I only need C0hat(√n+2) ≤ √n+2 for √n ≥ 316 — i.e., t ≥ 318 — where the bound is LOOSE. Maybe simpler: 148.5·C0hat(√n+2) ≤ (√n+2)² — directly: C0hat(√n+2) ≤ (√n+2)²/148.5 — with √n+2 ≥ 318: (√n+2)²/148.5 ≥ 318²/148.5 = 681 ≥ C0hat(√n+2)?? — need C0hat(√n+2) ≤ 681 — C0hat(√n+2) ≤ C0hat-ish bound — hmm — C0hat(t) ≤ t for t ≥ 318: 2.638√(t+2)+2.9 ≤ t at t = 318: 2.638·17.9+2.9 = 50.1 ≤ 318 ✓✓ — the square comparison at t ≥ 318 is even looser. OK — the plan for the pinch lemma (Lean): **pinch_horizontal**: for T ≥ √n (n ≥ 10⁵) — i.e., T ≥ 316...: 2·Z2hat(T)·log(148.5·C0hat(√n+2)) + 5.5·(10.5·log(T+2)+48) ≤ 7·Hhat T. Structure: (a) log(148.5·C0hat(√n+2)) ≤ 2·log(T+2): - C0hat(√n+2) ≤ √n+2 [needs: C0hat(t) ≤ t for t ≥ 14, applied at t := √n+2 ≥ 318] - 148.5·C0hat(√n+2) ≤ (√n+2)² [148.5 ≤ √n+2 since √n ≥ 316 ⟹ 148.5·C0hat ≤ (√n+2)·(√n+2) via C0hat ≤ √n+2 and 148.5 ≤ √n+2 — mul_le_mul (C0hat ≤ √n+2) (148.5 ≤ √n+2) (0 ≤ 148.5) (0 ≤ √n+2)] - log(148.5·C0hat(√n+2)) ≤ log((√n+2)²) = 2·log(√n+2) ≤ 2·log(T+2) [√n+2 ≤ T+2 ✓]. (b) 2·Z2hat(T)·log(148.5·C0hat(√n+2)) ≤ 2·Z2hat(T)·2·log(T+2) = 4·Z2hat(T)·log(T+2) [Z2hat ≥ 0]. (c) Z2hat(T) ≤ 2·C0hat(T) [elementary: 0.6L+0.886LL+2.7 ≤ 2·(0.433L+0.886LL+2.9) ⟺ 0 ≤ 0.266L+0.886LL+3.1 — L, LL ≥ 0 for T ≥ 1 ✓ (L = log(T+2) ≥ 1, LL ≥ 0 — as in Z2hat_mono's proof)]. (d) 4·Z2hat·log(T+2) ≤ 8·C0hat·log(T+2) ≤ 21·C0hat² ⟸ 8·log(T+2) ≤ 21·C0hat ⟸ log(T+2) ≤ (21/8)·C0hat ⟸ C0hat ≥ 0.433·log(T+2) and 1/0.433 ≤ 21/8 = 2.625 ⟺ 8 ≤ 2.625·0.433·8?? — let me redo: 8·C0hat·log(T+2) ≤ 21·C0hat² ⟺ 8·log(T+2) ≤ 21·C0hat (C0hat > 0) ⟸ log(T+2) ≤ (21/8)·C0hat(T) ⟸ C0hat(T) ≥ (8/21)·log(T+2) ⟸ 0.433·log(T+2) + 0.886LL + 2.9 ≥ (8/21)·log(T+2) ⟺ (0.433 − 0.381)·log(T+2) + 0.886LL + 2.9 ≥ 0 ✓ (0.433 > 8/21 = 0.380952... — certify 8/21 ≤ 0.433 ⟺ 8 ≤ 9.093 ⟺ 21·0.433 = 9.093 ✓ norm_num with 0.433·21 = 9.093 ≥ 8 ✓). So: log(T+2) ≤ C0hat(T)/0.433 hmm — cleaner: 0.433·log(T+2) ≤ C0hat(T) (since 0.886LL + 2.9 ≥ 0) ⟹ 8·log(T+2) = (8/0.433)·(0.433·log(T+2)) ≤ (8/0.433)·C0hat(T) ≤ 21·C0hat(T) (8/0.433 ≤ 21 ⟺ 8 ≤ 9.093 ✓). Then 8·C0hat·log(T+2) ≤ 21·C0hat·C0hat = 21·C0hat² — via mul_le_mul (8·log(T+2) ≤ 21·C0hat) (C0hat ≤ C0hat) (0 ≤ C0hat) (0 ≤ 21·C0hat)... — mul_le_mul (h₁ : 8·log(T+2) ≤ 21·C0hat) (h₂ : C0hat ≤ C0hat) (0 ≤ C0hat) (0 ≤ 21·C0hat) : 8·log(T+2)·C0hat ≤ 21·C0hat·C0hat ✓ then rearrange (ring). (e) 5.5·(10.5·log(T+2)+48) ≤ 7·(10.5·log(T+2)+48) ✓ (bracket ≥ 0 — 10.5·log(T+2)+48 ≥ 48 > 0 ✓). (f) Hhat T = 3·C0hat² + (10.5·log(T+2)+48): so (b)+(d) gives NI ≤ 21·C0hat² + ... wait — assemble: NI + 5.5R = 2·Z2hat·log(148.5·C0hat) + 5.5·R ≤ 21·C0hat² + 7·R [from (a)-(d) and (e)] ≤ 7·(3·C0hat² + R) = 7·Hhat T ⟺ 21·C0hat² ≤ 21·C0hat² ✓ and 7·R ≤ 7·R ✓. ✓✓ So pinch_horizontal is fully elementary modulo: - C0hat(t) ≤ t for t ≥ 14 (needs the log ≤ 2√ machine). - 148.5 ≤ √n+2 (√n ≥ 316). - Z2hat ≤ 2·C0hat (elementary L/LL nonneg). - C0hat ≥ 0.433·log(T+2) (LL, const nonneg). - 8 ≤ 21·0.433 (norm_num). The C0hat(t) ≤ t lemma: ```lean theorem C0hat_le_self {t : ℝ} (ht : 14 ≤ t) : C0hat t ≤ t := by -- log(t+2) ≤ 2·√(t+2); loglog(t+2) ≤ 2·√(t+2) [via log x ≤ x−1 at √x] have ht2 : (0:ℝ) < t + 2 := by linarith have hL : Real.log (t + 2) ≤ 2 * Real.sqrt (t + 2) := by have h1 := Real.log_le_sub_one_of_pos (Real.sqrt_pos.mpr ht2) -- h1 : log(√(t+2)) ≤ √(t+2) − 1 rw [Real.log_sqrt ht2.le] at h1 -- log(√x) = (log x)/2 -- (log(t+2))/2 ≤ √(t+2) − 1 ⟹ log(t+2) ≤ 2√(t+2) − 2 ≤ 2√(t+2) nlinarith [Real.sqrt_nonneg (t + 2)] have hLL : Real.log (Real.log (t + 2)) ≤ 2 * Real.sqrt (t + 2) := by have h1 := Real.log_le_sub_one_of_pos (Real.log_pos (show (1:ℝ) < t + 2 by linarith)) -- h1 : log(log(t+2)) ≤ log(t+2) − 1 ≤ log(t+2) ≤ 2√(t+2) linarith [hL, Real.log_le_sub_one_of_pos (Real.log_pos ...)] -- C0hat t = 0.433·L + 0.886·LL + 2.9 ≤ (0.433+0.886)·2√(t+2) + 2.9 ≤ t have hsq : 2.638 * Real.sqrt (t + 2) + 2.9 ≤ t := by -- square both sides have hsqrt_pos : 0 < Real.sqrt (t + 2) := Real.sqrt_pos.mpr ht2 have h1 : (2.638 * Real.sqrt (t + 2) + 2.9) ^ 2 ≤ t ^ 2 := by rw [Real.sq_sqrt ht2.le] -- hmm — need ((2.638√(t+2)+2.9)²) expansion ... ``` Hmm — proving 2.638·√(t+2) + 2.9 ≤ t for t ≥ 14: square: (2.638√(t+2)+2.9)² ≤ t² ⟺ 6.96(t+2) + 2·2.638·2.9·√(t+2) + 8.41 ≤ t² ⟺ 6.96t + 13.9 + 15.3·√(t+2) + 8.41 ≤ t² ⟺ 6.96t + 22.3 + 15.3·√(t+2) ≤ t². With √(t+2) ≤ (t+2)/2 (for t+2 ≥ 4 ⟺ t ≥ 2: √x ≤ x/2 ⟺ x ≥ 4): 15.3·√(t+2) ≤ 7.65(t+2) = 7.65t + 15.3. So LHS ≤ 6.96t + 22.3 + 7.65t + 15.3 = 14.61t + 37.6 ≤ t² ⟺ t² − 14.61t − 37.6 ≥ 0 — at t = 17: 289 − 248.4 − 37.6 = 3 > 0 ✓ — at t = 14: 196 − 204.5 − 37.6 = −46 ✗!! — hmm — so t ≥ 14 is NOT enough via this crude chain. Let me check the ACTUAL value at t = 14: C0hat(14) = 0.433·log(16) + 0.886·loglog(16) + 2.9 = 0.433·2.773 + 0.886·1.020 + 2.9 = 1.201 + 0.904 + 2.9 = 5.005 ≤ 14 ✓✓ — the bound holds easily at 14; my chain is just lossy. Since I only need t ≥ 318 (√n+2 for n ≥ 10⁵), let me just require t ≥ 100 (or 318): at t = 100: 2.638·√102 + 2.9 = 2.638·10.1 + 2.9 = 29.5 ≤ 100 ✓ via the square comparison: (2.638√(t+2)+2.9)² ≤ t² ⟸ 6.96(t+2) + 15.3√(t+2) + 8.41 ≤ t² — with √(t+2) ≤ (t+2)/4 at t ≥ 14?? — √x ≤ x/4 ⟺ x ≥ 16 ⟺ t ≥ 14 ✓ — 15.3·√(t+2) ≤ 15.3(t+2)/4 = 3.825(t+2): LHS ≤ 6.96(t+2) + 3.825(t+2) + 8.41 = 10.785(t+2) + 8.41 = 10.785t + 30 ≤ t² ⟺ t² − 10.785t − 30 ≥ 0 — at t = 14: 196 − 151 − 30 = 15 > 0 ✓ — and t² − 10.785t − 30 is increasing for t ≥ 14 (2t > 10.785 ✓). So C0hat(t) ≤ t for t ≥ 14 — with the auxiliary √x ≤ x/4 for x ≥ 16. Ugh — this is getting heavy. Let me simplify the whole C0hat ≤ t business: actually, what I ACTUALLY need is just `148.5·C0hat(√n+2) ≤ (√n+2)²`. Since √n ≥ 316 (n ≥ 10⁵), set u := √n+2 ≥ 318: 148.5·C0hat(u) ≤ u² ⟸ C0hat(u) ≤ u²/148.5 ⟸ C0hat(u) ≤ u/148.5·... — C0hat(u) ≤ u (for u ≥ 318) and 148.5 ≤ u ⟹ 148.5·C0hat(u) ≤ u·u = u² ✓ (mul_le_mul (C0hat(u) ≤ u) (148.5 ≤ u) (0 ≤ 148.5) (0 ≤ u)). So I need C0hat(u) ≤ u for u ≥ 318 — where it's VERY loose. At u = 318: C0hat ≈ 0.433·5.77 + 0.886·1.75 + 2.9 ≈ 6.95 ≤ 318 ✓✓✓. Simplest honest Lean route for C0hat(u) ≤ u at u ≥ 318: C0hat(u) = 0.433·log(u+2) + 0.886·loglog(u+2) + 2.9 ≤ 0.433·(u+1) + 0.886·(u) + 2.9?? — log(u+2) ≤ u+1 (log_le_sub_one), loglog(u+2) ≤ log(u+2) ≤ u+1: C0hat(u) ≤ (0.433+0.886)(u+1) + 2.9 = 1.319(u+1) + 2.9 ≤ u ⟺ 1.319u + 1.319 + 2.9 ≤ u ⟺ 4.219 ≤ −0.319u — FALSE again!! The crude log ≤ x−1 bound makes C0hat ~1.3u, not ≤ u. The issue: 0.433+0.886 > 1 coefficients... — wait no: log(u+2) ≤ u+1 has coefficient 1; both logs contribute ~u+1 each — total 1.319(u+1) — that exceeds u forever. The crude bound is hopeless; need log ≤ o(u): log(u+2) ≤ (u+2)/e·?? — log x ≤ x/e (at the optimum): log x ≤ x/e for all x > 0 (Real.log_le_sub_one_of_pos at x/e: log(x/e) ≤ x/e − 1 — log x − 1 ≤ x/e − 1 ⟹ log x ≤ x/e ✓ — the recipe mentions exactly this: "log x ≤ x/e (Real.log_le_sub_one_of_pos at x/e)"). log(u+2) ≤ (u+2)/e; loglog(u+2) ≤ log(u+2)/e ≤ (u+2)/e². So C0hat(u) ≤ (0.433/e + 0.886/e²)(u+2) + 2.9 = (0.433·0.3679 + 0.886·0.1353)(u+2) + 2.9 = (0.1593 + 0.1199)(u+2) + 2.9 = 0.279(u+2) + 2.9 ≤ u ⟺ 0.279u + 0.558 + 2.9 ≤ u ⟺ 3.458 ≤ 0.721u ⟺ u ≥ 4.8 ✓ — so C0hat(u) ≤ u for u ≥ ~5 via log ≤ x/e. With e ≤ 2.7183 certified (Real.exp_one_lt_d9 : e < 2.7182818286 — so 1/e ≥ 1/2.7183 — hmm — log x ≤ x/e needs e in the denominator — (u+2)/e ≤ (u+2)/2.7182?? — e > 2.7182 ⟹ 1/e < 1/2.7182 ⟹ (u+2)/e ≤ (u+2)·(1/2.7182) — hmm — 1/2.7182 ≈ 0.367885 vs 1/e ≈ 0.367879 — close but 1/e < 0.367885? — e > 2.7182 ⟹ 1/e < 1/2.7182 = 0.367885 ✓. So log(u+2) ≤ (u+2)/e ≤ 0.367885·(u+2)? — hmm — wait — log x ≤ x/e — the e is Euler's number: log x ≤ x/e ⟺ e·log x ≤ x ⟺ log(x^e) ≤ x — hmm — this is log x ≤ x/e — TRUE (standard: log x ≤ x/e since x/e is the tangent line to log at x = e). To prove in Lean: log(x/e) ≤ x/e − 1 (log_le_sub_one at x/e > 0) ⟹ log x − log e ≤ x/e − 1 ⟹ log x − 1 ≤ x/e − 1 ⟹ log x ≤ x/e ✓. With e ≈ 2.71828: log x ≤ x/e ≤ x/2.7182?? — NO — 1/e vs 1/2.7182: e ≥ 2.7182 ⟹ 1/e ≤ 1/2.7182 ⟹ x/e ≤ x/2.7182 ✓ — so log x ≤ x/e ≤ 0.36789·x... — hmm 1/2.7182 = 0.3678894... — and e ≤ 2.7183 gives 1/e ≥ 0.3678757 — the WRONG direction for upper-bounding 1/e. I need 1/e ≤ c for an upper bound on log x ≤ c·x — from e ≥ 2.7182 (lower bound on e): 1/e ≤ 1/2.7182 = 0.3678894. Do I have e ≥ 2.7182 certified? Real.exp_one_gt_d9?? — there might be `Real.exp_one_gt_d9 : 2.7182818283 < exp 1`? — hmm — exp_one_lt_d9 gives UPPER. For a lower bound on e: e > 2.7 trivially from exp_one_gt?? — or from add_one_le_exp: 1+1 ≤ e ✓ (e ≥ 2). e ≥ 2.7: from exp(1) ≥ sum of series terms: 1+1+1/2+1/6+1/24+1/120 = 2.7083 — via Real.add_one_le_exp?? — that gives x+1 ≤ e^x — at x=1: 2 ≤ e. For better: `Real.exp_one_gt_d9`? — let me check if it exists — likely yes (companion to lt_d9): `Real.exp_one_gt_d9 : 2.7182818283 < Real.exp 1`? — I'll grep. So C0hat(u) ≤ u for u ≥ 318 (much weaker than u ≥ 5 — plenty of room): C0hat(u) = 0.433L + 0.886LL + 2.9 with L := log(u+2), LL := log L. L ≤ (u+2)/e ≤ 0.3679·(u+2) [needs 1/e ≤ 0.3679 ⟺ e ≥ 1/0.3679 = 2.71816 — from e ≥ 2.7182?? — 1/2.7182 = 0.3678894 > 0.3679?? — hmm — 0.3679 vs 1/2.7182 = 0.367889 — 0.3679 > 0.367889 ✓ so 1/e ≤ 1/2.7182 = 0.367889 ≤ 0.3679 ✓]. LL = log L ≤ L/e ≤ (u+2)/e² ≤ 0.3679²·(u+2) = 0.13535·(u+2)?? — hmm — LL = log L ≤ L/e — wait — log L ≤ L/e (log x ≤ x/e at x := L) ✓ ≤ 0.3679·L ≤ 0.3679·0.3679·(u+2) = 0.13535·(u+2). C0hat(u) ≤ (0.433·0.3679 + 0.886·0.13535)(u+2) + 2.9 = (0.15930 + 0.11992)(u+2) + 2.9 = 0.27922(u+2) + 2.9 ≤ u ⟺ 0.27922u + 0.55844 + 2.9 ≤ u ⟺ 3.45844 ≤ 0.72078u ⟺ u ≥ 4.798 ✓ — at u ≥ 318 trivially ✓ — the final linarith with u ≥ 318: 0.27922u + 3.45844 ≤ 0.27922u + 0.72078u·?? — just: 0.27922(u+2) + 2.9 ≤ u ⟺ 2.9 + 0.55844 ≤ u − 0.27922u = 0.72078u ⟺ 3.45844 ≤ 0.72078u — at u ≥ 318: 0.72078·318 = 229.2 ≥ 3.45844 ✓✓ linarith closes with the coefficient arithmetic ✓ (linear in u once the 0.27922(u+2) bound is established). So the pinch is fully elementary with e ≥ 2.7182 (or even e ≥ 2.7 — let me redo with 1/e ≤ 1/2.7 = 0.3704: C0hat(u) ≤ (0.433·0.3704 + 0.886·0.13717)(u+2) + 2.9 = (0.16036 + 0.12153)(u+2) + 2.9 = 0.28189(u+2) + 2.9 ≤ u ⟺ 3.46378 ≤ 0.71811u ⟺ u ≥ 4.82 ✓ — works with just e ≥ 2.7!). e ≥ 2.7 from: exp(1) ≥ 1 + 1 + 1/2 = 2.5 via add_one_le_exp?? — no — add_one_le_exp x : x+1 ≤ exp x — at x := 1: 2 ≤ e only. For e ≥ 2.7 need a series lower bound: Real.add_one_le_exp is too weak. Options: Real.exp_one_gt_d9 (if exists) or `Real.exp_one_near_10`? — or Real.sum_... Let me check for exp_one_gt_d9. Alternatively — avoid e-lower-bounds: prove log x ≤ x/2.7 directly?? — log x ≤ x/2.7 ⟺ 2.7·log x ≤ x ⟺ log(x^2.7) ≤ x — hmm — or: log x ≤ x/e with e := Real.exp 1, then bound 1/e from ABOVE via e ≥ 2.7: 1/e ≤ 1/2.7: e ≥ 2.7 ⟸ 2.7 ≤ exp 1 ⟸ log 2.7 ≤ 1 ⟸ ... log 2.7 ≤ 1 ⟺ 2.7 ≤ e — circular. Hmm — e ≥ 2.7: exp(1) ≥ 2.7 — via the series: Real.exp 1 = ∑' 1/n! ≥ 1 + 1 + 1/2 + 1/6 = 2.6667 — not enough; +1/24 = 2.7083 ≥ 2.7 ✓ — proving exp(1) ≥ 2.7083 from the series definition needs the partial-sum bound for exp — `Real.sum_le_exp`? or `Real.exp_one_gt_...`? Let me check what exists: grep for exp_one_gt. Actually — check `Real.exp_one_gt_d9` exists. If not, alternative: e ≥ 2.7182 from Real.exp_one_lt_d9's companion. Let me grep.
All 6-d components located. The horizontal pinch needs a certified lower bound on `e` — checking what's available.
tool: Bash
input
{
  "command": "grep -rn \"exp_one_gt\\|exp_one_near\" \"D:/Python/Quantum/zeta-23-lean/.lake/packages/mathlib/Mathlib/Analysis/Complex/ExponentialBounds.lean\" | head -5; grep -rn \"theorem log_le_sub_one_of_pos\\|theorem log_le_rpow_div\\|nonrec theorem RectangleIntegral'\\|def RectangleIntegral'\\|theorem RectangleIntegral'\" \"D:/Python/Quantum/zeta-23-lean/\" 2>/dev/null | grep -v \".lake\" | head -8",
  "description": "Check exp_one_gt and RectangleIntegral' def"
}
#2302 assistant 2026-08-26 19:58:31.021 UTC · 481→1854 tok open ⤢
thinking (3782 chars)
`RectangleIntegral' f z w := (1 / (2 * π * I)) • RectangleIntegral f z w` ✓ — so ‖RectInt'‖ = ‖1/(2πi)‖·‖RectInt‖ = (1/(2π))·‖RectInt‖ — with ‖1/(2πi)‖ = 1/‖2πi‖ = 1/(2π·‖i‖) = 1/(2π): ‖(1/(2πi))‖ = 1/‖2 * π * I‖ — ‖2 * π * I‖ = 2·π·‖I‖ = 2π — so ‖1/(2πi)‖ = 1/(2π) — need: ‖2 * π * I‖ = 2π (norm_mul, Complex.norm_I, and π as a complex: ‖(π : ℂ)‖ = π — and the 2 : ℂ norm = 2). And norm_smul : ‖c • x‖ = ‖c‖·‖x‖ — norm_smul for ℂ smul on ℂ — `norm_smul (c : 𝕜) (x : E) : ‖c • x‖ = ‖c‖ * ‖x‖`? — for ℝ or ℂ scalars — `norm_smul` ✓ (used in VIntegral_norm: rw [norm_smul, Complex.norm_I, one_mul] ✓). Now the plan for 6-d. Statement: ```lean theorem sfBoundProp (hRH : RHUpTo (3 * 10 ^ 12)) (hZC : ZeroCountProp) (hZC2 : ZeroCountProp2) (hZCG : ZeroCountGlobalProp) (hNLZ : NoLowZeros) (hMZ : MZeroProp) : SfBoundProp := by intro n hn hN1 -- T from the wide pigeonhole at a := √n have hnR : (100000 : ℝ) ≤ (n : ℝ) := by exact_mod_cast hn have ha : (3 : ℝ) ≤ Real.sqrt (n : ℝ) := by ... (√n ≥ 316) obtain ⟨T, hTn, hTn1, hTwide⟩ := exists_zero_avoiding_T_wide hZC ha -- the sides identity have hsf := sf_abs_le_rectInt_add_zeroSum hZC (by omega : 1 ≤ n) (by ... : 1 ≤ T) hTn hTn1 hTwide -- the bridge have hTle : T ≤ 3 * 10 ^ 12 - 1 := by -- T ≤ √n+1 ≤ √(N₁)+1 = (3·10¹²−1)+1... hmm — n ≤ (3·10¹²−1)² ⟹ √n ≤ 3·10¹²−1 ⟹ T ≤ √n+1 ≤ 3·10¹² ✓ — need T ≤ 3·10¹²−1?? — the bridge needs hTle : T ≤ 3·10¹²−1!! T ≤ √n+1 ≤ (3·10¹²−1)+1 = 3·10¹² — NOT ≤ 3·10¹²−1!!! ``` WAIT — problem: the bridge zeroSum_norm_le_four_FAlt_add_one needs (hTle : T ≤ 3·10¹²−1). But T ∈ [√n, √n+1] with n ≤ N₁ = (3·10¹²−1)² — so √n ≤ 3·10¹²−1 and T ≤ √n+1 ≤ 3·10¹². hTle : T ≤ 3·10¹²−1 is NOT guaranteed when √n is within 1 of the endpoint!! Hmm — but the recipe says "bridge zeroSum_norm_le_four_FAlt_add_one hRH hZCG hNLZ (T ≤ 3×10¹²−1 from n ≤ N₁; 2 ≤ T)". Hmm — T ≤ √n+1 and √n ≤ 3·10¹²−1 gives T ≤ 3·10¹² — the recipe says T ≤ 3×10¹²−1 "from n ≤ N₁" — that's WRONG by 1 unless... — hmm — actually wait: maybe the bridge's hTle is used for the RHUpTo application: ρ.im ≤ T ⟹ re = 1/2 — hRH : RHUpTo (3·10¹²) := ∀ ρ, IsNontrivialZero ρ → |ρ.im| ≤ 3·10¹² → ρ.re = 1/2. The zeros in zeroBothWindow T have |γ| ≤ T+1 ≤ 3·10¹²+1?? — hmm — so actually maybe the bridge's hTle : T ≤ 3·10¹²−1 is used to get |γ| ≤ 3·10¹² for the window zeros: zeroBothWindow T covers |γ| ≤ T+1?? — with T ≤ 3·10¹²−1: |γ| ≤ T+1 ≤ 3·10¹² ✓ — so the bridge NEEDS T ≤ 3·10¹²−1 (so that T+1 ≤ 3·10¹² for RHUpTo). And T ≤ √n+1 with √n ≤ 3·10¹²−1 gives T ≤ 3·10¹² — INSUFFICIENT by the same slack. Hmm — but wait — actually — n ≤ N₁ = (3·10¹²−1)² — if n < N₁ slightly... — the issue is only at the very top of the range. Hmm — but actually — hold on: T ≤ √n + 1 where √n ≤ 3·10¹²−1 — T ≤ 3·10¹². For the bridge I need T ≤ 3·10¹²−1 — i.e., √n + 1 ≤ 3·10¹²−1 ⟺ √n ≤ 3·10¹²−2 ⟺ n ≤ (3·10¹²−2)². But n can be as large as (3·10¹²−1)² > (3·10¹²−2)². CONFLICT at the very top: for n ∈ ((3·10¹²−2)², (3·10¹²−1)²], the bridge's hTle fails?! Hmm wait — let me recheck: SfBoundProp's range: (n:ℝ) ≤ (3·10¹²−1)². T ∈ [√n, √n+1]. Bridge needs T ≤ 3·10¹²−1. Since T ≤ √n+1 ≤ (3·10¹²−1)+1 = 3·10¹², we have T ≤ 3·10¹² but need ≤ 3·10¹²−1. GAP of 1. Hmm — but the ii′-6-b bridge was DESIGNED for this fold (session-038) — surely this was accounted. Let me look at how the bridge's hTle is actually USED — maybe hTle : T ≤ 3·10¹²−1 is just one option and the actual requirement is |γ| ≤ 3·10¹² for window zeros: zeroBothWindow T — let me check zeroBothWindow's def — if it's |γ| ≤ T (not T+1), then |γ| ≤ T ≤ 3·10¹² needs T ≤ 3·10¹² ✓ — but the hypothesis is written as T ≤ 3·10¹²−1... — let me look at the bridge's proof to see what it really needs. Actually — hmm — simplest: check zeroBothWindow_finite/zeroBothWindow def.
Checking the bridge's actual T-range requirement vs what T ≤ √n+1 gives at the top of the range.
tool: Grep ok
input
{
  "pattern": "def zeroBothWindow|zeroBothWindow_finite|hTle",
  "path": "D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiAssembly.lean",
  "output_mode": "content",
  "-A": 6,
  "head_limit": 50
}
result 2026-08-26 19:58:31Z → 2026-08-26 19:58:31Z
mode='content' num_files=0 filenames=[] content='39:def zeroBothWindow (T : ℝ) : Set ℂ :=\n40-  {ρ | IsNontrivialZero ρ ∧ |ρ.im| ≤ T}\n41-\n42-/-- Finiteness of the both-signs window (via Zeta23 `zerosIn_finite`). -/\n43:theorem zeroBothWindow_finite (T : ℝ) : (zeroBothWindow T).Finite := by\n44-  apply Set.Finite.subset (zerosIn_finite (-T - 1) T)\n45-  intro ρ hρ\n46-  refine ⟨hρ.1, ?_, ?_⟩\n47-  · linarith [abs_le.mp hρ.2 |>.1]\n48-  · exact abs_le.mp hρ.2 |>.2\n49-\n--\n81:      ‖∑ ρ ∈ ((zeroBothWindow_finite T).toFinset ∪ {(-2 : ℂ)}),\n82-        (zeroMult ρ: ℂ) * (1 - (ρ / (ρ - 1)) ^ n)‖ := by\n83-  set z:= rectZ T\n84-  set w:= rectW n T\n85-  have hn1sqrt: (1 : ℝ) ≤ Real.sqrt n := Real.one_le_sqrt.mpr (by exact_mod_cast hn)\n86-  have hre: z.re ≤ w.re := by\n87-    rw [rectZ_re, rectW_re]\n--\n198:  set Sz : Finset ℂ := (zeroBothWindow_finite T).toFinset ∪ {(-2 : ℂ)} with hSz_def\n199-  have hSz: ∀ s : ℂ, s ∈ Sz ↔ s ∈ Rectangle z w ∧ zetaA s = 0 := by\n200-    intro s\n201-    constructor\n202-    · intro hs\n203-      rw [hSz_def, Finset.mem_union, Finset.mem_singleton] at hs\n204-      rcases hs with hs | rfl\n205:      · have hmem : s ∈ zeroBothWindow T := (zeroBothWindow_finite T).mem_toFinset.mp hs\n206-        obtain ⟨hnt, himT⟩:= hmem\n207-        have hs1: s ≠ 1 := by\n208-          intro h\n209-          have h22: (1 : ℂ).re < 1 := h ▸ hnt.2.2\n210-          rw [Complex.one_re] at h22\n211-          exact lt_irrefl 1 h22\n--\n259:        apply (zeroBothWindow_finite T).mem_toFinset.mpr\n260-        refine ⟨hnt, ?_⟩\n261-        have hi: s.im ∈ Set.uIcc z.im w.im := hmem.2\n262-        rw [Set.uIcc_of_le him, rectZ_im, rectW_im, Set.mem_Icc] at hi\n263-        exact abs_le.mpr hi\n264-      · have h00: zetaA 0 = 0 := by rwa [h0] at hzA\n265-        rw [zetaA_apply (by norm_num: (0 : ℂ) ≠ 1), riemannZeta_zero] at h00\n--\n336:    (hTle : T ≤ 3 * 10 ^ 12) :\n337:    ‖∑ ρ ∈ (zeroBothWindow_finite T).toFinset,\n338-      (zeroMult ρ: ℂ) * (1 - (ρ / (ρ - 1)) ^ n)‖ ≤\n339:      2 * ∑ ρ ∈ (zeroBothWindow_finite T).toFinset, (zeroMult ρ : ℝ) := by\n340:  calc ‖∑ ρ ∈ (zeroBothWindow_finite T).toFinset,\n341-        (zeroMult ρ: ℂ) * (1 - (ρ / (ρ - 1)) ^ n)‖\n342:      ≤ ∑ ρ ∈ (zeroBothWindow_finite T).toFinset,' num_lines=50 num_matches=None applied_limit=50 applied_offset=None
#2303 assistant 2026-08-26 20:12:05.336 UTC · 1867→20734 tok open ⤢
thinking (45526 chars)
Confirmed: `zeroSum_nontriv_norm_le` needs `(hTle : T ≤ 3 * 10 ^ 12)` — and the bridge proves it via `by linarith [hTle]` from the over-strong `T ≤ 3·10¹²−1`. So the bridge's PUBLIC hypothesis hTle : T ≤ 3·10¹²−1 is tighter than the internal need (T ≤ 3·10¹²). For MY fold: T ≤ √n+1 ≤ (3·10¹²−1)+1 = 3·10¹². So T ≤ 3·10¹² EXACTLY holds — the internal zeroSum_nontriv_norm_le is satisfied. But the bridge's hTle : T ≤ 3·10¹²−1 might fail when T ∈ (3·10¹²−1, 3·10¹²]. Hmm — wait — can T be in (3·10¹²−1, 3·10¹²]? T ∈ [√n, √n+1], √n ≤ 3·10¹²−1: T ≤ 3·10¹² ✓ and T ≥ √n could be up to 3·10¹²−1. So T ∈ [√n, √n+1] ⊆ [?, 3·10¹²] — T might land in (3·10¹²−1, 3·10¹²] (only if √n > 3·10¹²−2, i.e., n > (3·10¹²−2)², AND the pigeonhole picks T > 3·10¹²−1). So hTle is NOT guaranteed. RESOLUTION options: (a) Prove a variant bridge with hTle : T ≤ 3·10¹² (clone the ~30-line proof with the linarith adjusted — trivially, since the internal call already only needs T ≤ 3·10¹²). This is clean and honest: the bridge's content only ever used T ≤ 3·10¹². (b) Strengthen nothing — just inline the bridge's last mile in 6-d. (a) is best: prove `zeroSum_norm_le_four_FAlt_add_one_of_le` with (hTle : T ≤ 3 * 10 ^ 12) in LiAssembly — actually simpler: in LiSideBounds §6, prove the bridge variant by CALLING the same internal lemmas (zeroSum_nontriv_norm_le + zeroBothWindow_sum_eq_two_mul + trivTerm_norm_le_one + hZCG) — the bridge body is 30 lines, clone with hTle : T ≤ 3·10¹². Hmm wait — actually — even better: can I just get hTle : T ≤ 3·10¹²−1 directly?? T ≤ √n+1 — and I need T ≤ 3·10¹²−1. Hmm — what if n is EXACTLY N₁ = (3·10¹²−1)²? — SfBoundProp includes the endpoint. √n = 3·10¹²−1 (if n = N₁ exactly — but n : ℕ and (3·10¹²−1)² might not be a perfect square — (3·10¹²−1)² = 9·10²⁴ − 6·10¹² + 1 — is that a square? — 3·10¹²−1 = 2999999999999 — its square is (3·10¹²−1)² — n can equal that value (as a real bound (n:ℝ) ≤ (3·10¹²−1)² — the largest such n is ⌊(3·10¹²−1)²⌋ = (3·10¹²−1)² exactly since it's an integer ✓ (3·10¹²−1 is a natural number? — 3·10¹² = 3000000000000 ✓ integer, so 3·10¹²−1 = 2999999999999 ∈ ℕ ✓ and its square ∈ ℕ ✓). So n = (3·10¹²−1)² IS in range, with √n = 3·10¹²−1 EXACTLY (Real.sqrt of a perfect square — (√n)² = n = (3·10¹²−1)² ⟹ √n = 3·10¹²−1 ✓). Then T ∈ [3·10¹²−1, 3·10¹²] — the pigeonhole could pick T > 3·10¹²−1 ⟹ hTle FAILS. So I MUST have the T ≤ 3·10¹² variant of the bridge. Clone it in LiSideBounds §6 (or add to LiAssembly — but the freeze protocol... — the freeze is on PAPER files and existing proofs; adding NEW decls to LiAssembly is fine, but §6 of LiSideBounds is the natural home). I'll clone the bridge proof in LiSideBounds §6 with hTle : T ≤ 3 * 10 ^ 12 — calling zeroSum_nontriv_norm_le (T ≤ 3·10¹² ✓), zeroBothWindow_sum_eq_two_mul, trivTerm_norm_le_one, hZCG. Wait — the internal helpers h2not/hdisj2 etc. — all public? zeroSum_nontriv_norm_le is a `lemma` (public ✓), zeroBothWindow_sum_eq_two_mul ✓, trivTerm_norm_le_one ✓ — all accessible. So the 6-d theorem `sfBoundProp`: ```lean theorem sfBoundProp (hRH : RHUpTo (3 * 10 ^ 12)) (hZC : ZeroCountProp) (hZC2 : ZeroCountProp2) (hZCG : ZeroCountGlobalProp) (hNLZ : NoLowZeros) (hMZ : MZeroProp) : SfBoundProp := by intro n hn hN1 classical have hnR : (100000 : ℝ) ≤ (n : ℝ) := by exact_mod_cast hn have hn1 : 1 ≤ n := by omega have ha : (3 : ℝ) ≤ Real.sqrt (n : ℝ) := by have h1 : (10 : ℝ) ≤ Real.sqrt (n : ℝ) := by rw [Real.le_sqrt (by norm_num) (by positivity)] nlinarith [hnR] linarith obtain ⟨T, hTn, hTn1, hTwide⟩ := exists_zero_avoiding_T_wide hZC ha have hT1 : (1 : ℝ) ≤ T := by linarith [ha, hTn] have hT2 : (2 : ℝ) ≤ T := by linarith have h10T : (10 : ℝ) ≤ T := by linarith [show (10:ℝ) ≤ √(n:ℝ) from ...] -- T ≤ 3·10¹² (from n ≤ N₁) have hTle : T ≤ 3 * 10 ^ 12 := by have h1 : Real.sqrt (n : ℝ) ≤ 3 * 10 ^ 12 - 1 := by have h2 : (Real.sqrt (n : ℝ)) ^ 2 ≤ (3 * 10 ^ 12 - 1 : ℝ) ^ 2 := by rw [Real.sq_sqrt (by positivity : (0:ℝ) ≤ (n:ℝ))] -- (n:ℝ) ≤ (3·10¹²−1)² ✓ hN1 -- need ^2 form: (√n)² = n ≤ (3·10¹²−1)² — and RHS is already squared -- hmm — (3*10^12−1:ℝ)^2 vs hN1's (3 * 10 ^ 12 - 1 : ℝ) ^ 2 ✓ same exact hN1 -- √n ≤ 3·10¹²−1 from squares + nonneg exact (sq_le_sq₀ (Real.sqrt_nonneg _) (by norm_num : (0:ℝ) ≤ 3*10^12-1)).mp h2 linarith [hTn1] ``` Hmm — sq_le_sq₀ : 0 ≤ a → 0 ≤ b → (a^2 ≤ b^2 ↔ a ≤ b)?? or (a ≤ b ↔ a² ≤ b²)? — the name suggests sq_le_sq₀ (ha : 0 ≤ a) (hb : 0 ≤ b) : a ^ 2 ≤ b ^ 2 ↔ a ≤ b — the .mp direction: a² ≤ b² ⟹ a ≤ b ✓. Let me double check its exact form later (banked gotcha mentions sq_eq_sq₀ and pow_le_one₀ — and I used pow_le_pow_left₀ before). Alternative: use Real.sqrt_le_sqrt on hN1: √(n) ≤ √((3·10¹²−1)²) = |3·10¹²−1| = 3·10¹²−1: ```lean have hTle : T ≤ 3 * 10 ^ 12 := by have h1 : Real.sqrt (n : ℝ) ≤ 3 * 10 ^ 12 - 1 := by have h2 := Real.sqrt_le_sqrt hN1 rwa [Real.sqrt_sq (by norm_num : (0:ℝ) ≤ 3 * 10 ^ 12 - 1)] at h2 linarith [hTn1] ``` `Real.sqrt_le_sqrt (h : a ≤ b) : √a ≤ √b` ✓ (used in DL-5: Real.sqrt_le_sqrt (by exact_mod_cast hn) ✓ — takes the inequality directly). `Real.sqrt_sq (ha : 0 ≤ a) : √(a^2) = a` ✓ (Real.sqrt_sq, standard). hN1 : (n:ℝ) ≤ (3*10^12−1)^2 — Real.sqrt_le_sqrt hN1 : √(↑n) ≤ √((3·10¹²−1)²) — rwa [Real.sqrt_sq ...] at h2 : √((3·10¹²−1)²) → 3·10¹²−1 ✓ (norm_num proves 0 ≤ 3·10¹²−1 ✓). Then T ≤ √n+1 ≤ (3·10¹²−1)+1 = 3·10¹² ✓ linarith [hTn1]. -- the sides identity + bridge(variant) have hsf := sf_abs_le_rectInt_add_zeroSum hZC hn1 hT1 hTn hTn1 hTwide -- the bridge variant (T ≤ 3·10¹²) have hbridge : ‖∑ ρ ∈ ((zeroBothWindow_finite T).toFinset ∪ {(-2 : ℂ)}), (zeroMult ρ : ℂ) * (1 - (ρ / (ρ - 1)) ^ n)‖ ≤ 4 * FAlt T + 1 := by -- clone of zeroSum_norm_le_four_FAlt_add_one with hTle : T ≤ 3·10¹² have h2not : (-2 : ℂ) ∉ (zeroBothWindow_finite T).toFinset := by intro h have hnt := (zeroBothWindow_finite T).mem_toFinset.mp h |>.1 have h21 : (0 : ℝ) < (-2 : ℂ).re := hnt.2.1 norm_num at h21 have hdisj2 : Disjoint (zeroBothWindow_finite T).toFinset {(-2 : ℂ)} := by rw [Finset.disjoint_left] intro ρ hρ hρ2 rw [Finset.mem_singleton] at hρ2 exact h2not (hρ2 ▸ hρ) rw [Finset.sum_union hdisj2, Finset.sum_singleton] have htriv := trivTerm_norm_le_one n have hbr := zeroSum_nontriv_norm_le hRH (n := n) (T := T) hTle have hcount := zeroBothWindow_sum_eq_two_mul hNLZ T have hZ := hZCG T hT2 calc ... ``` (clone the calc verbatim.) -- the side bounds have hVl := vIntegral_Jlag_left_norm_le hZC hZC2 hNLZ hMZ hn hTn hTn1 have hVr := vIntegral_Jlag_right_le_exp_third_IIMaj hn hTn hTn1 have hH := hIntegral_Jlag_full_two_sides hZC hZC2 hNLZ hn hTwide hTn hTn1 -- ‖RectInt'‖ = (1/2π)·‖RectInt‖ have hnorm : ‖RectangleIntegral' (Jlag n) (rectZ T) (rectW n T)‖ = (1 / (2 * Real.pi)) * ‖RectangleIntegral (Jlag n) (rectZ T) (rectW n T)‖ := by unfold RectangleIntegral' rw [norm_smul] congr 1 -- ‖1/(2πi)‖ = 1/(2π) rw [norm_div, norm_one] have h1 : ‖(2 : ℂ) * Real.pi * Complex.I‖ = 2 * Real.pi := by rw [norm_mul, Complex.norm_I, mul_one] -- ‖(2:ℂ) * π‖ = 2π: norm_mul, ‖(2:ℂ)‖ = 2, ‖(π:ℂ)‖ = π rw [norm_mul] have h2 : ‖((2 : ℕ) : ℂ)‖ = 2 := by norm_num?... ``` Hmm — ‖(2 * π * I : ℂ)‖ — the elaborated form: (2 : ℂ) * (π : ℂ) * I — ‖·‖ = 2 * π * 1 = 2π. Note: `2 * π * I` in the RectangleIntegral' def — the 2 is a ℂ numeral, π : ℝ coerced to ℂ, I : ℂ. ‖2 * π * I‖ = ‖2‖·‖π‖·‖I‖ = 2·π·1 = 2π. In Lean: rw [norm_mul, norm_mul] : ‖(2:ℂ)‖·‖(π:ℂ)‖·‖I‖ — ‖(2:ℂ)‖ = 2 (norm_num or RCLike.norm_natCast... — hmm — `‖((2 : ℂ))‖` — via `norm_num`? or the natCast route (2:ℂ) = ((2:ℕ):ℂ): `show ((2:ℕ):ℂ) ...` — ugh. There's `Complex.norm_two`? — hmm. And ‖(π:ℂ)‖ = π : `Complex.norm_real`? — π is Real.pi coerced: ‖(↑π : ℂ)‖ = |π| = π (Complex.norm_real + abs_of_nonneg pi_nonneg). Let me look for existing norms: `Complex.norm_ofReal (r : ℝ) : ‖(↑r : ℂ)‖ = |r|` ✓ exists (norm_ofReal). And ‖(2:ℂ)‖ = 2 : via `show ((2:ℕ):ℂ)`... — simplest: `by norm_num [norm_mul, Complex.norm_I, Complex.norm_ofReal, abs_of_nonneg Real.pi_nonneg]`?? — hmm — norm_num with those simp lemmas might do it. Let me instead do it manually: ```lean have h1 : ‖(2 : ℂ) * (Real.pi : ℂ) * Complex.I‖ = 2 * Real.pi := by rw [norm_mul, norm_mul, Complex.norm_I, mul_one] have h2 : ‖((2 : ℂ) * (Real.pi : ℂ))‖ = 2 * Real.pi := by rw [norm_mul] have h3 : ‖(2 : ℂ)‖ = (2 : ℝ) := by rw [show (2 : ℂ) = ((2 : ℝ) : ℂ) by push_cast; ring, Complex.norm_real] exact abs_of_nonneg (by norm_num) have h4 : ‖(Real.pi : ℂ)‖ = Real.pi := by rw [Complex.norm_real] exact abs_of_nonneg Real.pi_nonneg rw [h3, h4] rw [h2] ``` Hmm — wait: ‖2 * π * I‖ — parse: ((2 * π) * I) — norm_mul → ‖2*π‖·‖I‖ — Complex.norm_I : ‖I‖ = 1 ✓ → ‖2*π‖ — norm_mul → ‖2‖·‖π‖ → 2·π via h3/h4. But — Complex.norm_real : ‖((r : ℝ) : ℂ)‖ = |r| ✓ (used in 6-c-ii ✓). (Real.pi : ℂ) — π is Real.pi : ℝ coerced — `(Real.pi : ℂ)` — hmm — how is π written in the def? `2 * π * I` — π here — in Zeta23's ResidueCalcOnRectangles, `π` is presumably `Real.pi` coerced to ℂ (they `open Complex` — π : ℂ via coercion or a local notation?). In the def `(1 / (2 * π * I))` — with Complex namespace open — `π` might be `Real.pi` with a coercion inserted automatically (since the context is ℂ). So the term is `(1 : ℂ) / ((2:ℂ) * (↑Real.pi) * I)` — the norm computation as above ✓. Complex.norm_real : ‖(↑r)‖ = |r| — with r := Real.pi ✓. Hmm — `Real.pi_nonneg : 0 ≤ π` ✓. Then ‖1/(2πI)‖ = 1/‖2πI‖ = 1/(2π) via norm_div + norm_one + h1: ```lean rw [norm_div, norm_one, h1] ``` ‖1/(2πI)‖ = ‖1‖/‖2πI‖ = 1/(2π) ✓. Wait — norm_smul : ‖c • x‖ = ‖c‖ * ‖x‖ — RectangleIntegral' = (1/(2πI)) • RectInt — the smul is ℂ-scalar on ℂ-vector ✓ norm_smul ✓ gives ‖1/(2πI)‖·‖RectInt‖ ✓. Hmm — the RectangleIntegral' is an `abbrev` — `unfold RectangleIntegral'` ✓ (or it's reducible — norm_smul might match directly without unfold — use unfold to be safe). -- the rectangle side decomposition + norm triangle have hrect := rectangleIntegral_Jlag_eq n T -- RectInt = H(−T) − H(T) + V(right) − V(left) have htri : ‖RectangleIntegral (Jlag n) (rectZ T) (rectW n T)‖ ≤ (‖HIntegral (Jlag n) (-5/2) (2 * Real.sqrt n) T‖ + ‖HIntegral (Jlag n) (-5/2) (2 * Real.sqrt n) (-T)‖) + (‖VIntegral (Jlag n) (2 * Real.sqrt n) (-T) T‖ + ‖VIntegral (Jlag n) (-5/2) (-T) T‖) := by rw [hrect] -- ‖(A − B) + (C − D)‖ ≤ ‖A‖ + ‖B‖ + ‖C‖ + ‖D‖ calc ‖(HIntegral (Jlag n) (-5/2) (2√n) (-T) − HIntegral (Jlag n) (-5/2) (2√n) T) + (VIntegral (Jlag n) (2√n) (-T) T − VIntegral (Jlag n) (-5/2) (-T) T)‖ ≤ ‖HIntegral (Jlag n) (-5/2) (2√n) (-T) − HIntegral (Jlag n) (-5/2) (2√n) T‖ + ‖VIntegral (Jlag n) (2√n) (-T) T − VIntegral (Jlag n) (-5/2) (-T) T‖ := norm_add_le _ _ _ ≤ (‖HIntegral (Jlag n) (-5/2) (2√n) (-T)‖ + ‖HIntegral (Jlag n) (-5/2) (2√n) T‖) + (‖VIntegral (Jlag n) (2√n) (-T) T‖ + ‖VIntegral (Jlag n) (-5/2) (-T) T‖) := add_le_add (norm_sub_le _ _) (norm_sub_le _ _) ``` Check rectangleIntegral_Jlag_eq's exact form (§1, lines 161-168): ``` RectangleIntegral (Jlag n) (rectZ T) (rectW n T) = HIntegral (Jlag n) (-5/2) (2*√n) (-T) − HIntegral (Jlag n) (-5/2) (2*√n) T + VIntegral (Jlag n) (2*√n) (-T) T − VIntegral (Jlag n) (-5/2) (-T) T ``` ✓ — the association: `H(−T) − H(T) + V(right) − V(left)` = `((H(−T) − H(T)) + V(right)) − V(left)`?? — the written form: `H - H + V - V` — infixl: `((H − H) + V) − V` — hmm — my calc assumed `(H−H) + (V−V)`. The def's association: `HIntegral ... - HIntegral ... + VIntegral ... - VIntegral ...` parses as `((H − H) + V) − V`. My norm split needs to match: ‖((H−H)+V)−V‖ ≤ ‖(H−H)+V‖ + ‖V‖ ≤ (‖H−H‖ + ‖V‖) + ‖V‖... Let me redo with the correct association: ‖((A − B) + C) − D‖ ≤ ‖(A−B)+C‖ + ‖D‖ ≤ ‖A−B‖ + ‖C‖ + ‖D‖ ≤ (‖A‖+‖B‖) + ‖C‖ + ‖D‖. ```lean calc ‖((H A − H B) + V C) − V D‖ ≤ ‖(H A − H B) + V C‖ + ‖V D‖ := norm_sub_le _ _ _ ≤ (‖H A − H B‖ + ‖V C‖) + ‖V D‖ := by linarith [norm_add_le _ _] _ ≤ (‖H A‖ + ‖H B‖) + ‖V C‖ + ‖V D‖ := by linarith [norm_sub_le _ _] ``` Hmm — the middle: ‖(A−B)+C‖ ≤ ‖A−B‖ + ‖C‖ := norm_add_le — then the linariths. Fine — but the exact association must match hrect's RHS. Let me write htri to match hrect exactly: hrect : RectInt = H(−T) − H(+T) + Vr − Vl — I'll structure: ```lean have htri : ‖RectangleIntegral (Jlag n) (rectZ T) (rectW n T)‖ ≤ ‖HIntegral (Jlag n) (-5/2) (2 * Real.sqrt (n:ℝ)) (-T)‖ + ‖HIntegral (Jlag n) (-5/2) (2 * Real.sqrt (n:ℝ)) T‖ + ‖VIntegral (Jlag n) (2 * Real.sqrt (n:ℝ)) (-T) T‖ + ‖VIntegral (Jlag n) (-5/2) (-T) T‖ := by rw [hrect] have h1 := norm_sub_le (HIntegral (Jlag n) (-5/2) (2*√n) (-T) − HIntegral (Jlag n) (-5/2) (2*√n) T + VIntegral (Jlag n) (2*√n) (-T) T) (VIntegral (Jlag n) (-5/2) (-T) T) have h2 := norm_add_le (HIntegral (Jlag n) (-5/2) (2*√n) (-T) − HIntegral (Jlag n) (-5/2) (2*√n) T) (VIntegral (Jlag n) (2*√n) (-T) T) have h3 := norm_sub_le (HIntegral (Jlag n) (-5/2) (2*√n) (-T)) (HIntegral (Jlag n) (-5/2) (2*√n) T) linarith [h1, h2, h3, norm_nonneg (VIntegral (Jlag n) (2*√n) (-T) T), norm_nonneg (VIntegral (Jlag n) (-5/2) (-T) T), norm_nonneg (HIntegral (Jlag n) (-5/2) (2*√n) (-T)), norm_nonneg (HIntegral (Jlag n) (-5/2) (2*√n) T)] ``` h1 : ‖(H1−H2+Vr) − Vl‖ ≤ ‖H1−H2+Vr‖ + ‖Vl‖; h2 : ‖H1−H2+Vr‖ ≤ ‖H1−H2‖ + ‖Vr‖; h3 : ‖H1−H2‖ ≤ ‖H1‖ + ‖H2‖. Sum: ‖RectInt‖ ≤ ‖H1‖+‖H2‖+‖Vr‖+‖Vl‖ ✓ linarith with atoms ✓ (no norm_nonneg needed — the inequalities suffice: h1+h2+h3 chain directly: ‖X‖ ≤ ‖H1−H2+Vr‖ + ‖Vl‖ ≤ (‖H1−H2‖+‖Vr‖)+‖Vl‖ ≤ ((‖H1‖+‖H2‖)+‖Vr‖)+‖Vl‖ ✓ pure linarith). -- the pinch (horizontal) — separate lemma have hpinch : 2 * Z2hat T * Real.log (148.5 * C0hat (Real.sqrt (n:ℝ) + 2)) + (3 - -5/2) * (10.5 * Real.log (T + 2) + 48) ≤ 7 * Hhat T := pinch_horizontal hTn hTn1 (the n/T facts) -- assemble -- ‖RectInt'‖ = (1/2π)·‖RectInt‖ ≤ (1/2π)·(‖H−‖+‖H+‖+‖Vr‖+‖Vl‖) -- ≤ (1/2π)·(2(mid+DLR) + e^{1/3}·IIMaj + I_III-charge) -- ≤ (1/2π)·(2·(e⁴+1)·(7·Hhat) + 2·DLR + 2π·IIMaj + I_III-charge) [pinch + e^{1/3} ≤ 2π] -- = (1/2π)·(2·(e⁴+1)·7·Hhat + 2·DLR) + IIMaj + (1/2π)·I_III-charge -- ≤ (1/2π)·(2·(e⁴+1)·7·Hhat + 2·DLR) + IIMaj + I_III-charge -- and Bn = 4F + I_III-charge + 40·1.3352 + 2(e⁴+1)·7Hhat + 2·DLR + IIMaj + 1 — so: -- |Sf| ≤ (1/2π)(...) + (4F+1) ≤ 4F + I_III-charge + 2(e⁴+1)·7Hhat + 2DLR + IIMaj + 1 = Bn − 40·1.3352?? ``` Hmm wait — careful with the Bn comparison. Bn FAlt n = 4·FAlt T' + 4·T'·((2/3)·Z2hat T' + 10.5·log(T'+2) + 48) + 40·1.3352 + 2·(e⁴+1)·(7·Hhat T') + 2·DogLegRight n + IIMaj n + 1 with T' := √n+1 (the LET in Bn's def). NOTE: Bn uses T' = √n + 1 (a DEFINITIONAL T), while my side bounds are at the PIGEONHOLE T ∈ [√n, √n+1] — DIFFERENT! The I_III charge is 4·T·((2/3)Z2hat T + 10.5 log(T+2) + 48) + 40·1.3352 at MY T — and Bn's I_III share is 4·T'·(...) + 40·1.3352 at T' = √n+1. Since T ≤ T' = √n+1: 4·T·X(T) ≤ 4·T'·X(T')?? — X(t) = (2/3)Z2hat(t) + 10.5 log(t+2) + 48 — increasing in t (Z2hat_mono + log mono) and T ≤ T' ⟹ 4·T·X(T) ≤ 4·T'·X(T') ✓ (need T·X(T) ≤ T'·X(T'): mul_le_mul (T ≤ T') (X(T) ≤ X(T')) (X(T) ≥ 0) (T' ≥ 0) ✓ — X(T) ≥ 0: Z2hat ≥ 0 (Z2hat_nonneg) + 10.5 log(T+2) ≥ 0 + 48 ✓). Similarly Hhat T ≤ Hhat T' (Hhat increasing — Hhat t = 3·C0hat(t)² + 10.5 log(t+2) + 48 — C0hat_mono + log mono + C0hat ≥ 0) ⟹ 7·Hhat T ≤ 7·Hhat T' ✓. And the horizontal pinch at MY T: NI(T) + 5.5·R(T) ≤ 7·Hhat T ≤ 7·Hhat T' ✓. And FAlt T ≤ FAlt T'?? — FAlt increasing (it's the Trudgian majorant — the perimeter ZeroCountGlobalProp is at the SAME T though — hZCG T hT2 : count(T) ≤ FAlt T — the bridge gives 4·FAlt T + 1. Bn has 4·FAlt T' + ... — need FAlt T ≤ FAlt T': FAlt t = (1/2π)·(t·log t) + 0.137·log t + 0.443·loglog t + (7/8+1.588) — increasing for t ≥ 316 ✓ (each term increasing: t·log t — its derivative log t + 1 > 0 ✓ — but for a Lean proof of FAlt T ≤ FAlt T' I'd need monotonicity — the ψ-block machinery in LiSkeleton PROVES FAlt-related monotonicity for the margin convexity — is there a ready `FAlt_mono`? — marginTilde_convex's support — hmm — there might not be a standalone FAlt_mono lemma. For the fold: 4·FAlt T + 1 ≤ 4·FAlt T' + 1 ⟸ FAlt T ≤ FAlt T' ⟸ (T·log T ≤ T'·log T' (mul_le_mul T≤T' logT≤logT' ...) + log T ≤ log T' + loglog T ≤ loglog T') — each term monotone — elementary via Real.log_le_log (T ≥ 316 > 0) + mul_le_mul. Let me just prove FAlt_le_FAlt inline: FAlt T ≤ FAlt T': - (1/2π)·(T·log T) ≤ (1/2π)·(T'·log T'): mul_le_mul (h : T ≤ T') (Real.log_le_log (T > 0) (T ≤ T')) (log T ≥ 0 — T ≥ 316 ⟹ log T ≥ 5 ✓) (T' ≥ 0) then mul by 1/2π ≥ 0. - 0.137·log T ≤ 0.137·log T' ✓ log mono. - 0.443·log(log T) ≤ 0.443·log(log T') ✓ log mono twice (log T ≥ 1 since T ≥ 316 > e ✓ — wait — Real.log_le_log needs 0 < log T — log T ≥ log 316 ≈ 5.76 > 0 ✓). - constants equal. So FAlt T ≤ FAlt T' — elementary linarith-assemblable. Hmm — but hold on — is it cleaner to just prove the comparison |Sf n| ≤ Bn FAlt n via Bn's UNFOLDED form and handle all the T vs T' comparisons in one giant linarith? The final comparison: |Sf n| ≤ ‖RectInt'‖ + ‖Σ‖ ≤ (1/2π)·‖RectInt‖ + (4·FAlt T + 1) ≤ (1/2π)·(‖H−‖+‖H+‖+‖Vr‖+‖Vl‖) + (4·FAlt T + 1) ≤ (1/2π)·([2(mid(T)+DLR)] + [e^{1/3}·IIMaj] + [4T·X(T) + 40·1.3352]) + (4·FAlt T + 1) where mid(T) = (e⁴+1)·(NI(T) + 5.5R(T)), and the horizontal two-sides bound is 2·(mid(T) + DLR) [6-c-v two-sides]. = (1/2π)·(2(e⁴+1)(NI+5.5R) + 2·DLR + e^{1/3}·IIMaj + 4T·X + 40·1.3352) + 4F + 1 Now: (1/2π)·(e^{1/3}·IIMaj) ≤ IIMaj ⟸ e^{1/3} ≤ 2π ✓ (e^{1/3} ≤ 1.4 ≤ 2π). (1/2π)·(2(e⁴+1)(NI+5.5R)) ≤ (1/2π)·(2(e⁴+1)·7·Hhat T) [pinch NI+5.5R ≤ 7·Hhat T; 2(e⁴+1) ≥ 0] (1/2π)·(2·DLR) ≤ 2·DLR ⟸ 1/2π ≤ 1 (π ≥ 1/2 — π > 3 ✓) and DLR ≥ 0. (1/2π)·(4T·X + 40·1.3352) ≤ 4T·X + 40·1.3352 ⟸ 1/2π ≤ 1 and the bracket ≥ 0. So: |Sf n| ≤ (1/2π)·(2(e⁴+1)·7·Hhat T) + 2·DLR + IIMaj + (4T·X(T) + 40·1.3352) + (4·FAlt T + 1) ≤ 2(e⁴+1)·7·Hhat T + 2·DLR + IIMaj + 4T·X(T) + 40·1.3352 + 4·FAlt T + 1 [(1/2π) ≤ 1 again on the Hhat term — Hhat ≥ 0] hmm wait — (1/2π)·(2(e⁴+1)·7·Hhat T) ≤ 2(e⁴+1)·7·Hhat T ⟸ 1/2π ≤ 1 ✓ (bracket ≥ 0). ≤ 2(e⁴+1)·7·Hhat T' + 2·DLR + IIMaj + 4T'·X(T') + 40·1.3352 + 4·FAlt T' + 1 [T ≤ T' monotonicity] = Bn FAlt n [Bn's def at T' = √n+1 — UNFOLD] The Bn-def match: Bn FAlt n = let T := √n+1; 4·FAlt T + 4·T·X(T) + 40·1.3352 + 2(e⁴+1)·(7·Hhat T) + 2·DLR n + IIMaj n + 1 — with X(T) = ((2/3)·Z2hat T + 10.5·log(T+2)+48) — Bn's exact def (line 2463-2467): `let T := Real.sqrt n + 1; 4 * F T + 4 * T * ((2/3) * Z2hat T + 10.5 * Real.log (T + 2) + 48) + 40 * 1.3352 + 2 * (Real.exp 4 + 1) * (7 * Hhat T) + 2 * DogLegRight n + IIMaj n + 1`. So the needed final form: 4·FAlt T' + 4·T'·X(T') + 40·1.3352 + 2(e⁴+1)·7·Hhat T' + 2·DLR + IIMaj + 1 — my assembled bound has EXACTLY these terms (modulo ordering — linarith/ring handles). So the proof structure: 1. T from the pigeonhole; T' := √n+1 comparisons (FAlt/X/Hhat monotonicity at T vs T'). 2. hsf (6-b identity) + hbridge (variant T ≤ 3·10¹²). 3. hnorm (‖RectInt'‖ = (1/2π)‖RectInt‖) + htri + hVl + hVr + hH. 4. hpinch : NI + 5.5R ≤ 7·Hhat T (the separate pinch_horizontal lemma). 5. he13 : e^{1/3} ≤ 1.4 ≤ 2π (certified via e ≤ 2.7183 ≤ 2.744 = 1.4³). 6. h2π : 1/(2π) ≤ 1 (π ≥ 1/2 — from π > 3? — Real.pi_gt_three : 3 < π ✓ — so 2π > 6 > 2 ⟹ 1/(2π) < 1/2 < 1 — actually 1/(2π) ≤ 1 ⟺ 1 ≤ 2π ⟸ 2·3 ≤ 2π ⟸ 3 ≤ π ✓ Real.pi_gt_three). 7. The giant final linarith assembling everything. Nonneg facts needed for the multiplicative comparisons: IIMaj ≥ 0? — hmm — IIMaj n = 2(√n+1)(e^{√n/2}+1)·0.57·2^{−(2√n−2)} ≥ 0 ✓ positivity. DLR ≥ 0 (sum of nonneg terms — ((√n+1)/2)(e^{5/2}+1)/2 + 8.38 + IIMaj/4 ≥ 0 ✓). Hhat T ≥ 0 (C0hat ≥ 2 > 0 ⟹ 3·C0hat² > 0 + bracket) ✓. X(T) ≥ 0 ✓. FAlt T ≥ 0 ✓ (each term nonneg for T ≥ 316: log T, loglog T ≥ 0, const > 0). 2π > 0 ✓. The mul-comparisons: - (1/2π)·(e^{1/3}·IIMaj) ≤ 1·IIMaj = IIMaj: (1/2π)·(e^{1/3}·IIMaj) ≤ IIMaj ⟸ (1/2π)·e^{1/3} ≤ 1 ⟸ e^{1/3} ≤ 2π ✓ IIMaj ≥ 0: mul_le_mul_of_nonneg_left-ish or: (1/2π)·(e^{1/3}·IIMaj) = ((1/2π)·e^{1/3})·IIMaj ≤ 1·IIMaj — mul_le_mul_of_nonneg_right ((1/2π)·e^{1/3} ≤ 1) (IIMaj ≥ 0) ✓. - (1/2π)·(2(e⁴+1)(NI+5.5R)) ≤ (1/2π)·(2(e⁴+1)·7·Hhat T): mul_le_mul_of_nonneg_left (NI+5.5R ≤ 7·Hhat T) ((1/2π)·2(e⁴+1) ≥ 0)... — break: 2(e⁴+1)·(NI+5.5R) ≤ 2(e⁴+1)·7·Hhat T := mul_le_mul_of_nonneg_left hpinch' (by positivity) where hpinch' : NI + 5.5R ≤ 7·Hhat T — wait — hpinch is stated as 2·Z2hat·log + 5.5R ≤ 7·Hhat — and NI + 5.5R = 2·Z2hat·log(148.5·C0hat) + 5.5·R — hmm — NI := 2·Z2hat(T)·log(148.5·C0hat(√n+2)) ✓ — and the mid bound's inner is NI + 5.5R with 5.5 written as (3 − −5/2) — (3 − −5/2) = 5.5 ✓ norm_num/ring for the coefficient match. - Then (1/2π)·(2(e⁴+1)·7·Hhat T) ≤ 2(e⁴+1)·7·Hhat T := mul_le_mul_of_nonneg_left?? — no — (1/2π)·Y ≤ Y ⟸ 1/2π ≤ 1, Y ≥ 0: `mul_le_mul_of_nonneg_right (h1/2π≤1) (Y ≥ 0)` : (1/2π)·Y ≤ 1·Y = Y ✓ — hmm — mul_le_mul_of_nonneg_right (hbc : b ≤ c) (ha : 0 ≤ a) : b * a ≤ c * a — with b := 1/2π, c := 1, a := Y ✓ (1/2π)·Y ≤ 1·Y ✓ then ring for 1·Y = Y. Let me now also nail the pinch lemma's exact statement: ```lean theorem pinch_horizontal {n : ℕ} (hn : 100000 ≤ n) {T : ℝ} (hTn : Real.sqrt (n : ℝ) ≤ T) (hTn1 : T ≤ Real.sqrt (n : ℝ) + 1) : 2 * Z2hat T * Real.log (148.5 * C0hat (Real.sqrt (n : ℝ) + 2)) + (3 - -5 / 2) * (10.5 * Real.log (T + 2) + 48) ≤ 7 * Hhat T ``` Proof components: (a) hL : Real.log (148.5 * C0hat (√n+2)) ≤ 2 * Real.log (T + 2): - hC0le : C0hat (√n+2) ≤ √n+2 — via C0hat_le_self (u ≥ 318 — actually make C0hat_le_self for t ≥ 100 to be safe: at t = 100: 0.27922·102 + 2.9 = 31.4 ≤ 100 ✓ — wait my coefficient chain gave C0hat(u) ≤ 0.28(u+2)+2.9 ≤ u ⟺ 3.46 ≤ 0.72u ⟺ u ≥ 4.8 ✓ — let me set C0hat_le_self for 5 ≤ t — then applies at √n+2 ≥ 318 ✓). - h148 : (148.5 : ℝ) ≤ √n + 2: √n ≥ 316 ⟹ √n+2 ≥ 318 ≥ 148.5 ✓. - hC0pos : 0 < C0hat (√n+2) (two_le_C0hat). - h1 : 148.5 * C0hat (√n+2) ≤ (√n+2)^2 := by have := mul_le_mul hC0le h148 (by norm_num : (0:ℝ) ≤ 148.5) (by positivity : (0:ℝ) ≤ √n+2) -- (148.5)·C0hat ≤ (√n+2)·(√n+2) = (√n+2)² nlinarith [this] hmm — mul_le_mul (h₁ : C0hat ≤ √n+2) (h₂ : 148.5 ≤ √n+2) (c0 : 0 ≤ 148.5) (b0 : 0 ≤ √n+2) : C0hat·148.5 ≤ (√n+2)·(√n+2) — ordering: mul_le_mul a≤b c≤d 0≤c 0≤b : a·c ≤ b·d — a := C0hat, b := √n+2, c := 148.5, d := √n+2 : C0hat·148.5 ≤ (√n+2)² — my target: 148.5·C0hat ≤ (√n+2)² — mul_comm ✓ nlinarith or rewrite. - h2 : Real.log (148.5 * C0hat (√n+2)) ≤ Real.log ((√n+2)^2) := Real.log_le_log (by positivity) h1 - h3 : Real.log ((√n+2)^2) = 2 * Real.log (√n+2) := by rw [Real.log_pow]; ring - h4 : Real.log (√n+2) ≤ Real.log (T+2) := Real.log_le_log (by positivity) (by linarith [hTn]) linarith [h2, h3, h4] (b) hZ2le : Z2hat T ≤ 2 * C0hat T: unfold both defs; L := log(T+2) ≥ 1, LL := log L ≥ 0 (T ≥ 316): Z2hat T = 0.6L + 0.886LL + 2.7; 2·C0hat T = 0.866L + 1.772LL + 5.8: diff = 0.266L + 0.886LL + 3.1 ≥ 0 ✓ — unfold Z2hat C0hat; have hL1 : 1 ≤ L (as in Z2hat_mono's hlog1); have hLL0 : 0 ≤ LL (Real.log_nonneg hL1); nlinarith. (c) hC0L : 0.433 * Real.log (T + 2) ≤ C0hat T: unfold C0hat; LL ≥ 0, 2.9 ≥ 0: 0.433L ≤ 0.433L + 0.886LL + 2.9 ✓ linarith. (d) the main: 2·Z2hat·log(148.5·C0hat) ≤ 2·Z2hat·2·log(T+2) [hL, Z2hat ≥ 0] = 4·Z2hat·log(T+2) ≤ 8·C0hat·log(T+2) [hZ2le, log(T+2) ≥ 0] ≤ 21·C0hat·C0hat [8·log(T+2) ≤ 21·C0hat from hC0L: 8·log(T+2) ≤ (8/0.433)·C0hat ≤ 21·C0hat ⟺ 8/0.433 ≤ 21 ⟺ 8 ≤ 9.093 ✓; then mul_le_mul (8·log(T+2) ≤ 21·C0hat) (le_refl C0hat) (C0hat ≥ 0) (by positivity : 0 ≤ 21·C0hat) : (8·log(T+2))·C0hat ≤ (21·C0hat)·C0hat ✓ rearrange ring]. (e) 5.5·R ≤ 7·R (R := 10.5·log(T+2)+48 ≥ 0): (3 − −5/2) = 5.5 ✓ norm_num; 5.5·R ≤ 7·R ⟸ R ≥ 0. (f) Hhat T = 3·C0hat T² + R: unfold Hhat ✓. 21·C0hat² + 7·R = 7·(3·C0hat² + R) = 7·Hhat ✓ ring. Assembly: 2·Z2hat·log148.5 + 5.5·R ≤ 21·C0hat² + 7·R = 7·Hhat ✓ linarith. Now — the C0hat_le_self lemma: ```lean theorem C0hat_le_self {t : ℝ} (ht : 5 ≤ t) : C0hat t ≤ t := by have ht2 : (0 : ℝ) < t + 2 := by linarith have ht1 : (1 : ℝ) < t + 2 := by linarith have he : (2.7182 : ℝ) ≤ Real.exp 1 := by have h := Real.exp_one_gt_d9.le -- 2.7182818283 < e norm_num at h ⊢ -- 2.7182818283 ≥ 2.7182 ✓ linarith [h] -- hmm — just linarith [Real.exp_one_gt_d9.le] -- log x ≤ x/e: log(x/e) ≤ x/e − 1 have hloge : ∀ x : ℝ, 0 < x → Real.log x ≤ x / Real.exp 1 := by intro x hx have h1 := Real.log_le_sub_one_of_pos (show (0:ℝ) < x / Real.exp 1 from div_pos hx (Real.exp_pos 1)) -- h1 : log(x/e) ≤ x/e − 1; log(x/e) = log x − log e = log x − 1 have h2 : Real.log (x / Real.exp 1) = Real.log x - 1 := by rw [Real.log_div (ne_of_gt hx) (Real.exp_pos 1).ne', Real.log_exp] ring?? -- log x − log(e^1) = log x − 1... wait Real.log_exp : log(exp x) = x — log(e^1) = 1 ✓ rw [h2] at h1 -- log x − 1 ≤ x/e − 1 ⟹ log x ≤ x/e linarith [h1] have hL : Real.log (t + 2) ≤ (t + 2) / Real.exp 1 := hloge (t + 2) ht2 have hLL : Real.log (Real.log (t + 2)) ≤ Real.log (t + 2) / Real.exp 1 := hloge (Real.log (t + 2)) (Real.log_pos ht1) -- 1/e ≤ 1/2.7182 have hinv : (1 : ℝ) / Real.exp 1 ≤ 1 / 2.7182 := one_div_le_one_div_of_le (by norm_num : (0:ℝ) < 2.7182) he -- assemble: C0hat t = 0.433L + 0.886LL + 2.9 ≤ (0.433·L + 0.886·L)/e?? ``` Hmm — assembling: C0hat t = 0.433·L + 0.886·LL + 2.9. With L ≤ (t+2)/e and LL ≤ L/e: 0.433L + 0.886LL ≤ 0.433(t+2)/e + 0.886·L/e = (0.433(t+2) + 0.886L)/e ≤ (0.433(t+2) + 0.886(t+2)/e)/e?? — ugh — nested. Cleaner bound: LL ≤ L (log x ≤ x − 1 ≤ x — actually LL = log L ≤ L/e ≤ L·0.368 hmm) — simplest: LL ≤ L/e (hLL) and L/e ≤ (t+2)/e² — I want a NUMERIC coefficient: C0hat t ≤ 0.433L + 0.886·(L/e) + 2.9. With L ≤ (t+2)/e ≤ (t+2)·0.36789 (1/e ≤ 1/2.7182 = 0.367889 — hmm — 1/2.7182 = ? — compute: 1/2.7182 = 0.3678894 — so L ≤ (t+2)/e ≤ (t+2)·0.36789?? — 1/e ≤ 0.36789 ⟺ e ≥ 1/0.36789 = 2.71821 — e > 2.7182818283 ≥ 2.71821 ✓. Let me use 0.36789 as the certified upper for 1/e: 1/e ≤ 0.36789 ⟺ 1 ≤ 0.36789·e ⟺ 1/0.36789 ≤ e ⟺ 2.71821 ≤ e ✓ from exp_one_gt_d9 (2.7182818283 ≤ e... wait — exp_one_gt_d9 : 2.7182818283 < exp 1 — so e > 2.7182818283 > 2.71821 ✓). Formally: 1/e ≤ 0.36789 ⟸ 1 ≤ 0.36789·e ⟸ 1/0.36789 ≤ e — hmm — 1/0.36789 = 2.718206... ≤ 2.7182818283 < e ✓. Prove: `have hinv : (1:ℝ)/Real.exp 1 ≤ 0.36789 := by rw [div_le_iff₀ (Real.exp_pos 1), one_mul]; norm_num1; nlinarith [Real.exp_one_gt_d9]` — goal after div_le_iff₀: 1 ≤ 0.36789·e¹ — hmm — div_le_iff₀ (hb : 0 < b) : a/b ≤ c ↔ a ≤ c·b: 1/e ≤ 0.36789 ⟺ 1 ≤ 0.36789·e ✓ — 0.36789·e ≥ 0.36789·2.7182818283 = 1.0000004... ≥ 1 ✓ — compute: 0.36789·2.7182818283 = 1.00000038... ✓ ≥ 1 — nlinarith [Real.exp_one_gt_d9]: 0.36789·e > 0.36789·2.7182818283 = 1.00000038 ≥ 1 ✓ — norm_num verifies 0.36789·2.7182818283 ≥ 1: 0.36789·2.7182818283 = 1.00000075... — let me compute precisely: 0.36789 × 2.7182818 = 0.36789×2.7 = 0.993303, 0.36789×0.0182818 = 0.006727 — total 1.000030 ≈ 1.00003 ≥ 1 ✓✓ — wait let me redo: 0.36789 × 2.7182818283: 0.36789×2 = 0.73578; ×0.7 = 0.257523; ×0.0182818283 = 0.0067272... — hmm — 0.73578 + 0.257523 = 0.993303 (that's ×2.7); + 0.36789×0.0182818283 = 0.006727... — total 1.000030 ✓ ≥ 1 ✓. So 0.36789·2.7182818283 ≈ 1.00003 — norm_num can verify 1 ≤ 0.36789 * 2.7182818283 ✓ (decimal arithmetic). Then: L ≤ (t+2)/e = (t+2)·(1/e) ≤ (t+2)·0.36789 (t+2 ≥ 0); LL ≤ L/e = L·(1/e) ≤ L·0.36789 ≤ (t+2)·0.36789² ≤ (t+2)·0.135343 (0.36789² = 0.135343...). Hmm — two-step: LL ≤ L·0.36789 and L ≤ (t+2)·0.36789: LL ≤ (t+2)·0.36789·0.36789 = (t+2)·0.36789². mul_le_mul (LL ≤ L·c) (L ≤ (t+2)·c) ... — or nlinarith with the products as hints. This is getting long but it's all mechanical. Let me bound LL more simply: LL = log L ≤ L − 1 ≤ L ≤ (t+2)·0.36789 — via Real.log_le_sub_one_of_pos (L > 0): LL ≤ L − 1 ≤ L ≤ (t+2)·0.36789 ✓ — NO e² needed!! LL ≤ L directly (log x ≤ x−1). So: C0hat t = 0.433L + 0.886LL + 2.9 ≤ 0.433L + 0.886L + 2.9 = 1.319·L + 2.9 ≤ 1.319·(t+2)·0.36789 + 2.9 = 0.485257·(t+2) + 2.9. ≤ t ⟺ 0.485257·(t+2) + 2.9 ≤ t ⟺ 0.485257t + 0.970514 + 2.9 ≤ t ⟺ 3.870514 ≤ 0.514743·t ⟺ t ≥ 7.519. So for t ≥ 8 (use ht : 8 ≤ t — or keep 14): 0.514743·t ≥ 0.514743·8 = 4.12 ≥ 3.87 ✓ — at t ≥ 8: 0.485257(t+2)+2.9 ≤ t ⟺ 3.870514 ≤ 0.514743t — at t = 8: 0.514743·8 = 4.118 ≥ 3.87 ✓✓ linarith closes. Hmm wait — 1.319·0.36789 = 0.485257... — verify: 1.319 × 0.36789 = 0.48524... ✓. And the final: 0.485257·(t+2) + 2.9 ≤ t: 0.485257t + 0.970514 + 2.9 ≤ t ⟺ 0.970514 + 2.9 ≤ (1 − 0.485257)t = 0.514743t ⟺ 3.870514 ≤ 0.514743t ⟺ t ≥ 7.5194 ✓ — take ht : 8 ≤ t. Wait — but I need LL ≤ L: Real.log_le_sub_one_of_pos (hL0 : 0 < L) : log L ≤ L − 1 — L = log(t+2) > 0 ⟸ t+2 > 1 ⟸ t > −1 ✓ (t ≥ 8). ✓ So C0hat_le_self (ht : 8 ≤ t): C0hat t ≤ t — clean: ```lean theorem C0hat_le_self {t : ℝ} (ht : 8 ≤ t) : C0hat t ≤ t := by have ht2 : (0 : ℝ) < t + 2 := by linarith have ht1 : (1 : ℝ) < t + 2 := by linarith have hinv : (1 : ℝ) / Real.exp 1 ≤ 0.36789 := by rw [div_le_iff₀ (Real.exp_pos 1), one_mul] nlinarith [Real.exp_one_gt_d9] have hL1 : Real.log (t + 2) ≤ (t + 2) / Real.exp 1 := by have h1 := Real.log_le_sub_one_of_pos (div_pos ht2 (Real.exp_pos 1)) have h2 : Real.log ((t + 2) / Real.exp 1) = Real.log (t + 2) - 1 := by rw [Real.log_div (ne_of_gt ht2) (Real.exp_pos 1).ne', Real.log_exp] rw [h2] at h1 linarith [h1] have hL2 : (t + 2) / Real.exp 1 ≤ (t + 2) * 0.36789 := by rw [div_eq_mul_one_div] exact mul_le_mul_of_nonneg_left hinv ht2.le have hL : Real.log (t + 2) ≤ (t + 2) * 0.36789 := hL1.trans hL2 have hLL : Real.log (Real.log (t + 2)) ≤ (t + 2) * 0.36789 := by have h1 := Real.log_le_sub_one_of_pos (Real.log_pos ht1) -- log(log(t+2)) ≤ log(t+2) − 1 ≤ log(t+2) ≤ (t+2)·0.36789 nlinarith [h1, hL, ht2] unfold C0hat nlinarith [hL, hLL, ht, Real.log_pos ht1] ``` The final nlinarith: goal: 0.433·L + 0.886·LL + 2.9 ≤ t (after unfold C0hat — wait — C0hat's def: `C0hat t = 0.433 * Real.log (t + 2) + 0.886 * Real.log (Real.log (t + 2)) + 2.9` — let me verify the def — from LiContour/audit-037: "C0hat t = 0.433·log(t+2) + 0.886·log log(t+2) + 2.9" ✓). With hL : L ≤ (t+2)·0.36789, hLL : LL ≤ (t+2)·0.36789, ht : 8 ≤ t: 0.433L + 0.886LL + 2.9 ≤ (0.433+0.886)·0.36789·(t+2) + 2.9 = 1.319·0.36789·(t+2) + 2.9 = 0.48525·(t+2) + 2.9 — nlinarith computes 1.319·0.36789 = 0.485247... hmm — 1.319 × 0.36789: 1.319×0.36789 = 0.485247 — and I claimed 0.485257 — whatever — nlinarith does the exact decimal arithmetic: 0.485247·(t+2) + 2.9 ≤ t ⟺ 0.485247t + 0.970494 + 2.9 ≤ t ⟺ 3.870494 ≤ 0.514753t ⟺ t ≥ 7.5194 — at t ≥ 8 ✓ — nlinarith [hL, hLL, ht] with products (hL, hLL as linear bounds on atoms L, LL): 0.433L ≤ 0.433·0.36789·(t+2), 0.886LL ≤ 0.886·0.36789·(t+2) — these are LINEAR in L, LL, t ✓ linarith suffices (the coefficient multiplication 0.433·0.36789 is scalar arithmetic ✓ linarith evaluates). ✓ Hmm — wait — Real.log_le_sub_one_of_pos : 0 < x → log x ≤ x − 1 — signature: `Real.log_le_sub_one_of_pos (hx : 0 < x) : Real.log x ≤ x - 1` ✓ (the recipe names it ✓). And `Real.log_exp (x) : Real.log (Real.exp x) = x` ✓. And `Real.log_pos (hx : 1 < x) : 0 < Real.log x` ✓. div_le_iff₀ (hb : 0 < b) : a / b ≤ c ↔ a ≤ c * b ✓ (used in 6-c-ii ✓). div_eq_mul_one_div ✓. Now — the LAST piece: he13 : Real.exp (1/3) ≤ 2·π: ```lean have he13 : Real.exp (1 / 3 : ℝ) ≤ 2 * Real.pi := by have h1 : Real.exp (1 / 3 : ℝ) ≤ 1.4 := by have h2 : (Real.exp (1 / 3 : ℝ)) ^ 3 ≤ (1.4 : ℝ) ^ 3 := by have h3 : (Real.exp (1 / 3 : ℝ)) ^ 3 = Real.exp 1 := by rw [← Real.exp_nat_mul (1/3 : ℝ) 3] norm_num rw [h3] -- e ≤ 1.4³ = 2.744 have h4 : Real.exp 1 ≤ 2.7183 := Real.exp_one_lt_d9.le norm_num at h4 ⊢ -- 1.4^3 = 2.744 linarith [h4] exact le_of_pow_le_pow_left₀ (by norm_num : (3 : ℕ) ≠ 0) (Real.exp_nonneg _) h2 have h5 : (3 : ℝ) ≤ Real.pi := Real.pi_gt_three.le nlinarith [h1, h5] ``` le_of_pow_le_pow_left₀ (hn : n ≠ 0) (ha : 0 ≤ a) (h : a^n ≤ b^n) : a ≤ b — signature check: `le_of_pow_le_pow_left₀ (hn : n ≠ 0) (ha : 0 ≤ a) (h : a ^ n ≤ b ^ n) : a ≤ b` — hmm — I think it's `le_of_pow_le_pow_left₀ (hn : n ≠ 0) (ha : 0 ≤ a) : a ^ n ≤ b ^ n → a ≤ b`? — hmm — actually the name pattern: le_of_pow_le_pow_left₀ (hn : n ≠ 0) (ha : 0 ≤ a) (h : a ^ n ≤ b ^ n) : a ≤ b ✓ plausible. Let me verify via grep later. (Real.exp (1/3))^3 = exp(3·(1/3)) = exp 1: `← Real.exp_nat_mul (1/3) 3` — Real.exp_nat_mul (x) (n) : exp(↑n·x) = (exp x)^n — ← direction: (exp(1/3))^3 → exp(3·(1/3)) — then 3·(1/3) = 1 — `norm_num` — hmm — the goal after rw [← Real.exp_nat_mul]: `Real.exp (↑3 * (1/3)) = Real.exp 1` — ↑3·(1/3) = 1 — `norm_num` proves exp(1) = exp(1)?? — need to simplify ↑3·(1/3) → 1: `norm_num` might close by reducing to exp 1 = exp 1 ✓ or `congr 1; norm_num`. Write `rw [← Real.exp_nat_mul (1/3 : ℝ) 3]; norm_num` — if norm_num leaves `exp 1 = exp 1` it closes ✓; if it reduces the exponent — the goal becomes Real.exp 1 = Real.exp 1 after norm_num simplifies ↑3*(1/3) — ✓ closed by rfl (norm_num closes rfl-goals ✓). Hmm — safer: `rw [← Real.exp_nat_mul (1/3 : ℝ) 3]; congr 1; norm_num` — hmm — wait — the rewrite direction: I want (exp(1/3))^3 = exp 1. h3's proof: `rw [← Real.exp_nat_mul (1/3 : ℝ) 3]` — the goal is `(exp (1/3))^3 = exp 1` — ← exp_nat_mul : pattern (exp ?x)^?n → exp(↑?n·?x) — matches with x := 1/3, n := 3 — goal becomes `exp (↑3·(1/3)) = exp 1` — `congr 1` → `↑3·(1/3) = 1` — norm_num ✓. Let me use that. Hmm — wait — Real.exp_nat_mul (x : ℝ) (n : ℕ) : Real.exp (↑n * x) = (Real.exp x) ^ n — the exponent is `↑n * x` with the NAT CAST — `(3 : ℝ) * (1/3)` where 3 = ↑3 — norm_num handles ↑3·(1/3) = 1 ✓. For the hnorm piece — the RectangleIntegral' norm — one more consideration: `unfold RectangleIntegral'` — it's an abbrev — unfold should work; alternatively `rw [RectangleIntegral']`?? — for abbrevs, `show` or `unfold` ✓. Actually — RectangleIntegral' f z w is DEFINED as `(1/(2πI)) • RectangleIntegral f z w` — the norm: ‖(1/(2πI)) • RectInt‖ = ‖1/(2πI)‖·‖RectInt‖ (norm_smul) — but wait — norm_smul for ℂ·ℂ: `norm_smul (a : α) (x : β) : ‖a • x‖ = ‖a‖ * ‖x‖` — needs [NormedSpace α β]-ish — for ℂ on ℂ ✓. ✓ Now — the perimeter hypotheses for sfBoundProp: (hRH : RHUpTo (3·10¹²)) (hZC : ZeroCountProp) (hZC2 : ZeroCountProp2) (hZCG : ZeroCountGlobalProp) (hNLZ : NoLowZeros) (hMZ : MZeroProp) — 6 hypotheses — matching the declared perimeter + RHUpTo. One more design point: the final Bn-unfold. After assembling |Sf n| ≤ [the T-form], unfold Bn FAlt n and close by linarith with the T ≤ T' monotonicity haves. The Bn def has `let T := Real.sqrt n + 1` — unfold Bn; simp only [Bn]? — `unfold Bn` produces the let — then `simp only []` or `show` to eliminate the let. Actually — the `let` in Bn's def: `let T := Real.sqrt n + 1; 4 * F T + ...` — after `unfold Bn`, the goal has the let-expression — `simp only [Bn]`? — hmm — Bn is a def with a let — `unfold Bn` might produce `let T := ...; ...` in the goal — Lean's `let` in the goal: `show` the beta-reduced form? — the let-expression `(let T := √n+1; body)` is defeq to body[√n+1/T] — a `show` with the substituted form should work: ```lean have hBn : Bn FAlt (n : ℝ) = 4 * FAlt (√n+1) + 4 * (√n+1) * ((2/3) * Z2hat (√n+1) + 10.5 * Real.log (√n+1 + 2) + 48) + 40 * 1.3352 + 2 * (Real.exp 4 + 1) * (7 * Hhat (√n+1)) + 2 * DogLegRight (n:ℝ) + IIMaj (n:ℝ) + 1 := by unfold Bn rfl?? or ring ``` Hmm — unfold Bn gives `let T := √↑n+1; ...` — is `let T := a; f T` rfl-equal to `f a`? YES (zeta reduction — defeq) — so `rfl` after unfold... — or even WITHOUT unfold: Bn FAlt ↑n is DEFEQ to the substituted body ✓ `rfl` directly might work. Use `by rfl`?? — if not, `by unfold Bn`. Let me write `by unfold Bn` and if the goal shows the let, hmm — actually in practice `unfold Bn` normalizes the let away (delta + beta/zeta reduce in the goal display) — then `rfl` might be needed — or the hBn's statement matches the unfolded form exactly and `unfold Bn` closes by rfl automatically? unfold closes goals that become rfl ✓ if the RHS matches. Write: ```lean have hBn : Bn FAlt (n : ℝ) = ... := by unfold Bn ``` — unfold Bn at goal `Bn FAlt ↑n = RHS` → goal becomes `(let T := √↑n+1; ...) = RHS` — hmm — unfold leaves the let — then need to zeta-reduce: `rfl`? or `show RHS = RHS`?? — the let-form is defeq to RHS — `unfold` tactic might fully whnf including zeta — uncertain. Robust: `by have : Bn FAlt (n:ℝ) = ... := rfl; exact this`?? — or just `by rfl`. Try `by rfl` first (Bn's body with let zeta-reduces to RHS by defeq ✓ rfl should work directly). If rfl fails: `by unfold Bn; rfl`? — hmm — `unfold Bn` then the goal still has let — `rfl` uses defeq incl zeta ✓. Actually simplest: `by simp only [Bn]` — simp only with the def-lemma unfolds and beta/zeta reduces ✓ then closes by rfl if RHS matches. Hmm — simp only [Bn] needs Bn to be unfoldable by simp (it's a def — simp only [Bn] uses the equation lemma Bn.eq_def or the definitional unfolding ✓ common pattern). I'll write `by unfold Bn` and add nothing; test. Hmm — actually — WAIT. Maybe cleaner: don't relate to Bn via an hBn equation at all — just `unfold Bn at *`?? no — unfold at goal. Final step: the goal is `|Sf ↑n| ≤ Bn FAlt ↑n` — `unfold Bn` at goal → `|Sf ↑n| ≤ (let ...)...` → hmm. Let me just plan: `rw [hBn]; exact the-assembled-linarith`? no — the assembled bound IS ≤ RHS-of-hBn. So: ```lean rw [hBn] -- goal: |Sf ↑n| ≤ (the T' = √n+1 form) -- then the giant linarith with all the pieces ``` Hmm — rw [hBn] rewrites Bn FAlt ↑n → the T'-form ✓ then linarith [everything]. The final linarith's logical content: - hsf : |Sf n| ≤ ‖RectInt'‖ + ‖Σ‖ - hnorm : ‖RectInt'‖ = (1/2π)·‖RectInt‖ - htri : ‖RectInt‖ ≤ ‖H−‖+‖H+‖+‖Vr‖+‖Vl‖ - hH : ‖H+‖+‖H−‖ ≤ 2·((e⁴+1)·(NI + 5.5R) + DLR) — wait — hIntegral_Jlag_full_two_sides gives ‖H(+)‖ + ‖H(−)‖ ≤ 2·(midBound + DLR) where midBound = (e⁴+1)·(2·Z2hat·log148.5 + (3−−5/2)·R) ✓ = (e⁴+1)·(NI + 5.5R) ✓. - hVr : ‖Vr‖ ≤ e^{1/3}·IIMaj - hVl : ‖Vl‖ ≤ 4T·X(T) + 40·1.3352 (I_III charge — where X(T) = (2/3)Z2hat T + 10.5 log(T+2) + 48 ✓ matches Bn's 4T·X(T) term ✓). - hbridge : ‖Σ‖ ≤ 4·FAlt T + 1 - hpinch : NI + 5.5R ≤ 7·Hhat T (with the (3−−5/2) = 5.5 coefficient matching — hpinch stated with (3 − −5/2)·R — and hH's midBound has (3−−5/2)·R too ✓ same atom). - he13 : e^{1/3} ≤ 2π - h2π1 : 1/(2π) ≤ 1 - T ≤ T' comparisons: hFAlt : FAlt T ≤ FAlt T'; hX : T·X(T) ≤ T'·X(T'); hHhat : Hhat T ≤ Hhat T'. - nonneg: IIMaj ≥ 0, DLR ≥ 0, Hhat T' ≥ 0?? — the linarith needs the multiplicative relations: (1/2π)·‖RectInt‖ ≤ (1/2π)·(‖H−‖+‖H+‖+‖Vr‖+‖Vl‖) — linear (1/2π ≥ 0 ✓ mul by scalar). Then (1/2π)·(2(mid+DLR) + e^{1/3}·IIMaj + (4TX + 40M0)): Expand: (1/2π)·2·mid + (1/2π)·2·DLR + (1/2π)·e^{1/3}·IIMaj + (1/2π)·(4TX + 40M0). Need: (1/2π)·e^{1/3}·IIMaj ≤ IIMaj ⟸ e^{1/3} ≤ 2π, IIMaj ≥ 0 — NONLINEAR (product (1/2π)·e13 — but both are CONSTANTS — linarith treats (1/2π)·e13·IIMaj as IIMaj·c with c a numeric-ish atom?? — the product (1/(2π))·(e^{1/3}) is a CONSTANT (no variables) — linarith can evaluate it?? — he13 : e^{1/3} ≤ 2π — so (1/2π)·e^{1/3} ≤ (1/2π)·2π = 1 — linarith with the atom IIMaj and the constant (1/(2π))·(e^{1/3}): from he13 (e13 ≤ 2π) and 2π > 0: (1/2π)·e13 ≤ 1 — is that LINEAR derivable? (1/2π)·e13 = e13/(2π) — with the DIVISION — linarith doesn't do division!! — need to precompute: have he13' : (1/(2·π))·(e^{1/3}) ≤ 1 — by: rw [div_le_iff₀ (2π > 0)]; ... e13 ≤ 1·(2π) = 2π ✓ he13. Provide as a have: ```lean have h2πpos : (0 : ℝ) < 2 * Real.pi := by positivity?? — pi_pos ✓ positivity with Real.pi_pos in context? — 2·π > 0 — by positivity — π > 0 is an atom-fact positivity knows (Real.pi_pos is a positivity extension? — `Real.pi_pos` — positivity proves 0 < π ✓ I believe). have he13' : (1 / (2 * Real.pi)) * Real.exp (1/3 : ℝ) ≤ 1 := by rw [div_le_iff₀ h2πpos, one_mul] exact he13 ``` Then (1/2π)·e13·IIMaj ≤ 1·IIMaj = IIMaj via mul_le_mul_of_nonneg_right he13' hIIMaj0 — provide hIIMaj0 : 0 ≤ IIMaj ↑n := by unfold IIMaj; positivity. Hmm — ((1/2π)·e13)·IIMaj ≤ 1·IIMaj — mul_le_mul_of_nonneg_right (he13' : (1/2π)e13 ≤ 1) (0 ≤ IIMaj) ✓ — the association ((1/2π)·e13)·IIMaj vs (1/2π)·(e13·IIMaj) — ring_nf in linarith ✓ fine if I provide the multiplied-out form. To be safe, provide: ```lean have hIIMajAbs : (1 / (2 * Real.pi)) * (Real.exp (1/3 : ℝ) * IIMaj (n:ℝ)) ≤ IIMaj (n:ℝ) := by have h1 := mul_le_mul_of_nonneg_right he13' hIIMaj0 -- h1 : ((1/2π)·e13) · IIMaj ≤ 1 · IIMaj calc (1/(2π))·(e13·IIMaj) = ((1/2π)·e13)·IIMaj := by ring _ ≤ 1·IIMaj := h1 _ = IIMaj := by ring ``` Similarly for the Hhat term: (1/2π)·(2(e⁴+1)·7·Hhat T) ≤ 2(e⁴+1)·7·Hhat T ⟸ 1/2π ≤ 1: ```lean have h2πle : (1 / (2 * Real.pi)) ≤ 1 := by rw [div_le_iff₀ h2πpos, one_mul] have h := Real.pi_gt_three.le linarith have hHhatAbs : (1/(2π))·(2·(e⁴+1)·(7·Hhat T)) ≤ 2·(e⁴+1)·(7·Hhat T) := by have hY : (0:ℝ) ≤ 2·(Real.exp 4+1)·(7·Hhat T) := by positivity?? — Hhat T ≥ 0 needed have h1 := mul_le_mul_of_nonneg_right h2πle hY calc (1/2π)·Y' = (1/2π)·Y' := rfl... — just: have := mul_le_mul_of_nonneg_right h2πle hY; rwa [one_mul] at this ``` Hhat T ≥ 0: Hhat T = 3·C0hat T² + 10.5 log(T+2)+48: 3·C0hat² ≥ 0 ✓, log(T+2) ≥ 1 > 0 ✓ ⟹ positivity-ish with two_le_C0hat... — `have hHhat0 : 0 ≤ Hhat T := by unfold Hhat; nlinarith [hC0two, hL1]` — hC0two : 2 ≤ C0hat T (two_le_C0hat hT1'), hL1 : 1 ≤ log(T+2) ✓ then 3·(≥0) + 10.5·(≥1) + 48 ≥ 0 ✓ nlinarith/linarith. And the remaining (1/2π)·(2·DLR) ≤ 2·DLR and (1/2π)·(4TX+40M0) ≤ 4TX+40M0 — same h2πle pattern with nonneg of each. This is a lot of small nonneg haves — let me just collect them: hDLR0 : 0 ≤ DogLegRight ↑n (unfold DogLegRight; positivity — needs IIMaj ≥ 0 for the /4 part ✓ unfold IIMaj; positivity — nested unfolds — `unfold DogLegRight IIMaj; positivity` — hmm — positivity can't always see through defs — DogLegRight's body: ((√n+1)/2)·(e^{5/2}+1)/2 + 8.38 + IIMaj n/4 — with IIMaj unfolded: 2(√n+1)(e^{√n/2}+1)·0.57·2^{...} — positivity: √n ≥ 0 ✓ (Real.sqrt_nonneg is a positivity extension ✓), exp > 0 ✓, 2^rpow ≥ 0 ✓ — should work). hX0 : 0 ≤ 4·T·X(T) + 40·1.3352 — X(T) ≥ 0: (2/3)·Z2hat T (Z2hat_nonneg hT1) + 10.5·log(T+2) (≥ 0) + 48 — positivity with Z2hat_nonneg and log_nonneg... — 4·T·X(T): T ≥ 0 ✓ — nlinarith [Z2hat_nonneg, log(T+2) ≥ 1] or positivity with the haves. Use: have hX0 : (0:ℝ) ≤ 4*T*((2/3)*Z2hat T + 10.5*Real.log (T+2)+48) + 40*1.3352 := by have h1 := Z2hat_nonneg hT1; have h2 := Real.log_nonneg (by linarith : (1:ℝ) ≤ T+2); nlinarith [hTn, Real.sqrt_nonneg (n:ℝ)]. The T ≤ T' comparisons: hFAlt : FAlt T ≤ FAlt (√n+1): unfold FAlt; T·log T ≤ (√n+1)·log(√n+1) [mul_le_mul hTn1 (Real.log_le_log hTpos (by linarith [hTn1])) (Real.log_nonneg (1 ≤ T)) (by positivity...)]; 0.137·log T ≤ 0.137·log(√n+1) [log_le_log]; 0.443·log(log T) ≤ 0.443·log(log(√n+1)) [log_le_log (log T > 0)...]; linarith. hX : 4·T·X(T) ≤ 4·(√n+1)·X(√n+1): X increasing: (2/3)·Z2hat T ≤ (2/3)·Z2hat(√n+1) [Z2hat_mono hT1 hTn1] + 10.5·log(T+2) ≤ 10.5·log(√n+1+2) [log_le_log] ⟹ X(T) ≤ X(√n+1); and 0 ≤ X(T); mul_le_mul hTn1 (that) (X(T) ≥ 0) ((√n+1) ≥ 0) then ×4. hHhat : Hhat T ≤ Hhat (√n+1): Hhat t = 3·C0hat t² + 10.5·log(t+2)+48: C0hat T ≤ C0hat(√n+1) [C0hat_mono hT1' hTn1'] ⟹ C0hat T² ≤ C0hat(√n+1)² [pow_le_pow_left₀ (C0hat ≥ 0) that 2] + log mono ⟹ linarith. C0hat_mono — signature: from LiGapCircles (cloned as Z2hat_mono in my §1): `C0hat_mono {a b : ℝ} (ha : 1 ≤ a) (hab : a ≤ b) : C0hat a ≤ C0hat b` ✓ (comment in my Z2hat_mono says "clone of C0hat_mono, LiGapCircles" ✓ exists). OK — this is a LOT of small pieces but all elementary. Total 6-d estimate: ~350-450 lines (the pinch ~120, the bridge clone ~35, hnorm ~25, htri ~25, monotonicity ~100, assembly ~60, small haves ~60). Let me also double check: `exists_zero_avoiding_T_wide hZC ha` gives ⟨T, hT1 : a ≤ T, hT2 : T ≤ a+1, hT3 : the clearance⟩ — obtain ⟨T, hTn, hTn1, hTwide⟩ with hTn : √n ≤ T, hTn1 : T ≤ √n+1 ✓. Also sf_abs_le_rectInt_add_zeroSum hZC hn1 hT1 hTn hTn1 hTwide — hT1 : 1 ≤ T ✓ (from ha : 3 ≤ √n ≤ T). hTwide's form for the 6-c theorems: `∀ ρ, IsNontrivialZero ρ → a − 1 < ρ.im → ρ.im < a + 2 → 1/(9·C0hat(a+2)) ≤ |ρ.im − T|` with a := √n ✓ matches hIntegral_Jlag_full_two_sides' hTwide parameter ✓. And the 6-c-ii's vIntegral_Jlag_left_norm_le hZC hZC2 hNLZ hMZ hn hTn hTn1 ✓; 6-c-iii's vIntegral_Jlag_right_le_exp_third_IIMaj hn hTn hTn1 ✓. Now — one subtle issue: the hH (two-sides) is stated as ‖H(+)‖ + ‖H(−)‖ ≤ 2·(mid + DLR) — and htri gives ‖RectInt‖ ≤ ‖H−‖+‖H+‖+‖Vr‖+‖Vl‖ — the sum ‖H−‖+‖H+‖ appears in a different ORDER (‖H(−T)‖ + ‖H(T)‖) vs hH's (‖H(T)‖ + ‖H(−T)‖) — linarith handles (atoms) ✓. Alright — also `Real.pi_gt_three : 3 < Real.pi` ✓ standard. `Real.exp_one_lt_d9 : exp 1 < 2.7182818286` ✓ used in Z2hat_mono. `le_of_pow_le_pow_left₀` — verify signature via grep. Let me write the whole §6 now. Statement name: `sfBoundProp` — hmm — a theorem named sfBoundProp proving the Prop `SfBoundProp` ✓ (per ii′-7's plan: "theorem sfBoundProp : SfBoundProp"). One more check — `Z2hat_nonneg (ht : 1 ≤ t)` ✓ (my §2 lemma). `two_le_C0hat (ht : 1 ≤ t)` ✓ (LiContour:169). `C0hat_mono` — verify its exact signature via grep (need (ha : 1 ≤ a) (hab : a ≤ b)).
All pieces mapped. The bridge's public `hTle : T ≤ 3·10¹²−1` is over-strong — internally only `T ≤ 3·10¹²` is used, which my T satisfies exactly. I'll clone the bridge's last mile with the right hypothesis. Verifying `C0hat_mono` and `le_of_pow_le_pow_left₀` signatures.
tool: Bash
input
{
  "command": "grep -rn -A2 \"theorem C0hat_mono\" \"D:/Python/Quantum/research/lean/NguyenRH1/NguyenRH1/\" | head -6; grep -rn -A2 \"theorem le_of_pow_le_pow_left\u2080\" \"D:/Python/Quantum/zeta-23-lean/.lake/packages/mathlib/Mathlib/\" | head -6",
  "description": "Verify C0hat_mono and le_of_pow_le_pow_left\u2080 signatures"
}
#2304 assistant 2026-08-26 20:19:43.892 UTC · 252→12435 tok open ⤢
thinking (9844 chars)
`le_of_pow_le_pow_left₀ (hn : n ≠ 0) (hb : 0 ≤ b) (h : a ^ n ≤ b ^ n) : a ≤ b` — note: the nonneg is on **b** (the RIGHT side), not a! So `le_of_pow_le_pow_left₀ (by norm_num : (3:ℕ) ≠ 0) (by norm_num : (0:ℝ) ≤ 1.4) h2` ✓ with b := 1.4. `C0hat_mono {a b} (ha : 1 ≤ a) (hab : a ≤ b) : C0hat a ≤ C0hat b` ✓. Now write §6 (6-d). Full draft: ```lean /-! ## 6. 6-d: the fold — `sfBoundProp : SfBoundProp` -/ /-- Ĉ₀(t) ≤ t for t ≥ 8 (via log x ≤ x/e and loglog x ≤ log x ≤ x/e). -/ theorem C0hat_le_self {t : ℝ} (ht : 8 ≤ t) : C0hat t ≤ t := by have ht2 : (0 : ℝ) < t + 2 := by linarith have ht1 : (1 : ℝ) < t + 2 := by linarith have hinv : (1 : ℝ) / Real.exp 1 ≤ 0.36789 := by rw [div_le_iff₀ (Real.exp_pos 1), one_mul] nlinarith [Real.exp_one_gt_d9] have hL1 : Real.log (t + 2) ≤ (t + 2) / Real.exp 1 := by have h1 := Real.log_le_sub_one_of_pos (div_pos ht2 (Real.exp_pos 1)) have h2 : Real.log ((t + 2) / Real.exp 1) = Real.log (t + 2) - 1 := by rw [Real.log_div (ne_of_gt ht2) (Real.exp_pos 1).ne', Real.log_exp] rw [h2] at h1 linarith [h1] have hL : Real.log (t + 2) ≤ (t + 2) * 0.36789 := by have h3 : (t + 2) / Real.exp 1 ≤ (t + 2) * 0.36789 := by rw [div_eq_mul_one_div] exact mul_le_mul_of_nonneg_left hinv ht2.le exact hL1.trans h3 have hLL : Real.log (Real.log (t + 2)) ≤ (t + 2) * 0.36789 := by have h1 := Real.log_le_sub_one_of_pos (Real.log_pos ht1) nlinarith [h1, hL] unfold C0hat nlinarith [hL, hLL, ht] /-- **The horizontal pinch** (E-030 V3 witness made analytic): the near-sum integral + the R-part ≤ 7·Ĥ(T). Chain (all elementary): C0hat(√n+2) ≤ √n+2 (t ≥ 318), so 148.5·C0hat(√n+2) ≤ (√n+2)² and the log ≤ 2·log(T+2); Z₂(T) ≤ 2·Ĉ₀(T); Ĉ₀(T) ≥ 0.433·log(T+2) and 8 ≤ 21·0.433. -/ theorem pinch_horizontal {n : ℕ} (hn : 100000 ≤ n) {T : ℝ} (hTn : Real.sqrt (n : ℝ) ≤ T) (hTn1 : T ≤ Real.sqrt (n : ℝ) + 1) : 2 * Z2hat T * Real.log (148.5 * C0hat (Real.sqrt (n : ℝ) + 2)) + (3 - -5 / 2) * (10.5 * Real.log (T + 2) + 48) ≤ 7 * Hhat T := by have hnR : (100000 : ℝ) ≤ (n : ℝ) := by exact_mod_cast hn have hsqrt316 : (316 : ℝ) ≤ Real.sqrt (n : ℝ) := by rw [Real.le_sqrt (by norm_num) (by positivity)] nlinarith [hnR] have hT1 : (1 : ℝ) ≤ T := by nlinarith [hsqrt316, hTn] have hT0 : (0 : ℝ) < T := by nlinarith -- (a) the log argument bound have hC0le : C0hat (Real.sqrt (n : ℝ) + 2) ≤ Real.sqrt (n : ℝ) + 2 := C0hat_le_self (by nlinarith [hsqrt316]) have h148 : (148.5 : ℝ) ≤ Real.sqrt (n : ℝ) + 2 := by nlinarith [hsqrt316] have hC0pos : (0 : ℝ) < C0hat (Real.sqrt (n : ℝ) + 2) := by have h2 := two_le_C0hat (by nlinarith [Real.sqrt_nonneg (n : ℝ)] : (1 : ℝ) ≤ Real.sqrt (n : ℝ) + 2) linarith have hsq1 : 148.5 * C0hat (Real.sqrt (n : ℝ) + 2) ≤ (Real.sqrt (n : ℝ) + 2) ^ 2 := by have h1 := mul_le_mul hC0le h148 (by norm_num : (0 : ℝ) ≤ (148.5 : ℝ)) (by positivity : (0 : ℝ) ≤ Real.sqrt (n : ℝ) + 2) nlinarith [h1, hC0pos.le] have hlogarg : Real.log (148.5 * C0hat (Real.sqrt (n : ℝ) + 2)) ≤ 2 * Real.log (T + 2) := by have h2 := Real.log_le_log (by positivity) hsq1 have h3 : Real.log ((Real.sqrt (n : ℝ) + 2) ^ 2) = 2 * Real.log (Real.sqrt (n : ℝ) + 2) := by rw [Real.log_pow] ring have h4 : Real.log (Real.sqrt (n : ℝ) + 2) ≤ Real.log (T + 2) := Real.log_le_log (by positivity) (by linarith [hTn]) linarith [h2, h3, h4] -- (b) Z₂ ≤ 2·Ĉ₀ have hZ2le : Z2hat T ≤ 2 * C0hat T := by have hL1 : (1 : ℝ) ≤ Real.log (T + 2) := by have ht2 : (0 : ℝ) < T + 2 := by linarith rw [Real.le_log_iff_exp_le ht2] have he := Real.exp_one_lt_d9.le linarith have hLL0 : (0 : ℝ) ≤ Real.log (Real.log (T + 2)) := Real.log_nonneg hL1 unfold Z2hat C0hat nlinarith [hL1, hLL0] -- (c) Ĉ₀ ≥ 0.433·log have hC0L : 0.433 * Real.log (T + 2) ≤ C0hat T := by have hL1 : (1 : ℝ) ≤ Real.log (T + 2) := by have ht2 : (0 : ℝ) < T + 2 := by linarith rw [Real.le_log_iff_exp_le ht2] have he := Real.exp_one_lt_d9.le linarith have hLL0 : (0 : ℝ) ≤ Real.log (Real.log (T + 2)) := Real.log_nonneg hL1 unfold C0hat nlinarith [hL1, hLL0] -- (d) the near part ≤ 21·Ĉ₀² have hZ2nn : (0 : ℝ) ≤ Z2hat T := Z2hat_nonneg hT1 have hlogT0 : (0 : ℝ) ≤ Real.log (T + 2) := Real.log_nonneg (by linarith) have hC0T0 : (0 : ℝ) < C0hat T := by have h2 := two_le_C0hat hT1 linarith have hnear : 2 * Z2hat T * Real.log (148.5 * C0hat (Real.sqrt (n : ℝ) + 2)) ≤ 21 * C0hat T * C0hat T := by have h1 : 2 * Z2hat T * Real.log (148.5 * C0hat (Real.sqrt (n : ℝ) + 2)) ≤ 2 * Z2hat T * (2 * Real.log (T + 2)) := mul_le_mul_of_nonneg_left hlogarg (by positivity : (0 : ℝ) ≤ 2 * Z2hat T) have h2 : 2 * Z2hat T * (2 * Real.log (T + 2)) ≤ 2 * (2 * C0hat T) * (2 * Real.log (T + 2)) := by have h3 := mul_le_mul hZ2le le_rfl hlogT0 (by positivity : (0:ℝ) ≤ 2 * (2:ℝ)) -- hmm — mul_le_mul (h₁ : Z2hat ≤ 2·C0hat) (h₂ : log ≤ log) (0 ≤ log) (0 ≤ 2·C0hat) ... ``` Hmm — getting the multiplications right: 4·Z2hat·log ≤ 8·C0hat·log: mul_le_mul (h₁ : Z2hat ≤ 2·C0hat) (h₂ : log(T+2) ≤ log(T+2)) (0 ≤ log) (0 ≤ 2·C0hat) : Z2hat·log ≤ 2·C0hat·log — then ×4. Let me restructure linearly: ```lean have hZ2L : Z2hat T * Real.log (T + 2) ≤ 2 * C0hat T * Real.log (T + 2) := mul_le_mul_of_nonneg_right hZ2le hlogT0 -- 8·log(T+2) ≤ 21·C0hat have h8 : 8 * Real.log (T + 2) ≤ 21 * C0hat T := by have h1 : (8 : ℝ) ≤ 21 * 0.433 := by norm_num nlinarith [hC0L, h1, hlogT0, hC0T0.le] ``` h8: 8·log(T+2) ≤ 21·C0hat T: from hC0L : 0.433·log(T+2) ≤ C0hat T: 8·log(T+2) = (8/0.433)·(0.433·log(T+2)) ≤ (8/0.433)·C0hat T ≤ 21·C0hat T ⟺ 8 ≤ 21·0.433 = 9.093 ✓ — linarith with the ATOMS log(T+2), C0hat T: 8·log = (8/0.433)·(0.433·log) — the coefficient division 8/0.433 — linarith does exact rational arithmetic ✓: 8·log(T+2) ≤ 21·C0hat T ⟺ 0.433·8·log ≤ 21·0.433·C0hat ⟸ 0.433·log ≤ C0hat (hC0L) and 8 ≤ 21·0.433 ✓ — linarith [hC0L] + norm_num-fact ✓: ```lean have h8 : 8 * Real.log (T + 2) ≤ 21 * C0hat T := by have h1 : (8 : ℝ) ≤ 21 * (0.433 : ℝ) := by norm_num nlinarith [hC0L, h1, hlogT0] ``` hmm — 8·log ≤ 21·C0hat: multiply hC0L by 21: 21·0.433·log ≤ 21·C0hat: 9.093·log ≤ 21·C0hat; and 8·log ≤ 9.093·log (log ≥ 0) ⟹ 8·log ≤ 21·C0hat ✓ linarith ✓. have h3 : (8 * Real.log (T + 2)) * C0hat T ≤ (21 * C0hat T) * C0hat T := mul_le_mul h8 le_rfl hC0T0.le (by positivity : (0:ℝ) ≤ 21 * C0hat T) -- assemble: 2·Z2hat·log148.5 ≤ 4·Z2hat·log ≤ 8·C0hat·log = (8·log)·C0hat ≤ 21·C0hat·C0hat nlinarith [h1, hZ2L, h3, hlogarg] ``` hmm — the assemble nlinarith: goal: 2·Z2hat·log148.5 ≤ 21·C0hat·C0hat. Facts: h1 : 2·Z2hat·log148.5 ≤ 2·Z2hat·(2·log) = 4·(Z2hat·log); hZ2L : Z2hat·log ≤ 2·C0hat·log; h3 : (8·log)·C0hat ≤ 21·C0hat·C0hat. Chain: 4·(Z2hat·log) ≤ 4·(2·C0hat·log) = 8·(C0hat·log) = (8·log)·C0hat ≤ 21·C0hat·C0hat ✓ — the products (Z2hat·log), (C0hat·log), (log·C0hat), (C0hat·C0hat) — ring_nf normalizes monomials ✓ linear ✓ nlinarith/linarith. Use linarith [h1, hZ2L, h3]. (e) the 5.5 part + Hhat unfold: ```lean have hR0 : (0 : ℝ) ≤ 10.5 * Real.log (T + 2) + 48 := by have h1 := Real.log_nonneg (by linarith : (1 : ℝ) ≤ T + 2) nlinarith have hR : (3 - -5 / 2 : ℝ) * (10.5 * Real.log (T + 2) + 48) ≤ 7 * (10.5 * Real.log (T + 2) + 48) := by norm_num1 -- 5.5·R ≤ 7·R ⟸ R ≥ 0 — after norm_num1 computes 3 − −5/2 = 5.5 nlinarith [hR0] ``` Hmm — norm_num1 on goal `(3 - -5/2)·R ≤ 7·R` — computes (3 − −5/2) → 5.5: goal `5.5·R ≤ 7·R` — then nlinarith [hR0] ✓ (linarith suffices: 5.5R ≤ 7R ⟺ 0 ≤ 1.5R ✓). Actually `norm_num1` might reformat the LHS — or use `have h55 : (3 - -5/2 : ℝ) = 5.5 := by norm_num; rw [h55]; nlinarith [hR0]` — deterministic. Use that. -- final unfold Hhat have hC0T02 : (0 : ℝ) ≤ C0hat T * C0hat T := by positivity linarith [hnear, hR, hC0T02] — goal: NI + 5.5R ≤ 7·(3·C0hat² + R) — from hnear (NI ≤ 21·C0hat²) and hR (5.5R ≤ 7R): NI + 5.5R ≤ 21·C0hat² + 7R = 7·(3·C0hat² + R) ✓ — after unfold Hhat, the goal's RHS is 7·(3·C0hat T^2 + 10.5·log(T+2)+48) — note C0hat T^2 vs my C0hat T * C0hat T — hnear has 21 * C0hat T * C0hat T — ring_nf normalizes ✓ linarith ✓. Hmm — Hhat's def: `Hhat (T: ℝ) : ℝ := 3 * C0hat T ^ 2 + 10.5 * Real.log (T + 2) + 48` ✓ (LiSkeleton:749). `unfold Hhat` at the goal ✓. Wait — the goal after unfold Hhat: `2·Z2hat·log148.5 + (3−−5/2)·R ≤ 7·(3·C0hat T^2 + 10.5·log(T+2)+48)`. linarith [hnear : 2·Z2hat·log148.5 ≤ 21·C0hat·C0hat, hR : 5.5·R ≤ 7·R, ...]: 2·Z2hat·log148.5 + 5.5R ≤ 21·C0hat·C0hat + 7R = 7·(3·C0hat·C0hat + R) — and 3·C0hat T^2 = 3·(C0hat·C0hat) ✓ ring_nf ✓ closes ✓. Then the bridge variant + hnorm + htri + the main theorem. Also `Real.le_log_iff_exp_le` used twice (hL1 in (b) and (c)) — fine. One concern in hZ2le/hC0L's unfold: `unfold Z2hat C0hat` — unfolds both ✓ (as in Z2hat_mono's `unfold Z2hat` ✓). For hnear's mul_le_mul h8 le_rfl hC0T0.le (by positivity : 0 ≤ 21·C0hat T) : (8·log)·C0hat ≤ (21·C0hat)·C0hat ✓ — mul_le_mul (h₁ : 8·log ≤ 21·C0hat) (h₂ : C0hat ≤ C0hat) (c0 : 0 ≤ C0hat) (b0 : 0 ≤ 21·C0hat) ✓. And h1's `mul_le_mul_of_nonneg_left hlogarg (by positivity : (0:ℝ) ≤ 2 * Z2hat T)` : 2·Z2hat·log148.5 ≤ 2·Z2hat·(2·log(T+2)) ✓ — mul_le_mul_of_nonneg_left (hbc : log148.5 ≤ 2·log(T+2)) (ha : 0 ≤ 2·Z2hat) ✓. Now the main sfBoundProp theorem. Let me also handle `Real.pi_gt_three` — exact name: `Real.pi_gt_three : 3 < Real.pi`? or `Real.pi_gt_three : 3 < π`? — I recall `Real.pi_gt_three : 3 < Real.pi` ✓ (in this Mathlib — session-033 banked "Real.pi_lt_d2 : π < 3.15" — so pi_gt_three likely exists too — let me trust it, fallback `Real.three_lt_pi`? — hmm — actually in recent Mathlib it's `Real.pi_gt_three : 3 < π` ✓). Also `Real.pi_pos : 0 < Real.pi` ✓. Now write the whole §6.
All signatures confirmed (`le_of_pow_le_pow_left₀` takes nonneg on the RIGHT side). Writing §6 (6-d) — pinch, bridge variant, norm machinery, and the fold.
tool: Edit
input
{
  "replace_all": false,
  "file_path": "D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiSideBounds.lean",
  "old_string": "  exact add_le_add (hIntegral_Jlag_full_norm_le hZC hZC2 hNLZ hn hTwide hTn hTn1)\n    (hIntegral_Jlag_full_neg_norm_le hZC hZC2 hNLZ hn hTwide hTn hTn1)\n\nend NguyenRH1",
  "new_string": "  exact add_le_add (hIntegral_Jlag_full_norm_le hZC hZC2 hNLZ hn hTwide hTn hTn1)\n    (hIntegral_Jlag_full_neg_norm_le hZC hZC2 hNLZ hn hTwide hTn hTn1)\n\n/-! ## 6. 6-d: the fold \u2014 `sfBoundProp : SfBoundProp` -/\n\n/-- \u0108\u2080(t) \u2264 t for t \u2265 8 (via log x \u2264 x/e with 1/e \u2264 0.36789 certified\nfrom e > 2.7182818283, and loglog x \u2264 log x \u2212 1). -/\ntheorem C0hat_le_self {t : \u211d} (ht : 8 \u2264 t) : C0hat t \u2264 t := by\n  have ht2 : (0 : \u211d) < t + 2 := by linarith\n  have ht1 : (1 : \u211d) < t + 2 := by linarith\n  have hinv : (1 : \u211d) / Real.exp 1 \u2264 0.36789 := by\n    rw [div_le_iff\u2080 (Real.exp_pos 1), one_mul]\n    nlinarith [Real.exp_one_gt_d9]\n  have hL1 : Real.log (t + 2) \u2264 (t + 2) / Real.exp 1 := by\n    have h1 := Real.log_le_sub_one_of_pos (div_pos ht2 (Real.exp_pos 1))\n    have h2 : Real.log ((t + 2) / Real.exp 1) = Real.log (t + 2) - 1 := by\n      rw [Real.log_div (ne_of_gt ht2) (Real.exp_pos 1).ne', Real.log_exp]\n    rw [h2] at h1\n    linarith [h1]\n  have hL : Real.log (t + 2) \u2264 (t + 2) * 0.36789 := by\n    have h3 : (t + 2) / Real.exp 1 \u2264 (t + 2) * 0.36789 := by\n      rw [div_eq_mul_one_div]\n      exact mul_le_mul_of_nonneg_left hinv ht2.le\n    exact hL1.trans h3\n  have hLL : Real.log (Real.log (t + 2)) \u2264 (t + 2) * 0.36789 := by\n    have h1 := Real.log_le_sub_one_of_pos (Real.log_pos ht1)\n    nlinarith [h1, hL]\n  unfold C0hat\n  nlinarith [hL, hLL, ht]\n\n/-- **The horizontal pinch** (the E-030 V3 \u2264 0.40 witness made analytic):\nthe near-sum integral + the R-part \u2264 7\u00b7H\u0302(T). Chain (all elementary, no\ncertified log numerics): \u0108\u2080(\u221an+2) \u2264 \u221an+2 (\u0108\u2080(t) \u2264 t, t \u2265 318), so\n148.5\u00b7\u0108\u2080(\u221an+2) \u2264 (\u221an+2)\u00b2 and its log \u2264 2\u00b7log(T+2); Z\u2082(T) \u2264 2\u00b7\u0108\u2080(T);\n\u0108\u2080(T) \u2265 0.433\u00b7log(T+2) with 8 \u2264 21\u00b70.433 = 9.093; 5.5 \u2264 7 on the R-part. -/\ntheorem pinch_horizontal {n : \u2115} (hn : 100000 \u2264 n) {T : \u211d}\n    (hTn : Real.sqrt (n : \u211d) \u2264 T) (hTn1 : T \u2264 Real.sqrt (n : \u211d) + 1) :\n    2 * Z2hat T * Real.log (148.5 * C0hat (Real.sqrt (n : \u211d) + 2)) +\n      (3 - -5 / 2) * (10.5 * Real.log (T + 2) + 48) \u2264 7 * Hhat T := by\n  have hnR : (100000 : \u211d) \u2264 (n : \u211d) := by exact_mod_cast hn\n  have hsqrt316 : (316 : \u211d) \u2264 Real.sqrt (n : \u211d) := by\n    rw [Real.le_sqrt (by norm_num) (by positivity)]\n    nlinarith [hnR]\n  have hT1 : (1 : \u211d) \u2264 T := by nlinarith [hsqrt316, hTn]\n  have hT0 : (0 : \u211d) < T := by nlinarith\n  -- (a) log(148.5\u00b7\u0108\u2080(\u221an+2)) \u2264 2\u00b7log(T+2)\n  have hC0le : C0hat (Real.sqrt (n : \u211d) + 2) \u2264 Real.sqrt (n : \u211d) + 2 :=\n    C0hat_le_self (by nlinarith [hsqrt316])\n  have h148 : (148.5 : \u211d) \u2264 Real.sqrt (n : \u211d) + 2 := by nlinarith [hsqrt316]\n  have hC0pos : (0 : \u211d) < C0hat (Real.sqrt (n : \u211d) + 2) := by\n    have h2 := two_le_C0hat (by nlinarith [Real.sqrt_nonneg (n : \u211d)] :\n      (1 : \u211d) \u2264 Real.sqrt (n : \u211d) + 2)\n    linarith\n  have hsq1 : 148.5 * C0hat (Real.sqrt (n : \u211d) + 2) \u2264 (Real.sqrt (n : \u211d) + 2) ^ 2 := by\n    have h1 := mul_le_mul hC0le h148 (by norm_num : (0 : \u211d) \u2264 (148.5 : \u211d))\n      (by positivity : (0 : \u211d) \u2264 Real.sqrt (n : \u211d) + 2)\n    nlinarith [h1, hC0pos.le]\n  have hlogarg : Real.log (148.5 * C0hat (Real.sqrt (n : \u211d) + 2)) \u2264\n      2 * Real.log (T + 2) := by\n    have h2 := Real.log_le_log (by positivity) hsq1\n    have h3 : Real.log ((Real.sqrt (n : \u211d) + 2) ^ 2) = 2 * Real.log (Real.sqrt (n : \u211d) + 2) := by\n      rw [Real.log_pow]\n      ring\n    have h4 : Real.log (Real.sqrt (n : \u211d) + 2) \u2264 Real.log (T + 2) :=\n      Real.log_le_log (by positivity) (by linarith [hTn])\n    linarith [h2, h3, h4]\n  -- (b) Z\u2082(T) \u2264 2\u00b7\u0108\u2080(T)\n  have hZ2le : Z2hat T \u2264 2 * C0hat T := by\n    have hL1 : (1 : \u211d) \u2264 Real.log (T + 2) := by\n      have ht2 : (0 : \u211d) < T + 2 := by linarith\n      rw [Real.le_log_iff_exp_le ht2]\n      have he := Real.exp_one_lt_d9.le\n      linarith\n    have hLL0 : (0 : \u211d) \u2264 Real.log (Real.log (T + 2)) := Real.log_nonneg hL1\n    unfold Z2hat C0hat\n    nlinarith [hL1, hLL0]\n  -- (c) \u0108\u2080(T) \u2265 0.433\u00b7log(T+2)\n  have hC0L : 0.433 * Real.log (T + 2) \u2264 C0hat T := by\n    have hL1 : (1 : \u211d) \u2264 Real.log (T + 2) := by\n      have ht2 : (0 : \u211d) < T + 2 := by linarith\n      rw [Real.le_log_iff_exp_le ht2]\n      have he := Real.exp_one_lt_d9.le\n      linarith\n    have hLL0 : (0 : \u211d) \u2264 Real.log (Real.log (T + 2)) := Real.log_nonneg hL1\n    unfold C0hat\n    nlinarith [hL1, hLL0]\n  -- (d) the near part \u2264 21\u00b7\u0108\u2080(T)\u00b2\n  have hlogT0 : (0 : \u211d) \u2264 Real.log (T + 2) := Real.log_nonneg (by linarith)\n  have hC0T0 : (0 : \u211d) < C0hat T := by\n    have h2 := two_le_C0hat hT1\n    linarith\n  have hnear : 2 * Z2hat T * Real.log (148.5 * C0hat (Real.sqrt (n : \u211d) + 2)) \u2264\n      21 * C0hat T * C0hat T := by\n    have h1 : 2 * Z2hat T * Real.log (148.5 * C0hat (Real.sqrt (n : \u211d) + 2)) \u2264\n        2 * Z2hat T * (2 * Real.log (T + 2)) := by\n      have hnn : (0 : \u211d) \u2264 2 * Z2hat T := by\n        have h := Z2hat_nonneg hT1\n        positivity\n      exact mul_le_mul_of_nonneg_left hlogarg hnn\n    have hZ2L : Z2hat T * Real.log (T + 2) \u2264 2 * C0hat T * Real.log (T + 2) :=\n      mul_le_mul_of_nonneg_right hZ2le hlogT0\n    have h8 : 8 * Real.log (T + 2) \u2264 21 * C0hat T := by\n      have h1 : (8 : \u211d) \u2264 21 * (0.433 : \u211d) := by norm_num\n      nlinarith [hC0L, h1, hlogT0]\n    have h3 : (8 * Real.log (T + 2)) * C0hat T \u2264 (21 * C0hat T) * C0hat T :=\n      mul_le_mul h8 le_rfl hC0T0.le (by positivity : (0 : \u211d) \u2264 21 * C0hat T)\n    nlinarith [h1, hZ2L, h3, hlogT0]\n  -- (e) the 5.5 \u2264 7 part\n  have hR0 : (0 : \u211d) \u2264 10.5 * Real.log (T + 2) + 48 := by\n    have h1 := Real.log_nonneg (by linarith : (1 : \u211d) \u2264 T + 2)\n    nlinarith\n  have hR : (3 - -5 / 2 : \u211d) * (10.5 * Real.log (T + 2) + 48) \u2264\n      7 * (10.5 * Real.log (T + 2) + 48) := by\n    have h55 : (3 - -5 / 2 : \u211d) = 5.5 := by norm_num\n    rw [h55]\n    nlinarith [hR0]\n  -- final: 21\u00b7\u0108\u2080\u00b2 + 7\u00b7R = 7\u00b7H\u0302\n  unfold Hhat\n  nlinarith [hnear, hR]\n\n/-- **THE FOLD (ii\u2032-6-d)**: `SfBoundProp` from the six declared perimeter\nhypotheses \u2014 the 6-b sides identity + the bridge (both-signs 4F + triv \u2264 1),\nthe rectangle side decomposition, the four side bounds of ii\u2032-6-c, the\nhorizontal pinch, and the F-E absorption e^{1/3} \u2264 1.4 \u2264 2\u03c0. -/\ntheorem sfBoundProp (hRH : RHUpTo (3 * 10 ^ 12)) (hZC : ZeroCountProp)\n    (hZC2 : ZeroCountProp2) (hZCG : ZeroCountGlobalProp) (hNLZ : NoLowZeros)\n    (hMZ : MZeroProp) : SfBoundProp := by\n  intro n hn hN1\n  classical\n  have hnR : (100000 : \u211d) \u2264 (n : \u211d) := by exact_mod_cast hn\n  have hn1 : 1 \u2264 n := by omega\n  have hn16 : 16 \u2264 n := by omega\n  have hsqrt316 : (316 : \u211d) \u2264 Real.sqrt (n : \u211d) := by\n    rw [Real.le_sqrt (by norm_num) (by positivity)]\n    nlinarith [hnR]\n  have ha : (3 : \u211d) \u2264 Real.sqrt (n : \u211d) := by nlinarith [hsqrt316]\n  obtain \u27e8T, hTn, hTn1, hTwide\u27e9 := exists_zero_avoiding_T_wide hZC ha\n  have hT1 : (1 : \u211d) \u2264 T := by nlinarith [hTn]\n  have hT2 : (2 : \u211d) \u2264 T := by nlinarith\n  have hT0 : (0 : \u211d) < T := by nlinarith\n  -- T \u2264 3\u00b710\u00b9\u00b2 (from n \u2264 N\u2081; the bridge's internal RHUpTo range)\n  have hTle : T \u2264 3 * 10 ^ 12 := by\n    have h1 : Real.sqrt (n : \u211d) \u2264 3 * 10 ^ 12 - 1 := by\n      have h2 := Real.sqrt_le_sqrt hN1\n      rwa [Real.sqrt_sq (by norm_num : (0 : \u211d) \u2264 3 * 10 ^ 12 - 1)] at h2\n    linarith [hTn1]\n  -- the sides identity\n  have hsf := sf_abs_le_rectInt_add_zeroSum hZC hn1 hT1 hTn hTn1 hTwide\n  -- the bridge (variant with T \u2264 3\u00b710\u00b9\u00b2 \u2014 the internal range actually used)\n  have hbridge : \u2016\u2211 \u03c1 \u2208 ((zeroBothWindow_finite T).toFinset \u222a {(-2 : \u2102)}),\n      (zeroMult \u03c1 : \u2102) * (1 - (\u03c1 / (\u03c1 - 1)) ^ n)\u2016 \u2264 4 * FAlt T + 1 := by\n    have h2not : (-2 : \u2102) \u2209 (zeroBothWindow_finite T).toFinset := by\n      intro h\n      have hnt := (zeroBothWindow_finite T).mem_toFinset.mp h |>.1\n      have h21 : (0 : \u211d) < (-2 : \u2102).re := hnt.2.1\n      norm_num at h21\n    have hdisj2 : Disjoint (zeroBothWindow_finite T).toFinset {(-2 : \u2102)} := by\n      rw [Finset.disjoint_left]\n      intro \u03c1 h\u03c1 h\u03c12\n      rw [Finset.mem_singleton] at h\u03c12\n      exact h2not (h\u03c12 \u25b8 h\u03c1)\n    rw [Finset.sum_union hdisj2, Finset.sum_singleton]\n    have htriv := trivTerm_norm_le_one n\n    have hbr := zeroSum_nontriv_norm_le hRH (n := n) (T := T) hTle\n    have hcount := zeroBothWindow_sum_eq_two_mul hNLZ T\n    have hZ := hZCG T hT2\n    calc \u2016\u2211 \u03c1 \u2208 (zeroBothWindow_finite T).toFinset,\n          (zeroMult \u03c1 : \u2102) * (1 - (\u03c1 / (\u03c1 - 1)) ^ n) +\n          (zeroMult (-2 : \u2102) : \u2102) * (1 - ((-2 : \u2102) / (-2 - 1)) ^ n)\u2016\n        \u2264 \u2016\u2211 \u03c1 \u2208 (zeroBothWindow_finite T).toFinset,\n          (zeroMult \u03c1 : \u2102) * (1 - (\u03c1 / (\u03c1 - 1)) ^ n)\u2016 +\n          \u2016(zeroMult (-2 : \u2102) : \u2102) * (1 - ((-2 : \u2102) / (-2 - 1)) ^ n)\u2016 := norm_add_le _ _\n      _ \u2264 (2 * \u2211 \u03c1 \u2208 (zeroBothWindow_finite T).toFinset, (zeroMult \u03c1 : \u211d)) + 1 := by\n          linarith [hbr, htriv]\n      _ = 2 * (2 * \u2211 \u03c1 \u2208 (zeroGlobalWindow_finite T).toFinset, (zeroMult \u03c1 : \u211d)) + 1 := by\n          rw [hcount]\n      _ \u2264 4 * FAlt T + 1 := by linarith [hZ]\n  -- the four side bounds\n  have hVl := vIntegral_Jlag_left_norm_le hZC hZC2 hNLZ hMZ hn hTn hTn1\n  have hVr := vIntegral_Jlag_right_le_exp_third_IIMaj hn hTn hTn1\n  have hH := hIntegral_Jlag_full_two_sides hZC hZC2 hNLZ hn hTwide hTn hTn1\n  -- \u2016RectInt'\u2016 = (1/2\u03c0)\u00b7\u2016RectInt\u2016\n  have hnorm : \u2016RectangleIntegral' (Jlag n) (rectZ T) (rectW n T)\u2016 =\n      (1 / (2 * Real.pi)) * \u2016RectangleIntegral (Jlag n) (rectZ T) (rectW n T)\u2016 := by\n    unfold RectangleIntegral'\n    rw [norm_smul]\n    congr 1\n    have h1 : \u2016(2 : \u2102) * (Real.pi : \u2102) * Complex.I\u2016 = 2 * Real.pi := by\n      rw [norm_mul, Complex.norm_I, mul_one, norm_mul]\n      have h3 : \u2016(2 : \u2102)\u2016 = (2 : \u211d) := by\n        rw [show (2 : \u2102) = ((2 : \u211d) : \u2102) by push_cast; ring, Complex.norm_real]\n        exact abs_of_nonneg (by norm_num)\n      have h4 : \u2016(Real.pi : \u2102)\u2016 = Real.pi := by\n        rw [Complex.norm_real]\n        exact abs_of_nonneg Real.pi_nonneg\n      rw [h3, h4]\n    rw [norm_div, norm_one, h1]\n  -- the rectangle norm triangle\n  have htri : \u2016RectangleIntegral (Jlag n) (rectZ T) (rectW n T)\u2016 \u2264\n      \u2016HIntegral (Jlag n) (-5 / 2) (2 * Real.sqrt (n : \u211d)) (-T)\u2016 +\n      \u2016HIntegral (Jlag n) (-5 / 2) (2 * Real.sqrt (n : \u211d)) T\u2016 +\n      \u2016VIntegral (Jlag n) (2 * Real.sqrt (n : \u211d)) (-T) T\u2016 +\n      \u2016VIntegral (Jlag n) (-5 / 2) (-T) T\u2016 := by\n    rw [rectangleIntegral_Jlag_eq n T]\n    have h1 := norm_sub_le (HIntegral (Jlag n) (-5 / 2) (2 * Real.sqrt (n : \u211d)) (-T) -\n      HIntegral (Jlag n) (-5 / 2) (2 * Real.sqrt (n : \u211d)) T +\n      VIntegral (Jlag n) (2 * Real.sqrt (n : \u211d)) (-T) T)\n      (VIntegral (Jlag n) (-5 / 2) (-T) T)\n    have h2 := norm_add_le (HIntegral (Jlag n) (-5 / 2) (2 * Real.sqrt (n : \u211d)) (-T) -\n      HIntegral (Jlag n) (-5 / 2) (2 * Real.sqrt (n : \u211d)) T)\n      (VIntegral (Jlag n) (2 * Real.sqrt (n : \u211d)) (-T) T)\n    have h3 := norm_sub_le (HIntegral (Jlag n) (-5 / 2) (2 * Real.sqrt (n : \u211d)) (-T))\n      (HIntegral (Jlag n) (-5 / 2) (2 * Real.sqrt (n : \u211d)) T)\n    linarith [h1, h2, h3]\n  -- the horizontal pinch\n  have hpinch := pinch_horizontal hn hTn hTn1\n  -- the F-E absorption: e^{1/3} \u2264 1.4 \u2264 2\u03c0\n  have he13 : Real.exp (1 / 3 : \u211d) \u2264 2 * Real.pi := by\n    have h1 : Real.exp (1 / 3 : \u211d) \u2264 1.4 := by\n      have h2 : (Real.exp (1 / 3 : \u211d)) ^ 3 \u2264 (1.4 : \u211d) ^ 3 := by\n        have h3 : (Real.exp (1 / 3 : \u211d)) ^ 3 = Real.exp 1 := by\n          rw [\u2190 Real.exp_nat_mul (1 / 3 : \u211d) 3]\n          congr 1\n          norm_num\n        rw [h3]\n        have h4 : Real.exp 1 \u2264 2.7183 := Real.exp_one_lt_d9.le\n        norm_num at h4 \u22a2\n        linarith [h4]\n      exact le_of_pow_le_pow_left\u2080 (by norm_num : (3 : \u2115) \u2260 0) (by norm_num : (0 : \u211d) \u2264 1.4) h2\n    have h5 : (3 : \u211d) \u2264 Real.pi := Real.pi_gt_three.le\n    nlinarith [h1, h5]\n  -- the 1/(2\u03c0) comparisons\n  have h2\u03c0pos : (0 : \u211d) < 2 * Real.pi := by positivity\n  have h2\u03c0le : (1 / (2 * Real.pi)) \u2264 1 := by\n    rw [div_le_iff\u2080 h2\u03c0pos, one_mul]\n    have h := Real.pi_gt_three.le\n    linarith\n  have he13' : (1 / (2 * Real.pi)) * Real.exp (1 / 3 : \u211d) \u2264 1 := by\n    rw [div_le_iff\u2080 h2\u03c0pos, one_mul]\n    exact he13\n  -- nonneg facts\n  have hIIMaj0 : (0 : \u211d) \u2264 IIMaj (n : \u211d) := by unfold IIMaj; positivity\n  have hDLR0 : (0 : \u211d) \u2264 DogLegRight (n : \u211d) := by unfold DogLegRight IIMaj; positivity\n  have hC0T0 : (0 : \u211d) < C0hat T := by\n    have h2 := two_le_C0hat hT1\n    linarith\n  have hHhat0 : (0 : \u211d) \u2264 Hhat T := by\n    unfold Hhat\n    have hL1 : (1 : \u211d) \u2264 Real.log (T + 2) := by\n      have ht2 : (0 : \u211d) < T + 2 := by linarith\n      rw [Real.le_log_iff_exp_le ht2]\n      have he := Real.exp_one_lt_d9.le\n      linarith\n    nlinarith [hC0T0.le, hL1]\n  have hX0 : (0 : \u211d) \u2264 4 * T * ((2 / 3) * Z2hat T + 10.5 * Real.log (T + 2) + 48) +\n      40 * 1.3352 := by\n    have h1 := Z2hat_nonneg hT1\n    have h2 := Real.log_nonneg (by linarith : (1 : \u211d) \u2264 T + 2)\n    nlinarith [hT0.le]\n  -- the T \u2264 \u221an+1 monotonicity comparisons\n  have hsqrt0 : (0 : \u211d) \u2264 Real.sqrt (n : \u211d) := Real.sqrt_nonneg _\n  have hlogT1 : (1 : \u211d) \u2264 Real.log (T + 2) := by\n    have ht2 : (0 : \u211d) < T + 2 := by linarith\n    rw [Real.le_log_iff_exp_le ht2]\n    have he := Real.exp_one_lt_d9.le\n    linarith\n  have hFAlt : FAlt T \u2264 FAlt (Real.sqrt (n : \u211d) + 1) := by\n    unfold FAlt\n    have hT0' : (0 : \u211d) < T := hT0\n    have hlog1 : (1 : \u211d) \u2264 Real.log T := by\n      have ht2 : (0 : \u211d) < T := hT0\n      rw [Real.le_log_iff_exp_le hT0]\n      have he := Real.exp_one_lt_d9.le\n      nlinarith [hTn, hsqrt316, he]\n    have hA : T * Real.log T \u2264 (Real.sqrt (n : \u211d) + 1) * Real.log (Real.sqrt (n : \u211d) + 1) := by\n      have h1 := Real.log_le_log hT0 (by linarith [hTn1] : T \u2264 Real.sqrt (n : \u211d) + 1)\n      exact mul_le_mul hTn1 h1 (Real.log_nonneg hlog1) (by linarith [hsqrt0])\n    have hB : Real.log T \u2264 Real.log (Real.sqrt (n : \u211d) + 1) :=\n      Real.log_le_log hT0 (by linarith [hTn1])\n    have hC : Real.log (Real.log T) \u2264 Real.log (Real.log (Real.sqrt (n : \u211d) + 1)) := by\n      apply Real.log_le_log (by linarith [hlog1] : (0 : \u211d) < Real.log T)\n      have h1 : (1 : \u211d) \u2264 Real.log (Real.sqrt (n : \u211d) + 1) := by\n        have ht2 : (0 : \u211d) < Real.sqrt (n : \u211d) + 1 := by positivity\n        rw [Real.le_log_iff_exp_le ht2]\n        have he := Real.exp_one_lt_d9.le\n        nlinarith [hsqrt316, he]\n      have h2 : Real.log T \u2264 Real.log (Real.sqrt (n : \u211d) + 1) := hB\n      linarith [h2, hlog1]\n    have hconst : (0 : \u211d) \u2264 1 / (2 * Real.pi) := by positivity\n    nlinarith [hA, hB, hC, hconst]\n  have hX : 4 * T * ((2 / 3) * Z2hat T + 10.5 * Real.log (T + 2) + 48) \u2264\n      4 * (Real.sqrt (n : \u211d) + 1) *\n        ((2 / 3) * Z2hat (Real.sqrt (n : \u211d) + 1) + 10.5 * Real.log (Real.sqrt (n : \u211d) + 1 + 2) + 48) := by\n    have hZ := Z2hat_mono hT1 hTn1\n    have hlog := Real.log_le_log (by linarith : (0 : \u211d) < T + 2)\n      (by linarith [hTn1] : T + 2 \u2264 Real.sqrt (n : \u211d) + 1 + 2)\n    have hXle : (2 / 3) * Z2hat T + 10.5 * Real.log (T + 2) + 48 \u2264\n        (2 / 3) * Z2hat (Real.sqrt (n : \u211d) + 1) + 10.5 * Real.log (Real.sqrt (n : \u211d) + 1 + 2) + 48 := by\n      nlinarith [hZ, hlog]\n    have hX0' : (0 : \u211d) \u2264 (2 / 3) * Z2hat T + 10.5 * Real.log (T + 2) + 48 := by\n      have h1 := Z2hat_nonneg hT1\n      nlinarith [hlogT1]\n    have h4 : T * ((2 / 3) * Z2hat T + 10.5 * Real.log (T + 2) + 48) \u2264\n        (Real.sqrt (n : \u211d) + 1) *\n          ((2 / 3) * Z2hat (Real.sqrt (n : \u211d) + 1) + 10.5 * Real.log (Real.sqrt (n : \u211d) + 1 + 2) + 48) :=\n      mul_le_mul hTn1 hXle hX0' (by linarith [hsqrt0])\n    nlinarith [h4]\n  have hHhat : Hhat T \u2264 Hhat (Real.sqrt (n : \u211d) + 1) := by\n    unfold Hhat\n    have hC := C0hat_mono hT1 hTn1\n    have hC2 : C0hat T ^ 2 \u2264 C0hat (Real.sqrt (n : \u211d) + 1) ^ 2 :=\n      pow_le_pow_left\u2080 hC0T0.le hC 2\n    have hlog := Real.log_le_log (by linarith : (0 : \u211d) < T + 2)\n      (by linarith [hTn1] : T + 2 \u2264 Real.sqrt (n : \u211d) + 1 + 2)\n    nlinarith [hC2, hlog]\n  -- the Bn unfold\n  have hBn : Bn FAlt (n : \u211d) = 4 * FAlt (Real.sqrt (n : \u211d) + 1) +\n      4 * (Real.sqrt (n : \u211d) + 1) *\n        ((2 / 3) * Z2hat (Real.sqrt (n : \u211d) + 1) + 10.5 * Real.log (Real.sqrt (n : \u211d) + 1 + 2) + 48) +\n      40 * 1.3352 + 2 * (Real.exp 4 + 1) * (7 * Hhat (Real.sqrt (n : \u211d) + 1)) +\n      2 * DogLegRight (n : \u211d) + IIMaj (n : \u211d) + 1 := by\n    unfold Bn\n    rfl\n  rw [hBn]\n  -- the IIMaj absorption: (1/2\u03c0)\u00b7e^{1/3}\u00b7IIMaj \u2264 IIMaj\n  have hIIMajAbs : (1 / (2 * Real.pi)) * (Real.exp (1 / 3 : \u211d) * IIMaj (n : \u211d)) \u2264\n      IIMaj (n : \u211d) := by\n    have h1 := mul_le_mul_of_nonneg_right he13' hIIMaj0\n    calc (1 / (2 * Real.pi)) * (Real.exp (1 / 3 : \u211d) * IIMaj (n : \u211d))\n        = ((1 / (2 * Real.pi)) * Real.exp (1 / 3 : \u211d)) * IIMaj (n : \u211d) := by ring\n      _ \u2264 1 * IIMaj (n : \u211d) := h1\n      _ = IIMaj (n : \u211d) := by ring\n  -- the Hhat absorption: (1/2\u03c0)\u00b7Y \u2264 Y for Y \u2265 0\n  have hHhatAbs : (1 / (2 * Real.pi)) * (2 * (Real.exp 4 + 1) * (7 * Hhat T)) \u2264\n      2 * (Real.exp 4 + 1) * (7 * Hhat T) := by\n    have hY : (0 : \u211d) \u2264 2 * (Real.exp 4 + 1) * (7 * Hhat T) := by\n      have h1 : (0 : \u211d) \u2264 7 * Hhat T := by nlinarith [hHhat0]\n      positivity\n    have h1 := mul_le_mul_of_nonneg_right h2\u03c0le hY\n    rwa [one_mul] at h1\n  have hDLRAbs : (1 / (2 * Real.pi)) * (2 * DogLegRight (n : \u211d)) \u2264 2 * DogLegRight (n : \u211d) := by\n    have h1 := mul_le_mul_of_nonneg_right h2\u03c0le hDLR0\n    rwa [one_mul] at h1\n  have hXAbs : (1 / (2 * Real.pi)) * (4 * T *\n        ((2 / 3) * Z2hat T + 10.5 * Real.log (T + 2) + 48) + 40 * 1.3352) \u2264\n      4 * T * ((2 / 3) * Z2hat T + 10.5 * Real.log (T + 2) + 48) + 40 * 1.3352 := by\n    have h1 := mul_le_mul_of_nonneg_right h2\u03c0le hX0\n    rwa [one_mul] at h1\n  -- the mid-bound pinch absorption: (1/2\u03c0)\u00b72(e\u2074+1)(NI+5.5R) \u2264 (1/2\u03c0)\u00b72(e\u2074+1)\u00b77\u00b7H\u0302\n  have hPinchAbs : (1 / (2 * Real.pi)) * (2 * (Real.exp 4 + 1) *\n        (2 * Z2hat T * Real.log (148.5 * C0hat (Real.sqrt (n : \u211d) + 2)) +\n          (3 - -5 / 2) * (10.5 * Real.log (T + 2) + 48))) \u2264\n      (1 / (2 * Real.pi)) * (2 * (Real.exp 4 + 1) * (7 * Hhat T)) := by\n    have h1 := mul_le_mul_of_nonneg_left hpinch (by positivity :\n      (0 : \u211d) \u2264 (1 / (2 * Real.pi)) * (2 * (Real.exp 4 + 1)))\n    convert h1 using 1\n    \u00b7 ring\n    \u00b7 ring\n  -- the giant assembly\n  have hnormRect : \u2016RectangleIntegral' (Jlag n) (rectZ T) (rectW n T)\u2016 \u2264\n      (1 / (2 * Real.pi)) *\n        (\u2016HIntegral (Jlag n) (-5 / 2) (2 * Real.sqrt (n : \u211d)) (-T)\u2016 +\n          \u2016HIntegral (Jlag n) (-5 / 2) (2 * Real.sqrt (n : \u211d)) T\u2016 +\n          \u2016VIntegral (Jlag n) (2 * Real.sqrt (n : \u211d)) (-T) T\u2016 +\n          \u2016VIntegral (Jlag n) (-5 / 2) (-T) T\u2016) := by\n    rw [hnorm]\n    have h1 := mul_le_mul_of_nonneg_left htri (by positivity : (0 : \u211d) \u2264 1 / (2 * Real.pi))\n    exact h1\n  -- |Sf n| \u2264 \u2016RectInt'\u2016 + \u2016\u03a3\u2016, now everything slots in\n  nlinarith [hsf, hnormRect, hVl, hVr, hH, hbridge, hIIMajAbs, hHhatAbs,\n    hDLRAbs, hXAbs, hPinchAbs, hFAlt, hX, hHhat, hIIMaj0, hDLR0, hHhat0, hX0]\n\nend NguyenRH1"
}
#2305 assistant 2026-08-26 20:25:13.620 UTC · 967→2879 tok open ⤢
thinking (6671 chars)
Six issues: 1. **1338**: `rw [div_le_iff₀ (Real.exp_pos 1), one_mul]` — goal `1 / Real.exp 1 ≤ 0.36789` — div_le_iff₀ : a/b ≤ c ↔ a ≤ c·b — with a := 1, b := e, c := 0.36789: goal becomes `1 ≤ 0.36789·e` — then `one_mul` — pattern `1 * ?a` — NOT found (the goal is `1 ≤ 0.36789·e` — the LHS `1` is not `1 * something`). My mistake — div_le_iff₀ gives `a ≤ c * b` = `1 ≤ 0.36789 * e` — no `1 *` anywhere. Remove `one_mul`: `rw [div_le_iff₀ (Real.exp_pos 1)]` then `nlinarith [Real.exp_one_gt_d9]`. The goal: `1 ≤ 0.36789·e` — from e > 2.7182818283: 0.36789·e > 0.36789·2.7182818283 = 1.000030... ≥ 1 ✓ nlinarith (linear: 0.36789·e atom ✓ norm_num computes 0.36789·2.7182818283 ≈ 1.00003 ✓). Wait — nlinarith [Real.exp_one_gt_d9]: Real.exp_one_gt_d9 : 2.7182818283 < exp 1. Goal: 1 ≤ 0.36789 * exp 1. From 0.36789·2.7182818283 ≥ 1: 0.36789 × 2.7182818283 — compute precisely: 0.36789 × 2.7182818283 = 0.36789×2 = 0.73578; ×0.7 = 0.257523; ×0.0182818283 = 0.0067271... — total 0.73578+0.257523+0.006727 = 1.00003 ✓ — nlinarith verifies 0.36789·2.7182818283 ≥ 1 as a decimal identity ✓ should close. 2. **1363 warning**: pinch_horizontal's hTn1 unused?? — wait — hTn1 IS used in (a) (h4 : log(√n+2) ≤ log(T+2)... no wait — (a) uses `by linarith [hTn]` for √n+2 ≤ T+2 — hmm — and hTn1 : T ≤ √n+1 — do I use it in pinch_horizontal? — the pinch is entirely about T (Z2hat T, C0hat T, Hhat T) and the log-arg bound uses √n+2 ≤ T+2 (hTn, not hTn1!). So hTn1 unused in pinch_horizontal ✓ — rename to _hTn1? — but the interface (6-d calls pinch_horizontal hn hTn hTn1 positionally) — renaming to _hTn1 keeps positional compat ✓ do it. 3. **1552**: `Real.exp_one_lt_d9.le : exp 1 ≤ 2.7182818286` vs expected `exp 1 ≤ 2.7183` — my h4 : Real.exp 1 ≤ 2.7183 := Real.exp_one_lt_d9.le — MISMATCH (2.7182818286 ≠ 2.7183). Fix: `have h4 : Real.exp 1 ≤ 2.7182818286 := Real.exp_one_lt_d9.le` and adjust the following norm_num/linarith: goal after rw [h3] was `Real.exp 1 ≤ (1.4:ℝ)^3` — with norm_num computing 1.4^3 = 2.744: goal `exp 1 ≤ 2.744` — from h4 : e ≤ 2.7182818286 ≤ 2.744 ✓ linarith. My code: `norm_num at h4 ⊢` — norm_num at ⊢ would compute 1.4^3 → 2.744 ✓ then `linarith [h4]` — with h4 : e ≤ 2.7182818286 ✓ — but the ERROR was the ascription mismatch itself. Fix the type ascription: `have h4 : Real.exp 1 ≤ 2.7182818286 := Real.exp_one_lt_d9.le`. Hmm wait — actually — `norm_num at h4 ⊢` — norm_num at h4 might rewrite 2.7182818286 — leave as is. Just fix the ascription. 4. **1565**: he13's `rw [div_le_iff₀ h2πpos, one_mul]` — goal `1/(2π)·exp(1/3) ≤ 1` — div_le_iff₀ : a/b ≤ c ↔ a ≤ c·b — hmm — the LHS is `1/(2π) * exp(1/3)` — a PRODUCT, not a single division!! div_le_iff₀'s pattern `?a / ?b ≤ ?c` — my LHS: `(1 / (2π)) * exp(1/3)` — the outer op is * — no match ✓ as reported. Fix: the goal `1/(2π)·exp(1/3) ≤ 1` — rewrite as exp(1/3)/(2π) ≤ 1 first: `rw [mul_comm, ← div_eq_mul_one_div]`?? — hmm — (1/(2π))·e13 = e13/(2π) via: `show Real.exp (1/3) / (2 * Real.pi) ≤ 1`?? — (1/(2π))·e13 vs e13/(2π) = e13·(1/(2π)) ✓ mul_comm-ish: `rw [mul_comm, div_eq_mul_one_div]`?? — div_eq_mul_one_div : a/b = a·(1/b) — e13/(2π) = e13·(1/(2π)) — so e13·(1/(2π)) = e13/(2π) — ← direction. Let me: `rw [show (1 / (2 * Real.pi)) * Real.exp (1/3 : ℝ) = Real.exp (1/3 : ℝ) / (2 * Real.pi) by rw [div_eq_mul_one_div, mul_comm], div_le_iff₀ h2πpos, one_mul]; exact he13` — hmm — div_eq_mul_one_div : a/b = a*(1/b) — rw [div_eq_mul_one_div] on goal `(1/(2π))·e13 = e13/(2π)` — pattern ?a/?b — matches RHS's e13/(2π)?? — rewrites it to e13·(1/(2π)) — then goal (1/(2π))·e13 = e13·(1/(2π)) — mul_comm closes?? — rw [mul_comm] — rewrites the FIRST product — (1/(2π))·e13 → e13·(1/(2π)) — goal becomes e13·(1/(2π)) = e13·(1/(2π)) ✓ rfl. Messy but works. SIMPLER: state he13' differently — as `Real.exp (1/3) / (2 * Real.pi) ≤ 1`?? — but I use it in hIIMajAbs as ((1/2π)·e13)·IIMaj ≤ 1·IIMaj — mul_le_mul_of_nonneg_right (he13' : (1/2π)·e13 ≤ 1) — needs the (1/2π)·e13 form. Alternative: prove he13' via `div_le_one_of_le (h : a ≤ b) (hb : 0 < b)?` — hmm — or `div_le_iff₀` with the division form: prove he13'/aux: exp(1/3)/(2π) ≤ 1 by rw [div_le_iff₀ h2πpos, one_mul]; exact he13 — then convert to the product form in hIIMajAbs: ```lean have he13' : Real.exp (1 / 3 : ℝ) / (2 * Real.pi) ≤ 1 := by rw [div_le_iff₀ h2πpos, one_mul] exact he13 ``` and in hIIMajAbs: `have h1 := mul_le_mul_of_nonneg_right he13' hIIMaj0` — h1 : (e13/(2π))·IIMaj ≤ 1·IIMaj — and the calc's step `_ ≤ 1 * IIMaj := h1` — the calc has `((1/(2π))·e13)·IIMaj` — vs h1's `(e13/(2π))·IIMaj` — mismatch — fix the calc's middle form to (e13/(2π))·IIMaj: `(1/(2π))·(e13·IIMaj) = (e13/(2π))·IIMaj := by ring`?? — (1/(2π))·(e13·IIMaj) vs (e13/(2π))·IIMaj — e13/(2π) = e13·(1/(2π)) — (1/(2π))·e13·IIMaj vs e13·(1/(2π))·IIMaj — commuted ✓ ring ✓. So: ```lean have hIIMajAbs : (1 / (2 * Real.pi)) * (Real.exp (1 / 3 : ℝ) * IIMaj (n : ℝ)) ≤ IIMaj (n : ℝ) := by have h1 := mul_le_mul_of_nonneg_right he13' hIIMaj0 calc (1 / (2 * Real.pi)) * (Real.exp (1 / 3 : ℝ) * IIMaj (n : ℝ)) = (Real.exp (1 / 3 : ℝ) / (2 * Real.pi)) * IIMaj (n : ℝ) := by ring _ ≤ 1 * IIMaj (n : ℝ) := h1 _ = IIMaj (n : ℝ) := by ring ``` 5. **1603**: `Real.log_nonneg hlog1` — hlog1 : 1 ≤ Real.log T — log_nonneg expects 1 ≤ T. My hA used `(Real.log_nonneg hlog1)` intending `0 ≤ log T` — hlog1 : 1 ≤ log T — so log_nonneg needs 1 ≤ T (not 1 ≤ log T). Fix: `Real.log_nonneg (by linarith [hTn, hsqrt316] : (1:ℝ) ≤ T)` — T ≥ 316 ≥ 1 ✓. In hA: `mul_le_mul hTn1 h1 (Real.log_nonneg ...) (by linarith [hsqrt0])` — the third arg: 0 ≤ log T := Real.log_nonneg hT1ge — provide hT1ge : 1 ≤ T := by linarith [hTn, hsqrt316]. Hmm — hA is in hFAlt: `exact mul_le_mul hTn1 h1 (Real.log_nonneg hlog1) (by linarith [hsqrt0])` — change to `(Real.log_nonneg (by linarith [hTn, hsqrt316] : (1 : ℝ) ≤ T))`. 6. **1612/1614/1616 timeouts** — deterministic timeout at isDefEq / tactic execution — at lines 1612-1616 — what's there? Probably the hX/hHhat or the final assembly region... 1612 — let me figure out: the file's §6 starts ~1330; 1612 is around hHhat or hBn. The isDefEq timeout at 1612:48 — col 48 — hmm — likely `unfold Bn; rfl` or one of the `unfold` calls exploding (unfold IIMaj positivity?). Line 1612 — let me count: 1330 (C0hat_le_self) + pinch ~80 + sfBoundProp ~... — 1612 is likely `have hBn : Bn FAlt (n : ℝ) = ... := by unfold Bn; rfl` region or hHhat. The timeout at isDefEq — the `rfl` after unfold Bn?? — the let-expression's defeq check exploding — OR `unfold DogLegRight IIMaj; positivity` in hDLR0 — unfold of a def with rpow — positivity traversing — hmm. Let me get the line contents at 1610-1620.
Six issues. Checking what's at lines 1610–1620 (the timeout region) before fixing.
tool: Read
input
{
  "file_path": "D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiSideBounds.lean",
  "offset": 1606,
  "limit": 24
}
#2306 assistant 2026-08-26 20:27:43.566 UTC · 3809→3850 tok open ⤢
thinking (8834 chars)
The timeouts are at 1612 (`nlinarith [hsqrt316, he]`), 1614 (`linarith [h2, hlog1]`), 1616 (hA' line or 1617's nlinarith). Line 1612's nlinarith: goal `Real.exp 1 ≤ Real.sqrt (n:ℝ) + 1` — from hsqrt316 (316 ≤ √n) and he (e ≤ 2.7182818286): e ≤ 2.72 ≤ 317 ≤ √n+1 ✓ — nlinarith TIMED OUT?? — that's a trivial linear fact! Why would nlinarith timeout there... — unless the isDefEq timeout is from `rw [Real.le_log_iff_exp_le ht2]` producing a goal whose form makes nlinarith choke. The goal after rw: `Real.exp 1 ≤ Real.sqrt (n:ℝ) + 1` — atoms: exp 1, √n — linear ✓ — hmm — isDefEq timeout at 1612:48 — col 48 — the `he` argument position?? — the `he := Real.exp_one_lt_d9.le` — its type: exp 1 ≤ 2.7182818286 — hmm. Hmm — wait — 1612:48 is the isDefEq timeout, 1614:6 tactic-execution timeout, 1616:4 tactic-execution timeout. The FIRST timeout (1612 isDefEq) might poison the rest (cascade). The isDefEq timeout could be nlinarith's preprocessing trying to unify something... — nlinarith calls ring_nf which does isDefEq checks — on WHAT here? The goal `exp 1 ≤ √↑n + 1` — tiny. UNLESS the context is HUGE (dozens of haves with big types — nlinarith's hypothesis collection... — nlinarith only uses the listed haves + context?? — nlinarith uses the LISTED haves AND tries products of hypotheses from the context?? — no — nlinarith [list] uses list + nothing else? — actually nlinarith ALSO considers hypotheses in the local context (all of them!) for products — with the huge context (hsf, hnormRect?? no — this is hFAlt's inner h1 — the context at that point has ~20 haves, some with massive types (hTwide, hHhat... no — this is inside hFAlt at line 1606 — the context has hnR..hTwide, hTle, hsf, hbridge(the calc!), hVl, hVr, hH, hnorm, htri, hpinch, he13, h2πpos, h2πle, he13', hIIMaj0, hDLR0, hC0T0, hHhat0, hX0, hsqrt0, hlogT1 — the bridge calc has a huge type — nlinarith's atomization of all context hypotheses could explode on those big types!! Actually — the isDefEq timeout at 1612 — nlinarith [hsqrt316, he] — with the giant context — nlinarith's preprocessing (ring_nf on all context hyps for atom collection?) — nlinarith considers pairwise products of context hypotheses too!! With huge types that explodes. FIX: use `linarith` (not nlinarith) for these trivially-linear goals — linarith also scans context but is much cheaper... — actually linarith also preprocesses context. Hmm — the safest: make the tiny sub-proofs use explicit `have`-chains with linarith and FEW facts, or restructure hFAlt to have smaller context. But hFAlt's context is genuinely big. Hmm — wait — actually — is the timeout REALLY context-size? The pinch (with a smaller context) worked... — the pinch's nlinarith calls also had ~15 haves in context but smaller types. The hFAlt region's context includes hsf (huge), hbridge (huge calc), hH, hnorm, htri — massive types. Standard fix: `clear` the irrelevant hypotheses before the expensive nlinarith calls, OR use `linarith only [list]` — `linarith only [h1, h2]` restricts linarith to the listed haves + local variables (NO context scan!) ✓ — use `linarith only [...]` for the small steps. Or nlinarith only [...]? — does nlinarith support `only`? — yes: `nlinarith only [h1, h2]`. Let me apply `only` to the nlinarith/linarith calls in the big-context regions (hFAlt, hX, hHhat, and the final assembly — the final nlinarith has an explicit list of ~20 haves — nlinarith scans context TOO on top of the list?? — the docs: nlinarith [h1, h2] uses the listed haves AND the local context hypotheses. With the giant context + 20 listed, that's a lot of products. Use `nlinarith only [the list]` for the final too.) Hmm — but wait — the final assembly NEEDS the context variables (T, n are variables, fine) but not the other haves. `nlinarith only [hsf, hnormRect, hVl', hVr', hH', hbridge, hIIMajAbs, hHhatAbs, hDLRAbs, hXAbs, hPinchAbs, hFAlt, hX, hHhat, hIIMaj0, hDLR0, hHhat0, hX0]` ✓. Also 1614: `linarith [h2, hlog1]` — goal `Real.log (Real.log T) ≤ Real.log (Real.log (√n+1))` — wait no — line 1614 is inside hC: after the `apply Real.log_le_log ...`, the remaining goal: `Real.log T ≤ Real.log (√n+1)` — with h2 := hB (exactly that!) — `linarith [h2, hlog1]` — should be `exact h2`!! — the goal after apply Real.log_le_log (pos) is `Real.log T ≤ Real.log (√n+1)` — h2 : Real.log T ≤ Real.log (√n+1) ✓ exact h2. But I wrote linarith — which scanned the context and timed out. Change to `exact h2`. And 1612's nlinarith → `nlinarith only [hsqrt316, he]` — or better a plain linarith only. Hmm — the goal `Real.exp 1 ≤ √↑n + 1` — from he : exp 1 ≤ 2.7182818286 and hsqrt316 : 316 ≤ √n: e ≤ 2.72 ≤ 317 ✓ — `linarith only [hsqrt316, he]`. And 1616-1617: `have hA' := mul_le_mul_of_nonneg_left hA hconst; nlinarith [hA', hB, hC]` — the nlinarith at 1617 → `nlinarith only [hA', hB, hC]` — hmm — does it need Real.pi? — the goal: (1/2π)·(T·log T) + 0.137·log T + 0.443·loglog T + const ≤ (1/2π)·(T'·log T') + 0.137·log T' + 0.443·loglog T' + const ✓ from hA' hB hC ✓ linarith only [hA', hB, hC] ✓ linear ✓ use linarith only. Also — the `Real.exp_one_lt_d9.le` fix (issue 3): change `have h4 : Real.exp 1 ≤ 2.7183 := Real.exp_one_lt_d9.le` to the correct bound. Wait — actually — hmm — `norm_num at h4 ⊢` with h4 : exp 1 ≤ 2.7182818286 — norm_num at ⊢ computes (1.4)^3 → 2.744 — goal `exp 1 ≤ 2.744` — then linarith [h4] — e ≤ 2.7183 ≤ 2.744 ✓ — with `linarith only [h4]`. And norm_num at h4 — leave (norm_num on h4's 2.7182818286 — harmless). Actually — simplify: `have h4 : Real.exp 1 ≤ 2.7182818286 := Real.exp_one_lt_d9.le; norm_num1 at ⊢; linarith only [h4]` — hmm — norm_num1 at ⊢ computes 1.4^3 = 2.744 ✓. Also — issue 1 (1338): remove `one_mul` from the rw — `rw [div_le_iff₀ (Real.exp_pos 1)]; nlinarith [Real.exp_one_gt_d9]` — hmm — also potential context-scan (in C0hat_le_self the context is small ✓ fine). Also — the `norm_num at h4 ⊢` pattern in h2's block (he13's h1): `have h2 : (Real.exp (1/3))^3 ≤ (1.4)^3 := by have h3 := ...; rw [h3]; have h4 : Real.exp 1 ≤ 2.7182818286 := Real.exp_one_lt_d9.le; norm_num at h4 ⊢; linarith [h4]` — the linarith [h4] with small context (inside he13's block — context is large again!) — `linarith only [h4]`. And issue 2: pinch_horizontal's hTn1 → _hTn1. And issue 4 (1565): he13' fix (the div form). And issue 5 (1603): Real.log_nonneg (1 ≤ T) fix. Let me also preemptively add `only` to the other large-context nlinarith/linarith calls in sfBoundProp's body (hFAlt's hA/hB/hC steps, hX's steps, hHhat, hBn?, hIIMajAbs rings (ring doesn't scan context ✓ fine), hHhatAbs/hDLRAbs/hXAbs (rwa — fine), the final nlinarith). Let me be systematic: in sfBoundProp, replace every `nlinarith [list]` and `linarith [list]` in the big-context zone with `nlinarith only [list]` / `linarith only [list]` where the goal is genuinely linear in the listed atoms. Which ones: - hTle's `linarith [hTn1]` — small context? — at that point context is moderate — but let me add only everywhere downstream of the bridge calc (the context bloats after hsf/hbridge). - hFAlt block: all its linarith/nlinarith → only-versions. - hX block: nlinarith [hZ, hlog] → nlinarith only [hZ, hlog]; nlinarith [hlogT1] → only; nlinarith [h4] → only. - hHhat: nlinarith [hC2, hlog] → only. - hHhat0's nlinarith [hC0T0.le, hL1] → only (context large). - hX0's nlinarith [hT0.le] → hmm — hX0's goal: 0 ≤ 4T·X(T) + 40·1.3352 — from h1 (Z2hat ≥ 0), h2 (log ≥ 0), T ≥ 0: 4·T·X ≥ 0 — NONLINEAR (T·X product!) — nlinarith only [hT0.le, h1, h2] — needs the product T·X ≥ 0 — nlinarith pairwise ✓. - he13's nlinarith [h1, h5] → nlinarith only [h1, h5]. - hlogT1's nlinarith [hTn, hsqrt316, he] → only. - hFAlt's hlog1's nlinarith [hTn, hsqrt316, he] → only. - Final: nlinarith only [big list]. - Also hTle's linarith [hTn1] → linarith only [hTn1] (safe). - The hX0 in the hX-block (hX0'): nlinarith [hlogT1] → only. - Also `hC0T0`'s linarith — small — skip (before the bloat). - hHhat0's nlinarith [hC0T0.le, hL1] → only. Hmm — also the `nlinarith [hsqrt316, hTn]` for hT1 (early, small context — fine). Also — WAIT — potential problem: `linarith only [list]` does NOT use local hypotheses outside the list — so I must make sure every needed fact is IN the list. Let me carefully re-audit each goal I'm changing... — I'll do the edits surgically. Also — h4's in hX's block: `nlinarith [h4]` — goal 4·T·X ≤ 4·T'·X' from h4 : T·X ≤ T'·X' — mul by 4 — linear ✓ `nlinarith only [h4]` — hmm — 4·(T·X) vs 4·T·X — association ✓ linarith normalizes ✓. Let me now write all the edits. First the six errors: 1. 1338: remove `, one_mul` from the rw chain. 2. 1363: hTn1 → _hTn1 in pinch_horizontal. 3. 1552: h4's ascription → 2.7182818286. 4. 1565: he13' restructure (div form) + hIIMajAbs calc form. 5. 1603: Real.log_nonneg arg fix. 6. timeouts: `only` everywhere in the big zone.
Timeouts are context-scan blowups — `linarith only`/`nlinarith only` needed throughout the big zone. Applying all fixes.
tool: Edit
input
{
  "replace_all": false,
  "file_path": "D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiSideBounds.lean",
  "old_string": "  have hinv : (1 : \u211d) / Real.exp 1 \u2264 0.36789 := by\n    rw [div_le_iff\u2080 (Real.exp_pos 1), one_mul]\n    nlinarith [Real.exp_one_gt_d9]",
  "new_string": "  have hinv : (1 : \u211d) / Real.exp 1 \u2264 0.36789 := by\n    rw [div_le_iff\u2080 (Real.exp_pos 1)]\n    nlinarith [Real.exp_one_gt_d9]"
}
tool: Edit
input
{
  "replace_all": false,
  "file_path": "D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiSideBounds.lean",
  "old_string": "theorem pinch_horizontal {n : \u2115} (hn : 100000 \u2264 n) {T : \u211d}\n    (hTn : Real.sqrt (n : \u211d) \u2264 T) (hTn1 : T \u2264 Real.sqrt (n : \u211d) + 1) :",
  "new_string": "theorem pinch_horizontal {n : \u2115} (hn : 100000 \u2264 n) {T : \u211d}\n    (hTn : Real.sqrt (n : \u211d) \u2264 T) (_hTn1 : T \u2264 Real.sqrt (n : \u211d) + 1) :"
}
tool: Edit
input
{
  "replace_all": false,
  "file_path": "D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiSideBounds.lean",
  "old_string": "        rw [h3]\n        have h4 : Real.exp 1 \u2264 2.7183 := Real.exp_one_lt_d9.le\n        norm_num at h4 \u22a2\n        linarith [h4]\n      exact le_of_pow_le_pow_left\u2080 (by norm_num : (3 : \u2115) \u2260 0) (by norm_num : (0 : \u211d) \u2264 1.4) h2\n    have h5 : (3 : \u211d) \u2264 Real.pi := Real.pi_gt_three.le\n    nlinarith [h1, h5]",
  "new_string": "        rw [h3]\n        have h4 : Real.exp 1 \u2264 2.7182818286 := Real.exp_one_lt_d9.le\n        norm_num1 at \u22a2\n        linarith only [h4]\n      exact le_of_pow_le_pow_left\u2080 (by norm_num : (3 : \u2115) \u2260 0) (by norm_num : (0 : \u211d) \u2264 1.4) h2\n    have h5 : (3 : \u211d) \u2264 Real.pi := Real.pi_gt_three.le\n    nlinarith only [h1, h5]"
}
#2307 assistant 2026-08-26 20:29:13.943 UTC · 1285→724 tok open ⤢
thinking (1603 chars)
Now the timeout fixes — replace linarith/nlinarith with `only` versions in the big-context zone of sfBoundProp. Let me find all of them. From the file (hFAlt block, hX block, hHhat, hHhat0, hX0, hlogT1, the he13 second nlinarith, final assembly). Let me list the spots from my written code: In sfBoundProp: 1. hT1/hT2/hT0's `nlinarith [hTn]` etc. — early, moderate context — leave. 2. hTle's `linarith [hTn1]` — leave (moderate). 3. hHhat0's `nlinarith [hC0T0.le, hL1]` — after hbridge (context bloats) → only. 4. hX0's `nlinarith [hT0.le]` → only [hT0.le, h1, h2]. 5. hlogT1's (the shared one after hsqrt0): `nlinarith [hTn, hsqrt316, he]` → only. 6. hFAlt's hlog1: `nlinarith [hTn, hsqrt316, he]` → only. 7. hFAlt's hC's inner nlinarith at 1612 → only [hsqrt316, he]; 1614's `linarith [h2, hlog1]` → `exact h2`. 8. hFAlt's final `nlinarith [hA', hB, hC]` → only. 9. hX's hXle `nlinarith [hZ, hlog]` → only; hX0' `nlinarith [hlogT1]` → only; final `nlinarith [h4]` → only. 10. hHhat's `nlinarith [hC2, hlog]` → only. 11. final `nlinarith [hsf, ...]` → only. Wait — I should double check which of these are past the hbridge/hH "big type" points. The context bloats progressively. Safest: add `only` to ALL nlinarith/linarith in sfBoundProp's body from hHhat0 onward. Also — hTle is BEFORE hsf/hbridge — moderate — but let me still convert the downstream ones. Actually — hmm — one more consideration: `linarith only [h]` — does `only` exist for linarith? Yes — `linarith only [h1, h2]` is standard syntax. And `nlinarith only [...]` ✓. Let me apply the edits. I'll grep for the line numbers of each.
Now the `only` conversions in the big-context zone.
tool: Bash
input
{
  "command": "cd \"D:/Python/Quantum\" && grep -n \"nlinarith \\|linarith \\[\" research/lean/NguyenRH1/NguyenRH1/LiSideBounds.lean | sed -n '1,80p' | awk -F: '$1 > 1500'",
  "description": "List linarith/nlinarith calls after line 1500"
}
#2308 assistant 2026-08-26 20:36:08.632 UTC · 2368→5254 tok open ⤢
thinking (10413 chars)
Now I can see the failing spots: 1. **1630**: hX0' : `0 ≤ (2/3)·Z2hat T + 10.5·log(T+2)+48` — `nlinarith only [hlogT1]` — hlogT1 : 1 ≤ log(T+2) — but h1 := Z2hat_nonneg hT1 : 0 ≤ Z2hat T is NOT in the only-list!! I dropped h1 by converting to `only`. Fix: `nlinarith only [hlogT1, h1]`. 2. **1671**: `rwa [one_mul] at h1` — h1 := mul_le_mul_of_nonneg_right h2πle hDLR0 : (1/(2π))·(2·DLR) ≤ 1·(2·DLR) — `rwa [one_mul] at h1` — rewrites `1 * ?a` → ?a — the RHS 1·(2·DLR) — hmm — one_mul pattern `1 * ?a` — the RHS of h1 is `1 * (2 * DogLegRight ↑n)` — should match — "assumption failed" at 1671:4 — rwa = rw + assumption — the rw [one_mul] happened?? and then assumption failed — meaning h1's rewritten type doesn't match the goal. The goal: `(1/2π)·(2·DLR) ≤ 2·DLR` — h1 after one_mul: `(1/2π)·(2·DLR) ≤ 2·DLR` — should match... — hmm — wait — mul_le_mul_of_nonneg_right (hbc : b ≤ c) (ha : 0 ≤ a) : b * a ≤ c * a — with b := 1/(2π), c := 1, a := 2·DLR: (1/(2π))·(2·DLR) ≤ 1·(2·DLR) ✓ — one_mul at h1 : rewrites `1 * (2·DLR)` → `2·DLR` ✓ then assumption vs goal `(1/2π)·(2·DLR) ≤ 2·DLR` — hmm — h1's form: `(1/(2·π)) * (2 * DogLegRight ↑n)` vs the goal's — the error says assumption FAILED. Maybe the one_mul rw didn't fire (pattern 1 * ?a — is there one? RHS `1 * (2·DLR)` ✓)... — if rw failed, rwa errors "pattern not found" not "assumption failed". It says assumption failed — so rw fired and h1 became `(1/2π)·(2·DLR) ≤ 2·DLR`... — hmm — but the goal might have `(1/(2·π)) * (2·DLR)` with the (2·DLR) associated differently? — my goal: `(1 / (2 * Real.pi)) * (2 * DogLegRight (n : ℝ)) ≤ 2 * DogLegRight (n : ℝ)` — h1 after one_mul: `(1/(2π)) * (2·DLR) ≤ 2·DLR` — IDENTICAL?? — unless the multiplication association inside h1's LHS is `(1/(2π)) * (2 * DLR)` where `2 * DLR` elaborated as `HMul.hMul 2 DLR` vs goal's `2 * DogLegRight ↑n` — same. Hmm — OR — the one_mul rewrote the WRONG `1 *`: h1's type: `1 / (2 * Real.pi) * (2 * DogLegRight ↑n) ≤ 1 * (2 * DogLegRight ↑n)` — one_mul pattern `1 * ?a` — FIRST match: could it match inside `1 / (2 * Real.pi)`?? — `1 / (2π)` is HDiv, not HMul — no `1 *` there. It rewrites the RHS `1 * (2·DLR)` ✓ → `2·DLR`. Then assumption: goal `(1/(2π))·(2·DLR) ≤ 2·DLR` — h1 now `(1/(2π))·(2·DLR) ≤ 2·DLR` — should close!!! Unless... — wait — maybe the elaborated h1 LHS has `1 / (2 * Real.pi) * (2 * DogLegRight ↑n)` parsed as `(1 / (2 * Real.pi)) * (2 * DogLegRight ↑n)` — and the GOAL has the same ✓ — hmm — OR the issue: mul_le_mul_of_nonneg_right's conclusion is `(1/(2π)) * (2·DLR) ≤ 1 * (2·DLR)` — hmm — actually WAIT — maybe h2πle has type `1 / (2 * Real.pi) ≤ 1` — mul_le_mul_of_nonneg_right h2πle hDLR0 : `1 / (2 * Real.pi) * (2 * DogLegRight ↑n) ≤ 1 * (2 * DogLegRight ↑n)`?? — b * a ≤ c * a with b := 1/(2π), a := (2·DLR) — the conclusion: `b * a ≤ c * a` = `(1/(2π)) * (2·DLR) ≤ 1 * (2·DLR)` ✓ — one_mul at h1 — hmm. Actually — let me reconsider: maybe the problem is that `rwa [one_mul] at h1` rewrote and the result is `(1/(2π)) * (2·DLR) ≤ 2·DLR` but the GOAL has `(1/(2π)) * (2 * DogLegRight (n:ℝ))` — with the term `2 * DogLegRight (n:ℝ)` — hmm — and h1's LHS `1 / (2 * Real.pi) * (2 * DogLegRight (n:ℝ))` — these should be syntactically identical... Unless the elaboration of `(1 / (2 * Real.pi)) * (2 * DogLegRight (n : ℝ))` in the GOAL vs in h1 differ in the multiplication grouping? — `1 / (2 * Real.pi) * (2 * DogLegRight ↑n)` — in h1 (from the lemma conclusion), the grouping is `(1 / (2 * Real.pi)) * (2 * DogLegRight ↑n)` — b * a where b := 1/(2π), a := 2·DLR — SAME as goal ✓. Hmm — wait — maybe the error is at line 1671 col 4 — `rwa [one_mul] at h1` — col 4 is the `rwa` — "assumption failed" — maybe the rw [one_mul] rewrote ALL `1 * ?a` including... — hmm — one_mul rewrites the first found instance and ALL its copies. If there's only the RHS one ✓. Hold on — let me reconsider: maybe mul_le_mul_of_nonneg_right's a is NOT `2·DLR` — hDLR0 : 0 ≤ DogLegRight ↑n — h2πle : 1/(2π) ≤ 1 — mul_le_mul_of_nonneg_right h2πle hDLR0 : `(1/(2π)) * DogLegRight ↑n ≤ 1 * DogLegRight ↑n` — NOT the 2·DLR version!! — mul_le_mul_of_nonneg_right (hbc : b ≤ c) (ha : 0 ≤ a) : b * a ≤ c * a — ha : 0 ≤ a with a := DogLegRight ↑n (from hDLR0's type!) — so h1 : (1/(2π))·DLR ≤ 1·DLR — but my GOAL: (1/(2π))·(2·DLR) ≤ 2·DLR — MISMATCH!! — h1 is about DLR, goal about 2·DLR. `rwa [one_mul] at h1` — rewrites h1's RHS to DLR — h1 : (1/(2π))·DLR ≤ DLR — assumption vs goal (1/(2π))·(2DLR) ≤ 2DLR — FAILS ✓ THAT's the bug. The intended: (1/(2π))·(2·DLR) ≤ 2·DLR ⟸ ((1/(2π))·2)·DLR ≤ 1·(2·DLR)?? — hmm — (1/(2π))·(2·DLR) vs 2·DLR: factor: (1/(2π))·(2·DLR) = ((1/(2π))·2)·DLR ≤ 1·(2·DLR)? — mul_le_mul_of_nonneg_right needs b ≤ c with a := 2·DLR and 0 ≤ 2·DLR — hDLR0 : 0 ≤ DLR — NOT 0 ≤ 2·DLR!! — I passed hDLR0 (0 ≤ DLR) where the a should be 2·DLR (0 ≤ 2·DLR). The lemma instantiated a := DLR — mismatch with my goal a := 2·DLR. FIX: provide h2DLR0 : 0 ≤ 2·DogLegRight ↑n := by positivity?? — from hDLR0 — `have h2DLR0 : (0:ℝ) ≤ 2 * DogLegRight (n:ℝ) := by nlinarith only [hDLR0]` (linear ✓) or `mul_nonneg (by norm_num) hDLR0` ✓ — then mul_le_mul_of_nonneg_right h2πle h2DLR0 : (1/(2π))·(2·DLR) ≤ 1·(2·DLR) ✓ rwa [one_mul]. Same bug pattern in hHhatAbs and hXAbs: - hHhatAbs: hY : 0 ≤ 2·(e⁴+1)·(7·Hhat T) — I DID provide hY correctly (via nlinarith only [hHhat0] + positivity) ✓ — mul_le_mul_of_nonneg_right h2πle hY : (1/(2π))·(2(e⁴+1)(7Hhat)) ≤ 1·(2(e⁴+1)(7Hhat)) ✓ — the a here is `2·(e⁴+1)·(7·Hhat T)` — hY's type matches ✓ — and the goal: (1/(2π))·(2(e⁴+1)·(7·Hhat T)) ≤ 2(e⁴+1)·(7·Hhat T) ✓ matches after one_mul ✓ — this one is CORRECT. - hXAbs: `mul_le_mul_of_nonneg_right h2πle hX0` — hX0 : 0 ≤ 4·T·X + 40·1.3352 — the goal's a := 4·T·X + 40·1.3352 ✓ matches ✓ CORRECT. - hIIMajAbs: mul_le_mul_of_nonneg_right he13' hIIMaj0 — hIIMaj0 : 0 ≤ IIMaj — a := IIMaj — h1 : (e13/(2π))·IIMaj ≤ 1·IIMaj ✓ — goal's middle step: (e13/(2π))·IIMaj ≤ 1·IIMaj ✓ CORRECT. So only hDLRAbs is wrong. Fix with h2DLR0. 3. **1685/1681**: hPinchAbs's `convert h1 using 1; · ring; · ring` — the first `ring` at 1685:6 — "ring_nf made no progress" — and 1681:69 "unsolved goals case e'_4" — hmm — 1681:69 is in the STATEMENT of hPinchAbs (col 69 — the `(7 * Hhat T))` region?). The convert produced goal(s) that ring can't close: h1 : ((1/(2π))·(2(e⁴+1)))·(NI+5.5R) ≤ ((1/(2π))·(2(e⁴+1)))·(7·Hhat T) — my hPinchAbs statement: (1/(2π))·(2(e⁴+1)·(NI+5.5R)) ≤ (1/(2π))·(2(e⁴+1)·(7·Hhat T)) — the association: ((1/(2π))·2(e⁴+1))·X vs (1/(2π))·(2(e⁴+1)·X) — convert using 1 should produce the association goal(s) — `ring` should close (pure associativity/commutativity of multiplication) — "ring_nf made no progress" — meaning convert DIDN'T produce an equality goal ring understands — maybe convert closed everything by defeq (association is defeq!) and the remaining goal is... — "unsolved goals case e'_4" at 1681 — hmm — convert left a goal `case e'_4` — a metavariable case? — maybe convert produced a goal of the form `NI+5.5R = NI'+5.5R'`?? — if hpinch's LHS (NI + 5.5R form) doesn't syntactically match my statement's (NI + 5.5R form) — hpinch : 2·Z2hat·log148.5 + (3−−5/2)·R ≤ 7·Hhat T — my hPinchAbs's inner LHS: (2·Z2hat·log148.5 + (3−−5/2)·R) ✓ same ✓. Hmm — "case e'_4" — a congr case — convert's decomposition produced an unexpected subgoal. Let me avoid convert: build hPinchAbs via explicit mul_le_mul with the exact association: ```lean have hPinchAbs : (1 / (2 * Real.pi)) * (2 * (Real.exp 4 + 1) * (2 * Z2hat T * Real.log (148.5 * C0hat (Real.sqrt (n : ℝ) + 2)) + (3 - -5 / 2) * (10.5 * Real.log (T + 2) + 48))) ≤ (1 / (2 * Real.pi)) * (2 * (Real.exp 4 + 1) * (7 * Hhat T)) := by have h1 : (0 : ℝ) ≤ (1 / (2 * Real.pi)) * (2 * (Real.exp 4 + 1)) := by positivity have h2 := mul_le_mul_of_nonneg_left hpinch h1 -- h2 : ((1/2π)·2(e⁴+1)) · (NI+5.5R) ≤ ((1/2π)·2(e⁴+1)) · (7·Hhat T) convert h2 using 1 · ring · ring ``` Same thing — the convert produced "case e'_4" — an eta-expansion case?? Hmm. Alternative — avoid convert: state h2's conclusion in MY form by rewriting first: ```lean have h2 := mul_le_mul_of_nonneg_left hpinch h1 rw [show (1 / (2 * Real.pi)) * (2 * (Real.exp 4 + 1)) * (2 * Z2hat T * Real.log (148.5 * C0hat (Real.sqrt (n : ℝ) + 2)) + (3 - -5 / 2) * (10.5 * Real.log (T + 2) + 48)) = (1 / (2 * Real.pi)) * (2 * (Real.exp 4 + 1) * (2 * Z2hat T * Real.log (148.5 * C0hat (Real.sqrt (n : ℝ) + 2)) + (3 - -5 / 2) * (10.5 * Real.log (T + 2) + 48))) by ring] at h2 rw [show (1 / (2 * Real.pi)) * (2 * (Real.exp 4 + 1)) * (7 * Hhat T) = (1 / (2 * Real.pi)) * (2 * (Real.exp 4 + 1) * (7 * Hhat T)) by ring] at h2 exact h2 ``` Clunky but deterministic. OR — cleaner — prove the association-agnostic version: skip the factorization: state hPinchAbs as the RAW h2 form ((1/2π)·2(e⁴+1))·X, and let the FINAL nlinarith handle associations via ring_nf (it normalizes products ✓!). Actually — the final assembly is nlinarith only — the monomial normalization by ring_nf handles ANY association ✓ — so I can state hPinchAbs as `((1/(2π))·(2(e⁴+1)))·(NI+5.5R) ≤ ((1/(2π))·(2(e⁴+1)))·(7·Hhat T)` := mul_le_mul_of_nonneg_left hpinch h1 — DIRECTLY, no convert! The final nlinarith only then sees the fact in whatever association and normalizes ✓. Let me do that — hPinchAbs := mul_le_mul_of_nonneg_left hpinch h1 (no conversion). 4. **1702**: the final nlinarith only failed — "linarith failed to find a contradiction". Possibly because hPinchAbs was broken (sorry'd, but TYPE correct — nlinarith uses the type ✓ should be fine?) — hmm — or genuinely insufficient/malformed. OR the hBn rw — line 1685's "ring_nf made no progress" at 1685:6 — that was hPinchAbs's ring. And 1681:69 "unsolved goals case e'_4" — also hPinchAbs. With hPinchAbs broken, the final nlinarith lacked the pinch fact (sorry'd hPinchAbs still has the right TYPE — nlinarith only [list] with hPinchAbs in the list — its TYPE is the claimed bound — sorry-backed facts still typecheck in the list ✓ — so the final failure is INDEPENDENT... — unless the final nlinarith's failure is genuinely due to a logical gap or an elaboration issue from the earlier errors. Let me fix 1630 (add h1), 1671 (h2DLR0), hPinchAbs (direct form), and re-run — then diagnose the final nlinarith if it still fails (with everything else clean, its error display will show the goal).
Four fixes: hX0' missing `h1` in the only-list; hDLRAbs wrong `a` (needs `0 ≤ 2·DLR`); hPinchAbs convert → direct form (final ring_nf handles association).
tool: Edit
input
{
  "replace_all": false,
  "file_path": "D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiSideBounds.lean",
  "old_string": "    have hX0' : (0 : \u211d) \u2264 (2 / 3) * Z2hat T + 10.5 * Real.log (T + 2) + 48 := by\n      have h1 := Z2hat_nonneg hT1\n      nlinarith only [hlogT1]",
  "new_string": "    have hX0' : (0 : \u211d) \u2264 (2 / 3) * Z2hat T + 10.5 * Real.log (T + 2) + 48 := by\n      have h1 := Z2hat_nonneg hT1\n      nlinarith only [hlogT1, h1]"
}
tool: Edit
input
{
  "replace_all": false,
  "file_path": "D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiSideBounds.lean",
  "old_string": "  have hDLRAbs : (1 / (2 * Real.pi)) * (2 * DogLegRight (n : \u211d)) \u2264 2 * DogLegRight (n : \u211d) := by\n    have h1 := mul_le_mul_of_nonneg_right h2\u03c0le hDLR0\n    rwa [one_mul] at h1",
  "new_string": "  have hDLRAbs : (1 / (2 * Real.pi)) * (2 * DogLegRight (n : \u211d)) \u2264 2 * DogLegRight (n : \u211d) := by\n    have h2DLR0 : (0 : \u211d) \u2264 2 * DogLegRight (n : \u211d) := by positivity\n    have h1 := mul_le_mul_of_nonneg_right h2\u03c0le h2DLR0\n    rwa [one_mul] at h1"
}
tool: Edit
input
{
  "replace_all": false,
  "file_path": "D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiSideBounds.lean",
  "old_string": "  -- the mid-bound pinch absorption: (1/2\u03c0)\u00b72(e\u2074+1)(NI+5.5R) \u2264 (1/2\u03c0)\u00b72(e\u2074+1)\u00b77\u00b7H\u0302\n  have hPinchAbs : (1 / (2 * Real.pi)) * (2 * (Real.exp 4 + 1) *\n        (2 * Z2hat T * Real.log (148.5 * C0hat (Real.sqrt (n : \u211d) + 2)) +\n          (3 - -5 / 2) * (10.5 * Real.log (T + 2) + 48))) \u2264\n      (1 / (2 * Real.pi)) * (2 * (Real.exp 4 + 1) * (7 * Hhat T)) := by\n    have h1 := mul_le_mul_of_nonneg_left hpinch (by positivity :\n      (0 : \u211d) \u2264 (1 / (2 * Real.pi)) * (2 * (Real.exp 4 + 1)))\n    convert h1 using 1\n    \u00b7 ring\n    \u00b7 ring",
  "new_string": "  -- the mid-bound pinch absorption: (1/2\u03c0)\u00b72(e\u2074+1)(NI+5.5R) \u2264 (1/2\u03c0)\u00b72(e\u2074+1)\u00b77\u00b7H\u0302\n  have hPinchAbs : ((1 / (2 * Real.pi)) * (2 * (Real.exp 4 + 1))) *\n      (2 * Z2hat T * Real.log (148.5 * C0hat (Real.sqrt (n : \u211d) + 2)) +\n        (3 - -5 / 2) * (10.5 * Real.log (T + 2) + 48)) \u2264\n      ((1 / (2 * Real.pi)) * (2 * (Real.exp 4 + 1))) * (7 * Hhat T) :=\n    mul_le_mul_of_nonneg_left hpinch (by positivity :\n      (0 : \u211d) \u2264 (1 / (2 * Real.pi)) * (2 * (Real.exp 4 + 1)))"
}
#2309 assistant 2026-08-26 20:39:22.599 UTC · 633→3991 tok open ⤢
thinking (8087 chars)
Only the final nlinarith (line 1700) fails now. Everything else elaborated. I need to see why the final assembly is insufficient. Let me trace the logical chain once more, VERY carefully, checking every hypothesis type against what the chain needs. Goal (after rw [hBn]): `|Sf ↑n| ≤ 4·FAlt(√n+1) + 4(√n+1)·X(√n+1) + 40·1.3352 + 2(e⁴+1)·7·Hhat(√n+1) + 2·DLR + IIMaj + 1` where X(t) = (2/3)·Z2hat t + 10.5·log(t+2)+48. Facts in the only-list: - hsf : |Sf n| ≤ ‖RectInt'‖ + ‖Σ‖ - hnormRect : ‖RectInt'‖ ≤ (1/2π)·(‖H−‖ + ‖H+‖ + ‖Vr‖ + ‖Vl‖) - hVl' : (1/2π)·‖Vl‖ ≤ (1/2π)·(4T·X(T) + 40·1.3352) - hVr' : (1/2π)·‖Vr‖ ≤ (1/2π)·(e13·IIMaj) - hH' : (1/2π)·(‖H+‖+‖H−‖) ≤ (1/2π)·(2·((e⁴+1)·(NI+5.5R)+DLR)) - hbridge : ‖Σ‖ ≤ 4·FAlt T + 1 - hIIMajAbs : (1/2π)·(e13·IIMaj) ≤ IIMaj - hHhatAbs : (1/2π)·(2(e⁴+1)·7·Hhat T) ≤ 2(e⁴+1)·7·Hhat T - hDLRAbs : (1/2π)·(2·DLR) ≤ 2·DLR - hXAbs : (1/2π)·(4T·X+40M0) ≤ 4T·X+40M0 - hPinchAbs : ((1/2π)·2(e⁴+1))·(NI+5.5R) ≤ ((1/2π)·2(e⁴+1))·(7·Hhat T) - hFAlt : FAlt T ≤ FAlt(√n+1) - hX : 4T·X(T) ≤ 4(√n+1)·X(√n+1) - hHhat : Hhat T ≤ Hhat(√n+1) - hIIMaj0, hDLR0, hHhat0, hX0 (nonnegs) The chain: |Sf| ≤ ‖RectInt'‖ + ‖Σ‖ ≤ (1/2π)(‖H−‖+‖H+‖+‖Vr‖+‖Vl‖) + (4F+1) [hsf, hnormRect, hbridge] ≤ (1/2π)(2((e⁴+1)(NI+5.5R)+DLR) + e13·IIMaj + (4TX+40M0)) + (4F+1) [hH', hVr', hVl'] The (1/2π)(‖H−‖+‖H+‖+‖Vr‖+‖Vl‖) distributes: (1/2π)‖H−‖ + (1/2π)‖H+‖ + (1/2π)‖Vr‖ + (1/2π)‖Vl‖. hH' bounds (1/2π)(‖H+‖+‖H−‖) — the DISTRIBUTED form is (1/2π)‖H+‖ + (1/2π)‖H−‖ — linarith's ring_nf connects (1/2π)·(A+B) ↔ (1/2π)A + (1/2π)B ✓ should be fine. = (1/2π)·2(e⁴+1)(NI+5.5R) + (1/2π)·2DLR + (1/2π)(e13·IIMaj) + (1/2π)(4TX+40M0) + 4F+1 hPinchAbs's LHS: ((1/2π)·2(e⁴+1))·(NI+5.5R) — and the distributed term from hH' is (1/2π)·(2·((e⁴+1)·(NI+5.5R)+DLR)) = (1/2π)·(2(e⁴+1)(NI+5.5R)) + (1/2π)·(2·DLR) — hPinchAbs's LHS form ((1/2π)·2(e⁴+1))·(NI+5.5R) vs (1/2π)·(2(e⁴+1)(NI+5.5R)) — ring_nf normalizes ✓ fine. ≤ (1/2π)·2(e⁴+1)·7·Hhat T + 2·DLR + IIMaj + (4TX+40M0) + 4F+1 [hPinchAbs, hDLRAbs, hIIMajAbs, hXAbs] ≤ 2(e⁴+1)·7·Hhat T + 2·DLR + IIMaj + 4TX+40M0 + 4F+1 [hHhatAbs] ≤ 2(e⁴+1)·7·Hhat(√n+1) + 2·DLR + IIMaj + 4(√n+1)·X(√n+1) + 40M0 + 4·FAlt(√n+1) + 1 [hHhat, hX, hFAlt] = Bn ✓ This looks airtight... — unless one of the haves' STATEMENTS doesn't match what I think. Possible mismatches: (i) hH's actual type (hIntegral_Jlag_full_two_sides): `‖H(+)‖ + ‖H(−)‖ ≤ 2·((e⁴+1)·(2·Z2hat T·log(148.5·C0hat(√n+2)) + (3−−5/2)·(10.5·log(T+2)+48)) + DogLegRight ↑n)` ✓ — and hH' = mul_le_mul_of_nonneg_left hH (0 ≤ 1/2π) : (1/2π)·(‖H+‖+‖H−‖) ≤ (1/2π)·(2·(...)) ✓. (ii) hVr's type: ‖VIntegral (Jlag n) (2·√n) (−T) T‖ ≤ e13·IIMaj ✓. (iii) hVl's type: ‖VIntegral (Jlag n) (−5/2) (−T) T‖ ≤ 4T·((2/3)·Z2hat T + 10.5·log(T+2)+48) + 40·1.3352 ✓. (iv) hnormRect's RHS: (1/2π)·(‖H−‖ + ‖H+‖ + ‖Vr‖ + ‖Vl‖) ✓. Hmm — the nlinarith might be failing because the products (1/2π)·X are treated as atoms (1/2π) times atom X — the distribution/normalization should hold... — OR — the issue: nlinarith's ring_nf might be UNABLE to normalize terms containing `Real.exp (1/3)`, `Real.pi`, divisions — the divisions: 1/(2·π) — ring_nf handles division by turning into inverse-power?? — ring_nf on `1 / (2 * Real.pi)` — normalizes to `(2*π)⁻¹` — and products like `(1/(2π))·e13` → `(2π)⁻¹·e13` ✓ fine — the ABSORPTION facts (hIIMajAbs etc.) have already-factored forms ✓. Hmm — what about `Hhat T` vs `Hhat (√n+1)` — different atoms ✓ hHhat relates them ✓. `Z2hat T` vs `Z2hat(√n+1)` ✓ via hX. `FAlt T` vs `FAlt(√n+1)` ✓ via hFAlt. `C0hat(√n+2)` inside log148.5 — that atom appears in hpinchAbs's NI term — and nowhere else — it just gets bounded via hPinchAbs ✓. Wait — actually — hmm — let me recheck the hHhatAbs statement: `(1/(2π))·(2·(e⁴+1)·(7·Hhat T)) ≤ 2·(e⁴+1)·(7·Hhat T)` ✓. And hPinchAbs : ((1/(2π))·2(e⁴+1))·(NI+5.5R) ≤ ((1/(2π))·2(e⁴+1))·(7·Hhat T) — so (1/2π)·2(e⁴+1)·(NI+5.5R) ≤ (1/2π)·2(e⁴+1)·7·Hhat T — chain to hHhatAbs ✓. Hmm — everything checks. So why does nlinarith fail? Possibilities: (a) A statement form mismatch — e.g., hnormRect's distribution: (1/2π)·(‖H−‖ + ‖H+‖ + ‖Vr‖ + ‖Vl‖) — the hH' needs (1/2π)·‖H+‖ + (1/2π)·‖H−‖ — the DISTRIBUTION of (1/2π)·(‖H−‖+‖H+‖+‖Vr‖+‖Vl‖) into 4 scaled terms — ring_nf does this ✓. (b) nlinarith's preprocessing TIMES OUT silently and reports failure? — it said "linarith failed to find a contradiction" — that's the linarith CORE failing (not timeout) — meaning after preprocessing, the linear problem is UNSOLVABLE with the given facts — a genuine LOGICAL GAP or atom mismatch. Let me look for an atom mismatch: the NI term: hH's inner: `2 * Z2hat T * Real.log (148.5 * C0hat (√↑n + 2)) + (3 - -5/2) * (10.5 * Real.log (T + 2) + 48)`. hpinch's LHS: `2 * Z2hat T * Real.log (148.5 * C0hat (√↑n + 2)) + (3 - -5/2) * (10.5 * Real.log (T + 2) + 48)` — wait — hpinch's statement: `2 * Z2hat T * Real.log (148.5 * C0hat (√n+2)) + (3 - -5/2) * (10.5 * Real.log (T + 2) + 48) ≤ 7 * Hhat T` ✓ SAME. And hPinchAbs = mul_le_mul_of_nonneg_left hpinch ... : ((1/2π)·2(e⁴+1))·(2·Z2hat·log148.5 + (3−−5/2)·R) ≤ ... ✓ consistent. Hmm — and hXAbs : (1/(2π))·(4T·X(T) + 40·1.3352) ≤ 4T·X(T) + 40·1.3352 ✓ and hVl' : (1/(2π))·‖Vl‖ ≤ (1/(2π))·(4T·X(T)+40·1.3352) ✓ chain ✓. WAIT — I see a possible issue: hnormRect's bound: (1/2π)·(‖H−‖ + ‖H+‖ + ‖Vr‖ + ‖Vl‖) — but hVr is about ‖VIntegral (Jlag n) (2·√n) (−T) T‖ and hVl about ‖VIntegral (Jlag n) (−5/2) (−T) T‖ ✓ match htri's forms ✓. Hmm — let me look at the ACTUAL goal once more. After `rw [hBn]` — the goal is `|Sf ↑n| ≤ (Bn-expanded)`. And hsf : |Sf n| ≤ ... — `Sf n` with n : ℕ — and the goal's `Sf ↑n`?? — Sf (n : ℕ) : ℝ — `Sf n` — the goal after intro n hn hN1: `|Sf n| ≤ Bn FAlt ↑n` — Sf n with n : ℕ ✓ — and hsf : |Sf n| ≤ ... ✓ same ✓. Hmm — one more: the hBn expansion — is it CORRECT? Bn's def: `let T := Real.sqrt n + 1; 4 * F T + 4 * T * ((2/3) * Z2hat T + 10.5 * Real.log (T + 2) + 48) + 40 * 1.3352 + 2 * (Real.exp 4 + 1) * (7 * Hhat T) + 2 * DogLegRight n + IIMaj n + 1` — my hBn's RHS: `4 * FAlt (√n+1) + 4*(√n+1)*((2/3)*Z2hat(√n+1) + 10.5*Real.log(√n+1+2) + 48) + 40*1.3352 + 2*(e⁴+1)*(7*Hhat(√n+1)) + 2*DogLegRight ↑n + IIMaj ↑n + 1` — with `Real.log (Real.sqrt (n:ℝ) + 1 + 2)` — Bn's def has `Real.log (T + 2)` with T := √n+1 — so log((√n+1) + 2) = log(√n + 1 + 2) ✓ same ✓. And the hBn rfl proved ✓ (no error at 1651!) ✓. Hmm — so the final goal IS the expanded Bn ✓. Then why does nlinarith fail?? — let me think about what could be UNSOLVABLE. Everything's linear modulo the atom products. Hmm — WAIT — one potential issue: hPinchAbs's RHS: ((1/2π)·2(e⁴+1))·(7·Hhat T) and hHhatAbs's LHS: (1/(2π))·(2·(e⁴+1)·(7·Hhat T)) — ring_nf should normalize both to the same monomial — (2π)⁻¹·2·(e⁴+1)·7·Hhat ✓ same ✓. Hmm — hold on — maybe the problem is the hH' distribution interacts with hPinchAbs: the hH' bound's RHS is (1/(2π))·(2·((e⁴+1)·(NI+5.5R)+DLR)) — distributing: (1/(2π))·(2(e⁴+1)(NI+5.5R) + 2·DLR) = (1/(2π))·(2(e⁴+1)(NI+5.5R)) + (1/(2π))·(2·DLR) — hPinchAbs's LHS: ((1/(2π))·2(e⁴+1))·(NI+5.5R) — normalized: 2·(2π)⁻¹·(e⁴+1)·(NI+5.5R)?? — hmm — (1/(2π))·(2(e⁴+1)(NI+5.5R)) vs ((1/(2π))·2(e⁴+1))·(NI+5.5R) — SAME monomial ✓. Everything normalizes identically. So the failure must be a MISSING link or an atom I overlooked. Let me actually TEST the assembly logic manually: maybe nlinarith can't handle the DISTRIBUTION of (1/(2π)) over the 4-term sum in hnormRect AND the subsequent bounding — it should. Hmm — wait — maybe the issue is `Sf n` vs the perimeter... — no. Let me try a different diagnosis: is it possible that nlinarith only [list] FAILS because one of the listed haves has a TYPE with a metavariable or is otherwise malformed (sorry'd from an earlier error that got fixed but the sorry persists)? — the earlier errors at hPinchAbs/convert — those got fixed — the current run has no other errors — so all haves are clean. Hmm — OR — the `rw [hBn]` happened and the goal is exactly Bn-expanded ✓. Let me look at the failing goal state directly — the error's goal display was truncated in my grep. Let me get the full goal at 1700.
Only the final assembly fails now. Let me see the full goal state at line 1700.
tool: Bash
input
{
  "command": "export PATH=\"$HOME/.elan/bin:$PATH\" && cd \"D:/Python/Quantum/research/lean/NguyenRH1\" && lake env lean NguyenRH1/LiSideBounds.lean 2>&1 | sed -n '/1700:2/,$p' | head -120",
  "description": "Show full final-assembly error context",
  "timeout": 600000
}
#2310 assistant 2026-08-26 20:42:13.124 UTC · 6436→3720 tok open ⤢
thinking (7734 chars)
Now I can see everything. The final nlinarith failed. Let me trace the chain with the EXACT types shown: - hsf : |Sf n| ≤ ‖RectInt'‖ + ‖Σ‖ - hbridge : ‖Σ‖ ≤ 4·FAlt T + 1 - hnormRect : ‖RectInt'‖ ≤ (1/2π)·(‖H(−T)‖ + ‖H(T)‖ + ‖Vr‖ + ‖Vl‖) - hVr' : (1/2π)·‖Vr‖ ≤ (1/2π)·(e13·IIMaj) - hVl' : (1/2π)·‖Vl‖ ≤ (1/2π)·(4T·X(T)+40M0) - hH' : (1/2π)·(‖H(T)‖+‖H(−T)‖) ≤ (1/2π)·(2·((e⁴+1)·(NI+5.5R)+DLR)) - hPinchAbs : (1/2π)·2(e⁴+1)·(NI+5.5R) ≤ (1/2π)·2(e⁴+1)·(7·Hhat T) - hHhatAbs : (1/2π)·(2(e⁴+1)·7·Hhat T) ≤ 2(e⁴+1)·7·Hhat T - hDLRAbs : (1/2π)·(2·DLR) ≤ 2·DLR - hXAbs : (1/2π)·(4T·X+40M0) ≤ 4T·X+40M0 - hIIMajAbs : (1/2π)·(e13·IIMaj) ≤ IIMaj - hFAlt : FAlt T ≤ FAlt(√n+1) - hX : 4T·X(T) ≤ 4(√n+1)·X(√n+1) - hHhat : Hhat T ≤ Hhat(√n+1) Chain: |Sf| ≤ ‖RectInt'‖ + ‖Σ‖ ≤ (1/2π)·(‖H−‖+‖H+‖+‖Vr‖+‖Vl‖) + (4F+1). The horizontal part: (1/2π)·(‖H−‖+‖H+‖) = (1/2π)·‖H−‖ + (1/2π)·‖H+‖ ≤ hH's RHS via hH' : (1/2π)·(‖H+‖+‖H−‖) ≤ (1/2π)·(2·(MID+DLR)) where MID = (e⁴+1)·(NI+5.5R). So: (1/2π)·(‖H−‖+‖H+‖) ≤ (1/2π)·(2·(MID+DLR)) = (1/2π)·(2·MID) + (1/2π)·(2·DLR). (1/2π)·(2·MID) = (1/2π)·2·(e⁴+1)·(NI+5.5R) ≤ (1/2π)·2·(e⁴+1)·7·Hhat T [hPinchAbs] ≤ 2·(e⁴+1)·7·Hhat T [hHhatAbs]. (1/2π)·(2·DLR) ≤ 2·DLR [hDLRAbs]. (1/2π)·‖Vr‖ ≤ (1/2π)·(e13·IIMaj) ≤ IIMaj [hVr', hIIMajAbs]. (1/2π)·‖Vl‖ ≤ (1/2π)·(4TX+40M0) ≤ 4TX+40M0 [hVl', hXAbs]. Total: |Sf| ≤ 2(e⁴+1)·7·Hhat T + 2·DLR + IIMaj + 4TX+40M0 + 4F+1 ≤ 2(e⁴+1)·7·Hhat(√n+1) + 2·DLR + IIMaj + 4(√n+1)·X(√n+1) + 40M0 + 4·FAlt(√n+1) + 1 [hHhat, hX, hFAlt] = Bn ✓ This is airtight LINEAR reasoning. Why does nlinarith fail?! Possible culprit: nlinarith's ring_nf preprocessing might be normalizing the terms in a way that breaks atom-matching. E.g., `Real.exp (1/3)` — ring_nf might expand `exp (1/3)`... no, exp is an atom. `Real.pi` — atom ✓. The DIVISIONS `1 / (2 * Real.pi)` — ring_nf normalizes division to `(2·π)⁻¹`?? — if ring_nf turns `1/(2·π)` into `(2·π)⁻¹`, then the monomial `(1/(2π))·e13·IIMaj` becomes `(2π)⁻¹·e13·IIMaj` ✓ consistent across all haves ✓. Hmm — WAIT. Look at hH''s RHS vs hPinchAbs's LHS once more, VERY carefully: hH' RHS: `1/(2π) * (2 * ((e⁴+1) * (NI + 5.5R) + DLR))` hPinchAbs LHS: `1/(2π) * (2*(e⁴+1)) * (NI + 5.5R)` Expand hH' RHS: (1/(2π))·(2·(e⁴+1)·(NI+5.5R) + 2·DLR) — ring_nf: `2·(2π)⁻¹·(e⁴+1)·NI' + 2·(2π)⁻¹·DLR` where NI' = NI+5.5R. hPinchAbs LHS: `2·(2π)⁻¹·(e⁴+1)·NI'` ✓ MATCHES ✓. hPinchAbs RHS: `2·(2π)⁻¹·(e⁴+1)·(7·Hhat T)` — hHhatAbs LHS: `(2π)⁻¹·(2·(e⁴+1)·(7·Hhat T))` = `2·(2π)⁻¹·(e⁴+1)·7·Hhat T` ✓ MATCHES ✓. hXAbs LHS: `(2π)⁻¹·(4T·X+40M0)` ✓ hVl' RHS ✓. hVr' RHS: `(2π)⁻¹·(e13·IIMaj)` ✓ hIIMajAbs LHS ✓. Everything matches!!! So why?? — hmm — let me look at the goal once more: the goal is the NEGATED form (a✝ : Bn < |Sf n| ⊢ False) — linarith works by contradiction ✓ normal. Hmm — WAIT. What about `Hhat T` vs `Hhat (√n+1)` — hHhat relates ✓. What about `7 * Hhat T` — hPinchAbs RHS has `(7 * Hhat T)` and hHhatAbs has `(7 * Hhat T)` ✓. What about `FAlt T` — hbridge's `4 * FAlt T + 1` and hFAlt's `FAlt T ≤ FAlt (√n+1)` — so 4·FAlt T + 1 ≤ 4·FAlt(√n+1) + 1 ✓ linear ✓. `X(T) = 2/3·Z2hat T + 10.5·log(T+2)+48` — hVl'/hXAbs have `4*T*(2/3*Z2hat T + ...) + 40·1.3352` and hX has the comparison to the √n+1 version ✓. Hmm — everything links. UNLESS — nlinarith is choking on the NORMS: the goal has `|Sf n|` — abs — and hsf has `|Sf n|` — the ABS is an atom ✓ consistent. WAIT — I see it!!! Look at hnormRect's RHS association: `1/(2π) * (‖H(−T)‖ + ‖H(T)‖ + ‖Vr‖ + ‖Vl‖)` — as displayed: ``` 1 / (2 * Real.pi) * (‖HIntegral (Jlag n) (-5 / 2) (2 * √↑n) (-T)‖ + ‖HIntegral (Jlag n) (-5 / 2) (2 * √↑n) T‖ + ‖VIntegral (Jlag n) (2 * √↑n) (-T) T‖ + ‖VIntegral (Jlag n) (-5 / 2) (-T) T‖) ``` The `‖H(−T)‖ + ‖H(T)‖ + ‖Vr‖ + ‖Vl‖` — displayed with `+ ‖Vr‖ +` on a new line indented DEEPER — `A + (B + C)`-style?? — the display shows `‖H(−T)‖ + ‖H(T)‖ + ‖Vr‖ + ‖Vl‖` with the last `+` at lower indent — per the infixl convention this is `((A+B)+C)+D` ✓ left-nested, fine — and htri's RHS: `‖H(−T)‖ + ‖H(T)‖ + ‖Vr‖ + ‖Vl‖` — displayed as `‖H(−T)‖ + ‖H(T)‖ + ‖Vr‖ + ‖Vl‖` with the SAME structure?? — htri's display: ``` ‖HIntegral (Jlag n) (-5 / 2) (2 * √↑n) (-T)‖ + ‖HIntegral (Jlag n) (-5 / 2) (2 * √↑n) T‖ + ‖VIntegral (Jlag n) (2 * √↑n) (-T) T‖ + ‖VIntegral (Jlag n) (-5 / 2) (-T) T‖ ``` Hmm — same shape ✓. hnormRect was PROVED ✓ (no error) — so its statement is fine. OK so the chain is logically airtight but nlinarith fails. The likely real issue: nlinarith's preprocessing is failing to normalize the ABS/NORM-containing and division-containing terms consistently, OR the linarith core's atomization differs between hypotheses (e.g., `Real.exp (1/3 : ℝ)` vs `Real.exp (1/3)` — display shows `Real.exp (1 / 3)` everywhere ✓ same). Hmm — hold on — what about `(3 - -5 / 2)` — in hpinch/hH'/hPinchAbs — displayed as `(3 - -5 / 2)` — an ATOM-level term `3 - -5/2` — ring_nf normalizes to `5.5` — CONSISTENTLY everywhere ✓. What about `40 * 1.3352` — atom product of numerals — normalized to 53.408 everywhere ✓. Hmm hmm. Let me look at the hX statement ONE more time: ``` hX : 4 * T * (2 / 3 * Z2hat T + 10.5 * Real.log (T + 2) + 48) ≤ 4 * (√↑n + 1) * (2 / 3 * Z2hat (√↑n + 1) + 10.5 * Real.log (√↑n + 1 + 2) + 48) ``` ✓. And hVl'/hXAbs: `4 * T * (2/3·Z2hat T + ...) + 40·1.3352` ✓ consistent atoms: `Z2hat T`, `Z2hat (√n+1)`, `Real.log (T+2)`, `Real.log (√n+1+2)` ✓. So the chain should close. The failure might be linarith's ring_nf on `1 / (2 * Real.pi)` — hmm — what if ring_nf CANNOT normalize `1 / (2 * Real.pi)` and treats `1 / (2 * Real.pi)` as an opaque atom DIFFERENTLY across hypotheses?? — it's the same syntactic term everywhere ✓ same atom. OK — different theory: maybe the problem is that nlinarith runs its FULL nonlinear search (products of hypotheses) on this large fact set and blows through some resource bound, reporting failure. nlinarith's default does multiple rounds of products. With ~20 haves, pairwise products ~200, some huge. The error is "linarith failed" not timeout — hmm — but nlinarith reports "linarith failed" when its search concludes unsolvable — after its product rounds. Test: does plain `linarith only [list]` (NOT nlinarith) suffice? The chain is FULLY LINEAR (all the nonlinear facts are pre-packaged). The only products in the goal/hypotheses are atom·atom monomials — which linarith handles as atoms ✓ — the ring_nf normalization is shared. Let me try `linarith only [same list]`. If linarith only fails too, the issue is a genuine atom mismatch — then I'd restructure with an explicit calc (more work but deterministic). Actually — let me ALSO prepare the deterministic calc as the likely-needed route. The calc: ```lean have hsum1 : ‖RectangleIntegral' (Jlag n) (rectZ T) (rectW n T)‖ + ‖Σ‖ ≤ (1 / (2 * Real.pi)) * ((2 * (Real.exp 4 + 1) * (7 * Hhat T)) + 2 * DogLegRight (n:ℝ) + (Real.exp (1/3 : ℝ) * IIMaj (n:ℝ)) + (4 * T * ((2/3) * Z2hat T + 10.5 * Real.log (T + 2) + 48) + 40 * 1.3352)) + (4 * FAlt T + 1) := by -- from hsf's RHS + hnormRect + the per-side bounds nlinarith only [hsf, hnormRect, hVl', hVr', hH', hPinchAbs, hbridge] -- hmm same issue ``` Hmm — maybe the issue is subtler. Let me test the hypothesis that linarith-only (not nlinarith) closes it, since the chain is linear. If linarith only fails, I'll bisect by feeding the facts incrementally in a calc. Actually — WAIT. I just realized — maybe the problem is that some hypothesis in my list has an INTERNAL contradiction-irrelevant issue — like `hHhat0 : 0 ≤ Hhat T` — fine — or hX0/hIIMaj0/hDLR0 — fine. Hmm — OR — the `he13'` in the context — `Real.exp (1/3)/(2·π) ≤ 1` — not in the list (good, unused). Let me just try `linarith only` first.
The chain is airtight on paper — testing whether plain `linarith only` (no nonlinear search) closes it.
tool: Edit
input
{
  "replace_all": false,
  "file_path": "D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiSideBounds.lean",
  "old_string": "  -- |Sf n| \u2264 \u2016RectInt'\u2016 + \u2016\u03a3\u2016, now everything slots in\n  nlinarith only [hsf, hnormRect, hVl', hVr', hH', hbridge, hIIMajAbs, hHhatAbs,\n    hDLRAbs, hXAbs, hPinchAbs, hFAlt, hX, hHhat, hIIMaj0, hDLR0, hHhat0, hX0]",
  "new_string": "  -- |Sf n| \u2264 \u2016RectInt'\u2016 + \u2016\u03a3\u2016, now everything slots in\n  linarith only [hsf, hnormRect, hVl', hVr', hH', hbridge, hIIMajAbs, hHhatAbs,\n    hDLRAbs, hXAbs, hPinchAbs, hFAlt, hX, hHhat, hIIMaj0, hDLR0, hHhat0, hX0]"
}
#2311 assistant 2026-08-26 20:50:37.362 UTC · 258→923 tok open ⤢
thinking (1426 chars)
CLEAN — zero errors/warnings. **6-d (the fold) elaborates green — `sfBoundProp : SfBoundProp` is PROVED.** This is the ii′-6 capstone. Now: header markers, full build, audit, commit. Then update NGUYEN.md checklist + STATE.md per the END protocol... — wait — the session isn't over; the user asked to "continue with the rest of the 6-c, starting with 6-c-ii" — 6-c AND 6-d are now done. The remaining per the plan is ii′-7 (closure: lambda_pos_finite_height_closed with the 3 perimeter hypotheses only, full axiom audit, paper Appendix B + tag upgrade, archive closure). That's a separate, larger stage — I should report 6-c + 6-d completion to the user and ask/note about ii′-7. Actually — let me check the scope: "Let's continue with the rest of the 6-c, starting with 6-c-ii" — the user asked for "the rest of the 6-c" — 6-c-i through 6-c-v. 6-d was in my todo list as part of "the rest of the 6-c" reading (the sub-stage map "6-c-i ✓ → 6-c-ii → 6-c-iii → 6-c-iv → 6-c-v → 6-d fold ⟹ sfBoundProp → ii′-7"). I've now completed through 6-d — a natural major milestone (sfBoundProp PROVED — the entire ii′-6 is done). Let me finish the protocol for 6-d: header comment for §6, full build, audit (all new decls), commit. Then report to the user with the state and the ii′-7 question. New decls in §6: C0hat_le_self, pinch_horizontal, sfBoundProp. Plus the 6-c decls already audited. Audit: add §6 decls to AuditSession041.
**6-d GREEN — `sfBoundProp : SfBoundProp` is PROVED.** Header marker, full build, audit, commit.
tool: Edit
input
{
  "replace_all": false,
  "file_path": "D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiSideBounds.lean",
  "old_string": "\u00a74 (6-c-iv) DONE: the mid-horizontal segment machine [\u22125/2, 3].\n\u00a75 (6-c-v) DONE: the full horizontal assembly (mid + DL-5 tails, both sides).",
  "new_string": "\u00a74 (6-c-iv) DONE: the mid-horizontal segment machine [\u22125/2, 3].\n\u00a75 (6-c-v) DONE: the full horizontal assembly (mid + DL-5 tails, both sides).\n\u00a76 (6-d) DONE: the fold \u2014 `sfBoundProp : SfBoundProp` (six perimeter hyps)."
}
tool: Edit
input
{
  "replace_all": false,
  "file_path": "D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\AuditSession041.lean",
  "old_string": "-- \u00a75 (6-c-v): the full horizontal assembly\n#print axioms hIntegral_Jlag_full_norm_le\n#print axioms hIntegral_Jlag_full_neg_norm_le\n#print axioms hIntegral_Jlag_full_two_sides",
  "new_string": "-- \u00a75 (6-c-v): the full horizontal assembly\n#print axioms hIntegral_Jlag_full_norm_le\n#print axioms hIntegral_Jlag_full_neg_norm_le\n#print axioms hIntegral_Jlag_full_two_sides\n\n-- \u00a76 (6-d): the fold \u2014 sfBoundProp : SfBoundProp\n#print axioms C0hat_le_self\n#print axioms pinch_horizontal\n#print axioms sfBoundProp"
}
#2312 assistant 2026-08-26 20:54:23.045 UTC · 1122→1816 tok open ⤢
thinking (3764 chars)
6-d committed (774fec5). **ii′-6 is COMPLETE** — sfBoundProp : SfBoundProp proved from exactly the six declared perimeter hypotheses. Now — what's the state? Per the plan: - ii′-6 DONE (6-a through 6-d). - Remaining: **ii′-7** (closure: `lambda_pos_finite_height_closed` with the 3 perimeter hypotheses only — wait — the perimeter is now SIX hypotheses (RHUpTo, LiVerifiedUpTo + ZeroCountProp, ZeroCountProp2, ZeroCountGlobalProp, NoLowZeros, MZeroProp) — hmm — the "3 perimeter hypotheses only" refers to the ORIGINAL three (RHUpTo, LiVerifiedUpTo, SfBoundProp) — with SfBoundProp now PROVED from the 6, the final `lambda_pos_finite_height_closed` would take (hRH : RHUpTo) (hfinite : LiVerifiedUpTo) + the 6 analytic/counting Props. Let me check what lambda_pos_finite_height_verified takes: (hRH : RHUpTo 3×10¹²) (hfinite : LiVerifiedUpTo 5×10⁵) (hSf : SfBoundProp). So the closed version: (hRH) (hfinite) + the 6 = 8 hypotheses total, of which RHUpTo/LiVerifiedUpTo are the computational certificates and the other 6 are the declared analytic perimeter (Trudgian N(T)/S(T) forms + Platt-class NoLowZeros + M₀). - ii′-7 also: full axiom audit; paper Appendix B + tag upgrade (D-062 §4 endpoint); archive closure + D-037 line-63 erratum pointer; the paper correction note (F-A both-signs + F-B two-unit + F-D dog-leg + F-E I_I kernel footnote). That's a substantial next stage. The user asked for "the rest of the 6-c" — I've now done 6-c AND 6-d (the whole ii′-6). Time to report and ask about proceeding to ii′-7. Actually — per the pacing directive and the natural milestone, I should report this milestone clearly and ask whether to continue into ii′-7. The session has been long and productive. Let me also update the todos. Actually — before reporting — should I do the immediate assembly `lambda_pos_finite_height_closed`? That's small (instantiate lambda_pos_finite_height_verified with sfBoundProp) — but it's ii′-7 territory per the plan. The user's directive was "the rest of the 6-c" — I've completed it plus 6-d. The natural stop: report and ask. Let me update the todo list and write the summary. Also per the session protocol, I should consider whether to update STATE.md/NGUYEN.md now — those are part of the END protocol. The session isn't necessarily over (user might say continue to ii′-7). But the crash-safe protocol says commit per sub-stage — done (each sub-stage committed). STATE.md updates are part of session END. I'll report the milestone and ask about ii′-7, noting that STATE.md/journal updates will happen at session end (or when the user directs). Actually — hmm — let me reconsider. The user's directive was specifically "the rest of the 6-c". The 6-d fold was the logical completion of the side bounds (without it, the side bounds are inert). Completing ii′-6 is the right milestone. ii′-7 is a NEW stage (closure + paper + audit + errata) that the user should green-light, especially since it touches the paper (freeze protocol lifts at ii′-7). Report: - 6-c complete: 6-c-ii (I_III), 6-c-iii (I_I F-E), 6-c-iv (mid-horizontal arcsinh machine), 6-c-v (horizontal assembly). - 6-d complete: sfBoundProp : SfBoundProp PROVED from exactly the six declared perimeter hypotheses. - Build green 8774 jobs; axiom audits: 3 standard axioms only, 0 sorryAx on everything. - Commits: 6e5a98f, 9bd8ba9, 1fed1df, 5d3dc0a, 774fec5. - What this means: ii′-6 (the L0 assembly) is complete — |Sf n| ≤ Bn FAlt n on [10⁵, N₁ = (3×10¹²−1)²] is now Lean-proved; the ORIGINAL N₁ ≈ 9×10²⁴ preserved (D-074 dog-leg endpoint). - Next: ii′-7 (closure: lambda_pos_finite_height_closed, full axiom audit, paper Appendix B + tag upgrade + the F-A/F-B/F-D/F-E correction note, archive closure + D-037 line-63 erratum pointer). Ask whether to proceed.
**ii′-6 COMPLETE.** The full arc is committed. Final status: ## Milestone: `sfBoundProp : SfBoundProp` PROVED All of ii′-6 is done — five commits this session: | Commit | Sub-stage | Content | |--------|-----------|---------| | `6e5a98f` | 6-c-ii | I_III left-vertical ≤ 4T((2/3)Z₂(T) + 10.5·log(T+2) + 48) + 40·1.3352 | | `9bd8ba9` | 6-c-iii | I_I right-vertical ≤ e^{1/3}·IIMaj (F-E kernel, uniform in t) | | `1fed1df` | 6-c-iv | mid-horizontal segment machine (arcsinh-FTC from scratch) | | `5d3dc0a` | 6-c-v | horizontal assembly (mid + DL-5 dog-leg tails, both sides) | | `774fec5` | 6-d | **the fold: `sfBoundProp : SfBoundProp`** | Build green (8774 jobs); axiom audits: **[propext, Classical.choice, Quot.sound] only, 0 sorryAx** on all 27 new declarations. **What this closes:** |Sf n| ≤ Bn FAlt n on [10⁵, N₁] is now Lean-proved from exactly the six declared perimeter hypotheses (RHUpTo, ZeroCountProp, ZeroCountProp2, ZeroCountGlobalProp, NoLowZeros, MZeroProp) — with **N₁ = (3×10¹²−1)² ≈ 9×10²⁴ PRESERVED** (the D-074 dog-leg endpoint). Notable: the pinch (NI + 5.5R ≤ 7·Ĥ) discharged by a fully elementary log-mono chain — no certified log numerics needed; the bridge needed a variant at T ≤ 3·10¹² (the public T ≤ 3·10¹²−1 hypothesis fails at the top of the range incl. n = N₁ exactly — handled). **Remaining: ii′-7** (closure) — `lambda_pos_finite_height_closed` with the perimeter assembled, full axiom audit, paper Appendix B + tag upgrade + the correction note (F-A both-signs, F-B two-unit, F-D dog-leg, F-E I_I kernel), archive closure + D-037 line-63 erratum pointer. This stage touches the paper (freeze lifts at ii′-7).
tool: AskUserQuestion ok
input
{
  "questions": [
    {
      "question": "ii\u2032-6 is complete (sfBoundProp proved). Proceed to ii\u2032-7 (closure + paper) now?",
      "header": "Next stage",
      "options": [
        {
          "label": "Continue to ii\u2032-7 (Recommended)",
          "description": "Assemble lambda_pos_finite_height_closed, full axiom audit, paper Appendix B + tag upgrade + correction note, archive closure."
        },
        {
          "label": "Stop at this milestone",
          "description": "Run the session-END protocol (journal, STATE.md, NGUYEN.md updates) and stop here; ii\u2032-7 next session."
        },
        {
          "label": "Only the Lean closure",
          "description": "Do just lambda_pos_finite_height_closed + axiom audit now; paper/errata later."
        }
      ]
    }
  ]
}
result 2026-08-26 20:56:24Z → 2026-08-26 20:56:24Z
{
  "questions": [
    "question='ii\u2032-6 is complete (sfBoundProp proved). Proceed to ii\u2032-7 (closure + paper) now?' header='Next stage' options=[QuestionOption(label='Continue to ii\u2032-7 (Recommended)', description='Assemble lambda_pos_finite_height_closed, full axiom audit, paper Appendix B + tag upgrade + correction note, archive closure.', preview=None), QuestionOption(label='Stop at this milestone', description='Run the session-END protocol (journal, STATE.md, NGUYEN.md updates) and stop here; ii\u2032-7 next session.', preview=None), QuestionOption(label='Only the Lean closure', description='Do just lambda_pos_finite_height_closed + axiom audit now; paper/errata later.', preview=None)] multi_select=False"
  ],
  "answers": {
    "ii\u2032-6 is complete (sfBoundProp proved). Proceed to ii\u2032-7 (closure + paper) now?": "Continue to ii\u2032-7 (Recommended)"
  }
}
#2313 assistant 2026-08-26 20:57:19.394 UTC · 2003→953 tok open ⤢
thinking (1959 chars)
The user approved continuing to ii′-7. Let me plan ii′-7's components: **ii′-7 (per NGUYEN.md):** Closure: theorem sfBoundProp : SfBoundProp ✓ DONE (6-d); final `lambda_pos_finite_height_closed` (3 perimeter hypotheses only — now the 8: hRH, hfinite + the 6); full axiom audit; paper Appendix B + tag upgrade (D-062 §4 endpoint); archive closure + D-037 line-63 erratum pointer; the correction note (F-A/F-B/F-D/F-E). Sub-steps: 1. **lambda_pos_finite_height_closed**: instantiate `lambda_pos_finite_height_verified (hRH : RHUpTo 3×10¹²) (hfinite : LiVerifiedUpTo 500000) (hSf : SfBoundProp)` with `sfBoundProp` — the closed theorem takes (hRH) (hfinite) + the 6 perimeter Props as hypotheses. Check lambda_pos_finite_height_verified's exact signature first (LiBasePoints.lean). 2. **Full axiom audit** of the closed theorem + the whole chain (should be 3 standard axioms). 3. **Paper updates** (freeze lifts): paperA.tex — the formal-status paragraph at "3 hypotheses" → update to the new perimeter (8 hypotheses: RHUpTo, LiVerifiedUpTo, + the 6 analytic Props — which are the declared Trudgian N(T)/S(T), Platt-class zero-location, M₀ certificate, and the strengthened counting forms); Appendix B update (the Lean appendix describing SfBoundProp as a named hypothesis → now a proved theorem with the perimeter enumerated); the correction note: F-A (bridge 4F both-signs — the paper's B(n) account says 2F?), F-B (I_III two-unit count (2/3)Z₂), F-D (the dog-leg repair — the right-tail charge), F-E (I_I kernel footnote). Also the D-037 line-63 erratum pointer (archive-side, in research/derivations/D-037-*.md). 4. **Archive closure**: update NGUYEN.md checklist (ii′-6 → complete, ii′-7 → complete as items land), the Result Ledger (R-004 tag upgrade?), STATE.md, journal entry. This is a big stage. Let me start with the Lean closure (small), then the audit, then the paper. First — check lambda_pos_finite_height_verified's exact statement/signature.
Proceeding to ii′-7. First the Lean closure — checking `lambda_pos_finite_height_verified`'s exact signature.
tool: TodoWrite
input
{
  "todos": [
    {
      "content": "6-c + 6-d: four side bounds + fold \u27f9 sfBoundProp",
      "status": "completed",
      "activeForm": "Proving side bounds and fold"
    },
    {
      "content": "ii\u2032-7a: lambda_pos_finite_height_closed assembly (Lean)",
      "status": "in_progress",
      "activeForm": "Assembling lambda_pos_finite_height_closed"
    },
    {
      "content": "ii\u2032-7b: full axiom audit of the closed chain",
      "status": "pending",
      "activeForm": "Running full axiom audit"
    },
    {
      "content": "ii\u2032-7c: paper Appendix B + tag upgrade + correction note (F-A/F-B/F-D/F-E)",
      "status": "pending",
      "activeForm": "Updating paper with closure + corrections"
    },
    {
      "content": "ii\u2032-7d: archive closure (D-037 erratum pointer, NGUYEN.md, STATE.md, journal)",
      "status": "pending",
      "activeForm": "Closing the archive"
    }
  ]
}
tool: Grep ok
input
{
  "pattern": "theorem lambda_pos_finite_height_verified|theorem lambda_pos_finite_height\\b",
  "path": "D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1",
  "output_mode": "content",
  "-B": 4,
  "-A": 12
}
result 2026-08-26 20:57:19Z → 2026-08-26 20:57:19Z
mode='content' num_files=0 filenames=[] content="D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiBasePoints.lean-702-analytic hypothesis (`SfBoundProp`, the traced contour bound) plus the\nD:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiBasePoints.lean-703-declared computational perimeter (Platt–Trudgian `RHUpTo`; the finite Li\nD:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiBasePoints.lean-704-verification `LiVerifiedUpTo`). The two elementary margin evaluations\nD:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiBasePoints.lean-705-are discharged in this file. -/\nresearch\\lean\\NguyenRH1\\NguyenRH1\\LiBasePoints.lean:706:theorem lambda_pos_finite_height_verified\nresearch\\lean\\NguyenRH1\\NguyenRH1\\LiBasePoints.lean-707-    (hRH: RHUpTo (3 * 10 ^ 12)) (hfinite : LiVerifiedUpTo (5 * 10 ^ 5))\nresearch\\lean\\NguyenRH1\\NguyenRH1\\LiBasePoints.lean-708-    (hSf: SfBoundProp) :\nresearch\\lean\\NguyenRH1\\NguyenRH1\\LiBasePoints.lean-709-    ∀ n: ℕ, 1 ≤ n → (n : ℝ) ≤ (3 * 10 ^ 12 - 1 : ℝ) ^ 2 →\nresearch\\lean\\NguyenRH1\\NguyenRH1\\LiBasePoints.lean-710-      0 < (liCoeff n).re:=\nD:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiBasePoints.lean-711-  fun n hn1 hnN => lambda_pos_finite_height hRH hfinite\nD:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiBasePoints.lean-712-    marginTilde_FAlt_80000_pos marginTilde_FAlt_80000_lt_100000 hSf n hn1 hnN\nD:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiBasePoints.lean-713-\nD:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiBasePoints.lean-714-end NguyenRH1\nD:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiBasePoints.lean-715-\nD:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiBasePoints.lean-716-end\n--\nD:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiSkeleton.lean-2795-= margiñ(n) > 0 by convexity (`marginTilde_convex` + `FAlt_psi_neg`)\nD:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiSkeleton.lean-2796-and the two base points (`pos_of_convex_of_two_points`).\nresearch\\lean\\NguyenRH1\\NguyenRH1\\LiSkeleton.lean-2797-`_hRH` documents the bridge input: it is consumed by `SfBoundProp`'s\nD:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiSkeleton.lean-2798-eventual proof (D-037 §6), not by the assembly itself. -/\nresearch\\lean\\NguyenRH1\\NguyenRH1\\LiSkeleton.lean:2799:theorem lambda_pos_finite_height\nresearch\\lean\\NguyenRH1\\NguyenRH1\\LiSkeleton.lean-2800-    (_hRH: RHUpTo (3 * 10 ^ 12)) (hfinite : LiVerifiedUpTo (5 * 10 ^ 5))\nresearch\\lean\\NguyenRH1\\NguyenRH1\\LiSkeleton.lean-2801-    (hbase: 0 < marginTilde FAlt 80000)\nresearch\\lean\\NguyenRH1\\NguyenRH1\\LiSkeleton.lean-2802-    (hbase2: marginTilde FAlt 80000 < marginTilde FAlt 100000)\nresearch\\lean\\NguyenRH1\\NguyenRH1\\LiSkeleton.lean-2803-    (hSf: SfBoundProp) :\nresearch\\lean\\NguyenRH1\\NguyenRH1\\LiSkeleton.lean-2804-    ∀ n: ℕ, 1 ≤ n → (n : ℝ) ≤ (3 * 10 ^ 12 - 1 : ℝ) ^ 2 →\nresearch\\lean\\NguyenRH1\\NguyenRH1\\LiSkeleton.lean-2805-      0 < (liCoeff n).re:= by\nD:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiSkeleton.lean-2806-  intro n hn1 hnN\nD:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiSkeleton.lean-2807-  rcases lt_or_ge n 100000 with hlt | hge\nD:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiSkeleton.lean-2808-  · exact hfinite n hn1 (by omega)\nresearch\\lean\\NguyenRH1\\NguyenRH1\\LiSkeleton.lean-2809-  · have hn20: 20 ≤ n := by omega\nresearch\\lean\\NguyenRH1\\NguyenRH1\\LiSkeleton.lean-2810-    have h1:= sInfty_lower_bound n hn20\nresearch\\lean\\NguyenRH1\\NguyenRH1\\LiSkeleton.lean-2811-    have h2:= hSf n hge hnN" num_lines=33 num_matches=None applied_limit=None applied_offset=None
#2314 assistant 2026-08-26 20:58:39.969 UTC · 2788→1899 tok open ⤢
thinking (4176 chars)
`lambda_pos_finite_height_verified (hRH : RHUpTo (3*10^12)) (hfinite : LiVerifiedUpTo (5*10^5)) (hSf : SfBoundProp) : ∀ n : ℕ, 1 ≤ n → (n:ℝ) ≤ (3*10^12−1)^2 → 0 < (liCoeff n).re` So the closed theorem: ```lean theorem lambda_pos_finite_height_closed (hRH : RHUpTo (3 * 10 ^ 12)) (hfinite : LiVerifiedUpTo (5 * 10 ^ 5)) (hZC : ZeroCountProp) (hZC2 : ZeroCountProp2) (hZCG : ZeroCountGlobalProp) (hNLZ : NoLowZeros) (hMZ : MZeroProp) : ∀ n : ℕ, 1 ≤ n → (n : ℝ) ≤ (3 * 10 ^ 12 - 1 : ℝ) ^ 2 → 0 < (liCoeff n).re := lambda_pos_finite_height_verified hRH hfinite (sfBoundProp hRH hZC hZC2 hZCG hNLZ hMZ) ``` 8 hypotheses total: the two computational certificates (RHUpTo = Platt–Trudgian RH-to-height; LiVerifiedUpTo = finite Li verification) + the six analytic perimeter Props (Trudgian N(T)/S(T) forms: ZeroCountProp unit-window, ZeroCountProp2 two-unit, ZeroCountGlobalProp global; Platt-class NoLowZeros; MZeroProp M₀-certificate). Where to put it? Options: LiSideBounds.lean (it imports LiAssembly + LiDogLeg — but does it import LiBasePoints? No — LiBasePoints is imported by the root). The closed theorem needs lambda_pos_finite_height_verified (LiBasePoints) + sfBoundProp (LiSideBounds). LiSideBounds doesn't import LiBasePoints. Options: (a) add to LiSideBounds with an import of LiBasePoints — check the import graph: LiBasePoints imports LiSkeleton (presumably); LiSideBounds imports LiAssembly, LiDogLeg (which transitively import everything including LiSkeleton). Adding `import NguyenRH1.LiBasePoints` to LiSideBounds — no cycle (LiBasePoints doesn't import LiSideBounds). (b) A new small file LiClosed.lean for ii′-7 — cleaner separation (the closure artifact). Per the project's file-per-stage convention, a new file LiClosed.lean is cleanest. Let me check LiBasePoints' imports to make sure no cycle: LiBasePoints presumably imports LiSkeleton + Mathlib. LiSideBounds imports LiAssembly + LiDogLeg. A new file LiClosed importing LiBasePoints + LiSideBounds — no cycle ✓. Write LiClosed.lean: ```lean /- LiClosed.lean — tier ii′-7: THE CLOSED finite-height Li theorem (R-004 endpoint, D-062 §4). `lambda_pos_finite_height_closed`: λ_n > 0 for all n ≤ N₁ = (3×10¹²−1)² ≈ 9×10²⁴ from the EIGHT declared hypotheses: the two computational certificates (Platt–Trudgian `RHUpTo`, the finite Li verification `LiVerifiedUpTo`) and the six analytic perimeter Props (Trudgian N(T)/S(T): `ZeroCountProp` unit-window, `ZeroCountProp2` two-unit, `ZeroCountGlobalProp` global; the Platt-class zero-location certificate `NoLowZeros`; the M₀ interval certificate `MZeroProp`) — the SfBoundProp analytic hypothesis of sessions 012–040 is now PROVED (ii′-6, LiSideBounds). -/ import Mathlib import NguyenRH1.LiBasePoints import NguyenRH1.LiSideBounds open Zeta23 Complex Filter Topology open scoped Nat noncomputable section namespace NguyenRH1 /-- **THE CLOSED FINITE-HEIGHT LI THEOREM** (R-004, ii′-7 endpoint): λ_n > 0 for ALL 1 ≤ n ≤ N₁ = (3×10¹²−1)² ≈ 9×10²⁴, from the eight declared hypotheses (two computational certificates + six analytic perimeter Props). The session-012 named analytic hypothesis `SfBoundProp` is discharged by ii′-6's `sfBoundProp`. -/ theorem lambda_pos_finite_height_closed (hRH : RHUpTo (3 * 10 ^ 12)) (hfinite : LiVerifiedUpTo (5 * 10 ^ 5)) (hZC : ZeroCountProp) (hZC2 : ZeroCountProp2) (hZCG : ZeroCountGlobalProp) (hNLZ : NoLowZeros) (hMZ : MZeroProp) : ∀ n : ℕ, 1 ≤ n → (n : ℝ) ≤ (3 * 10 ^ 12 - 1 : ℝ) ^ 2 → 0 < (liCoeff n).re := lambda_pos_finite_height_verified hRH hfinite (sfBoundProp hRH hZC hZC2 hZCG hNLZ hMZ) end NguyenRH1 end ``` Then root-import it, build, audit (lambda_pos_finite_height_closed + re-audit sfBoundProp — full chain). Then the paper. Let me also double check `open scoped Nat` isn't needed here — no factorial use — drop it? — the other files have it; harmless to include or drop. Keep minimal: skip `open scoped Nat`. Also `noncomputable section` not needed for a plain theorem — skip. Match house style anyway? — LiBasePoints ends with `end NguyenRH1` + `end`. Keep it simple. Then `import NguyenRH1.LiClosed` in the root NguyenRH1.lean + doc bullet.
`lambda_pos_finite_height_verified` takes exactly the three expected hypotheses. Writing the closure file `LiClosed.lean`.
tool: Write
input
{
  "file_path": "D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiClosed.lean",
  "content": "/-\nLiClosed.lean \u2014 tier ii\u2032-7: THE CLOSED finite-height Li theorem (R-004\nendpoint, D-062 \u00a74). `lambda_pos_finite_height_closed`: \u03bb_n > 0 for all\n1 \u2264 n \u2264 N\u2081 = (3\u00d710\u00b9\u00b2\u22121)\u00b2 \u2248 9\u00d710\u00b2\u2074 from the EIGHT declared hypotheses:\nthe two computational certificates (Platt\u2013Trudgian `RHUpTo`, the finite\nLi verification `LiVerifiedUpTo`) and the six analytic perimeter Props\n(Trudgian N(T)/S(T): `ZeroCountProp` unit-window, `ZeroCountProp2`\ntwo-unit, `ZeroCountGlobalProp` global; the Platt-class zero-location\ncertificate `NoLowZeros`; the M\u2080 interval certificate `MZeroProp`) \u2014\nthe session-012 named analytic hypothesis `SfBoundProp` is now PROVED\n(tier ii\u2032-6, LiSideBounds).\n-/\nimport Mathlib\nimport NguyenRH1.LiBasePoints\nimport NguyenRH1.LiSideBounds\n\nopen Zeta23 Complex Filter Topology\n\nnamespace NguyenRH1\n\n/-- **THE CLOSED FINITE-HEIGHT LI THEOREM** (R-004, ii\u2032-7 endpoint):\n\u03bb_n > 0 for ALL 1 \u2264 n \u2264 N\u2081 = (3\u00d710\u00b9\u00b2\u22121)\u00b2 \u2248 9\u00d710\u00b2\u2074, from the eight\ndeclared hypotheses (two computational certificates + six analytic\nperimeter Props). The session-012 named analytic hypothesis\n`SfBoundProp` is discharged by ii\u2032-6's `sfBoundProp`. -/\ntheorem lambda_pos_finite_height_closed\n    (hRH : RHUpTo (3 * 10 ^ 12)) (hfinite : LiVerifiedUpTo (5 * 10 ^ 5))\n    (hZC : ZeroCountProp) (hZC2 : ZeroCountProp2) (hZCG : ZeroCountGlobalProp)\n    (hNLZ : NoLowZeros) (hMZ : MZeroProp) :\n    \u2200 n : \u2115, 1 \u2264 n \u2192 (n : \u211d) \u2264 (3 * 10 ^ 12 - 1 : \u211d) ^ 2 \u2192\n      0 < (liCoeff n).re :=\n  lambda_pos_finite_height_verified hRH hfinite\n    (sfBoundProp hRH hZC hZC2 hZCG hNLZ hMZ)\n\nend NguyenRH1\n\nend\n"
}
#2315 assistant 2026-08-26 21:03:48.268 UTC · 397→300 tok open ⤢
thinking (180 chars)
Build GREEN (8775 jobs) with LiClosed. Now the full axiom audit (ii′-7b): audit lambda_pos_finite_height_closed + sfBoundProp + the whole new chain. Add to AuditSession041 and run.
Build green (8775 jobs). Now the full axiom audit of the closed chain.
tool: Edit
input
{
  "replace_all": false,
  "file_path": "D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\AuditSession041.lean",
  "old_string": "-- \u00a76 (6-d): the fold \u2014 sfBoundProp : SfBoundProp\n#print axioms C0hat_le_self\n#print axioms pinch_horizontal\n#print axioms sfBoundProp",
  "new_string": "-- \u00a76 (6-d): the fold \u2014 sfBoundProp : SfBoundProp\n#print axioms C0hat_le_self\n#print axioms pinch_horizontal\n#print axioms sfBoundProp\n\n-- ii\u2032-7: THE CLOSED finite-height Li theorem (the full chain)\n#print axioms lambda_pos_finite_height_closed\n#print axioms lambda_pos_finite_height_verified"
}
#2316 assistant 2026-08-26 21:05:21.367 UTC · 1101→247 tok open ⤢
thinking (386 chars)
Committed (a0943a6). Now ii′-7c — the paper. Let me read paperA.tex to find the formal-verification appendix + the formal-status paragraph that needs updating (the "3 hypotheses" claim → now the closed theorem with 8 hypotheses) + where the correction note goes (F-A/F-B/F-D/F-E affect the B(n) account in §3 / the R-004 theorem statement). Let me look at paperA.tex's structure first.
Now the paper (freeze lifts at ii′-7). Reading paperA.tex to find the formal-status paragraph and Appendix B.
tool: Bash
input
{
  "command": "cd \"D:/Python/Quantum\" && grep -n \"hypotheses\\|SfBoundProp\\|appendix\\|Appendix\\|formal\\|Formal\\|Lean\" research/paper/paperA.tex | head -40",
  "description": "Locate formal-status mentions in paperA.tex"
}
#2317 assistant 2026-08-26 21:09:09.208 UTC · 2678→4470 tok open ⤢
thinking (13057 chars)
Now I have the full picture. The paper edits needed (ii′-7c): **1. Abstract (lines 43-50)**: "the theorem is proved from exactly three named hypotheses --- the two computational certificates above and a single named analytic hypothesis, the traced contour bound $|S_f(n)| \le B(n)$ on $10^5 \le n \le (T_0-1)^2$" → the contour bound is now PROVED (ii′-6/7). New form: "proved from exactly eight named hypotheses — the two computational certificates above and six declared analytic inputs (Trudgian's explicit zero-counting bounds in three forms and two interval certificates), with the traced contour bound $|S_f(n)| \le B(n)$ itself machine-proved from those six". **2. Intro paragraph (line 75)**: "machine-verified in Lean~4 modulo one named analytic hypothesis and a declared computational perimeter" → "machine-verified in Lean~4 modulo a declared computational and analytic perimeter (no remaining named hypothesis)". **3. §3 Step 2 B(n) display (lines 199-207)**: the F-A/F-B/F-D/F-E corrections: - `2N(\sqrt n{+}1)` → `4F(T)` (F-A: the bridge is 4F, both-signs; the paper's 2N was the factor-2 slip — D-037 line-63 erratum). - `\tfrac23 C_0(T)` → `\tfrac23 Z_2(T)` (F-B: the two-unit count, D-069 Finding 2). - `2(e^4{+}1)\bigl[7H(T) + 3.3\bigr]` → `2(e^4{+}1)\,7H(T) + 2\,\mathrm{DL}(n)` (F-D: the "3.3" right-tail was broken — Lagarias's (6.12) stepping at T = √n grows ~e^{0.066√n} (E-031); the zero-free dog-leg repair (D-074) replaces the charge by DL(n) = ((√n+1)/2)(e^{5/2}+1)/2 + 8.38 + IIMaj/4, keeping T_cut = √n and hence N₁ unchanged). - `I_I` stays but with the F-E footnote (charged e^{1/3}·IIMaj; the uniform right-vertical kernel is e^{√n/2+1/3}+1, one eighth-power above the t² ≥ n charge; absorbed by the 2π normalization). **4. §3 Step 3 (bridge)**: "so $|1-(1-1/\rho)^n| \le 2$ per zero and $|\lambda_n(T)| \le 2N(T)$" → "$|\lambda_n(T)| \le 4F(T)$ (both signs)" — the bridge runs over both signs of γ. **5. Appendix B (lines 345-378)**: the "second tier is complete in its assembly form" paragraph → "second tier COMPLETE: SfBoundProp is now machine-proved (tier ii′), so the closed theorem lambda_pos_finite_height_closed takes exactly eight named hypotheses (two computational certificates + six analytic perimeter inputs)". The "remaining formalization target (ii′)" paragraph (lines 370-378) → replaced: the ii′ target is COMPLETE (the contour estimate inside SfBoundProp is machine-proved — residue machinery on rectangles, the partial-fraction identity for ζ′/ζ, the tight R-bound, the four side bounds, the fold). **6. The B(n) description text (lines 203-207)**: "C₀(t) the explicit unit-interval zero count" → Z₂ the two-unit count; "H(T) = ..." stays; add DogLegRight definition. Let me also check the paper's theorem-environment count for "conditional on computation" — the theorem is STILL conditional on computation (Platt–Trudgian + Li verifications) — the tag stays [PROVED-CONDITIONAL(computational RH-to-T₀; finite verification)] — the ANALYTIC part is now fully machine-proved. The theorem environment header "conditional on computation" remains correct. Now — how much detail to put in the correction note? The honest approach: a "Correction note" paragraph in §3 (after the B(n) display or in the remark) documenting the four corrections with pointers (D-071/D-072/D-074, E-031/E-032). The B(n) display gets the corrected formula inline. Let me write the edits now. I'll be careful with LaTeX escaping. Edit plan: **Edit A — abstract (lines 43-50):** Replace: ``` Moreover the full reduction is machine-verified in Lean~4 (package \texttt{NguyenRH1}): the theorem is proved from exactly three named hypotheses --- the two computational certificates above and a single named analytic hypothesis, the traced contour bound $|S_f(n)| \le B(n)$ on $10^5 \le n \le (T_0-1)^2$ --- with axiom profile exactly the three standard axioms (\texttt{propext}, \texttt{Classical.choice}, \texttt{Quot.sound}) and no \texttt{sorry} anywhere in the development's Li chain. ``` With: ``` Moreover the full reduction is machine-verified in Lean~4 (package \texttt{NguyenRH1}): the theorem is proved from exactly eight named hypotheses --- the two computational certificates above and six declared analytic inputs (Trudgian's explicit zero-counting bounds in their unit-window, two-unit, and global forms, a Platt-class zero-location certificate, and one interval-arithmetic constant), with the traced contour bound $|S_f(n)| \le B(n)$ on $10^5 \le n \le (T_0-1)^2$ itself machine-proved from those six --- with axiom profile exactly the three standard axioms (\texttt{propext}, \texttt{Classical.choice}, \texttt{Quot.sound}) and no \texttt{sorry} anywhere in the development's Li chain. ``` **Edit B — intro (line 75-76):** "and with its reduction machine-verified in Lean~4 modulo one named analytic hypothesis and a declared computational perimeter" → "and with its full reduction --- including the contour estimate --- machine-verified in Lean~4 modulo a declared perimeter (computational certificates and explicit zero-counting bounds)". **Edit C — Step 2 B(n) display (lines 198-207):** Replace the B(n) formula and description with the corrected one + correction note. New: ``` $|S_f(n)| \le B(n)$ with \[ B(n) = 4F(T) + 4T\bigl[\tfrac23 Z_2(T) + 10.5\log(T{+}2) + 48\bigr] + 40M_0 + 2(e^4{+}1)\,7H(T) + 2\,\mathrm{DL}(n) + I_{\mathrm{Maj}}(n) + 1, \] $T = \sqrt n + 1$, $F$ the honest zero-count majorant (the formal development's mildly enlarged Trudgian shape), $Z_2(t)$ the explicit two-unit-interval zero count (including the $\log\log$ term), $H(T) = 3C_0(T)^2 + 10.5\log(T{+}2) + 48$ with $C_0$ the unit-interval count, $\mathrm{DL}(n) = \tfrac{\sqrt n + 1}{4}(e^{5/2}+1) + 8.38 + \tfrac14 I_{\mathrm{Maj}}(n)$ the zero-free dog-leg charge for the right tail, $I_{\mathrm{Maj}}(n) = 2(\sqrt n+1)(e^{\sqrt n/2}+1)\cdot 0.57 \cdot 2^{-(2\sqrt n - 2)}$ the right-vertical majorant, and the final $1$ the trivial-zero contribution.\footnote{Correction note (2026-08, D-071/D-072/D-074 with E-031/E-032): four adjustments to the $B(n)$ account were made during the machine closure of Step 2, and the displayed formula is the corrected one. (i) The bridge is $4F$, not $2N$: the residue sum runs over both signs of $\gamma$ (a factor-2 accounting slip in the first tracing; the theorem statement is unaffected). (ii) The left-vertical near-sum charges the two-unit count $Z_2$ (its window is $|\gamma - T| \le 1$, two unit intervals), not $C_0$. (iii) The right-tail charge ``$3.3$'' inherited from Lagarias's Thm~6.1 is broken at $T = \sqrt n$: the (6.12) stepping product ratio is $e/2 > 1$, so the horizontal right-tail envelope grows like $e^{0.066\sqrt n}$ ($\sim 6\times 10^{10}$ at $n = 10^5$ measured); the repaired proof replaces that segment by a zero-free dog-leg at height $(3/2)\sqrt n$ (where the stepping is repaired and certified), keeping $T = \sqrt n$ and hence the endpoint $N_1 = (T_0-1)^2$ unchanged. (iv) The right-vertical kernel is charged honestly as $e^{1/3} I_{\mathrm{Maj}}(n)$ (the uniform bound is $e^{\sqrt n/2 + 1/3} + 1$); the factor $e^{1/3} \le 1.4$ is absorbed by the $2\pi$ normalization of the residue theorem. All four are machine-verified in \texttt{NguyenRH1}.} ``` Hmm — this is long for a footnote but appropriate. Also the text after the display mentions "I_I an explicitly bounded boundary-integral term (D-037/D-056)" — covered by the new text. Wait — the original text at lines 203-207: ``` $T = \sqrt n + 1$, $C_0(t)$ the explicit unit-interval zero count (the honest Trudgian form, including the $\log\log$ term), $H(T) = 3C_0(T)^2 + 10.5\log(T{+}2) + 48$, $I_I$ an explicitly bounded boundary-integral term (D-037/D-056) and the final $1$ the trivial-zero contribution. ``` **Edit D — Step 3 bridge (lines 209-213):** ``` \emph{Step 3 (bridge).} For $n \le (T_0-1)^2$ every zero with $|\gamma| \le \sqrt n + 1$ is on the line, so $|1-(1-1/\rho)^n| \le 2$ per zero and $|\lambda_n(T)| \le 2N(T)$; all other contour terms are \emph{unconditional}. ``` → ``` \emph{Step 3 (bridge).} For $n \le (T_0-1)^2$ every zero with $|\gamma| \le \sqrt n + 1$ is on the line, so $|1-(1-1/\rho)^n| \le 2$ per zero and $|\lambda_n(T)| \le 4F(T)$ (the sum runs over both signs of $\gamma$; $F$ the zero-count majorant); all other contour terms are \emph{unconditional}. ``` **Edit E — Appendix B (lines 345-378):** Replace the "second tier" paragraph + "remaining formalization target" paragraph with the closure account: ``` The second tier is complete. The named analytic hypothesis \texttt{SfBoundProp} of the assembly --- the traced contour bound $|S_f(n)| \le B(n)$ --- is machine-proved (tier ii$'$), so the closed headline theorem \texttt{lambda\_pos\_finite\_height\_closed} is machine-proved from exactly eight named hypotheses: the two computational-perimeter inputs of Theorem~\ref{thm:finite-height-li} (Platt--Trudgian; the Li verifications) and six declared analytic inputs --- Trudgian's explicit zero-counting bounds \cite{trudgian2014} in their unit-window, two-unit-window, and global forms (\texttt{ZeroCountProp}, \texttt{ZeroCountProp2}, \texttt{ZeroCountGlobalProp}), a Platt-class zero-location certificate (\texttt{NoLowZeros}), and the short-segment constant $M_0 = 1.3352$ (\texttt{MZeroProp}, an interval-arithmetic certificate in the same trust class). The proof of \texttt{SfBoundProp} contains the full complex-analytic content of Step 2: meromorphic residue machinery on rectangles (\texttt{rectInt'\_Jlag}), the partial-fraction identity for $\zeta'/\zeta$ (\texttt{partialFraction\_identity}), the reference-point bound $|R(s)| \le 10.5\log(|t|{+}2) + 48$ (\texttt{logDeriv\_riemannZeta\_sub\_nearZeros\_le\_tight}), the exact identity $S_f(n) = \mathrm{Res}_1(n)$ (\texttt{sf\_eq\_resW1}), the zero-free dog-leg repair of the right tail (\texttt{upper\_right\_horizontal\_eq\_dogleg} and the \texttt{LiDogLeg} bounds), the four side bounds (\texttt{LiSideBounds}), and the final fold (\texttt{sfBoundProp}). The two elementary evaluations of the explicit margin function that the assembly consumes ($\mathrm{margin}(8{\cdot}10^4) > 0$ and $\mathrm{margin}(10^5) > \mathrm{margin}(8{\cdot}10^4)$) are discharged in the development itself, by certified rational interval bounds on $\log$, $\exp$, and $\sqrt{\cdot}$ (\texttt{LiBasePoints.lean}). The margin majorant in the development is a mildly enlarged Trudgian-shape function (a negative-weight term dropped, at a cost of $\approx 0.9\sqrt n$ in the bound) chosen so that every convexity block has provably negative $\psi$; the axiom profile is unchanged --- the three standard axioms, no \texttt{sorryAx} (audited declaration-by-declaration on the full chain, \texttt{AuditSession041}). The finite-height Li theorem is, by design, \emph{conditional} on computations (Platt--Trudgian's interval arithmetic; the Li-coefficient verifications); the six analytic inputs are explicit published bounds of the same literature class \cite{trudgian2014,platt2021riemann}. The formal reduction leaves exactly those as trusted inputs, which we regard as the honest formal target: every other step of the proof is machine-checked. ``` Also the AI Contribution Statement could add the closure, but that's the user's authorship domain — the statement says "sessions 001--013" — I should NOT edit the AI contribution statement (user's call, per the authorship directive — the statement is their text; updating session ranges is beyond my remit? — hmm — actually the statement references "project ledger: sessions 001--013" which is now stale (we're at 041) and the Lean list. The authorship directive says papers list the user as sole author and never add AI assistants — editing the AI Contribution Statement's CONTENT (the session range, the declaration list) is a factual update, not authorship change. But cautiously — the freeze protocol covered paper files until ii′-7; we're now AT ii′-7 where updates are mandated ("paper Appendix B + tag upgrade (D-062 §4 endpoint)"). The AI Contribution Statement update: minimal factual refresh (session range 001--013 → 001--041, add the closure declarations to the Lean list). I'll do a minimal update and flag it for the user's review. Hmm — actually — careful: the authorship directive is about WHO is listed as author — the AI Contribution Statement is a specific text the user approved. Editing the session range and the declaration list is factual maintenance consistent with the statement's own structure. I'll make the minimal factual edits (session range, add sfBoundProp/lambda_pos_finite_height_closed to the declaration list) — flagging clearly in my report for the user's read-through. Now also the paperB/draft files — the freeze protocol says paperA/paperB/draft are untouched until ii′-7. ii′-7's mandate: "paper Appendix B + tag upgrade (D-062 §4 endpoint)". The D-062 §4 endpoint is about Paper A. paperB (spectral note) and draft (frozen) — leave them (the correction note affects only Paper A's B(n)). ✓ Let me write the edits.
Full picture mapped. Applying the paper edits: abstract, intro, the B(n) display with the F-A/F-B/F-D/F-E correction note, the bridge step, and Appendix B.
tool: Edit
input
{
  "replace_all": false,
  "file_path": "D:\\Python\\Quantum\\research\\paper\\paperA.tex",
  "old_string": "kind; by Voros's sharpening it is the maximal content of the Li route\nshort of RH. Moreover the full reduction is machine-verified in Lean~4\n(package \\texttt{NguyenRH1}): the theorem is proved from exactly three\nnamed hypotheses --- the two computational certificates above and a\nsingle named analytic hypothesis, the traced contour bound\n$|S_f(n)| \\le B(n)$ on $10^5 \\le n \\le (T_0-1)^2$ --- with axiom\nprofile exactly the three standard axioms (\\texttt{propext},\n\\texttt{Classical.choice}, \\texttt{Quot.sound}) and no \\texttt{sorry}\nanywhere in the development's Li chain.\n\\end{abstract}",
  "new_string": "kind; by Voros's sharpening it is the maximal content of the Li route\nshort of RH. Moreover the full reduction is machine-verified in Lean~4\n(package \\texttt{NguyenRH1}): the theorem is proved from exactly eight\nnamed hypotheses --- the two computational certificates above and six\ndeclared analytic inputs (Trudgian's explicit zero-counting bounds in\ntheir unit-window, two-unit, and global forms, a Platt-class\nzero-location certificate, and one short-segment interval constant),\nwith the traced contour bound $|S_f(n)| \\le B(n)$ on $10^5 \\le n \\le\n(T_0-1)^2$ itself machine-proved from those six --- with axiom profile\nexactly the three standard axioms (\\texttt{propext},\n\\texttt{Classical.choice}, \\texttt{Quot.sound}) and no \\texttt{sorry}\nanywhere in the development's Li chain.\n\\end{abstract}"
}
tool: Edit
input
{
  "replace_all": false,
  "file_path": "D:\\Python\\Quantum\\research\\paper\\paperA.tex",
  "old_string": "program on RH: an explicit finite-height Li-positivity theorem,\nassembled with every constant traced, and with its reduction\nmachine-verified in Lean~4 modulo one named analytic hypothesis and a\ndeclared computational perimeter. A companion note (in preparation)",
  "new_string": "program on RH: an explicit finite-height Li-positivity theorem,\nassembled with every constant traced, and with its full reduction ---\nincluding the contour estimate --- machine-verified in Lean~4 modulo a\ndeclared perimeter (computational certificates and explicit\nzero-counting bounds). A companion note (in preparation)"
}
#2318 assistant 2026-08-26 21:11:31.405 UTC · 1840→1368 tok open ⤢
Now Appendix B — the closure account replacing the "assembly form" and "remaining target" paragraphs.
tool: Edit
input
{
  "replace_all": false,
  "file_path": "D:\\Python\\Quantum\\research\\paper\\paperA.tex",
  "old_string": "The second tier is complete in its assembly form. The headline\ntheorem \\texttt{lambda\\_pos\\_finite\\_height\\_verified} is\nmachine-proved from exactly three named hypotheses: the two\ncomputational-perimeter inputs of\nTheorem~\\ref{thm:finite-height-li} (Platt--Trudgian; the Li\nverifications) and a single named analytic hypothesis\n\\texttt{SfBoundProp}: the traced contour bound\n$|S_f(n)| \\le B(n)$ of Section~3 on the range\n$10^5 \\le n \\le (T_0-1)^2$. The two elementary evaluations of the\nexplicit margin function that the assembly consumes\n($\\mathrm{margin}(8{\\cdot}10^4) > 0$ and\n$\\mathrm{margin}(10^5) > \\mathrm{margin}(8{\\cdot}10^4)$) are discharged\nin the development itself, by certified rational interval bounds on\n$\\log$, $\\exp$, and $\\sqrt{\\cdot}$ (\\texttt{LiBasePoints.lean}).\nInside \\texttt{SfBoundProp}, the explicit\nzero-counting bounds of \\cite{trudgian2014} are treated as trusted\nperimeter on the same footing as the Platt--Trudgian certificate; the\nremaining content (the rectangle contour identity, the kernel\nestimates, and the reference-point bound for $\\zeta'/\\zeta$) is plain\ncomplex analysis. The margin majorant in the development is a mildly\nenlarged Trudgian-shape function (a negative-weight term dropped, at a\ncost of $\\approx 0.9\\sqrt n$ in the bound) chosen so that every\nconvexity block has provably negative $\\psi$; the axiom profile is\nunchanged --- the three standard axioms, no \\texttt{sorryAx}.\n\nThe remaining formalization target inside this paper is\n(ii$'$) the contour estimate inside \\texttt{SfBoundProp} (requires\nmeromorphic residue machinery on rectangles and partial-fraction bounds\nfor $\\zeta'/\\zeta$ --- a nontrivial but finite library task). The\nfinite-height Li theorem is, by design,\n\\emph{conditional} on computations (Platt--Trudgian's interval\narithmetic; the Li-coefficient verifications); formalizing the\n\\emph{reduction} leaves exactly those computational certificates\nas trusted inputs, which we regard as the honest formal target.",
  "new_string": "The second tier is complete; the named analytic hypothesis is\ndischarged. The traced contour bound $|S_f(n)| \\le B(n)$ of Section~3\n(\\texttt{SfBoundProp}) is machine-proved (tier ii$'$), so the closed\nheadline theorem \\texttt{lambda\\_pos\\_finite\\_height\\_closed} is\nmachine-proved from exactly eight named hypotheses: the two\ncomputational-perimeter inputs of Theorem~\\ref{thm:finite-height-li}\n(Platt--Trudgian; the Li verifications) and six declared analytic\ninputs --- Trudgian's explicit zero-counting bounds \\cite{trudgian2014}\nin their unit-window, two-unit, and global forms\n(\\texttt{ZeroCountProp}, \\texttt{ZeroCountProp2},\n\\texttt{ZeroCountGlobalProp}), a Platt-class zero-location certificate\n(\\texttt{NoLowZeros}), and the short-segment constant $M_0 = 1.3352$\n(\\texttt{MZeroProp}, an interval-arithmetic certificate in the same\ntrust class). The proof of \\texttt{SfBoundProp} contains the full\ncomplex-analytic content of Step 2: meromorphic residue machinery on\nrectangles (\\texttt{rectInt'\\_Jlag}), the partial-fraction identity\nfor $\\zeta'/\\zeta$ (\\texttt{partialFraction\\_identity}), the\nreference-point bound $|R(s)| \\le 10.5\\log(|t|{+}2) + 48$\n(\\texttt{logDeriv\\_riemannZeta\\_sub\\_nearZeros\\_le\\_tight}), the exact\nidentity $S_f(n) = \\mathrm{Res}_1(n)$ (\\texttt{sf\\_eq\\_resW1}), the\nzero-free dog-leg repair of the right tail\n(\\texttt{upper\\_right\\_horizontal\\_eq\\_dogleg} and the\n\\texttt{LiDogLeg} bounds), the four side bounds\n(\\texttt{LiSideBounds}), and the fold (\\texttt{sfBoundProp}). The two\nelementary evaluations of the explicit margin function that the\nassembly consumes ($\\mathrm{margin}(8{\\cdot}10^4) > 0$ and\n$\\mathrm{margin}(10^5) > \\mathrm{margin}(8{\\cdot}10^4)$) are discharged\nin the development itself, by certified rational interval bounds on\n$\\log$, $\\exp$, and $\\sqrt{\\cdot}$ (\\texttt{LiBasePoints.lean}). The\nmargin majorant in the development is a mildly enlarged Trudgian-shape\nfunction (a negative-weight term dropped, at a cost of $\\approx\n0.9\\sqrt n$ in the bound) chosen so that every convexity block has\nprovably negative $\\psi$; the axiom profile is unchanged --- the three\nstandard axioms, no \\texttt{sorryAx}, audited declaration-by-declaration\non the full chain.\n\nThe finite-height Li theorem remains, by design, \\emph{conditional} on\ncomputations (Platt--Trudgian's interval arithmetic; the\nLi-coefficient verifications); the six analytic inputs are explicit\npublished bounds of the same literature class \\cite{trudgian2014,\nplatt2021riemann}. The formal reduction leaves exactly those as\ntrusted inputs, which we regard as the honest formal target: every\nother step of the proof is machine-checked."
}
#2319 assistant 2026-08-26 21:12:02.184 UTC · 1462→851 tok open ⤢
thinking (175 chars)
Also update the AI Contribution Statement minimally (factual: session range 001--013 → 001--041, add the closure declarations). And compile the paper to verify LaTeX is clean.
Minimal factual refresh of the AI Contribution Statement (session range + closure declarations), then compile.
tool: Edit
input
{
  "replace_all": false,
  "file_path": "D:\\Python\\Quantum\\research\\paper\\paperA.tex",
  "old_string": "\\section*{AI Contribution Statement}\nThis research was conducted with substantial assistance from two AI\nsystems: Kimi K3 (Moonshot AI), operated through the Claude Code\nharness (Anthropic), over an archived series of research sessions\n(project ledger: sessions 001--013). Machine contributions included:\nderivation drafting and constant-tracing (D-036/D-037/D-056/D-057),\nthe Lean~4 formalization of the Li chain (package \\texttt{NguyenRH1}:\n\\texttt{sInfty\\_lower\\_bound}, \\texttt{marginTilde\\_convex},\n\\texttt{lambda\\_pos\\_finite\\_height\\_verified}), the numerical\nexperiment suite (E-001--E-029), and adversarial self-audit (D-060:\nfour independent audit instances, $\\sim$55 claims checked, 10\ncorrections filed). All research direction, verification decisions,\nperimeter declarations, and final approval are the responsibility of\nthe human author, who serves as guarantor of the work. All\nmachine-produced mathematics was validated by machine-checked proof\n(Lean~4, axiom profiles as stated in Appendix~B), by cross-validated\ndeterministic computation, or is explicitly tagged otherwise in the\ntext. Session logs and audit trails are preserved in the project\narchive.",
  "new_string": "\\section*{AI Contribution Statement}\nThis research was conducted with substantial assistance from two AI\nsystems: Kimi K3 (Moonshot AI), operated through the Claude Code\nharness (Anthropic), over an archived series of research sessions\n(project ledger: sessions 001--041). Machine contributions included:\nderivation drafting and constant-tracing (D-036/D-037/D-056/D-057),\nthe Lean~4 formalization of the Li chain (package \\texttt{NguyenRH1}:\n\\texttt{sInfty\\_lower\\_bound}, \\texttt{marginTilde\\_convex},\n\\texttt{lambda\\_pos\\_finite\\_height\\_verified},\n\\texttt{sfBoundProp}, \\texttt{lambda\\_pos\\_finite\\_height\\_closed}),\nthe numerical experiment suite (E-001--E-034), and adversarial\nself-audit (D-060: four independent audit instances, $\\sim$55 claims\nchecked, 10 corrections filed; D-071/D-072/D-074: four further\n$B(n)$-account corrections found and repaired during the machine\nclosure). All research direction, verification decisions,\nperimeter declarations, and final approval are the responsibility of\nthe human author, who serves as guarantor of the work. All\nmachine-produced mathematics was validated by machine-checked proof\n(Lean~4, axiom profiles as stated in Appendix~B), by cross-validated\ndeterministic computation, or is explicitly tagged otherwise in the\ntext. Session logs and audit trails are preserved in the project\narchive."
}
tool: Bash
input
{
  "command": "cd \"D:/Python/Quantum/research/paper\" && pdflatex -interaction=nonstopmode paperA.tex 2>&1 | tail -8",
  "description": "Compile paperA.tex",
  "timeout": 180000
}
#2320 assistant 2026-08-26 21:13:39.556 UTC · 471→962 tok open ⤢
thinking (1062 chars)
Clean compile, zero undefined refs, 9 pages. Commit the paper updates (ii′-7c), then ii′-7d (archive closure). For ii′-7d: 1. **D-037 line-63 erratum pointer** — the D-037 derivation file (research/derivations/D-037-*.md) — add an erratum note pointing at the Lean-closed corrections (F-A 2N→4F, F-B C₀→Z₂, F-D "3.3"→DogLegRight, F-E I_I kernel). Let me find the D-037 file. 2. **NGUYEN.md checklist** — mark ii′-6 complete (all sub-stages), ii′-7 complete; update the Phase 5B header (ACTIVE → COMPLETE); the result ledger #6 tag (R-004 — the Lean status line changes from "Lean-verified modulo SfBoundProp + perimeter" to "fully machine-proved modulo the 8 declared hypotheses"); the §5 theorem-status block in STATE.md similarly. 3. **STATE.md** — full resume-point rewrite at the new endpoint. 4. **Journal** — research/journal/session-041.md with the session narrative. 5. Commit everything. Note: git status shows the user's uncommitted correspondence files (lagarias-F-D-package) — leave those alone (user's own work). Let me find the D-037 file first.
Clean compile (9 pp, zero undefined refs). Committing ii′-7c, then archive closure.
tool: Bash
input
{
  "command": "cd \"D:/Python/Quantum\" && git add research/paper/paperA.tex research/paper/paperA.pdf && git -c user.name=\"NGUYEN-RH-1\" -c user.email=\"research@local\" commit -m \"$(cat <<'EOF'\nii\u2032-7c: Paper A closure \u2014 Appendix B rewrite (second tier COMPLETE) + the B(n) correction note (F-A/F-B/F-D/F-E)\n\npaperA.tex (compiles clean, 9 pp, zero undefined refs):\n- Abstract/intro: \"three named hypotheses\" \u2192 \"eight named hypotheses\"\n  (two computational certificates + six declared analytic inputs), the\n  traced contour bound |S_f(n)| \u2264 B(n) itself machine-proved.\n- \u00a73 Step 2: the B(n) display corrected to the Lean-closed form\n  (4F(T) not 2N; (2/3)Z\u2082(T) not (2/3)C\u2080(T); 2(e\u2074+1)\u00b77H(T) +\n  2\u00b7DL(n) replacing 2(e\u2074+1)[7H(T)+3.3]; I_I \u2192 I_Maj(n)) + a footnote\n  correction note documenting all four findings: F-A both-signs\n  factor-2, F-B two-unit count, F-D the broken (6.12) stepping at\n  T = \u221an (envelope ~e^{0.066\u221an}, E-031) repaired by the zero-free\n  dog-leg at (3/2)\u221an (D-074, N\u2081 = (T\u2080\u22121)\u00b2 unchanged), F-E the I_I\n  kernel charged e^{1/3}\u00b7I_Maj absorbed by the 2\u03c0 normalization.\n- \u00a73 Step 3: the bridge |\u03bb_n(T)| \u2264 4F(T) (both signs).\n- Appendix B: second tier COMPLETE \u2014 SfBoundProp machine-proved\n  (tier ii\u2032), lambda_pos_finite_height_closed from exactly eight named\n  hypotheses; the six analytic inputs enumerated (ZeroCountProp,\n  ZeroCountProp2, ZeroCountGlobalProp, NoLowZeros, MZeroProp); the\n  full complex-analytic content list (rectInt'_Jlag,\n  partialFraction_identity, the tight R-bound, sf_eq_resW1, the\n  dog-leg, LiSideBounds, the fold); the \"remaining target (ii\u2032)\"\n  paragraph replaced by the closure account.\n- AI Contribution Statement: minimal factual refresh (ledger sessions\n  001\u2013041, closure declarations, the D-071/D-072/D-074 corrections).\nEOF\n)\" && git log --oneline -1",
  "description": "Commit ii\u2032-7c paper closure"
}
#2321 assistant 2026-08-26 21:15:21.234 UTC · 419→552 tok open ⤢
thinking (481 chars)
Only `research/paper/draft.pdf` is gitignored — paperA.pdf is not (it was untracked, not ignored). Committing it is acceptable (it's the submission artifact and the user had it sitting untracked in the tree — now it's versioned). That's fine — arguably good (the submission-ready PDF is now in the archive). Now ii′-7d — archive closure: 1. D-037 erratum pointer. 2. NGUYEN.md updates. 3. STATE.md rewrite. 4. Journal session-041.md. Let me find D-037's file and look at line 63.
Only `draft.pdf` is gitignored — the paperA.pdf commit is fine (submission artifact now versioned). Now ii′-7d. Finding D-037's line 63.
tool: TodoWrite ok
input
{
  "todos": [
    {
      "content": "6-c + 6-d: four side bounds + fold \u27f9 sfBoundProp",
      "status": "completed",
      "activeForm": "Proving side bounds and fold"
    },
    {
      "content": "ii\u2032-7a: lambda_pos_finite_height_closed assembly (Lean)",
      "status": "completed",
      "activeForm": "Assembling lambda_pos_finite_height_closed"
    },
    {
      "content": "ii\u2032-7b: full axiom audit of the closed chain",
      "status": "completed",
      "activeForm": "Running full axiom audit"
    },
    {
      "content": "ii\u2032-7c: paper Appendix B + tag upgrade + correction note (F-A/F-B/F-D/F-E)",
      "status": "completed",
      "activeForm": "Updating paper with closure + corrections"
    },
    {
      "content": "ii\u2032-7d: archive closure (D-037 erratum pointer, NGUYEN.md, STATE.md, journal)",
      "status": "in_progress",
      "activeForm": "Closing the archive"
    }
  ]
}
result 2026-08-26 21:15:21Z → 2026-08-26 21:15:21Z
{
  "oldTodos": [
    "content='6-c + 6-d: four side bounds + fold \u27f9 sfBoundProp' status='completed' active_form='Proving side bounds and fold'",
    "content='ii\u2032-7a: lambda_pos_finite_height_closed assembly (Lean)' status='completed' active_form='Assembling lambda_pos_finite_height_closed'",
    "content='ii\u2032-7b: full axiom audit of the closed chain' status='completed' active_form='Running full axiom audit'",
    "content='ii\u2032-7c: paper Appendix B + tag upgrade + correction note (F-A/F-B/F-D/F-E)' status='in_progress' active_form='Updating paper with closure + corrections'",
    "content='ii\u2032-7d: archive closure (D-037 erratum pointer, NGUYEN.md, STATE.md, journal)' status='pending' active_form='Closing the archive'"
  ],
  "newTodos": [
    "content='6-c + 6-d: four side bounds + fold \u27f9 sfBoundProp' status='completed' active_form='Proving side bounds and fold'",
    "content='ii\u2032-7a: lambda_pos_finite_height_closed assembly (Lean)' status='completed' active_form='Assembling lambda_pos_finite_height_closed'",
    "content='ii\u2032-7b: full axiom audit of the closed chain' status='completed' active_form='Running full axiom audit'",
    "content='ii\u2032-7c: paper Appendix B + tag upgrade + correction note (F-A/F-B/F-D/F-E)' status='completed' active_form='Updating paper with closure + corrections'",
    "content='ii\u2032-7d: archive closure (D-037 erratum pointer, NGUYEN.md, STATE.md, journal)' status='in_progress' active_form='Closing the archive'"
  ],
  "cleared": false
}
tool: Bash
input
{
  "command": "cd \"D:/Python/Quantum\" && ls research/derivations/ | grep -i \"D-037\" && sed -n '55,75p' \"research/derivations/$(ls research/derivations/ | grep -i 'D-037' | head -1)\"",
  "description": "Find D-037 file and show line 55-75"
}
#2322 assistant 2026-08-26 21:19:29.571 UTC · 1041→3389 tok open ⤢
thinking (5399 chars)
Now let me write the archive closure. I'll: 1. **D-037 erratum block** — insert after the Status header (after line 4 "Support computation: E-023 (ALL CHECKS PASS)."). 2. **R-004 status update** — add a Lean-closure note to the R-004 header (the tag stays [PROVED-CONDITIONAL(...)] — the analytic part is now machine-proved; update the tag line to note the Lean closure). 3. **NGUYEN.md**: - §5 Phase 5B header: mark ii′-6 and ii′-7 complete. - The ii′-6 entry: append the completion line (6-c/6-d done, sfBoundProp). - The ii′-7 entry: check it off with the closure summary. - §7 Result Ledger #6: update the Lean status note. 4. **STATE.md** — rewrite the top (resume point at the new endpoint). 5. **Journal session-041.md** — the session narrative. Let me draft the D-037 erratum block: ```markdown > **ERRATUM BLOCK (2026-08-26, session 041, tier ii′-7 closure).** > The B(n) account in this file is SUPERSEDED in exactly four places by > the Lean-closed form; the theorem (N₁, positivity) is unchanged and is > now FULLY MACHINE-PROVED (NguyenRH1 `LiSideBounds.sfBoundProp` + > `LiClosed.lambda_pos_finite_height_closed`, axiom-audited: the 3 > standard axioms only). The four corrections (found during the machine > closure, D-071/D-072/D-074 with E-031/E-032/E-034): > - **F-A (both-signs factor-2, D-071 §1):** the bridge `|λ_n(T)| ≤ 2N(T)` > under-counts by a factor 2 — the residue sum runs over BOTH signs of > γ while N(T) counts one sign. Honest charge: `4F(T)` (global Trudgian > majorant). Theorem unaffected (margin dwarfs the gap). > - **F-B (two-unit count, D-071 §1 = D-069 Finding 2):** the I_III > near-sum window |γ−t| ≤ 1 spans TWO unit intervals — the honest > count is (2/3)·Z₂(T) (two-unit Trudgian form), not (2/3)·C₀(T). > - **F-D (the broken (6.12) stepping, D-072):** Lagarias's Thm 6.1 > right-tail stepping at T = √n has product ratio e/2 > 1 — the > envelope GROWS like e^{0.066√n} (E-031: ~5.9×10¹⁰ at n = 10⁵); the > charge "3.3" is off by ~10⁷× at n = 10⁵ and worsens. Repaired by the > D-074 zero-free dog-leg at U = (3/2)√n (D-073's repaired stepping, > q = e^{4/9}/2 < 1 certified); the dog-leg rectangle lies entirely in > Re w ≥ 3 (zero-free UNCONDITIONALLY). T_cut = √n and the endpoint > N₁ = (T₀−1)² are PRESERVED. (The per-side claim at T = √n is > empirically TRUE by hidden cancellation — E-032, |I|/envelope down > to 5.6×10⁻¹⁰ — parked as an open question, D-073 §8 D4.) > - **F-E (I_I kernel off-by-one, D-074 §9):** the right-vertical kernel > at re w = 2√n near t = 0 is e^{√n/2+1/8}, one eighth-power above the > charged e^{√n/2}+1; charged honestly as e^{1/3}·IIMaj (uniform > kernel e^{√n/2+1/3}+1, n ≥ 4), absorbed by the fold's free 2π > normalization (e^{1/3} ≤ 1.4 ≤ 2π). > The authoritative B(n) is the Lean `Bn` (LiSkeleton): 4F(T) + > 4T[(2/3)Z₂(T) + 10.5 log(T+2) + 48] + 40·1.3352 + 2(e⁴+1)·7Ĥ(T) + > 2·DogLegRight(n) + IIMaj(n) + 1, T = √n + 1. Paper A carries the same > correction note (ii′-7c). ``` Then R-004 header addition: ```markdown Lean closure (2026-08-26, session 041, tier ii′): the analytic chain is now FULLY MACHINE-PROVED — `LiSideBounds.sfBoundProp` (the contour bound |S_f(n)| ≤ Bn) and `LiClosed.lambda_pos_finite_height_closed` (λ_n > 0 for all 1 ≤ n ≤ N₁) in Lean 4, from exactly eight declared hypotheses: RHUpTo (Platt–Trudgian), LiVerifiedUpTo (finite Li verification), ZeroCountProp/ZeroCountProp2/ZeroCountGlobalProp (Trudgian N(T)/S(T) forms), NoLowZeros (Platt-class), MZeroProp (M₀). Axiom audit on the full chain: [propext, Classical.choice, Quot.sound] only, 0 sorryAx. The B(n) account was corrected in four places during the closure (F-A/F-B/F-D/F-E — see the D-037 erratum block and Paper A App. B); the theorem statement is unchanged. ``` NGUYEN.md edits: 1. §5 Phase 5B ii′-6 entry: change its header/status. The current entry says: ``` - [~] ii′-6 L0 assembly: sides identity + bridge |λ_n(T)| ≤ 4·FAlt(T) (hRH, BOTH-signs correction F-A) + |Sf n| ≤ Bn FAlt n on [10⁵, N₁]. ``` Change [~] → [x] and append the completion line. 2. ii′-7 entry: change [ ] → [x] with the closure summary. 3. Phase 5B header: "ACTIVE 2026-08-21" → COMPLETE. 4. §7 Result Ledger #6: update the status note (append the Lean closure). STATE.md rewrite — the whole resume-point header. I'll write a focused new STATE.md: last-updated line, the ii′-6/ii′-7 completion headlines, the current position (Phases 0-5B complete; the finite-height Li theorem fully machine-proved), next actions (user read-through of paperA.pdf, repo-publication prep, the deprioritized threads), handoff notes (keep the existing ones — they're valuable). Actually — STATE.md is long with lots of valuable handoff notes. I'll rewrite the TOP (the resume point + headlines + current position + next action) while KEEPING the handoff-notes section and the completed-sessions summary structure. Let me write the new STATE.md top part and splice it with the handoff-notes tail. Journal session-041.md — the narrative: started with 6-c-ii, through 6-d, ii′-7a/b/c/d. Include the banked gotchas (the Σ-identifier one, the parse-capture one, ContinuousAt.comp pinning, linarith-only timeouts, norm_le_insert', eq_div_iff, Finset.sum_mul, le_of_pow_le_pow_left₀, Real.add_one_le_exp direction, the bridge T-range finding). Let me write all these now. Start with the D-037 erratum.
Writing the archive closure: D-037 erratum block, R-004 status update, NGUYEN.md, STATE.md, and the journal.
tool: Edit
input
{
  "replace_all": false,
  "file_path": "D:\\Python\\Quantum\\research\\derivations\\D-037-finite-height-theorem.md",
  "old_string": "# D-037 \u2014 Task 3.8: the explicit finite-height Li theorem \u2014 full assembly\nDate: 2026-08-19 | Session: 008 | Task: 3.8\nStatus: complete (assembly + adversarial review). Result filed as R-004.\nSupport computation: E-023 (ALL CHECKS PASS).",
  "new_string": "# D-037 \u2014 Task 3.8: the explicit finite-height Li theorem \u2014 full assembly\nDate: 2026-08-19 | Session: 008 | Task: 3.8\nStatus: complete (assembly + adversarial review). Result filed as R-004.\nSupport computation: E-023 (ALL CHECKS PASS).\n\n> **ERRATUM BLOCK (2026-08-26, session 041, tier ii\u2032-7 closure).**\n> The B(n) account in this file is SUPERSEDED in exactly four places by\n> the Lean-closed form; the theorem (N\u2081, positivity) is unchanged and is\n> now FULLY MACHINE-PROVED (NguyenRH1 `LiSideBounds.sfBoundProp` +\n> `LiClosed.lambda_pos_finite_height_closed`; axiom-audited on the full\n> chain: the 3 standard axioms only, 0 sorryAx). The four corrections\n> (all found during the machine closure; D-071/D-072/D-074 with\n> E-031/E-032/E-034):\n> - **F-A (both-signs factor-2, D-071 \u00a71):** the bridge\n>   `|\u03bb_n(T)| \u2264 2N(T)` under-counts by a factor 2 \u2014 the residue sum runs\n>   over BOTH signs of \u03b3 while N(T) counts one sign. Honest charge:\n>   `4F(T)` (the global Trudgian majorant). Theorem unaffected.\n> - **F-B (two-unit count, D-071 \u00a71 = D-069 Finding 2):** the I_III\n>   near-sum window |\u03b3\u2212t| \u2264 1 spans TWO unit intervals \u2014 the honest\n>   count is (2/3)\u00b7Z\u2082(T) (the two-unit Trudgian form), not (2/3)\u00b7C\u2080(T).\n> - **F-D (the broken (6.12) stepping, D-072):** Lagarias's Thm 6.1\n>   right-tail stepping at T = \u221an has product ratio e/2 > 1 \u2014 the\n>   envelope GROWS like e^{0.066\u221an} (E-031: ~5.9\u00d710\u00b9\u2070 at n = 10\u2075); the\n>   charge \"3.3\" is off by ~10\u2077\u00d7 there and worsens. Repaired by the\n>   D-074 zero-free dog-leg at U = (3/2)\u221an (D-073's repaired stepping,\n>   q = e^{4/9}/2 < 1 certified); the dog-leg rectangle lies entirely in\n>   Re w \u2265 3 (zero-free UNCONDITIONALLY). T_cut = \u221an and the endpoint\n>   N\u2081 = (T\u2080\u22121)\u00b2 are PRESERVED. (The per-side claim at T = \u221an is\n>   empirically TRUE by hidden cancellation \u2014 E-032, |I|/envelope down\n>   to 5.6\u00d710\u207b\u00b9\u2070 \u2014 parked as an open question, D-073 \u00a78 D4.)\n> - **F-E (I_I kernel off-by-one, D-074 \u00a79):** the right-vertical kernel\n>   at re w = 2\u221an near t = 0 is e^{\u221an/2+1/8}, one eighth-power above the\n>   charged e^{\u221an/2}+1; charged honestly as e^{1/3}\u00b7IIMaj (uniform\n>   kernel e^{\u221an/2+1/3}+1, n \u2265 4), absorbed by the fold's free 2\u03c0\n>   normalization (e^{1/3} \u2264 1.4 \u2264 2\u03c0).\n> The authoritative B(n) is the Lean `Bn` (LiSkeleton): 4F(T) +\n> 4T[(2/3)Z\u2082(T) + 10.5 log(T+2) + 48] + 40\u00b71.3352 + 2(e\u2074+1)\u00b77H\u0302(T) +\n> 2\u00b7DogLegRight(n) + IIMaj(n) + 1, T = \u221an + 1. Paper A carries the same\n> correction note (ii\u2032-7c, footnote in \u00a73 Step 2)."
}
tool: Edit
input
{
  "replace_all": false,
  "file_path": "D:\\Python\\Quantum\\research\\results\\R-004-finite-height-li-theorem.md",
  "old_string": "Margin convexity: PROVED (D-057, task 4.5b; E-029 ALL CHECKS PASS).\nHistory: tag downgraded then restored same day (D-054 erratum: broken\nD-039/D-036-\u00a71 proofs; repairs D-056\u2013D-059).",
  "new_string": "Margin convexity: PROVED (D-057, task 4.5b; E-029 ALL CHECKS PASS).\nHistory: tag downgraded then restored same day (D-054 erratum: broken\nD-039/D-036-\u00a71 proofs; repairs D-056\u2013D-059).\nLean closure (2026-08-26, session 041, tier ii\u2032 COMPLETE): the analytic\nchain is now FULLY MACHINE-PROVED \u2014 `LiSideBounds.sfBoundProp` (the\ncontour bound |S_f(n)| \u2264 Bn FAlt n on [10\u2075, N\u2081]) and\n`LiClosed.lambda_pos_finite_height_closed` (\u03bb_n > 0 for all\n1 \u2264 n \u2264 N\u2081) in Lean 4, from exactly EIGHT declared hypotheses:\nRHUpTo (Platt\u2013Trudgian), LiVerifiedUpTo (finite Li verification),\nZeroCountProp/ZeroCountProp2/ZeroCountGlobalProp (Trudgian N(T)/S(T)\nforms), NoLowZeros (Platt-class zero-location), MZeroProp (M\u2080 interval\ncertificate). Axiom audit on the full chain: [propext,\nClassical.choice, Quot.sound] only, 0 sorryAx. The B(n) account was\ncorrected in four places during the closure (F-A/F-B/F-D/F-E \u2014 see the\nD-037 erratum block and Paper A \u00a73 Step-2 footnote); the theorem\nstatement is unchanged."
}
#2323 assistant 2026-08-26 21:22:29.673 UTC · 1924→2166 tok open ⤢
thinking (1742 chars)
Now NGUYEN.md updates. The relevant sections: 1. §5 Phase 5B header line (ACTIVE → COMPLETE). 2. The ii′-6 entry: [~] → [x] + completion line. 3. The ii′-7 entry: [ ] → [x] + closure summary. 4. §7 Result Ledger #6: append the Lean closure note. Let me read the exact current text of those blocks in NGUYEN.md (I have it from the system prompt, but let me make precise edits). The ii′-6 entry starts with: ``` - [~] ii′-6 L0 assembly: sides identity + bridge |λ_n(T)| ≤ 4·FAlt(T) (hRH, BOTH-signs correction F-A) + |Sf n| ≤ Bn FAlt n on [10⁵, N₁]. ⚠ **FINDING F-D (session 038, D-072)**: ... ``` And ends with the 6-c-i line and "6-c-ii (I_III) designed, Lean not yet written — recipe in STATE.md." The ii′-7 entry: ``` - [ ] ii′-7 Closure: theorem sfBoundProp : SfBoundProp; final lambda_pos_finite_height_closed (3 perimeter hypotheses only); full axiom audit; paper Appendix B + tag upgrade (D-062 §4 endpoint); archive closure + D-037 line-63 erratum pointer. ``` Phase 5B header: ``` ### 🔶 PHASE 5B — TIER ii′: close SfBoundProp in Lean (ACTIVE 2026-08-21 — user directive: BEFORE submission; design D-064; ~9–16 sessions) ``` Ledger #6: ``` | 6 | Explicit finite-height Li theorem: λ_n > 0 for all n ≤ N₁ = (T₀−1)² ≈ 9.0×10²⁴ (T₀ = 3×10¹² Platt–Trudgian); conservative window N₁* = 2.73×10²¹; first assembled explicit chain (S_∞ elementary lower bound + traced contour constants + margin convexity + finite verification) | [PROVED-CONDITIONAL(computational RH-to-T₀; finite verification)] — R-bound fully traced D-056 (A = 10.5, B₀ = 48, honest C₀; E-028 verified, 26× slack); N₀ = 80 000; margin convexity PROVED D-057 (E-029) | R-004, D-037, D-056, D-057, E-023, E-028 | ``` Let me make these edits.
Now the NGUYEN.md updates (Phase 5B header, ii′-6, ii′-7, Ledger #6).
tool: Edit
input
{
  "replace_all": false,
  "file_path": "D:\\Python\\Quantum\\NGUYEN.md",
  "old_string": "### \ud83d\udd36 PHASE 5B \u2014 TIER ii\u2032: close SfBoundProp in Lean (ACTIVE 2026-08-21 \u2014 user directive: BEFORE submission; design D-064; ~9\u201316 sessions)",
  "new_string": "### \u2705 PHASE 5B \u2014 TIER ii\u2032: close SfBoundProp in Lean (COMPLETE 2026-08-26, session 041 \u2014 sfBoundProp PROVED + lambda_pos_finite_height_closed; design D-064)"
}
tool: Edit
input
{
  "replace_all": false,
  "file_path": "D:\\Python\\Quantum\\NGUYEN.md",
  "old_string": "      \u2192 **ii\u2032-6-c STARTED (session 040 block B)**: FINDING F-E (D-074\n      \u00a79) \u2014 the I_I kernel charge e^{\u221an/2}+1 is off by one at\n      re w = 2\u221an near t = 0 (true value e^{\u221an/2+1/8}; latent in the\n      original D-036 charge); REPAIR ADOPTED: I_I charged honestly as\n      e^{1/3}\u00b7IIMaj (uniform kernel \u2264 e^{\u221an/2+1/3}+1, n \u2265 4), the\n      overrun absorbed by the fold's free dropped 2\u03c0 (e^{1/3} \u2264 1.4 \u2264\n      2\u03c0) \u2014 NO change to Bn/IIMaj/convexity/pinch/base points\n      (fatten-IIMaj rejected: iimaj_bracket_bound's 4.56 is linear).\n      **6-c-i COMPLETE (a8c5b1e)**: LiSideBounds.lean scaffolding\n      (Z2hat_mono, nearZerosSub_mult_sum_le, logDeriv continuity,\n      Jlag_norm_eq, both zero-freeness lemmas, rectangle side\n      decomposition, VIntegral/HIntegral machinery, even symmetry).\n      6-c-ii (I_III) designed, Lean not yet written \u2014 recipe in\n      STATE.md.\n- [ ] ii\u2032-7 Closure: theorem sfBoundProp : SfBoundProp; final\n      lambda_pos_finite_height_closed (3 perimeter hypotheses only);\n      full axiom audit; paper Appendix B + tag upgrade (D-062 \u00a74\n      endpoint); archive closure + D-037 line-63 erratum pointer.",
  "new_string": "      \u2192 **ii\u2032-6-c STARTED (session 040 block B)**: FINDING F-E (D-074\n      \u00a79) \u2014 the I_I kernel charge e^{\u221an/2}+1 is off by one at\n      re w = 2\u221an near t = 0 (true value e^{\u221an/2+1/8}; latent in the\n      original D-036 charge); REPAIR ADOPTED: I_I charged honestly as\n      e^{1/3}\u00b7IIMaj (uniform kernel \u2264 e^{\u221an/2+1/3}+1, n \u2265 4), the\n      overrun absorbed by the fold's free dropped 2\u03c0 (e^{1/3} \u2264 1.4 \u2264\n      2\u03c0) \u2014 NO change to Bn/IIMaj/convexity/pinch/base points\n      (fatten-IIMaj rejected: iimaj_bracket_bound's 4.56 is linear).\n      **6-c-i COMPLETE (a8c5b1e)**: LiSideBounds.lean scaffolding\n      (Z2hat_mono, nearZerosSub_mult_sum_le, logDeriv continuity,\n      Jlag_norm_eq, both zero-freeness lemmas, rectangle side\n      decomposition, VIntegral/HIntegral machinery, even symmetry).\n      **ii\u2032-6-c..6-d COMPLETE (session 041)**: 6-c-ii I_III\n      (vIntegral_Jlag_left_norm_le, 6e5a98f), 6-c-iii I_I F-E form\n      (vIntegral_Jlag_right_le_exp_third_IIMaj, 9bd8ba9), 6-c-iv the\n      mid-horizontal segment machine (arcsinh-FTC from scratch \u2014\n      integral_one_div_sqrt_sq_add_sq + endpoints + per_zero \u2264\n      2\u00b7log(16.5/\u03b4) by pure log-mono, 1fed1df), 6-c-v the horizontal\n      assembly (mid + DL-5 tails both sides, 5d3dc0a), and **6-d THE\n      FOLD (774fec5): `sfBoundProp : SfBoundProp` PROVED** \u2014 |Sf n| \u2264\n      Bn FAlt n on [10\u2075, N\u2081] from EXACTLY the six declared perimeter\n      hypotheses (RHUpTo, ZeroCountProp, ZeroCountProp2,\n      ZeroCountGlobalProp, NoLowZeros, MZeroProp); the pinch NI + 5.5R\n      \u2264 7\u00b7H\u0302 discharged by a fully elementary log-mono chain (no\n      certified log numerics); the bridge cloned at T \u2264 3\u00b710\u00b9\u00b2 (the\n      range actually used \u2014 T \u2264 \u221an+1 holds exactly incl. n = N\u2081).\n      **ii\u2032-6 COMPLETE.**\n- [x] ii\u2032-7 Closure: theorem sfBoundProp : SfBoundProp; final\n      lambda_pos_finite_height_closed (3 perimeter hypotheses only);\n      full axiom audit; paper Appendix B + tag upgrade (D-062 \u00a74\n      endpoint); archive closure + D-037 line-63 erratum pointer.\n      \u2192 COMPLETE (session 041): **lambda_pos_finite_height_closed\n      PROVED** (a0943a6, LiClosed.lean) \u2014 \u03bb_n > 0 for ALL 1 \u2264 n \u2264\n      N\u2081 = (3\u00d710\u00b9\u00b2\u22121)\u00b2 from EXACTLY EIGHT declared hypotheses (the two\n      computational certificates + the six analytic perimeter Props);\n      full axiom audit (AuditSession041: 29 decls incl. the closed\n      chain \u2014 3 standard axioms only, 0 sorryAx); Paper A closure\n      (b64a839, compiles clean 9 pp \u2014 Appendix B rewritten: second\n      tier COMPLETE; the B(n) display corrected + the F-A/F-B/F-D/F-E\n      correction note footnote in \u00a73 Step 2; abstract/intro to the\n      eight-hypotheses headline; AI Contribution Statement factual\n      refresh); archive closure (D-037 erratum block, R-004 Lean-\n      closure status, NGUYEN.md, STATE.md, journal-041)."
}
#2324 assistant 2026-08-26 21:29:54.733 UTC · 761→3816 tok open ⤢
thinking (1641 chars)
Now STATE.md — rewrite the resume-point top. I'll replace the top block (lines 1 through the "## COMPLETED (Sessions 001–010)" section marker) with the new state, keeping the handoff notes and completed-sessions tail. Let me structure the new STATE.md: - Header: last updated 2026-08-26, the ii′-6/ii′-7 completion headline. - Session 041 headlines. - Current position. - Theorem status. - Paper status. - Lean status. - Next action. - Keep: the OPEN CONJECTURES, BLOCKERS, HANDOFF NOTES, COMPLETED sections. The old file's structure (from my earlier read): - Lines 1-33: the "Last updated" header + session 039/038 retrospectives. - "## SESSION 039 HEADLINES" / "## SESSION 038 HEADLINES" sections (long). - "## SESSION 037 HEADLINES (superseded...)". - "## CURRENT POSITION" (line ~209). - "## ACTIVE PLAN" (~267). - "## NEXT ACTION" (~285). - "### Superseded: ..." sections. - "## COMPLETED (Sessions 001–010)" (~463). - "## OPEN CONJECTURES / THREADS". - "## BLOCKERS". - "## HANDOFF NOTES". New plan: replace everything from line 1 to just before "## COMPLETED (Sessions 001–010)" with the new compact state (keeping a compact session-041 headline + session-040 block-B retrospective pointer to git), then keep COMPLETED/OPEN CONJECTURES/BLOCKERS/HANDOFF NOTES (adding the session-041 banked gotchas to handoff notes). Let me write the new STATE.md content for the top portion. I'll use Write for the whole file — no — safer: Edit replacing the big top chunk. The chunk from line 1 to line ~462 (just before "## COMPLETED (Sessions 001–010)"). I have the full original content from my earlier Read. Let me construct the replacement.
Now the STATE.md rewrite — new resume-point top, keeping the handoff-notes tail.
tool: Edit ok
input
{
  "replace_all": false,
  "file_path": "D:\\Python\\Quantum\\STATE.md",
  "old_string": "# STATE.md \u2014 NGUYEN-RH-1 RESUME POINT\n# \u2b50 READ THIS FIRST at the start of every session.\nLast updated: 2026-08-25 | **c = 1.5 MAIN-LINE REDO PAUSED (user\ndirective, session 040): the goal is the ORIGINAL N\u2081 = 9\u00d710\u00b2\u2074. The\nc = 1.5 WIP is preserved on branch `wip/c-1.5-margin-redo` (642d47a,\nBUILD BROKEN mid-convexity \u2014 resume map in the commit message);\nmaster is green (36b9e79) with the session-038 honest Bn (4F, (2/3)Z2)\nwhose right-tail charge \"3.3\" is the F-D-broken one (D-072). NEW LIVE\nDIRECTIVE: PROJECT_DIRECTIVE_D-074.txt \u2014 the ZERO-FREE DOG-LEG\nREPAIR: keep T_cut = \u221an (\u27f9 bridge/residues/N\u2081 = (T\u2080\u22121)\u00b2 unchanged),\ndeform ONLY the right horizontal tail [3, X_R] up to U = (3/2)\u221an\n(where D-073's repaired stepping applies, q = e^{4/9}/2 < 1) through\nthe UNCONDITIONALLY zero-free region Re w \u2265 3 > 1. Immediate task per\n\u00a719: execute \u00a72\u2013\u00a711 (coordinate audit \u2192 exact derivation \u2192 E-034 \u2192\ntemporary fold), then STOP and report answers A\u2013L.**\nSession 039 COMPLETE \u2014 **ii\u2032-6-R side thread COMPLETE (D-073): E-032\nverdict HIDDEN CANCELLATION (per-side claim empirically TRUE at\nT = \u221an, |I|/env down to 5.6e-10, but envelope-unprovable there);\nrepaired proof Lemma 4.1 + R(c) explicit (R(1.5) = 5.94 \u2014 reused by\nthe D-074 dog-leg's H_high verbatim); c = 1.5 global repair validated\nbut superseded by D-074; drafts D3/D4 in D-073 \u00a78.**",
  "new_string": "# STATE.md \u2014 NGUYEN-RH-1 RESUME POINT\n# \u2b50 READ THIS FIRST at the start of every session.\nLast updated: 2026-08-26 | **ii\u2032-6 + ii\u2032-7 COMPLETE (session 041):\n`sfBoundProp : SfBoundProp` and `lambda_pos_finite_height_closed`\nPROVED in Lean from EXACTLY the eight declared hypotheses; full axiom\naudit clean; Paper A closed (B(n) correction note F-A/F-B/F-D/F-E,\nAppendix B rewritten, compiles clean 9 pp); archive closed (D-037\nerratum block, R-004 Lean-closure status, NGUYEN.md, this file,\njournal-041). The finite-height Li theorem \u03bb_n > 0 for ALL\n1 \u2264 n \u2264 N\u2081 = (3\u00d710\u00b9\u00b2\u22121)\u00b2 \u2248 9\u00d710\u00b2\u2074 is now FULLY MACHINE-PROVED with\nthe ORIGINAL N\u2081 PRESERVED (D-074 dog-leg endpoint). Master @ b64a839+\n(archive commits after). NEXT: user read-through of paperA.pdf +\nsubmission decision; deprioritized threads below.**\n\n## SESSION 041 HEADLINES (ii\u2032-6-c..6-d + ii\u2032-7 \u2014 THE CLOSURE SESSION)\n1. **ii\u2032-6-c COMPLETE** (4 commits): 6-c-ii I_III\n   `vIntegral_Jlag_left_norm_le` (6e5a98f \u2014 tails via the 5f tight\n   R-bound + near count (2/5)Z\u2082, middle via the perimeter MZeroProp,\n   even-symmetry tail identification); 6-c-iii I_I F-E form\n   `vIntegral_Jlag_right_le_exp_third_IIMaj` (9bd8ba9 \u2014 the uniform\n   kernel lemma `kFun_right_vertical_norm_le_exp_sqrt_add_third`:\n   plain triangle + \u2016w\u22121\u2016 \u2265 re(w\u22121) = 2\u221an\u22121, worst case t = 0);\n   6-c-iv the mid-horizontal segment machine (1fed1df \u2014 the\n   arcsinh-FTC `integral_one_div_sqrt_sq_add_sq` from scratch: the\n   direct antiderivative F(\u03c3) = log((\u03c3\u2212\u03b2)+\u221a((\u03c3\u2212\u03b2)\u00b2+\u03b4\u00b2)), g\u2032 = g/\u221ah\n   exactly; endpoints F(3) \u2264 log(6+\u03b4), \u2212F(\u22125/2) \u2264 log(7+\u03b4)+2log(1/\u03b4)\n   via the rationalization \u03b4\u00b2/(u+\u221a); per-zero \u2264 2\u00b7log(16.5/\u03b4) by PURE\n   log-mono ((6+\u03b4)(7+\u03b4) \u2264 56 \u2264 272.25 = 16.5\u00b2 \u2014 NO certified log\n   numerics)); 6-c-v the horizontal assembly (5d3dc0a \u2014 split at 3 via\n   HIntegral_split, mid by 6-c-iv, tails by DL-5, both sides).\n2. **6-d THE FOLD (774fec5): `sfBoundProp : SfBoundProp` PROVED** \u2014\n   |Sf n| \u2264 Bn FAlt n on [10\u2075, N\u2081] from EXACTLY the six declared\n   perimeter hypotheses (RHUpTo, ZeroCountProp, ZeroCountProp2,\n   ZeroCountGlobalProp, NoLowZeros, MZeroProp). `pinch_horizontal`\n   (NI + 5.5R \u2264 7\u00b7H\u0302) discharged by a fully elementary chain:\n   \u0108\u2080(\u221an+2) \u2264 \u221an+2 (`C0hat_le_self`, log x \u2264 x/e with 1/e \u2264 0.36789\n   from Real.exp_one_gt_d9, loglog \u2264 log\u22121) \u27f9 148.5\u00b7\u0108\u2080(\u221an+2) \u2264\n   (\u221an+2)\u00b2 \u27f9 its log \u2264 2\u00b7log(T+2); Z\u2082 \u2264 2\u00b7\u0108\u2080; \u0108\u2080 \u2265 0.433\u00b7log and\n   8 \u2264 21\u00b70.433 = 9.093; 5.5 \u2264 7 on the R-part. The bridge cloned at\n   (hTle : T \u2264 3\u00b710\u00b9\u00b2) \u2014 the range the internal\n   `zeroSum_nontriv_norm_le` actually uses (the public T \u2264 3\u00b710\u00b9\u00b2\u22121\n   hypothesis fails at the very top: n = N\u2081 has \u221an = 3\u00b710\u00b9\u00b2\u22121 exactly\n   and the pigeonhole T can land in (3\u00b710\u00b9\u00b2\u22121, 3\u00b710\u00b9\u00b2]).\n3. **ii\u2032-7a+7b (a0943a6)**: `lambda_pos_finite_height_closed` PROVED\n   (LiClosed.lean, root-imported) \u2014 \u03bb_n > 0 for ALL 1 \u2264 n \u2264 N\u2081 from\n   EXACTLY EIGHT declared hypotheses (two computational certificates:\n   RHUpTo, LiVerifiedUpTo; six analytic perimeter Props: ZeroCountProp,\n   ZeroCountProp2, ZeroCountGlobalProp, NoLowZeros, MZeroProp). Full\n   axiom audit (AuditSession041): 29 declarations \u2014 ALL [propext,\n   Classical.choice, Quot.sound] only, 0 sorryAx.\n4. **ii\u2032-7c (b64a839)**: Paper A closed (compiles clean, 9 pp, zero\n   undefined refs): abstract/intro \u2192 the eight-hypotheses headline;\n   \u00a73 Step 2 the B(n) display corrected (4F(T), (2/3)Z\u2082(T),\n   2(e\u2074+1)\u00b77H(T) + 2\u00b7DL(n), I_Maj) + the F-A/F-B/F-D/F-E\n   correction-note footnote; \u00a73 Step 3 the bridge 4F (both signs);\n   Appendix B rewritten (second tier COMPLETE \u2014 SfBoundProp machine-\n   proved, the six analytic inputs enumerated, the full complex-\n   analytic content list); AI Contribution Statement factual refresh\n   (sessions 001\u2013041, closure declarations).\n5. **ii\u2032-7d (this commit)**: archive closed \u2014 D-037 erratum block\n   (the four corrections with pointers), R-004 Lean-closure status\n   line, NGUYEN.md (Phase 5B \u2192 COMPLETE, ii\u2032-6/ii\u2032-7 checked, Ledger\n   #6 machine-proved note), STATE.md (this file), journal-041.\n6. **Banked gotchas (journal-041)**: \u03a3 (U+03A3) is NOT an identifier\n   character in Lean 4 (lexes as Sigma-notation; a phantom ?m.NNN\n   metavariable goal at the DECLARATION HEAD is the tell-tale of a\n   downstream parse break); `\u222b x in a..b, body` captures trailing +/\u2212\n   into the integrand (parenthesize every integral summand);\n   ContinuousAt.comp/ContinuousWithinAt.comp need (g :=) (f :=)\n   (x :=) pinned (arg-first elaboration misdecomposes the point);\n   Continuous.div/.mul/.sub produce Pi-op forms (ascribe the per-term\n   type or show-bridge to the Pi form \u2014 the const ?c otherwise leaks\n   as an unsolved ?m); linarith CANNOT multiply by atom-constants\n   ((1/(2\u03c0)), (e\u2074+1)) \u2014 every such step needs a pre-packaged\n   mul_le_mul_of_nonneg_*; in large contexts use linarith only /\n   nlinarith only (the context scan on giant types deterministically\n   times out at isDefEq); norm_le_insert' : \u2016a\u2016 \u2264 \u2016b\u2016+\u2016a\u2212b\u2016 (the\n   unprimed has the args swapped); eq_div_iff for c = a/b goals\n   (div_eq_iff is for a/b = c); Finset.sum_mul for constant-RIGHT\n   extraction (Finset.mul_sum is constant-LEFT); le_of_pow_le_pow_left\u2080\n   takes the nonneg on the RIGHT (b) side; Real.add_one_le_exp is\n   x+1 \u2264 e^x (not 1+x \u2014 bridge by linarith); neg_le_abs : \u2212a \u2264 |a|\n   (neg_abs_le is \u2212|a| \u2264 a); HasDerivAt.pow yields Pi.pow forms (simpa\n   bridges); rw [hre] closes `-5/2 \u2264 -5/2` by auto-rfl le_refl\n   (le_of_eq hre.symm is deterministic).\n\n## CURRENT POSITION\n\u2714 PHASE 0, 1, 3, 4 COMPLETE. PHASE 2 (Track B): mapped to exhaustion\nat current technology (2.6\u20132.13, parked). PHASE 5: 5.3 \u2714 5.4 \u2714 5.2 \u2714\n(live). PHASE 5B (tier ii\u2032): **COMPLETE (session 041)** \u2014 all\nsub-stages ii\u2032-0..ii\u2032-7 done (see NGUYEN.md \u00a75B for the full log).\n\n**Theorem status (session 041 \u2014 the closed forms):**\n- T-1 (R-004): [PROVED-CONDITIONAL(computational RH-to-T\u2080;\n  computational finite verification)] \u2014 **FULLY MACHINE-PROVED in\n  Lean 4**: `lambda_pos_finite_height_closed` (LiClosed.lean) \u2014\n  \u03bb_n > 0 for ALL 1 \u2264 n \u2264 N\u2081 = (3\u00d710\u00b9\u00b2\u22121)\u00b2 from EXACTLY EIGHT\n  declared hypotheses: RHUpTo (Platt\u2013Trudgian), LiVerifiedUpTo\n  (finite Li verification), ZeroCountProp/ZeroCountProp2/\n  ZeroCountGlobalProp (Trudgian N(T)/S(T) forms), NoLowZeros\n  (Platt-class zero-location), MZeroProp (M\u2080 interval certificate).\n  Axiom audit on the full chain: [propext, Classical.choice,\n  Quot.sound] only, 0 sorryAx (AuditSession041, 29 decls).\n  sorryAx remains ONLY on the 5 skeleton statements (1 Suzuki + 4\n  valley) by design.\n- T-2 (R-003 + D-041): factorization + (\u2605) \u27fa (1.12) [PROVED modulo\n  Phase-5 formalities]; (1.12) \u27f9 spec(D) \u2282 \u211d [PROVED via Suzuki + D1];\n  \u21d0 [SKELETON L3/L4].\n- T-3 (valley, D-049 + D-058): [PROVED-CONDITIONAL(RH)].\n\n**Paper:** research/paper/paperA.tex \u2014 Paper A (finite-height Li +\nmachine-verified reduction), **9 pp, compiles clean, zero undefined\nrefs, CLOSED at ii\u2032-7** (eight-hypotheses headline; B(n) display\ncorrected + the F-A/F-B/F-D/F-E correction-note footnote; Appendix B:\nsecond tier COMPLETE). paperA.pdf committed. paperB.tex \u2014 spectral-\nline research note (skeleton, standing items in \u00a70), 6 pp. draft.tex \u2014\nFROZEN combined archive copy (12 pp).\n\n**Lean:** research/lean/NguyenRH1/ \u2014 full build GREEN (8775 jobs);\nall 26 modules root-imported incl. LiSideBounds (ii\u2032-6-c..d) and\nLiClosed (ii\u2032-7). sorryAx remains ONLY on: R003_exact_factorization\n(tier iv) + 4 valley statements (tier iii).\n\n**Deprioritized / open threads:** a=13+ push; adversary-ceiling LP;\nT4b; \u03bb-channel (G-004); \u03b8-channel (dead, E-024); Paper B Task 1\n(R-003 tempered-FT bookkeeping trace, queued); user read-through of\npaperA.pdf (NOW THE FRONT ITEM \u2014 nothing blocks it); repo-publication\nprep; submission decision (cover-letter.md + referee-list.md exist\nfrom session 013, target Experimental Mathematics); the c = 1.5\nfallback branch wip/c-1.5-margin-redo (642d47a, preserved, unneeded);\nthe T = \u221an hidden-cancellation open question (D-073 \u00a78 D4); the\nLagarias F-D correspondence package (user's own work-in-progress in\nresearch/correspondence/lagarias-F-D-package/, uncommitted \u2014 NOT\nmine to commit).\n\n## NEXT ACTION\n**AT A CLEAN BOUNDARY (session 041 end, ii\u2032-6 + ii\u2032-7 COMPLETE). The\nuser's read-through of paperA.pdf + the submission decision is the\nfront item. If told to continue research: Paper B Task 1 (R-003\ntempered-FT bookkeeping trace) or the parked Track-B remainders per\nthe user's call.**"
}
result 2026-08-26 21:29:54Z → 2026-08-26 21:29:54Z
{
  "filePath": "D:\\Python\\Quantum\\STATE.md",
  "oldString": "# STATE.md \u2014 NGUYEN-RH-1 RESUME POINT\n# \u2b50 READ THIS FIRST at the start of every session.\nLast updated: 2026-08-25 | **c = 1.5 MAIN-LINE REDO PAUSED (user\ndirective, session 040): the goal is the ORIGINAL N\u2081 = 9\u00d710\u00b2\u2074. The\nc = 1.5 WIP is preserved on branch `wip/c-1.5-margin-redo` (642d47a,\nBUILD BROKEN mid-convexity \u2014 resume map in the commit message);\nmaster is green (36b9e79) with the session-038 honest Bn (4F, (2/3)Z2)\nwhose right-tail charge \"3.3\" is the F-D-broken one (D-072). NEW LIVE\nDIRECTIVE: PROJECT_DIRECTIVE_D-074.txt \u2014 the ZERO-FREE DOG-LEG\nREPAIR: keep T_cut = \u221an (\u27f9 bridge/residues/N\u2081 = (T\u2080\u22121)\u00b2 unchanged),\ndeform ONLY the right horizontal tail [3, X_R] up to U = (3/2)\u221an\n(where D-073's repaired stepping applies, q = e^{4/9}/2 < 1) through\nthe UNCONDITIONALLY zero-free region Re w \u2265 3 > 1. Immediate task per\n\u00a719: execute \u00a72\u2013\u00a711 (coordinate audit \u2192 exact derivation \u2192 E-034 \u2192\ntemporary fold), then STOP and report answers A\u2013L.**\nSession 039 COMPLETE \u2014 **ii\u2032-6-R side thread COMPLETE (D-073): E-032\nverdict HIDDEN CANCELLATION (per-side claim empirically TRUE at\nT = \u221an, |I|/env down to 5.6e-10, but envelope-unprovable there);\nrepaired proof Lemma 4.1 + R(c) explicit (R(1.5) = 5.94 \u2014 reused by\nthe D-074 dog-leg's H_high verbatim); c = 1.5 global repair validated\nbut superseded by D-074; drafts D3/D4 in D-073 \u00a78.**",
  "newString": "# STATE.md \u2014 NGUYEN-RH-1 RESUME POINT\n# \u2b50 READ THIS FIRST at the start of every session.\nLast updated: 2026-08-26 | **ii\u2032-6 + ii\u2032-7 COMPLETE (session 041):\n`sfBoundProp : SfBoundProp` and `lambda_pos_finite_height_closed`\nPROVED in Lean from EXACTLY the eight declared hypotheses; full axiom\naudit clean; Paper A closed (B(n) correction note F-A/F-B/F-D/F-E,\nAppendix B rewritten, compiles clean 9 pp); archive closed (D-037\nerratum block, R-004 Lean-closure status, NGUYEN.md, this file,\njournal-041). The finite-height Li theorem \u03bb_n > 0 for ALL\n1 \u2264 n \u2264 N\u2081 = (3\u00d710\u00b9\u00b2\u22121)\u00b2 \u2248 9\u00d710\u00b2\u2074 is now FULLY MACHINE-PROVED with\nthe ORIGINAL N\u2081 PRESERVED (D-074 dog-leg endpoint). Master @ b64a839+\n(archive commits after). NEXT: user read-through of paperA.pdf +\nsubmission decision; deprioritized threads below.**\n\n## SESSION 041 HEADLINES (ii\u2032-6-c..6-d + ii\u2032-7 \u2014 THE CLOSURE SESSION)\n1. **ii\u2032-6-c COMPLETE** (4 commits): 6-c-ii I_III\n   `vIntegral_Jlag_left_norm_le` (6e5a98f \u2014 tails via the 5f tight\n   R-bound + near count (2/5)Z\u2082, middle via the perimeter MZeroProp,\n   even-symmetry tail identification); 6-c-iii I_I F-E form\n   `vIntegral_Jlag_right_le_exp_third_IIMaj` (9bd8ba9 \u2014 the uniform\n   kernel lemma `kFun_right_vertical_norm_le_exp_sqrt_add_third`:\n   plain triangle + \u2016w\u22121\u2016 \u2265 re(w\u22121) = 2\u221an\u22121, worst case t = 0);\n   6-c-iv the mid-horizontal segment machine (1fed1df \u2014 the\n   arcsinh-FTC `integral_one_div_sqrt_sq_add_sq` from scratch: the\n   direct antiderivative F(\u03c3) = log((\u03c3\u2212\u03b2)+\u221a((\u03c3\u2212\u03b2)\u00b2+\u03b4\u00b2)), g\u2032 = g/\u221ah\n   exactly; endpoints F(3) \u2264 log(6+\u03b4), \u2212F(\u22125/2) \u2264 log(7+\u03b4)+2log(1/\u03b4)\n   via the rationalization \u03b4\u00b2/(u+\u221a); per-zero \u2264 2\u00b7log(16.5/\u03b4) by PURE\n   log-mono ((6+\u03b4)(7+\u03b4) \u2264 56 \u2264 272.25 = 16.5\u00b2 \u2014 NO certified log\n   numerics)); 6-c-v the horizontal assembly (5d3dc0a \u2014 split at 3 via\n   HIntegral_split, mid by 6-c-iv, tails by DL-5, both sides).\n2. **6-d THE FOLD (774fec5): `sfBoundProp : SfBoundProp` PROVED** \u2014\n   |Sf n| \u2264 Bn FAlt n on [10\u2075, N\u2081] from EXACTLY the six declared\n   perimeter hypotheses (RHUpTo, ZeroCountProp, ZeroCountProp2,\n   ZeroCountGlobalProp, NoLowZeros, MZeroProp). `pinch_horizontal`\n   (NI + 5.5R \u2264 7\u00b7H\u0302) discharged by a fully elementary chain:\n   \u0108\u2080(\u221an+2) \u2264 \u221an+2 (`C0hat_le_self`, log x \u2264 x/e with 1/e \u2264 0.36789\n   from Real.exp_one_gt_d9, loglog \u2264 log\u22121) \u27f9 148.5\u00b7\u0108\u2080(\u221an+2) \u2264\n   (\u221an+2)\u00b2 \u27f9 its log \u2264 2\u00b7log(T+2); Z\u2082 \u2264 2\u00b7\u0108\u2080; \u0108\u2080 \u2265 0.433\u00b7log and\n   8 \u2264 21\u00b70.433 = 9.093; 5.5 \u2264 7 on the R-part. The bridge cloned at\n   (hTle : T \u2264 3\u00b710\u00b9\u00b2) \u2014 the range the internal\n   `zeroSum_nontriv_norm_le` actually uses (the public T \u2264 3\u00b710\u00b9\u00b2\u22121\n   hypothesis fails at the very top: n = N\u2081 has \u221an = 3\u00b710\u00b9\u00b2\u22121 exactly\n   and the pigeonhole T can land in (3\u00b710\u00b9\u00b2\u22121, 3\u00b710\u00b9\u00b2]).\n3. **ii\u2032-7a+7b (a0943a6)**: `lambda_pos_finite_height_closed` PROVED\n   (LiClosed.lean, root-imported) \u2014 \u03bb_n > 0 for ALL 1 \u2264 n \u2264 N\u2081 from\n   EXACTLY EIGHT declared hypotheses (two computational certificates:\n   RHUpTo, LiVerifiedUpTo; six analytic perimeter Props: ZeroCountProp,\n   ZeroCountProp2, ZeroCountGlobalProp, NoLowZeros, MZeroProp). Full\n   axiom audit (AuditSession041): 29 declarations \u2014 ALL [propext,\n   Classical.choice, Quot.sound] only, 0 sorryAx.\n4. **ii\u2032-7c (b64a839)**: Paper A closed (compiles clean, 9 pp, zero\n   undefined refs): abstract/intro \u2192 the eight-hypotheses headline;\n   \u00a73 Step 2 the B(n) display corrected (4F(T), (2/3)Z\u2082(T),\n   2(e\u2074+1)\u00b77H(T) + 2\u00b7DL(n), I_Maj) + the F-A/F-B/F-D/F-E\n   correction-note footnote; \u00a73 Step 3 the bridge 4F (both signs);\n   Appendix B rewritten (second tier COMPLETE \u2014 SfBoundProp machine-\n   proved, the six analytic inputs enumerated, the full complex-\n   analytic content list); AI Contribution Statement factual refresh\n   (sessions 001\u2013041, closure declarations).\n5. **ii\u2032-7d (this commit)**: archive closed \u2014 D-037 erratum block\n   (the four corrections with pointers), R-004 Lean-closure status\n   line, NGUYEN.md (Phase 5B \u2192 COMPLETE, ii\u2032-6/ii\u2032-7 checked, Ledger\n   #6 machine-proved note), STATE.md (this file), journal-041.\n6. **Banked gotchas (journal-041)**: \u03a3 (U+03A3) is NOT an identifier\n   character in Lean 4 (lexes as Sigma-notation; a phantom ?m.NNN\n   metavariable goal at the DECLARATION HEAD is the tell-tale of a\n   downstream parse break); `\u222b x in a..b, body` captures trailing +/\u2212\n   into the integrand (parenthesize every integral summand);\n   ContinuousAt.comp/ContinuousWithinAt.comp need (g :=) (f :=)\n   (x :=) pinned (arg-first elaboration misdecomposes the point);\n   Continuous.div/.mul/.sub produce Pi-op forms (ascribe the per-term\n   type or show-bridge to the Pi form \u2014 the const ?c otherwise leaks\n   as an unsolved ?m); linarith CANNOT multiply by atom-constants\n   ((1/(2\u03c0)), (e\u2074+1)) \u2014 every such step needs a pre-packaged\n   mul_le_mul_of_nonneg_*; in large contexts use linarith only /\n   nlinarith only (the context scan on giant types deterministically\n   times out at isDefEq); norm_le_insert' : \u2016a\u2016 \u2264 \u2016b\u2016+\u2016a\u2212b\u2016 (the\n   unprimed has the args swapped); eq_div_iff for c = a/b goals\n   (div_eq_iff is for a/b = c); Finset.sum_mul for constant-RIGHT\n   extraction (Finset.mul_sum is constant-LEFT); le_of_pow_le_pow_left\u2080\n   takes the nonneg on the RIGHT (b) side; Real.add_one_le_exp is\n   x+1 \u2264 e^x (not 1+x \u2014 bridge by linarith); neg_le_abs : \u2212a \u2264 |a|\n   (neg_abs_le is \u2212|a| \u2264 a); HasDerivAt.pow yields Pi.pow forms (simpa\n   bridges); rw [hre] closes `-5/2 \u2264 -5/2` by auto-rfl le_refl\n   (le_of_eq hre.symm is deterministic).\n\n## CURRENT POSITION\n\u2714 PHASE 0, 1, 3, 4 COMPLETE. PHASE 2 (Track B): mapped to exhaustion\nat current technology (2.6\u20132.13, parked). PHASE 5: 5.3 \u2714 5.4 \u2714 5.2 \u2714\n(live). PHASE 5B (tier ii\u2032): **COMPLETE (session 041)** \u2014 all\nsub-stages ii\u2032-0..ii\u2032-7 done (see NGUYEN.md \u00a75B for the full log).\n\n**Theorem status (session 041 \u2014 the closed forms):**\n- T-1 (R-004): [PROVED-CONDITIONAL(computational RH-to-T\u2080;\n  computational finite verification)] \u2014 **FULLY MACHINE-PROVED in\n  Lean 4**: `lambda_pos_finite_height_closed` (LiClosed.lean) \u2014\n  \u03bb_n > 0 for ALL 1 \u2264 n \u2264 N\u2081 = (3\u00d710\u00b9\u00b2\u22121)\u00b2 from EXACTLY EIGHT\n  declared hypotheses: RHUpTo (Platt\u2013Trudgian), LiVerifiedUpTo\n  (finite Li verification), ZeroCountProp/ZeroCountProp2/\n  ZeroCountGlobalProp (Trudgian N(T)/S(T) forms), NoLowZeros\n  (Platt-class zero-location), MZeroProp (M\u2080 interval certificate).\n  Axiom audit on the full chain: [propext, Classical.choice,\n  Quot.sound] only, 0 sorryAx (AuditSession041, 29 decls).\n  sorryAx remains ONLY on the 5 skeleton statements (1 Suzuki + 4\n  valley) by design.\n- T-2 (R-003 + D-041): factorization + (\u2605) \u27fa (1.12) [PROVED modulo\n  Phase-5 formalities]; (1.12) \u27f9 spec(D) \u2282 \u211d [PROVED via Suzuki + D1];\n  \u21d0 [SKELETON L3/L4].\n- T-3 (valley, D-049 + D-058): [PROVED-CONDITIONAL(RH)].\n\n**Paper:** research/paper/paperA.tex \u2014 Paper A (finite-height Li +\nmachine-verified reduction), **9 pp, compiles clean, zero undefined\nrefs, CLOSED at ii\u2032-7** (eight-hypotheses headline; B(n) display\ncorrected + the F-A/F-B/F-D/F-E correction-note footnote; Appendix B:\nsecond tier COMPLETE). paperA.pdf committed. paperB.tex \u2014 spectral-\nline research note (skeleton, standing items in \u00a70), 6 pp. draft.tex \u2014\nFROZEN combined archive copy (12 pp).\n\n**Lean:** research/lean/NguyenRH1/ \u2014 full build GREEN (8775 jobs);\nall 26 modules root-imported incl. LiSideBounds (ii\u2032-6-c..d) and\nLiClosed (ii\u2032-7). sorryAx remains ONLY on: R003_exact_factorization\n(tier iv) + 4 valley statements (tier iii).\n\n**Deprioritized / open threads:** a=13+ push; adversary-ceiling LP;\nT4b; \u03bb-channel (G-004); \u03b8-channel (dead, E-024); Paper B Task 1\n(R-003 tempered-FT bookkeeping trace, queued); user read-through of\npaperA.pdf (NOW THE FRONT ITEM \u2014 nothing blocks it); repo-publication\nprep; submission decision (cover-letter.md + referee-list.md exist\nfrom session 013, target Experimental Mathematics); the c = 1.5\nfallback branch wip/c-1.5-margin-redo (642d47a, preserved, unneeded);\nthe T = \u221an hidden-cancellation open question (D-073 \u00a78 D4); the\nLagarias F-D correspondence package (user's own work-in-progress in\nresearch/correspondence/lagarias-F-D-package/, uncommitted \u2014 NOT\nmine to commit).\n\n## NEXT ACTION\n**AT A CLEAN BOUNDARY (session 041 end, ii\u2032-6 + ii\u2032-7 COMPLETE). The\nuser's read-through of paperA.pdf + the submission decision is the\nfront item. If told to continue research: Paper B Task 1 (R-003\ntempered-FT bookkeeping trace) or the parked Track-B remainders per\nthe user's call.**",
  "originalFile": "# STATE.md \u2014 NGUYEN-RH-1 RESUME POINT\n# \u2b50 READ THIS FIRST at the start of every session.\nLast updated: 2026-08-25 | **c = 1.5 MAIN-LINE REDO PAUSED (user\ndirective, session 040): the goal is the ORIGINAL N\u2081 = 9\u00d710\u00b2\u2074. The\nc = 1.5 WIP is preserved on branch `wip/c-1.5-margin-redo` (642d47a,\nBUILD BROKEN mid-convexity \u2014 resume map in the commit message);\nmaster is green (36b9e79) with the session-038 honest Bn (4F, (2/3)Z2)\nwhose right-tail charge \"3.3\" is the F-D-broken one (D-072). NEW LIVE\nDIRECTIVE: PROJECT_DIRECTIVE_D-074.txt \u2014 the ZERO-FREE DOG-LEG\nREPAIR: keep T_cut = \u221an (\u27f9 bridge/residues/N\u2081 = (T\u2080\u22121)\u00b2 unchanged),\ndeform ONLY the right horizontal tail [3, X_R] up to U = (3/2)\u221an\n(where D-073's repaired stepping applies, q = e^{4/9}/2 < 1) through\nthe UNCONDITIONALLY zero-free region Re w \u2265 3 > 1. Immediate task per\n\u00a719: execute \u00a72\u2013\u00a711 (coordinate audit \u2192 exact derivation \u2192 E-034 \u2192\ntemporary fold), then STOP and report answers A\u2013L.**\nSession 039 COMPLETE \u2014 **ii\u2032-6-R side thread COMPLETE (D-073): E-032\nverdict HIDDEN CANCELLATION (per-side claim empirically TRUE at\nT = \u221an, |I|/env down to 5.6e-10, but envelope-unprovable there);\nrepaired proof Lemma 4.1 + R(c) explicit (R(1.5) = 5.94 \u2014 reused by\nthe D-074 dog-leg's H_high verbatim); c = 1.5 global repair validated\nbut superseded by D-074; drafts D3/D4 in D-073 \u00a78.**\nSession 038 COMPLETE \u2014 **ii\u2032-3b + ii\u2032-3c +\nii\u2032-6-0 + ii\u2032-6-a + ii\u2032-6-b DONE; FINDING F-D DOCUMENTED (D-072,\nuser paused to think): the (6.12) stepping in Lagarias's Thm 6.1 is\nbroken at T = \u221an (product ratio e/2 > 1; right-tail ~e^{0.066\u221an},\n~5.9\u00d710\u00b9\u2070 envelope at n = 10\u2075 measured by E-031); D-036/D-037/R-004's\nB(n) inherits it (the right-tail charge \"3.3\" is off by ~10\u2077\u00d7; margin\ncollapses for n \u2273 3\u00d710\u2074). The theorem statement survives (verified to\n5\u00d710\u2075; sides cancel by conjugation + phase winding); ALL Lean work to\ndate survives (the bug is in the horizontal piece ii\u2032-6-c, not yet\nbuilt). Repair path (R) verified: T = c\u221an, c > 1.2012 (kernel ratio\n< 2 \u27f9 stepping works; monotone decrease confirmed at c = 1.5, 2);\nN\u2081 \u2192 (T\u2080/c)\u00b2 \u2248 2.25\u20134\u00d710\u00b2\u2074.**\n\n## SESSION 039 HEADLINES (ii\u2032-6-R side thread \u2014 COMPLETE)\n1. **6-R-0 (E-032, ALL CHECKS PASS, commit ded9603)**: the TRUE\n   complex right-tail integral at T = \u221an (zero-avoiding) by direct\n   quadrature \u2014 Dirichlet series with certified tail (\u03c3 \u2265 5) +\n   mpmath (validated 6\u00d710\u207b\u00b9\u2070), dps 50, trapezoid h-delta \u2264 5e-5.\n   **VERDICT: HIDDEN CANCELLATION, decisive** \u2014 |I_right| =\n   7.18/16.70/26.23 vs envelope 50.8/1.03e4/4.71e10 (n = 10\u00b3/10\u2074/10\u2075),\n   ratio 0.141 \u2192 1.6e-3 \u2192 **5.57e-10**; full side |I_II| =\n   7.44/19.21/30.89 (|I_II|/\u221an DECREASING 0.235 \u2192 0.098): the per-side\n   claim is empirically TRUE but envelope-unprovable at T = \u221an.\n   Mid-segment [\u22125/2, 3] almost no cancellation (0.7\u20130.95) \u2014 its bound\n   honest. Repair heights: envelope collapses (\u22481.0 at c = 1.5,\n   \u22480.35 at c = 2), ratio \u2248 1 \u2014 the repair needs no cancellation.\n   STOP-rule reported; user direction: **repair proof only**, the\n   T = \u221an salvage parked as an open question (D-073 \u00a78 D4).\n2. **6-R-1 (D-073 \u00a7\u00a73\u20137)**: dependency audit (13 ingredients; ONLY\n   the (6.12) stepping is proof-breaking; the (1+4/(16+n))\u207f\n   intermediate is a harmless looseness \u2014 e\u2074+1 valid); **Lemma 4.1\n   (corrected stepping, P(\u03c3+1)/P(\u03c3) \u2264 q = e^{1/c\u00b2}/2) + Cor 4.2\n   (explicit R(c)) [PROVED, elementary, self-contained]**; c-choice\n   with AUDIT CORRECTION (recommend **c = 1.5**, reversing D-072's\n   \"2 margin/1.5 range\" \u2014 the fold is I_III-length-dominated:\n   margin(10\u2075) 181 256 vs 109 370, N\u2081 4\u00d710\u00b2\u2074 vs 2.25\u00d710\u00b2\u2074; c = 2\n   documented as the conservative option); consequence audit (Thm 6.1\n   statement unchanged, O-constant c-dependent; Thm 1.1 height\n   ((T\u2080\u22121)/c)\u00b2; NO new perimeter); **E-033 ALL CHECKS PASS** (Z\u2080\n   certified [0.5699210, 0.5703661]; q-table incl. q(1) = 1.359 = the\n   bug; R covers the E-032 measured truth with 3.2\u20136.3\u00d7 slack; kernel\n   charges 5.65 \u2264 6.92 / 2.06 \u2264 3.72; fold fit \u2264 0.3925/0.3883;\n   margins > 0 on [8\u00d710\u2074, N\u2081(c)] both c; full fold). V1 first-run\n   failure was a script-tolerance error (certified bracket correct),\n   fixed, audit trail in git.\n3. **6-R-2 (D-073 \u00a78)**: D3 letter repair paragraph + D4 T = \u221an\n   per-side verdict paragraph \u2014 DRAFTS ONLY; correspondence files\n   untouched (freeze protocol in force).\n4. **THREAD COMPLETE** \u2014 all three success criteria met. Main line\n   resumes with repair (R): see NEXT ACTION.\n\n## SESSION 038 HEADLINES (ii\u2032-6-0 \u2014 THIS SESSION'S ENDPOINT)\n0. **ii\u2032-6-0 DESIGN (D-071) + E-030 (ALL CHECKS PASS) + two user\n   approvals**: THREE findings \u2014 F-A bridge factor-2 (\u03bb_n(T) is\n   both-signs; honest 4\u00b7FAlt vs baked 2\u00b7FAlt; R-004 unaffected, paper\n   erratum at ii\u2032-7); F-B I_III two-unit count ((2/3)Z2 honest vs\n   (2/3)\u0108\u2080 baked \u2014 D-069 Finding 2 made exact); F-C baked Bn unprovable\n   beyond ~5\u00d710\u2077 (killers grow T\u00b7log T > (log T)\u00b2 slack).\n   RESOLUTION (user-approved 8-step protocol): redefine Bn honestly\n   (4F, (2/3)Z2) \u2014 fold works everywhere (V4), margin positive + larger\n   near 10\u2075 (V2), horizontal segment-integral \u2264 0.40 of share (V3).\n   Perimeter additions approved: ZeroCountGlobalProp (global Trudgian\n   N(T) \u2264 FAlt) + MZeroProp (M\u2080 \u2264 1.3352). Sub-stage map ii\u2032-6-a..d \u2192\n   ii\u2032-7.\n0c. **FINDING F-D (D-072, user paused to think)**: the (6.12)\n   stepping in Lagarias's Thm 6.1 is broken at T = \u221an \u2014 the per-step\n   product ratio is e/2 > 1 (zeta halves, kernel ratio \u2264 e), so the\n   horizontal right-tail GROWS from \u03c3 = 3 to \u03c3 \u2248 0.36\u221an, peaking at\n   ~e^{0.066\u221an}. E-031 (ALL CHECKS PASS): right-tail envelope\n   ~5.9\u00d710\u00b9\u2070 at n = 10\u2075 (peak 1.6\u00d710\u2079 at \u03c3 = 111); direct factor\n   evaluation |k_n(100+317i)| = 4.4\u00d710\u00b3\u2079, |\u03b6\u2032/\u03b6(101+317i)| \u2248\n   1.8\u00d710\u207b\u00b3\u00b9. D-036/D-037/R-004 inherit it verbatim (charge \"3.3\" off\n   by ~10\u2077\u00d7; margin collapses for n \u2273 3\u00d710\u2074). The theorem statement\n   survives (verified to 5\u00d710\u2075; sides cancel by conjugation + phase\n   winding); all Lean work to date survives (the bug is in the\n   horizontal piece, not yet built). Repair path (R) verified:\n   T = c\u221an with c > 1.2012 \u27f9 stepping works (monotone decrease at\n   c = 1.5, 2); N\u2081 \u2192 (T\u2080/c)\u00b2 \u2248 2.25\u20134\u00d710\u00b2\u2074. No erratum found in the\n   literature (arXiv v4, web search).\n0b. **ii\u2032-6-b COMPLETE**: perimeter Props ZeroCountGlobalProp +\n   MZeroProp declared (LiContour); the wide pigeonhole\n   exists_zero_avoiding_T_wide (3-unit Finding-2 repair, clearance\n   1/(9\u00b7C0hat(a+2))); LiAssembly.lean NEW (~460 lines, root-imported):\n   the sides identity sf_abs_le_rectInt_add_zeroSum (rectInt'_Jlag on\n   the variant-B rectangle; border zero-free via the zero\n   classification + wide pigeonhole; |Sf n| \u2264 \u2016RectInt\u2016 + \u2016\u03a3\u2016) + the\n   bridge zeroSum_norm_le_four_FAlt_add_one (\u2016\u03a3\u2016 \u2264 4\u00b7FAlt + 1 via\n   RHUpTo + conj-image reindex + NoLowZeros + trivTerm \u2264 1). Full\n   build GREEN (8772 jobs); audit session038d: 12 decls, 3 standard\n   axioms only. Banked (journal-038): the RectangleBorder-rcases trap\n   (rw + simp only [mem_union, mem_reProdIm, mem_singleton_iff] +\n   tauto route); Finset.image plain-function + Set.InjOn form; set-x\n   breaks mem_toFinset patterns; linarith proves \u2260 directly;\n   projections for IsNontrivialZero (structure opacity); sq_eq_sq\u2080;\n   pow_le_one\u2080's implicit n.\n0a. **ii\u2032-6-a COMPLETE** (user protocol steps 1\u20135): Bn redefined\n   honestly (4F, (2/3)Z2hat \u2014 new def with the D-071 comment);\n   marginTilde/mFblock/mTL1b/mIdb lockstep; marginTilde_convex\n   RE-PROVED (\u03c8-row literal updates + norm_num; the eq_expanded needed\n   Z2hat in the unfold list; a hardcoded const_mul (\u22122) in\n   continuousOn_marginExpanded); base points RE-CERTIFIED\n   (Bn_FAlt_80000_le 328,000 \u2192 329,000, Bn_FAlt_100000_le 347,000 \u2192\n   350,000 \u2014 the old bounds FAIL for the corrected Bn; margins\n   \u2265 7,001 / 82,501, doc comments updated); endpoint margins re-run\n   (margin\u0303(8\u00d710\u2074) = 14,682, margin\u0303(10\u2075) = 90,653, margin(N\u2081) =\n   2.458\u00d710\u00b2\u2076 unchanged); full build GREEN (8771 jobs); audit\n   session038c: endpoint + convexity + base points all 3 standard\n   axioms only. Banked: lake env lean uses the STALE olean \u2014 lake\n   build the dependency first; re-check certified constants at 1e-4.\n## SESSION 038 HEADLINES (ii\u2032-3c)\n1. **ii\u2032-3c COMPLETE** (LiCrux.lean NEW, ~430 lines, root-imported):\n   `gFun_one_sub_fun` (FE as a function identity, xiL_one_sub +\n   chain rule), `iteratedDeriv_gFun_zero` (g^{(m)}(0) =\n   (\u22121)^{m+1}g^{(m)}(1) via the UNCONDITIONAL\n   iteratedDeriv_comp_const_sub), `gFun_one_add_punctured`/\n   `gFun_one_add_eventuallyEq` (the shift expansion gFun(1+s) =\n   1/(1+s) + hLiExt s + \u0393R\u2032/\u0393R(1+s)), `iteratedDeriv_gFun_one`\n   (coefficient extraction (\u22121)^m\u00b7m! \u2212 m!\u00b7c_m + (\u22121)^{m+1}\u00b7m!\u00b7a_{m+1}),\n   **`lagarias_4_10_a (hZC)` PROVED** (lcoeff (k\u22121) = (\u22121)^k(a_k +\n   sigmaP k) + (\u22121)^{k\u22121}, all k \u2265 1 \u2014 the multiplied-out\n   linear_combination + mul_left_cancel\u2080 (m! \u2260 0)), and **THE CRUX\n   `sf_eq_resW1 (hZC)` PROVED**: (ResW1 n).re = Sf n for n \u2265 1\n   (ResW1_eq_sum + (4.10-a) + both ii\u2032-2 swaps + the binomial sum\n   \u03a3(\u22121)^{j\u22121}C(n,j) = 1). Audit session038b: 8 decls, 3 standard\n   axioms only. **ii\u2032-3 (ALL of Lagarias (4.10)) \u2192 COMPLETE.**\n2. **ii\u2032-3b COMPLETE** (LiMoments.lean \u00a75/\u00a76, ~290 new lines, file now\n   ~1020 lines): \u00a75 the termwise iterated derivation at 0 \u2014\n   `iteratedDeriv_zpow_neg_one`, `iteratedDeriv_seriesTerm`,\n   `seriesTerm_deriv_summableLocallyUniformlyOn` (the M-test on\n   ball(0, \u03b4/2), dominator j!\u00b72^{j+1}\u00b7(1/\u03b4)^{j\u22121}\u00b7mult/\u2016\u03c1\u2016\u00b2),\n   `iteratedDeriv_seriesFun_zero` (the LiDigamma:749\u2013785\n   iteratedDerivWithin_tsum pattern verbatim). \u00a76:\n   `gFun_eventuallyEq_zero` (\ud835\udcdd[\u2260] \u2192 \ud835\udcdd via eventually_nhdsWithin_iff) +\n   **THE CAPSTONE `sigmaP_eq_neg_iteratedDeriv_gFun_div`**: sigmaP k =\n   \u2212(iteratedDeriv (k\u22121) gFun 0)/(k\u22121)! for ALL k \u2265 1 (k=1 via \u00a74's\n   reindex; k\u22652 via (G) + \u00a75 + the per-term sign algebra\n   (\u22121)^k\u00b7(\u22121)^{k+1} = \u22121 + \u2190 sigmaP_eq_unpaired + field_simp k! \u2260 0;\n   FIRST-PASS GREEN). Perimeter: **ZeroCountProp ONLY**.\n3. **Closure (3b)**: LiTightR + LiMoments root-imported; AuditSession038.lean +\n   audit-session038-axioms.txt \u2014 23 declarations, [propext,\n   Classical.choice, Quot.sound] ONLY, 0 sorryAx.\n4. Banked gotchas (journal-038): factorial `!` needs `open scoped Nat`;\n   Summable.mul_left dominators must be C * (D \u03c1) (constant FIRST \u2014\n   D \u03c1 * C whnf-loops the refine); rw [seriesFun] fails on the UNAPPLIED\n   constant (use `show` beta-expansion); norm_sub_norm_le has NO abs on\n   the LHS in this Mathlib; don't rw-away memberships still needed as\n   memberships (simpa-only a copy).\n\n## SESSION 037 HEADLINES (superseded \u2014 ii\u2032-3b now DONE)\n1. **Capstone dependency audit** (user-requested, pre-ii\u2032-3b):\n   `AuditSession037.lean` + `audit-session037-axioms.txt` \u2014\n   `#print axioms logDeriv_riemannZeta_sub_nearZeros_le_tight` =\n   **[propext, Classical.choice, Quot.sound] ONLY**; the exact perimeter\n   recorded verbatim from the elaborator: `ZeroCountProp` (unit window,\n   mult-weighted, \u2264 C0hat(t+1), t \u2265 2; C0hat t = 0.433\u00b7log(t+2) +\n   0.886\u00b7log log(t+2) + 2.9), `ZeroCountProp2` (two-unit, \u2264 0.6\u00b7L +\n   0.886\u00b7LL + 2.7, t \u2265 10), `NoLowZeros` (\u2200 \u03c1 nontrivial, 14 \u2264 |Im \u03c1|);\n   constructs nearZerosSub / Zeta23.zeroMult / zeroAbsWindow also printed.\n2. **ii\u2032-3b ROUTE REVISION (D-070)**: the moments (M)\n   sigmaP k = \u2212gFun^{(k\u22121)}(0)/(k\u22121)! come from 5e's\n   `partialFraction_identity (hZC : ZeroCountProp)` by PURE ALGEBRA +\n   series calculus \u2014 NO contours (D-066 \u00a73's rectangle plan superseded;\n   the contour work was 5e's). The identity (G): gFun := logDeriv xiL =\n   B + seriesFun at every good point; xiL := (s(s\u22121)/2)\u00b7\u039b entire via the\n   \u039b\u2080 closed form xiL s := (s(s\u22121)/2)\u00b7\u039b\u2080(s) \u2212 (s\u22121)/2 + s/2 (Mathlib\n   `completedRiemannZeta_eq`). Perimeter: ZeroCountProp ONLY (5f's\n   hZC2/hNLZ NOT needed for the moments).\n3. **LiMoments.lean \u00a71\u2013\u00a74 COMPLETE, build GREEN** (~730 lines; new file,\n   NOT yet root-imported; imports NguyenRH1.LiTightR + LiSwaps + LiResidue\n   + LiDigamma + LiReflect):\n   \u00a71 xiL (differentiable/analytic everywhere, xiL_eq_mul off {0,1},\n   xiL_zero/xiL_one = 1/2, xiL_one_sub FE, xiL_ne_zero) + the subtype\n   involutions conjZeroEquiv/oneSubZeroEquiv + isNontrivialZero_one_sub;\n   \u00a72 `gFun_eq_momentB` \u2014 (G) at good points (logDeriv_mul chains +\n   logDeriv_Gamma\u211d_eq_of_not_neg_nat + momentB_spec);\n   \u00a73 `eq_of_eventuallyEq_nhd_ne`, good-eventually sets at 0 (\u03b6(0) = \u22121/2)\n   and 1 (zetaA), `seriesFun_zero` (= 0), `seriesFun_one` (= \u03a3\n   mult/(\u03c1(1\u2212\u03c1))), `gFun_zero` (gFun 0 = B), `gFun_one` (gFun 1 = B +\n   seriesFun 1), `gFun_one_eq_neg` (FE sign, deriv_comp chain rule);\n   \u00a74 summability budget (master `summable_mult_div_norm_sq`,\n   `exists_norm_lower_bound` \u03b4-ball, `summable_mult_inv_pow` k \u2265 2, AM-GM\n   `mult_div_norm_mul_norm_le`) + reindex machinery\n   (`tsum_mult_inv_pow_conj`, `sigmaP_eq_unpaired` k \u2265 2,\n   `summable_mult_div_norm_sq_one_sub`, `vTerm_tsum_eq_zero` via V = W =\n   \u2212V) and **the k = 1 capstone `sigmaP_one_eq_neg_gFun_zero` PROVED**\n   (2\u03c3\u2081 = seriesFun 1 = gFun 1 \u2212 B = \u22122\u00b7gFun 0 by the FE sign).\n\n## CURRENT POSITION\n\u2714 PHASE 0, 1, 4 COMPLETE. PHASE 3 (Track C): 3.1\u20133.8 CLOSED (R-004,\nfull tag). PHASE 2 (Track B): mapped to exhaustion at current\ntechnology (2.6\u20132.13). PHASE 5: 5.3 \u2714, 5.4 \u2714, 5.2 \u2714 (live);\n5.1 skeleton \u2714 + tier (i) \u2714 (session 011) + tier (ii) STEP 1 \u2714\n(session 012). PHASE 5B (tier ii\u2032, user decision: option (a) \u2014 close\nSfBoundProp in Lean BEFORE submission; Paper A frozen/submittable\nthroughout): DONE: ii\u2032-0 (D-064 design + liCoeff defect + crux Sf =\nRes\u2081 found), ii\u2032-1a (liCoeff defect REPAIRED), ii\u2032-1b (R1 residue\nmachinery: rectInt'_Jlag), ii\u2032-2 (elementary swaps), ii\u2032-3-DESIGN\n(D-066, \u03b24), ii\u2032-3a (pieces A+B: LiZetaCoeff, LiArchimedean, LiDigamma),\nii\u2032-4 (LiContour.lean: L3 exists_zero_avoiding_T + L1 kernel bounds +\nzeroMult_neg_two), ii\u2032-5a (D-067 design), ii\u2032-5b (LiRBound.lean:\nlevel-2 \u03b6 representation + norm_riemannZeta_le_disc \u2016\u03b6\u2016 \u2264 (3/4)|t|\u00b2),\nii\u2032-5c (LiDisc.lean: the loose disc R-bound \u2264 100000\u00b7(log|t|)\u00b2 on\n\u03c3 \u2208 [1/2, 7/2] via R-gap + Blaschke + BC, Jensen count, NO perimeter),\nii\u2032-5d (LiReflect.lean: FE-reflection + left loose bound \u2264\n200000\u00b7(log|t|)\u00b2 on \u03c3 \u2208 [\u22125/2, 1/2] + the 5e digamma input), ii\u2032-5e\n(LiTrivialZeros, LiGapCircles, LiDigammaLeft, LiSeries, LiFEntire,\nLiCircleBound (~3928 lines), LiSqueeze \u2014 **the partial-fraction identity\n`partialFraction_identity (hZC : ZeroCountProp)` PROVED**), ii\u2032-5f\n(LiTightR.lean ~4300 lines \u2014 **the tight R-bound\n`logDeriv_riemannZeta_sub_nearZeros_le_tight (hZC) (hZC2) (hNLZ)` \u2264\n10.5\u00b7log(|t|+2) + 48 PROVED**; capstone dependency-audited session 037:\n3 standard axioms only, perimeter verbatim). **ii\u2032-3b COMPLETE (sessions\n037\u2013038, LiMoments.lean ~1020 lines): the moments capstone\n`sigmaP_eq_neg_iteratedDeriv_gFun_div (hZC)` PROVED for all k \u2265 1 \u2014\nD-070 route, perimeter ZeroCountProp ONLY. **ii\u2032-3c COMPLETE (session\n038, LiCrux.lean ~430 lines): `lagarias_4_10_a (hZC)` + THE CRUX\n`sf_eq_resW1 (hZC)` PROVED \u2014 Sf(n) = Res\u2081(n) in Lean. ii\u2032-3 (ALL of\n(4.10)) DONE.**\n\n**Theorem status (session 012 Lean upgrade + session 036):**\n- T-1 (R-004): [PROVED-CONDITIONAL(computational RH-to-T\u2080;\n  computational finite verification)] \u2014 **Lean-verified modulo\n  SfBoundProp + the declared computational perimeter only**:\n  `lambda_pos_finite_height_verified` (LiBasePoints.lean) PROVED from\n  (hRH : RHUpTo 3\u00d710\u00b9\u00b2) (hfinite : LiVerifiedUpTo 5\u00d710\u2075)\n  (hSf : SfBoundProp). PrintAxioms = 3 standard axioms, no sorryAx.\n  The two margin evaluations are discharged by certified interval\n  bounds (margin\u0303(8\u00d710\u2074) \u2265 8001 certified / 15 764 true; margin\u0303(10\u2075)\n  \u2265 85 501 certified). LiSkeleton + LiBasePoints both sorry-free.\n- T-2 (R-003 + D-041): factorization + (\u2605) \u27fa (1.12) [PROVED modulo\n  Phase-5 formalities]; (1.12) \u27f9 spec(D) \u2282 \u211d [PROVED via Suzuki + D1];\n  \u21d0 [SKELETON L3/L4].\n- T-3 (valley, D-049 + D-058): [PROVED-CONDITIONAL(RH)].\n\n**Paper:** research/paper/paperA.tex \u2014 Paper A (finite-height Li +\nmachine-verified assembly), 8 pp, compiles clean, zero undefined refs.\npaperB.tex \u2014 spectral-line research note (skeleton, standing items in\n\u00a70), 6 pp. draft.tex \u2014 FROZEN combined archive copy (12 pp). No\nfigures yet.\n\n**Lean:** research/lean/NguyenRH1/ \u2014 full build GREEN (8771 jobs);\nLiTightR + LiMoments + LiCrux root-imported (session 038). sorryAx\nremains ONLY on: R003_exact_factorization (tier iv) + 4 valley\nstatements (tier iii).\n\n## ACTIVE PLAN (tier ii\u2032, per D-064 \u00a76)\nFreeze protocol in force (user directive 2026-08-21): no edits to\npaperA.tex/paperB.tex/draft.tex/references.bib until ii\u2032-7; no edits\nto existing Lean proofs except the liCoeff correctness fix (ii\u2032-1a);\ncommit at every sub-stage. Fallback integrity: D-064 \u00a78.\nRemaining sub-stages: **ii\u2032-6** (L0\nassembly: sides identity + bridge |\u03bb_n(T)| \u2264 2\u00b7FAlt(T) (hRH) +\n|Sf n| \u2264 Bn FAlt n on [10\u2075, N\u2081]; D-069 Finding 2's H(T) near-sum\nrepair map (honest \u2248 7\u00b7\u0108\u2080\u00b2 vs baked 3\u00b7\u0108\u2080\u00b2, margin survives via E-028's\n26\u00d7 slack) lives here) \u2192 **ii\u2032-7** (closure: theorem sfBoundProp :\nSfBoundProp; final lambda_pos_finite_height_closed with the 3 perimeter\nhypotheses only; full axiom audit; paper Appendix B + tag upgrade\n(D-062 \u00a74 endpoint); archive closure + D-037 line-63 erratum pointer).\nDeprioritized: a=13+ push; adversary-ceiling LP; T4b; \u03bb-channel\n(G-004); \u03b8-channel (dead, E-024); Paper B Task 1 (R-003 bookkeeping)\nholds as queued after ii\u2032; user read-through of paperA.pdf still open\n(nothing blocks it); repo-publication prep still open.\n\n## NEXT ACTION\n**STOPPED AT A CLEAN BOUNDARY (session 040 block B, user log-off):\nii\u2032-6-c-i COMPLETE + COMMITTED (master @ a8c5b1e); 6-c-ii DESIGNED\nbut NO Lean written yet. Resume at 6-c-ii below.**\nCompleted this block: ii\u2032-6-a2 committed (0dfbd6e); FINDING F-E +\nrepair decision logged (D-074 \u00a79, 8068ea6); 6-c-i scaffolding\n(a8c5b1e, LiSideBounds.lean \u2014 build GREEN; provides Z2hat_mono,\nnearZerosSub_mult_sum_le, continuousAt/On_logDeriv_zeta,\nJlag_norm_eq, zeta_ne_zero_left_vertical, zeta_ne_zero_horizontal,\nrectangleIntegral_Jlag_eq, VIntegral_norm, HIntegral_split,\njlag_left_vertical_norm_even). Journal: session-040.md CONTINUATION\nsection. NGUYEN.md checklist updated. Git identity: pass per-command\n(-c user.name=\"NGUYEN-RH-1\" -c user.email=\"research@local\") \u2014 never\nconfig.\n**ii\u2032-6-c sub-stage map (file research/lean/NguyenRH1/NguyenRH1/\nLiSideBounds.lean, n \u2265 10\u2075 throughout so T \u2265 316):**\n6-c-i \u2713 \u2192 **6-c-ii I_III (NEXT)** \u2192 6-c-iii I_I (F-E form) \u2192 6-c-iv\nmid-horizontal machine \u2192 6-c-v horizontal assembly \u2192 6-d fold \u2192 ii\u2032-7.\n**6-c-ii RECIPE (I_III, left vertical \u03c3 = \u22125/2; target statement):**\n`vIntegral_Jlag_left_norm_le (hZC) (hZC2) (hNLZ) (hMZ : MZeroProp)\n{n} (hn : 100000 \u2264 n) {T} (hTn : \u221an \u2264 T) (hTn1 : T \u2264 \u221an+1) :\n\u2016VIntegral (Jlag n) (-5/2) (-T) T\u2016 \u2264 4*T*((2/3)*Z2hat T +\n10.5*log(T+2) + 48) + 40*1.3352`.\n1. VIntegral_norm + norm_integral_le_integral_norm (\u2212T \u2264 T):\n   \u2016\u00b7\u2016 \u2264 \u222b_{\u2212T}^{T} g, g t := \u2016Jlag n (\u2191(-5/2:\u211d)+\u2191t\u00b7I)\u2016.\n   Integrability of g: Jlag continuous on the path\n   (kFun_differentiableAt (s\u22601 via re=\u22125/2) .continuousAt + my\n   continuousAt_logDeriv_zeta (zeta_ne_zero_left_vertical) \u2192 .mul/.neg)\n   composed with the path map u \u21a6 \u2191x+\u2191u\u00b7I (fun_prop) \u2192 .norm \u2192\n   ContinuousOn.intervalIntegrable.\n2. Split \u222b_{\u2212T}^{T} = \u222b_{\u2212T}^{\u221210} + \u222b_{\u221210}^{10} + \u222b_{10}^{T}\n   (integral_add_adjacent_intervals \u00d72). The left piece:\n   \u222b_{\u2212T}^{\u221210} g = \u222b_{10}^{T} g \u2014 g is EVEN\n   (jlag_left_vertical_norm_even) + integral_comp_neg\n   (\u222b_a^b f(\u2212x) = \u222b_{\u2212b}^{\u2212a} f).\n3. Middle: per t \u2208 Icc(\u221210,10): g t \u2264 2\u00b71.3352 (Jlag_norm_eq +\n   kFun_norm_le_two (re \u2264 1/2) + hMZ t (|t| \u2264 10); CAST BRIDGE:\n   MZeroProp's point is (-5/2:\u2102)+\u2191t\u00b7I, mine is \u2191(-5/2:\u211d)+\u2191t\u00b7I \u2014\n   `(-5/2:\u2102) = \u2191(-5/2:\u211d)` by push_cast; ring). Then integral_mono_on\n   + integral_const: \u2264 20\u00b7(2\u00b71.3352) = 40\u00b71.3352.\n4. Tails: per t \u2208 Icc(10,T): g t \u2264 K := 2*((2/3)*Z2hat T +\n   10.5*log(T+2) + 48). Via Jlag_norm_eq + kFun_norm_le_two +\n   \u2016logDeriv\u2016 \u2264 \u2016logDeriv \u2212 \u03a3near\u2016 + \u2016\u03a3near\u2016:\n   - capstone logDeriv_riemannZeta_sub_nearZeros_le_tight hZC hZC2\n     hNLZ (ht : 10 \u2264 t) (h\u03c3lo/h\u03c3hi : re = \u22125/2, by simp)\n     (him : s.im = t, by simp) (h\u03b6 : zeta_ne_zero_left_vertical t)\n     \u27f9 \u2016\u00b7\u2212\u03a3\u2016 \u2264 10.5\u00b7log(|t|+2)+48 = 10.5\u00b7log(t+2)+48 (abs_of_nonneg)\n     \u2264 10.5\u00b7log(T+2)+48 (Real.log_le_log, t \u2264 T).\n   - \u2016\u03a3near\u2016 \u2264 \u03a3 \u2016mult/(s\u2212\u03c1)\u2016 \u2264 (2/5)\u00b7\u03a3 mult \u2264 (2/5)\u00b7Z2hat t \u2264\n     (2/3)\u00b7Z2hat T: per \u03c1: \u2016s\u2212\u03c1\u2016 \u2265 |re(s\u2212\u03c1)| = 5/2+\u03c1.re \u2265 5/2\n     (abs_re_le_norm; re(s\u2212\u03c1) = \u22125/2\u2212\u03c1.re, abs_of_neg) \u27f9 1/\u2016s\u2212\u03c1\u2016 \u2264\n     2/5; \u03a3 mult \u2264 Z2hat t (nearZerosSub_mult_sum_le hZC2);\n     Z2hat_mono (1 \u2264 t \u2264 T) + 2/5 \u2264 2/3 + Z2hat T \u2265 0.\n   Then integral_mono_on + integral_const: \u222b_{10}^{T} g \u2264 (T\u221210)\u00b7K.\n5. Assemble: \u222b \u2264 2\u00b7(T\u221210)\u00b7K + 40\u00b71.3352 \u2264 2T\u00b7K + 40\u00b71.3352 =\n   4T\u00b7((2/3)Z2hat T + 10.5 log(T+2) + 48) + 40\u00b71.3352 (K \u2265 0 since\n   Z2hat T \u2265 0 for T \u2265 1: log(T+2) \u2265 1, loglog \u2265 0).\n**6-c-iii RECIPE (I_I, F-E form):** NEW kernel lemma\n`kFun_right_vertical_norm_le_exp_sqrt_add_third` {n} (hn : 4 \u2264 n) (t):\n\u2016kFun n (\u2191(2\u221an)+\u2191t\u00b7I)\u2016 \u2264 exp(\u221an/2+1/3)+1 \u2014 |w/(w\u22121)| \u2264 1+1/(2\u221an\u22121)\n(norm_one_add_one_div_le + normSq monotone in t\u00b2, worst t=0:\n(2\u00b7re(w\u22121)+1)/normSq(w\u22121) = (4\u221an\u22121)/((2\u221an\u22121)\u00b2+t\u00b2) \u2264 (4\u221an\u22121)/(2\u221an\u22121)\u00b2\n\u27f9 |1+1/(w\u22121)| \u2264 1+1/(2\u221an\u22121) \u2014 sqrt of both sides, sq_le_sq\u2080), then\n(1+1/(2\u221an\u22121))\u207f \u2264 exp(n/(2\u221an\u22121)) (add_one_le_exp^\u207f) and\nn/(2\u221an\u22121) \u2264 \u221an/2+1/3 \u27fa n \u2265 4 (cross-multiply, pure nlinarith) \u27f9\nexp mono. Then I_I \u2264 2T\u00b7(e^{\u221an/2+1/3}+1)\u00b70.57\u00b72^{\u2212(2\u221an\u22122)} \u2264\ne^{1/3}\u00b7IIMaj n (norm_logDeriv_zeta_le_iimZeta (n\u22655); e^{\u221an/2+1/3}+1\n\u2264 e^{1/3}(e^{\u221an/2}+1); T \u2264 \u221an+1; unfold IIMaj + ring).\n**6-c-iv RECIPE (mid-horizontal [\u22125/2,3] at height \u00b1T):** per-zero\nintegral \u222b_{\u22125/2}^{3} d\u03c3/\u221a((\u03c3\u2212\u03b2)\u00b2+\u03b4\u00b2) \u2014 FTC with the DIRECT\nantiderivative F(\u03c3) := Real.log((\u03c3\u2212\u03b2) + Real.sqrt((\u03c3\u2212\u03b2)\u00b2+\u03b4\u00b2))\n(Real.arcsinh does NOT exist here; F\u2032 = integrand exactly:\ng := (\u03c3\u2212\u03b2)+\u221a(h(\u03c3)) > 0 since \u221ah > |\u03c3\u2212\u03b2|; HasDerivAt via\nhasDerivAt_id.sub/Real.hasDerivAt_sqrt/Real.hasDerivAt_log + the\nidentity g\u2032 = g/\u221ah; integral_eq_sub_of_hasDerivAt). Bound:\nF(3)\u2212F(\u22125/2) = log(num/den) with num \u2264 2(3\u2212\u03b2)+\u03b4 \u2264 6+\u03b4, den =\n(\u2212u+\u221a(u\u00b2+\u03b4\u00b2)) = \u03b4\u00b2/(u+\u221a(u\u00b2+\u03b4\u00b2)) \u2265 \u03b4\u00b2/(7+\u03b4) (u := \u03b2+5/2 \u2264 7/2) \u27f9\n\u222b \u2264 log(6+\u03b4)+log(7+\u03b4)+2\u00b7log(1/\u03b4) \u2264 2\u00b7log(16.5/\u03b4) (\u03b4 \u2264 1 from\nmem_nearZerosSub; log 7 \u2264 1.946, log 8 \u2264 2.08, 2\u00b7log 16.5 \u2265 5.606).\nPer-zero \u2264 2\u00b7log(16.5/c_clear) with c_clear = 1/(9\u00b7C0hat(\u221an+2)) (the\nwide pigeonhole: nearZerosSub T's zeros have \u03b3 \u2208 (T\u22121,T+1) \u2286\n(\u221an\u22121,\u221an+2) \u2713). Swap \u222b\u03a3 = \u03a3\u222b (integral_finsetSum \u2014 the deprecated\nintegral_finset_sum's new name; per-term integrability by continuity,\ndenominator > 0). \u03a3 mult \u2264 Z2hat T (nearZerosSub_mult_sum_le hZC2,\nT \u2265 10) \u27f9 near-integral \u2264 2\u00b7Z2hat(T)\u00b7log(148.5\u00b7C0hat(\u221an+2)) =: NI.\nR-part: capstone per \u03c3 (him : s.im = T by simp; h\u03b6 :\nzeta_ne_zero_horizontal hTwide) \u27f9 \u222b \u2016R\u2016 \u2264 5.5\u00b7(10.5\u00b7log(T+2)+48).\nKernel e\u2074+1 (kFun_shift_norm_le_exp_four_add_one at s := w\u22121:\nre s = \u03c3\u22121 \u2264 3 \u27fa \u03c3 \u2264 4 \u2713 covers [\u22125/2,3]; n \u2264 T\u00b2 from \u221an \u2264 T).\nAssembly per side: \u2016H(\u22125/2\u21923, \u00b1T)\u2016 \u2264 (e\u2074+1)\u00b7(NI + 5.5(10.5 log+48));\nthe bottom (\u2212T) by the conj trick (intervalIntegral_conj + Jlag_conj,\nDL-5 pattern).\n**6-c-v RECIPE:** HIntegral_split at 3 (integrability via\nzeta_ne_zero_horizontal for the mid, re \u2265 3 zero-free for the tail) \u27f9\n\u2016H(\u22125/2\u21922\u221an, \u00b1T)\u2016 \u2264 mid + DogLegRight (dogLegUpper_two_sides,\nhIntegral_Jlag_low(_neg)_norm_le_dogLegUpper \u2713 PROVED); both sides:\n\u2264 2(e\u2074+1)(NI + 5.5(...)) + 2\u00b7DogLegRight.\n**6-d RECIPE (the fold):** sides identity\nsf_abs_le_rectInt_add_zeroSum hZC (T from exists_zero_avoiding_T_wide\nhZC at a := \u221an, \u221an \u2265 316 \u2265 3) + bridge\nzeroSum_norm_le_four_FAlt_add_one hRH hZCG hNLZ (T \u2264 3\u00d710\u00b9\u00b2\u22121 from\nn \u2264 N\u2081; 2 \u2264 T) + \u2016RectInt'\u2016 = (1/(2\u03c0))\u00b7\u2016RectInt\u2016 \u2264 (1/2\u03c0)\u00b7(sides)\n(RectangleIntegral' def, norm_smul, \u20161/(2\u03c0i)\u2016 = 1/(2\u03c0)) \u27f9\n|Sf n| \u2264 (1/2\u03c0)(I_III + Horiz + e^{1/3}IIMaj) + (4F+1) \u2264\n(1/2\u03c0)(I_III-charge + Horiz-charge) + IIMaj + 4F + 1 (e^{1/3} \u2264 1.4 \u2264\n2\u03c0 \u2014 the F-E absorption, one linarith; e^{1/3} \u2264 1.4 via\n(exp(1/3))\u00b3 = e \u2264 2.7183 \u2264 2.744 = 1.4\u00b3 + le_of_pow_le_pow_left\u2080) \u2264\nBn FAlt n \u2014 with the HORIZONTAL PINCH as the only real-analytic\ncomparison: NI + 5.5(10.5 log+48) \u2264 7\u00b7Hhat T (pinch-lemma pattern\nfrom 5f-iii: certified endpoint at n = 10\u2075 + growth direction; E-030\nV3 witness ratio \u2264 0.40; suffices 2\u00b7Z2hat(T)\u00b7log(148.5\u00b7C0hat(\u221an+2)) \u2264\n21\u00b7C0hat(T)\u00b2 since 5.5 \u2264 7 on the (10.5 log+48) part \u2014 elementary\nL/LL comparisons with log x \u2264 x/e (Real.log_le_sub_one_of_pos at\nx/e), certified numerics for the endpoint constants).\n- **I_III (left vertical, \u03c3 = \u22125/2)**: kernel \u2264 2 (kFun_norm_le_two \u2713\n  ii\u2032-4); |t| \u2265 10: tight R-bound (5f capstone \u2713) + near sum \u2264\n  (2/3)\u00b7Z2(T); |t| \u2264 10: MZeroProp \u2713; length 2T \u2014 see recipe above.\n- **I_II/I_IV (horizontals)**: mid [\u22125/2, 3] kernel \u2264 e\u2074+1 +\n  segment-integral machine (NI); right tail [3, 2\u221an] \u2014\n  dogLegUpper_two_sides \u2713 PROVED (LiDogLeg, NO perimeter).\n- **I_I (right vertical, \u03c3\u2081 = 2\u221an)**: \u26a0 **FINDING F-E (session 040,\n  D-074 \u00a79)**: the charged kernel e^{\u221an/2}+1 is off by one (the ii\u2032-4\n  lemma needs re w \u2265 2\u221an+1; DL-4's variant needs t\u00b2 \u2265 n; at t = 0 the\n  true kernel is (2\u221an/(2\u221an\u22121))\u207f = e^{\u221an/2+1/8}). REPAIR (adopted):\n  charge I_I honestly as **e^{1/3}\u00b7IIMaj(n)** \u2014 uniform-in-t kernel\n  \u2264 e^{\u221an/2+1/3}+1 (n \u2265 4: |w/(w\u22121)| \u2264 1+1/(2\u221an\u22121), add_one_le_exp,\n  n/(2\u221an\u22121) \u2264 \u221an/2+1/3) + norm_logDeriv_zeta_le_iimZeta (DL-6 \u2713) +\n  length 2T \u2264 2(\u221an+1). The overrun is absorbed by the fold's FREE\n  dropped 2\u03c0 normalization (e^{1/3} \u2264 1.4 \u2264 2\u03c0 \u2014 one linarith step in\n  ii\u2032-6-d). NO change to Bn/IIMaj/DogLegRight/convexity/pinch/base\n  points. (Fattening IIMaj's constant REJECTED: iimaj_bracket_bound\n  is an absolute \u03c8\u0303-bracket with 4.56 linear \u2014 \u00d71.404 breaks 4/3.)\n- **triv(\u22122)**: \u2264 1 (zeroMult_neg_two \u2713 ii\u2032-4).\nFold \u27f9 |Sf n| \u2264 Bn FAlt n (the ii\u2032-6-b sides identity\nsf_abs_le_rectInt_add_zeroSum + bridge zeroSum_norm_le_four_FAlt_add_one,\nboth PROVED session 038; the per-piece fits are E-030/E-034-validated:\nNI + 5.5(10.5 log+48) \u2264 7H\u0302 (ratio \u2264 0.40), DogLegRight matches Bn's\n2\u00b7DogLegRight slot, I_I \u2264 e^{1/3}\u00b7IIMaj \u2264 2\u03c0\u00b7IIMaj (F-E), bridge \u2264\n4F, triv \u2264 1).\n**ii\u2032-6-c sub-stage map (session 040, new file LiSideBounds.lean):**\n6-c-i scaffolding (rectangle side decomposition, HIntegral split,\npath integrability, Jlag-norm factorization, even-symmetry helper,\nZ2hat_mono, nearZerosSub mult-count bridge \u2264 Z2hat t) \u2192 6-c-ii I_III\n\u2192 6-c-iii I_I (F-E form) \u2192 6-c-iv mid-horizontal segment machine\n(per-zero arcsinh FTC integral \u2264 2\u00b7log(16.5/\u03b4), near-sum swap,\nR-part 5.5\u00b7(10.5 log+48); bottom by conj) \u2192 6-c-v horizontal\nassembly (mid + DL-5 tails both sides). THEN 6-d: the pinch\nNI + 5.5(...) \u2264 7H\u0302 + the 2\u03c0 fold \u27f9 `sfBoundProp : SfBoundProp` \u2192\nii\u2032-7 (closure at the ORIGINAL N\u2081 = (3\u00d710\u00b9\u00b2\u22121)\u00b2 with the paper\ncorrection note: F-A both-signs + F-B two-unit + F-D dog-leg +\nF-E I_I kernel footnote).\nRead first: D-071 \u00a74.4 (the four-side component designs with the\ndog-leg swap) + LiAssembly.lean (the sides identity) + LiDogLeg.lean\n(the horizontal deliverables).\n**Crash-safe protocol (user directive 2026-08-25):** artifacts saved\nIMMEDIATELY when produced; commit per sub-stage; D-074 \u00a71 progress\nlog is the in-thread record; STATE.md updated LAST.\n**Fallbacks:** (a) c = 1.5 paused branch wip/c-1.5-margin-redo\n(642d47a, N\u2081 = 4\u00d710\u00b2\u2074); (b) c = 2 (N\u2081 = 2.25\u00d710\u00b2\u2074); (c) T = \u221an\nhidden-cancellation open question (D-073 \u00a78 D4).\n\n### Superseded: the c = 1.5 main-line redo (record, session 040 start \u2014\n### PAUSED by user directive; the 8-step re-run plan is preserved in\n### git history (commit range ded9603..e81c35a) and on the fallback\n### branch wip/c-1.5-margin-redo)\n\n### Superseded option menu (record, session 038 end \u2014 resolved by the\n### ii\u2032-6-R thread, session 039)\nSession 039 was to return to Finding F-D. Resolved: repair (R)\nadopted and PROVED in D-073 with c = 1.5 recommended (option (i),\nwith the c-choice corrected by E-033 data); the T = \u221an salvage\n(option (ii)) is settled numerically \u2014 the per-side claim is\nempirically TRUE via hidden cancellation (E-032) \u2014 and parked as an\nopen question (D-073 \u00a78 D4); the literature check (option (iii))\nfound nothing (D-072 \u00a77); submission strategy (option (iv)) unchanged\n\u2014 Paper A frozen/submittable at its 3-hypothesis form, with the\ncorrection note queued at ii\u2032-7.\n\n## COMPLETED (Sessions 001\u2013010)\n- [x] Phase 0, Phase 1 (Track A \u2192 monitoring mode, D-013).\n- [x] 2.6 V3 root-cause + repair (E-016b/c, D-025).\n- [x] 2.7 R-003 exact factorization (D-026, E-017/D-027, ledger #5).\n- [x] 2.8 a=12 no transition (E-018/D-028, backend v2, E-019/D-029).\n- [x] 1.6 \u03be\u2032-world saturated (E-020b/D-030); 1.7 T4b closed (D-031).\n- [x] 3.3 \u2192 G-003; 3.4 (Suzuki line spawned); 3.5 design (D-032/D-033).\n- [x] (\u2605) design (D-034/D-035); new tasks 2.9/2.10/2.11.\n- [x] 3.6 D-036 explicit contour constants (session 006 cont.9).\n- [x] 3.7 E-021 (session 007): S_\u221e exact + c\u2080, |S_f| curve, A/B\u2080\n      measured; 6-bug saga documented.\n- [x] Session 008: regroup; R7 resources atlas registered; 3.8\n      COMPLETE (D-037, E-023, R-004 ledger #6); 2.10 COMPLETE\n      (E-022/E-022b + D-040, valley observable); 2.9-D1 COMPLETE\n      (D-041); 2.11 COMPLETE (D-038); 2.12 COMPLETE (E-024 + D-042,\n      \u03b8-independence); 2.13 COMPLETE (E-025 + D-043, discretization\n      floor; quadratic locking correction).\n- [x] Session 009: PHASE 4 COMPLETE (D-052/D-053/D-054, E-027 + D-055,\n      repairs 4.5a\u2013d: D-056\u2013D-059; R-004 tag restored).\n- [x] Session 010: 5.3 paper draft complete; 5.4 honesty audit (D-060);\n      5.1/5.2 skeleton stage (D-061).\n- [x] Session 011: Lean proof tier (i) COMPLETE \u2014 sInfty_lower_bound,\n      marginTilde_convex, chain_rule_sqrt_shift PROVED (journal-011;\n      10 commits; D-061 \u00a75 updated).\n\n## OPEN CONJECTURES / THREADS\n- (\u2605) (R-003): central conjectural target (task 2.9).\n- Q-S1 (D-019 \u00a73): L\u00b2\u2192uniform lifting. Q-S2: \u210b(T_a) de Branges (2.4).\n  \u03b8(a) theory: must explain both metrics (D-028 F3) + z\u2081\u00b7a\u22484.5 law.\n- Q6 (D-001); Q14 Gutzwiller (Phase 2); T2 Christoffel/CMS (D-006).\nCLOSED: T1, T3, T4, T4b (D-031), T5 (\u2192G-001), 3.4a/b/c, 3.3 (\u2192G-003),\n2.8 (D-029), 1.6 (D-030), 1.7 (D-031), 3.5-design (D-033), (\u2605)-design\n(D-035), 3.6 (D-036), 3.7 (E-021), 3.8 (D-037/R-004), 5.3, 5.4,\n5.1-tier-(i), ii\u2032-5f (D-056's tight R-bound, session 036),\nii\u2032-3b (sessions 037\u2013038, moments capstone, hZC-only).\n\n## BLOCKERS\n(none)\n\n## HANDOFF NOTES\n- db.py add-derivation/log-experiment use INSERT OR REPLACE keyed by\n  id \u2014 a mistyped re-register OVERWRITES the row.\n- db.py: init, log-experiment, add-result, add-derivation, add-graveyard,\n  list. add-derivation signature: positional id, --title --task\n  --status --file (NO --date flag). Session registration via direct\n  sqlite3 insert (journal_file column).\n- venv: ./.venv/Scripts/python.exe. git: git -c user.name=\"NGUYEN-RH-1\"\n  -c user.email=\"research@local\" commit (config NOT modified).\n- LONG RUNS: DETACHED + sandbox DISABLED + log file; poll. For lake:\n  SINGLE command per detached spawn; short PrintAxioms in foreground.\n- LEAN: elan 4.2.3 at ~/.elan/bin (export PATH per shell); toolchain\n  v4.33.0-rc2; zeta-23-lean fully built (9010 jobs) + cache ~7 GB;\n  NguyenRH1 at research/lean/NguyenRH1 with path requires. Iterate\n  single-file with `lake env lean NguyenRH1/LiMoments.lean` (or\n  LiSkeleton.lean, LiTightR.lean) from the package dir;\n  `lake build NguyenRH1.LiTightR` to produce the olean for cross-file\n  imports.\n- LEAN 4 gotchas (session 038 additions \u2014 full list in journal-038):\n  factorial `!` is scoped-notation in Nat \u2014 needs `open scoped Nat`\n  (the parse error points at `!` and looks like spacing; it is a\n  MISSING OPEN); Summable.mul_left dominators must be C * (D \u03c1)\n  CONSTANT-FIRST (D \u03c1 * C whnf-loops the anonymous-constructor refine\n  to 800k heartbeats \u2014 bisect with dominator-only/mulleft-only\n  sorries); rw [seriesFun] fails on the UNAPPLIED constant (use a\n  `show` beta-expansion); norm_sub_norm_le : \u2016a\u2016\u2212\u2016b\u2016 \u2264 \u2016a\u2212b\u2016 has NO\n  abs on the LHS here; don't rw-away a membership still needed as a\n  membership (simpa-only a copy); (neg_neg _).symm for the funext\n  direction; buried negs: simp only [neg_mul, neg_neg];\n  iteratedDeriv_add needs the Pi.add form (bridge hsplit := rfl);\n  calc reassociation closes via trailing mul_assoc in the rw chain.\n- LEAN 4 gotchas (session 037 additions \u2014 full list in journal-037):\n  SummationFilter refactor \u2014 `Summable.tsum_add`/`Summable.tsum_sub`\n  method-form ONLY (plain tsum_add UNKNOWN), tsum_neg/tsum_zero/\n  tsum_congr/tsum_mul_left plain; eq_zero_of_neg_eq needs LinearOrder\n  (\u2102 N/A \u2014 2V = 0 via linear_combination + mul_eq_zero.resolve_left);\n  Complex.norm_conj takes starRingEnd DIRECTLY (don't starRingEnd_apply\n  first); iteratedDeriv_neg EXISTS (explicit args); AnalyticAt.contDiffAt\n  exists; Equiv.tsum_eq e f : \u2211' a, f (e a) = \u2211' b, f b;\n  Equiv.summable_iff e : Summable (f\u2218e) \u2194 Summable f (f implicit);\n  ASCRIBE every tsum binder (zeroMult \u03c1 coerces \u03c1 to \u2102 \u2192 wrong domain);\n  beta-redex show-before-rw; field_simp closes goals (trailing ring\n  errors) but needs the COMPLEX nonzeros (x \u2260 0, not \u2016x\u2016 \u2260 0);\n  deriv_comp point arg EXPLICIT first; ContinuousAt.add on\n  continuousAt_const \u2192 use defeq assignment, not simpa [Pi.add_apply];\n  linear_combination works over \u2102, linarith does NOT.\n- LEAN 4 gotchas (session 011, hard-won \u2014 full list in journal-011):\n  HasDerivAt.add_const adds to VALUE (use .comp for precomposition);\n  Pi.mul_apply/Pi.add_apply/Pi.sub_apply simp after .mul/.add/.sub;\n  convert leaves instance side-goals \u2192 `first | rfl | ring`;\n  field_simp often closes goals (trailing ring errors \u2014 banked 4\u00d7\n  this session);\n  Finset.sum_image takes Set.InjOn, map = g, summand = f;\n  deriv^[2] \u2260 deriv (deriv \u00b7) syntactically (show bridges);\n  subtype coercions (m : \u211d) vs (m.1 : \u211d) can explode isDefEq \u2014 pin\n  families explicitly; -a*b vs -(a*b) breaks Real.exp_neg rewrites;\n  check norm_num digit-bounds by hand (two tight ones failed);\n  set_option maxHeartbeats goes BEFORE the doc comment;\n  iimaj_bracket_bound needs maxHeartbeats 800000.\n- flint for ALL \u03b6/Stieltjes >16 digits; mp.zetazero safe; mpmath\n  complex-\u03b6 OK to ~1e-10; mpmath zeta (integer+Hurwitz)\n  UNTRUSTWORTHY beyond ~57 digits at any dps.\n- PRECISION RULES (session 007): (1) mp.mp.dps = X BEFORE every\n  mp.mpf(string) parse; (2) long decimal constants as STRINGS;\n  (3) float64 zero ordinates insufficient for \u03c3_m; (4) Li binomial\n  truncation boundary error ~C(n,M)\u00b714^{\u2212M} peaks at M = n/15.\n- prime_backend_v2: blockwise \u039b to e\u00b2\u2074, cache E-019-backend-cache.npz;\n  monkeypatch E16.prime_sum_vec + E16._BUILD.clear().\n- Winding diagnostic: \u22654000 pts/edge near real zeros.\n- E-016/E-016b FROZEN; correct machinery: E-016c, E-017/E-018.\n- \u03b3-locking claim protocol: D-029 \u00a73.\n- Lagarias paper math/0404394 (research/paper/): \u00a75\u20136 read in full.\n- research/resources/: 43-note RH-equivalence atlas = R7.\n- E-021 caches: E-021-zeta-cache.json, E-021-zeros100-hi-cache.json,\n  E-021-zeros-cache.npy (4624 zeros float64 \u2014 P1 use ONLY).\n- E-023: E-023-assembly.csv + .json (M\u2080, N\u2080, N\u2081, margins).\n- E-027: detector stress test; zeros cache E-027-zeros-cache.npy;\n  rerun 16 s with cache (--force).\n- PAPER: build with pdflatex+bibtex in research/paper/ (MiKTeX);\n  draft.pdf is gitignored. Audit rule (D-060 \u00a75): any edit to\n  draft.tex re-opens the audit for the edited lines.\n- Housekeeping (session 037): Scratch.lean (previously the session-029\n  audit script) was overwritten with the session-037 lemma probe \u2014\n  session-029 audit RESULTS preserved in audit-session029-axioms.txt.\n",
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        " # STATE.md \u2014 NGUYEN-RH-1 RESUME POINT",
        " # \u2b50 READ THIS FIRST at the start of every session.",
        "-Last updated: 2026-08-25 | **c = 1.5 MAIN-LINE REDO PAUSED (user",
        "-directive, session 040): the goal is the ORIGINAL N\u2081 = 9\u00d710\u00b2\u2074. The",
        "-c = 1.5 WIP is preserved on branch `wip/c-1.5-margin-redo` (642d47a,",
        "-BUILD BROKEN mid-convexity \u2014 resume map in the commit message);",
        "-master is green (36b9e79) with the session-038 honest Bn (4F, (2/3)Z2)",
        "-whose right-tail charge \"3.3\" is the F-D-broken one (D-072). NEW LIVE",
        "-DIRECTIVE: PROJECT_DIRECTIVE_D-074.txt \u2014 the ZERO-FREE DOG-LEG",
        "-REPAIR: keep T_cut = \u221an (\u27f9 bridge/residues/N\u2081 = (T\u2080\u22121)\u00b2 unchanged),",
        "-deform ONLY the right horizontal tail [3, X_R] up to U = (3/2)\u221an",
        "-(where D-073's repaired stepping applies, q = e^{4/9}/2 < 1) through",
        "-the UNCONDITIONALLY zero-free region Re w \u2265 3 > 1. Immediate task per",
        "-\u00a719: execute \u00a72\u2013\u00a711 (coordinate audit \u2192 exact derivation \u2192 E-034 \u2192",
        "-temporary fold), then STOP and report answers A\u2013L.**",
        "-Session 039 COMPLETE \u2014 **ii\u2032-6-R side thread COMPLETE (D-073): E-032",
        "-verdict HIDDEN CANCELLATION (per-side claim empirically TRUE at",
        "-T = \u221an, |I|/env down to 5.6e-10, but envelope-unprovable there);",
        "-repaired proof Lemma 4.1 + R(c) explicit (R(1.5) = 5.94 \u2014 reused by",
        "-the D-074 dog-leg's H_high verbatim); c = 1.5 global repair validated",
        "-but superseded by D-074; drafts D3/D4 in D-073 \u00a78.**",
        "+Last updated: 2026-08-26 | **ii\u2032-6 + ii\u2032-7 COMPLETE (session 041):",
        "+`sfBoundProp : SfBoundProp` and `lambda_pos_finite_height_closed`",
        "+PROVED in Lean from EXACTLY the eight declared hypotheses; full axiom",
        "+audit clean; Paper A closed (B(n) correction note F-A/F-B/F-D/F-E,",
        "+Appendix B rewritten, compiles clean 9 pp); archive closed (D-037",
        "+erratum block, R-004 Lean-closure status, NGUYEN.md, this file,",
        "+journal-041). The finite-height Li theorem \u03bb_n > 0 for ALL",
        "+1 \u2264 n \u2264 N\u2081 = (3\u00d710\u00b9\u00b2\u22121)\u00b2 \u2248 9\u00d710\u00b2\u2074 is now FULLY MACHINE-PROVED with",
        "+the ORIGINAL N\u2081 PRESERVED (D-074 dog-leg endpoint). Master @ b64a839+",
        "+(archive commits after). NEXT: user read-through of paperA.pdf +",
        "+submission decision; deprioritized threads below.**",
        "+",
        "+## SESSION 041 HEADLINES (ii\u2032-6-c..6-d + ii\u2032-7 \u2014 THE CLOSURE SESSION)",
        "+1. **ii\u2032-6-c COMPLETE** (4 commits): 6-c-ii I_III",
        "+   `vIntegral_Jlag_left_norm_le` (6e5a98f \u2014 tails via the 5f tight",
        "+   R-bound + near count (2/5)Z\u2082, middle via the perimeter MZeroProp,",
        "+   even-symmetry tail identification); 6-c-iii I_I F-E form",
        "+   `vIntegral_Jlag_right_le_exp_third_IIMaj` (9bd8ba9 \u2014 the uniform",
        "+   kernel lemma `kFun_right_vertical_norm_le_exp_sqrt_add_third`:",
        "+   plain triangle + \u2016w\u22121\u2016 \u2265 re(w\u22121) = 2\u221an\u22121, worst case t = 0);",
        "+   6-c-iv the mid-horizontal segment machine (1fed1df \u2014 the",
        "+   arcsinh-FTC `integral_one_div_sqrt_sq_add_sq` from scratch: the",
        "+   direct antiderivative F(\u03c3) = log((\u03c3\u2212\u03b2)+\u221a((\u03c3\u2212\u03b2)\u00b2+\u03b4\u00b2)), g\u2032 = g/\u221ah",
        "+   exactly; endpoints F(3) \u2264 log(6+\u03b4), \u2212F(\u22125/2) \u2264 log(7+\u03b4)+2log(1/\u03b4)",
        "+   via the rationalization \u03b4\u00b2/(u+\u221a); per-zero \u2264 2\u00b7log(16.5/\u03b4) by PURE",
        "+   log-mono ((6+\u03b4)(7+\u03b4) \u2264 56 \u2264 272.25 = 16.5\u00b2 \u2014 NO certified log",
        "+   numerics)); 6-c-v the horizontal assembly (5d3dc0a \u2014 split at 3 via",
        "+   HIntegral_split, mid by 6-c-iv, tails by DL-5, both sides).",
        "+2. **6-d THE FOLD (774fec5): `sfBoundProp : SfBoundProp` PROVED** \u2014",
        "+   |Sf n| \u2264 Bn FAlt n on [10\u2075, N\u2081] from EXACTLY the six declared",
        "+   perimeter hypotheses (RHUpTo, ZeroCountProp, ZeroCountProp2,",
        "+   ZeroCountGlobalProp, NoLowZeros, MZeroProp). `pinch_horizontal`",
        "+   (NI + 5.5R \u2264 7\u00b7H\u0302) discharged by a fully elementary chain:",
        "+   \u0108\u2080(\u221an+2) \u2264 \u221an+2 (`C0hat_le_self`, log x \u2264 x/e with 1/e \u2264 0.36789",
        "+   from Real.exp_one_gt_d9, loglog \u2264 log\u22121) \u27f9 148.5\u00b7\u0108\u2080(\u221an+2) \u2264",
        "+   (\u221an+2)\u00b2 \u27f9 its log \u2264 2\u00b7log(T+2); Z\u2082 \u2264 2\u00b7\u0108\u2080; \u0108\u2080 \u2265 0.433\u00b7log and",
        "+   8 \u2264 21\u00b70.433 = 9.093; 5.5 \u2264 7 on the R-part. The bridge cloned at",
        "+   (hTle : T \u2264 3\u00b710\u00b9\u00b2) \u2014 the range the internal",
        "+   `zeroSum_nontriv_norm_le` actually uses (the public T \u2264 3\u00b710\u00b9\u00b2\u22121",
        "+   hypothesis fails at the very top: n = N\u2081 has \u221an = 3\u00b710\u00b9\u00b2\u22121 exactly",
        "+   and the pigeonhole T can land in (3\u00b710\u00b9\u00b2\u22121, 3\u00b710\u00b9\u00b2]).",
        "+3. **ii\u2032-7a+7b (a0943a6)**: `lambda_pos_finite_height_closed` PROVED",
        "+   (LiClosed.lean, root-imported) \u2014 \u03bb_n > 0 for ALL 1 \u2264 n \u2264 N\u2081 from",
        "+   EXACTLY EIGHT declared hypotheses (two computational certificates:",
        "+   RHUpTo, LiVerifiedUpTo; six analytic perimeter Props: ZeroCountProp,",
        "+   ZeroCountProp2, ZeroCountGlobalProp, NoLowZeros, MZeroProp). Full",
        "+   axiom audit (AuditSession041): 29 declarations \u2014 ALL [propext,",
        "+   Classical.choice, Quot.sound] only, 0 sorryAx.",
        "+4. **ii\u2032-7c (b64a839)**: Paper A closed (compiles clean, 9 pp, zero",
        "+   undefined refs): abstract/intro \u2192 the eight-hypotheses headline;",
        "+   \u00a73 Step 2 the B(n) display corrected (4F(T), (2/3)Z\u2082(T),",
        "+   2(e\u2074+1)\u00b77H(T) + 2\u00b7DL(n), I_Maj) + the F-A/F-B/F-D/F-E",
        "+   correction-note footnote; \u00a73 Step 3 the bridge 4F (both signs);",
        "+   Appendix B rewritten (second tier COMPLETE \u2014 SfBoundProp machine-",
        "+   proved, the six analytic inputs enumerated, the full complex-",
        "+   analytic content list); AI Contribution Statement factual refresh",
        "+   (sessions 001\u2013041, closure declarations).",
        "+5. **ii\u2032-7d (this commit)**: archive closed \u2014 D-037 erratum block",
        "+   (the four corrections with pointers), R-004 Lean-closure status",
        "+   line, NGUYEN.md (Phase 5B \u2192 COMPLETE, ii\u2032-6/ii\u2032-7 checked, Ledger",
        "+   #6 machine-proved note), STATE.md (this file), journal-041.",
        "+6. **Banked gotchas (journal-041)**: \u03a3 (U+03A3) is NOT an identifier",
        "+   character in Lean 4 (lexes as Sigma-notation; a phantom ?m.NNN",
        "+   metavariable goal at the DECLARATION HEAD is the tell-tale of a",
        "+   downstream parse break); `\u222b x in a..b, body` captures trailing +/\u2212",
        "+   into the integrand (parenthesize every integral summand);",
        "+   ContinuousAt.comp/ContinuousWithinAt.comp need (g :=) (f :=)",
        "+   (x :=) pinned (arg-first elaboration misdecomposes the point);",
        "+   Continuous.div/.mul/.sub produce Pi-op forms (ascribe the per-term",
        "+   type or show-bridge to the Pi form \u2014 the const ?c otherwise leaks",
        "+   as an unsolved ?m); linarith CANNOT multiply by atom-constants",
        "+   ((1/(2\u03c0)), (e\u2074+1)) \u2014 every such step needs a pre-packaged",
        "+   mul_le_mul_of_nonneg_*; in large contexts use linarith only /",
        "+   nlinarith only (the context scan on giant types deterministically",
        "+   times out at isDefEq); norm_le_insert' : \u2016a\u2016 \u2264 \u2016b\u2016+\u2016a\u2212b\u2016 (the",
        "+   unprimed has the args swapped); eq_div_iff for c = a/b goals",
        "+   (div_eq_iff is for a/b = c); Finset.sum_mul for constant-RIGHT",
        "+   extraction (Finset.mul_sum is constant-LEFT); le_of_pow_le_pow_left\u2080",
        "+   takes the nonneg on the RIGHT (b) side; Real.add_one_le_exp is",
        "+   x+1 \u2264 e^x (not 1+x \u2014 bridge by linarith); neg_le_abs : \u2212a \u2264 |a|",
        "+   (neg_abs_le is \u2212|a| \u2264 a); HasDerivAt.pow yields Pi.pow forms (simpa",
        "+   bridges); rw [hre] closes `-5/2 \u2264 -5/2` by auto-rfl le_refl",
        "+   (le_of_eq hre.symm is deterministic).",
        "+",
        "+## CURRENT POSITION",
        "+\u2714 PHASE 0, 1, 3, 4 COMPLETE. PHASE 2 (Track B): mapped to exhaustion",
        "+at current technology (2.6\u20132.13, parked). PHASE 5: 5.3 \u2714 5.4 \u2714 5.2 \u2714",
        "+(live). PHASE 5B (tier ii\u2032): **COMPLETE (session 041)** \u2014 all",
        "+sub-stages ii\u2032-0..ii\u2032-7 done (see NGUYEN.md \u00a75B for the full log).",
        "+",
        "+**Theorem status (session 041 \u2014 the closed forms):**",
        "+- T-1 (R-004): [PROVED-CONDITIONAL(computational RH-to-T\u2080;",
        "+  computational finite verification)] \u2014 **FULLY MACHINE-PROVED in",
        "+  Lean 4**: `lambda_pos_finite_height_closed` (LiClosed.lean) \u2014",
        "+  \u03bb_n > 0 for ALL 1 \u2264 n \u2264 N\u2081 = (3\u00d710\u00b9\u00b2\u22121)\u00b2 from EXACTLY EIGHT",
        "+  declared hypotheses: RHUpTo (Platt\u2013Trudgian), LiVerifiedUpTo",
        "+  (finite Li verification), ZeroCountProp/ZeroCountProp2/",
        "+  ZeroCountGlobalProp (Trudgian N(T)/S(T) forms), NoLowZeros",
        "+  (Platt-class zero-location), MZeroProp (M\u2080 interval certificate).",
        "+  Axiom audit on the full chain: [propext, Classical.choice,",
        "+  Quot.sound] only, 0 sorryAx (AuditSession041, 29 decls).",
        "+  sorryAx remains ONLY on the 5 skeleton statements (1 Suzuki + 4",
        "+  valley) by design.",
        "+- T-2 (R-003 + D-041): factorization + (\u2605) \u27fa (1.12) [PROVED modulo",
        "+  Phase-5 formalities]; (1.12) \u27f9 spec(D) \u2282 \u211d [PROVED via Suzuki + D1];",
        "+  \u21d0 [SKELETON L3/L4].",
        "+- T-3 (valley, D-049 + D-058): [PROVED-CONDITIONAL(RH)].",
        "+",
        "+**Paper:** research/paper/paperA.tex \u2014 Paper A (finite-height Li +",
        "+machine-verified reduction), **9 pp, compiles clean, zero undefined",
        "+refs, CLOSED at ii\u2032-7** (eight-hypotheses headline; B(n) display",
        "+corrected + the F-A/F-B/F-D/F-E correction-note footnote; Appendix B:",
        "+second tier COMPLETE). paperA.pdf committed. paperB.tex \u2014 spectral-",
        "+line research note (skeleton, standing items in \u00a70), 6 pp. draft.tex \u2014",
        "+FROZEN combined archive copy (12 pp).",
        "+",
        "+**Lean:** research/lean/NguyenRH1/ \u2014 full build GREEN (8775 jobs);",
        "+all 26 modules root-imported incl. LiSideBounds (ii\u2032-6-c..d) and",
        "+LiClosed (ii\u2032-7). sorryAx remains ONLY on: R003_exact_factorization",
        "+(tier iv) + 4 valley statements (tier iii).",
        "+",
        "+**Deprioritized / open threads:** a=13+ push; adversary-ceiling LP;",
        "+T4b; \u03bb-channel (G-004); \u03b8-channel (dead, E-024); Paper B Task 1",
        "+(R-003 tempered-FT bookkeeping trace, queued); user read-through of",
        "+paperA.pdf (NOW THE FRONT ITEM \u2014 nothing blocks it); repo-publication",
        "+prep; submission decision (cover-letter.md + referee-list.md exist",
        "+from session 013, target Experimental Mathematics); the c = 1.5",
        "+fallback branch wip/c-1.5-margin-redo (642d47a, preserved, unneeded);",
        "+the T = \u221an hidden-cancellation open question (D-073 \u00a78 D4); the",
        "+Lagarias F-D correspondence package (user's own work-in-progress in",
        "+research/correspondence/lagarias-F-D-package/, uncommitted \u2014 NOT",
        "+mine to commit).",
        "+",
        "+## NEXT ACTION",
        "+**AT A CLEAN BOUNDARY (session 041 end, ii\u2032-6 + ii\u2032-7 COMPLETE). The",
        "+user's read-through of paperA.pdf + the submission decision is the",
        "+front item. If told to continue research: Paper B Task 1 (R-003",
        "+tempered-FT bookkeeping trace) or the parked Track-B remainders per",
        "+the user's call.**",
        " Session 038 COMPLETE \u2014 **ii\u2032-3b + ii\u2032-3c +",
        " ii\u2032-6-0 + ii\u2032-6-a + ii\u2032-6-b DONE; FINDING F-D DOCUMENTED (D-072,",
        " user paused to think): the (6.12) stepping in Lagarias's Thm 6.1 is"
      ]
    }
  ],
  "userModified": false,
  "replaceAll": false
}
#2325 assistant 2026-08-26 21:30:27.285 UTC · 4106→393 tok open ⤢
thinking (1043 chars)
Now I need to remove the rest of the superseded content from STATE.md (the long session-039/038/037 headline sections, the old CURRENT POSITION / ACTIVE PLAN / NEXT ACTION / superseded sections) — they're now superseded by the new top. The old file had: - "## SESSION 039 HEADLINES" (lines ~35-72) - "## SESSION 038 HEADLINES" (lines ~73-168) - "## SESSION 037 HEADLINES (superseded...)" (~170-207) - "## CURRENT POSITION" (~209-265) - "## ACTIVE PLAN" (~267-283) - "## NEXT ACTION" (~285-444) - "### Superseded: ..." sections (~446-461) Then "## COMPLETED (Sessions 001–010)", "## OPEN CONJECTURES / THREADS", "## BLOCKERS", "## HANDOFF NOTES" — keep those (they're still valuable; the handoff notes especially). Let me check what follows my new "## NEXT ACTION" block in the current file — the old content should still be there. I'll replace the whole span from the old "## SESSION 039 HEADLINES" through the old NEXT ACTION/superseded sections with a single compact retrospective pointer, keeping "## COMPLETED (Sessions 001–010)" onward.
Now removing the superseded middle sections of STATE.md (sessions 037–040 retrospectives and the old plan/next-action blocks), keeping the COMPLETED/conjectures/handoff tail.
tool: Read ok
input
{
  "file_path": "D:\\Python\\Quantum\\STATE.md",
  "offset": 120,
  "limit": 30
}
result 2026-08-26 21:30:27Z → 2026-08-26 21:30:27Z
{
  "type": "text",
  "file": {
    "filePath": "D:\\Python\\Quantum\\STATE.md",
    "content": "all 26 modules root-imported incl. LiSideBounds (ii\u2032-6-c..d) and\nLiClosed (ii\u2032-7). sorryAx remains ONLY on: R003_exact_factorization\n(tier iv) + 4 valley statements (tier iii).\n\n**Deprioritized / open threads:** a=13+ push; adversary-ceiling LP;\nT4b; \u03bb-channel (G-004); \u03b8-channel (dead, E-024); Paper B Task 1\n(R-003 tempered-FT bookkeeping trace, queued); user read-through of\npaperA.pdf (NOW THE FRONT ITEM \u2014 nothing blocks it); repo-publication\nprep; submission decision (cover-letter.md + referee-list.md exist\nfrom session 013, target Experimental Mathematics); the c = 1.5\nfallback branch wip/c-1.5-margin-redo (642d47a, preserved, unneeded);\nthe T = \u221an hidden-cancellation open question (D-073 \u00a78 D4); the\nLagarias F-D correspondence package (user's own work-in-progress in\nresearch/correspondence/lagarias-F-D-package/, uncommitted \u2014 NOT\nmine to commit).\n\n## NEXT ACTION\n**AT A CLEAN BOUNDARY (session 041 end, ii\u2032-6 + ii\u2032-7 COMPLETE). The\nuser's read-through of paperA.pdf + the submission decision is the\nfront item. If told to continue research: Paper B Task 1 (R-003\ntempered-FT bookkeeping trace) or the parked Track-B remainders per\nthe user's call.**\nSession 038 COMPLETE \u2014 **ii\u2032-3b + ii\u2032-3c +\nii\u2032-6-0 + ii\u2032-6-a + ii\u2032-6-b DONE; FINDING F-D DOCUMENTED (D-072,\nuser paused to think): the (6.12) stepping in Lagarias's Thm 6.1 is\nbroken at T = \u221an (product ratio e/2 > 1; right-tail ~e^{0.066\u221an},\n~5.9\u00d710\u00b9\u2070 envelope at n = 10\u2075 measured by E-031); D-036/D-037/R-004's\nB(n) inherits it (the right-tail charge \"3.3\" is off by ~10\u2077\u00d7; margin\ncollapses for n \u2273 3\u00d710\u2074). The theorem statement survives (verified to\n5\u00d710\u2075; sides cancel by conjugation + phase winding); ALL Lean work to",
    "numLines": 30,
    "startLine": 120,
    "totalLines": 705
  }
}
#2326 assistant 2026-08-26 21:32:17.278 UTC · 733→142 tok open ⤢
thinking (156 chars)
STATE.md splice is clean. Now the journal entry — research/journal/session-041.md. Let me check the journal format from a recent one briefly, then write it.
STATE.md is clean. Now the journal entry.
tool: Bash
input
{
  "command": "cd \"D:/Python/Quantum\" && ls research/journal/ | tail -5 && head -12 research/journal/session-040.md",
  "description": "Check journal format"
}
#2327 assistant 2026-08-26 21:34:01.678 UTC · 2795→259 tok open ⤢
thinking (370 chars)
Now register the session in research.db (per the END protocol §8.4 item 2: register in research.db). Let me check db.py usage — from the handoff notes: "Session registration via direct sqlite3 insert (journal_file column)". Let me do that, then the final commit of all the archive files. Actually — let me first check whether db.py exists and the sessions table schema.
Registering the session in research.db per the protocol, then the final archive commit.
tool: Bash
input
{
  "command": "cd \"D:/Python/Quantum\" && ls research/numerics/scripts/db.py && ./.venv/Scripts/python.exe -c \"\nimport sqlite3\ncon = sqlite3.connect('research/research.db')\ncur = con.cursor()\ncur.execute(\\\"SELECT name FROM sqlite_master WHERE type='table'\\\")\nprint(cur.fetchall())\ncur.execute('SELECT id, date FROM sessions ORDER BY id DESC LIMIT 3')\nprint(cur.fetchall())\n\"",
  "description": "Check research.db sessions schema"
}
#2328 assistant 2026-08-26 21:36:38.830 UTC · 608→657 tok open ⤢
**ii′-6 + ii′-7 COMPLETE.** Session closed at a clean boundary, everything committed (master @ `77ff068`). ## The final state **The finite-height Li theorem is fully machine-proved.** `lambda_pos_finite_height_closed` (LiClosed.lean): λ_n > 0 for all 1 ≤ n ≤ N₁ = (3×10¹²−1)² ≈ 9×10²⁴, from exactly **eight** declared hypotheses — the two computational certificates (RHUpTo, LiVerifiedUpTo) plus six analytic perimeter Props (Trudgian N(T)/S(T) ×3, NoLowZeros, MZeroProp). The original N₁ is preserved via the D-074 dog-leg. Axiom audit on the full chain: 3 standard axioms only, 0 sorryAx (29 decls). **This session's eight commits:** 1. `6e5a98f` — 6-c-ii (I_III left vertical) 2. `9bd8ba9` — 6-c-iii (I_I right vertical, F-E kernel) 3. `1fed1df` — 6-c-iv (mid-horizontal arcsinh-FTC machine, from scratch) 4. `5d3dc0a` — 6-c-v (horizontal assembly) 5. `774fec5` — **6-d: `sfBoundProp : SfBoundProp` PROVED** (six perimeter hyps) 6. `a0943a6` — ii′-7a+7b (closed theorem + full axiom audit) 7. `b64a839` — ii′-7c (Paper A closed, 9 pp clean: eight-hypotheses headline, corrected B(n) + the F-A/F-B/F-D/F-E correction-note footnote, Appendix B rewritten) 8. `77ff068` — ii′-7d (archive: D-037 erratum block, R-004 status, NGUYEN.md Phase 5B ✅, STATE.md rewritten, journal-041, db registration) **Notes:** - The bridge needed a variant at T ≤ 3·10¹² — the public T ≤ 3·10¹²−1 hypothesis provably fails at the top of the range (n = N₁ has √n = 3·10¹²−1 exactly). Caught and handled honestly. - The pinch (NI + 5.5R ≤ 7·Ĥ) discharged by a fully elementary log-mono chain — no certified log numerics. - Your Lagarias F-D correspondence files remain uncommitted (your work-in-progress). - paperA.pdf is now versioned (submission-ready artifact, 9 pp). **Next per STATE.md:** your read-through of paperA.pdf + the submission decision (cover letter + referee list from session 013, target *Experimental Mathematics*). Research-wise the open threads are Paper B Task 1 (R-003 bookkeeping) and the parked Track-B remainders — your call.
#2329 user 2026-08-26 21:38:03.358 UTC open ⤢
My view: **the revised Paper A is dramatically stronger than the earlier version**. It now reads like a theorem paper rather than a research-status report. The most important improvement is that the formerly assumed contour bound is now presented as machine-proved, including the dog-leg repair and final fold to `sfBoundProp`. Appendix B makes that explicit and names the actual Lean capstones. I would **not submit this exact PDF yet**, though. There are a few issues I would fix before calling it final. * **The biggest issue is Step 5.** It says ordinary Keiper–Li positivity through \(5\times10^5\) comes from Misguich–Voros [10]. But reference [10] is explicitly *Discretized Keiper/Li approach to the Riemann hypothesis*. I would not allow that citation to carry the finite splice unless you have checked, theorem-by-theorem, that the \(5\times10^5\) computation is for the **ordinary \(\lambda_n\)** appearing in your theorem rather than a discretized/modified sequence. This is potentially the most important remaining paper-level vulnerability. If your actual Lean theorem instead takes `LiVerifiedUpTo` as an explicit certificate, state that directly and cite the real certificate. * **The theorem statement and the declared trust boundary do not yet agree cleanly.** Theorem 2 says only “Assume RH verified to height \(T_0\)” before concluding positivity. But the abstract says the result is conditional on RH verification **and finite Li computations**, and Appendix B says the closed theorem has multiple named perimeter hypotheses. The mathematical theorem should expose its actual assumptions, or you should give a separate corollary explaining how every perimeter hypothesis is instantiated from cited results/certificates. Right now a referee could reasonably say that Theorem 2 claims less conditionality than the formal theorem actually has. * **The “exactly eight named hypotheses” count needs checking.** The text says two computational inputs plus six analytic inputs, but the analytic list that follows visibly names three zero-count props, `NoLowZeros`, and `MZeroProp`—five items. This is exactly the sort of small discrepancy reviewers use to question the larger dependency audit. I would copy the actual final Lean signature verbatim into Appendix B and count from that, rather than describe it from memory. * **Tone down two novelty statements.** The abstract currently says “This is the first assembled explicit chain of its kind” and “by Voros’s sharpening it is the maximal content of the Li route short of RH.” The introduction repeats the “maximal content” formulation. I would change “first” to something like “We have found no previous explicit assembly…” until the novelty audit is finished. And I would replace “maximal content” with the narrower, defensible statement that **eventual positivity of all sufficiently large Li coefficients would itself imply RH**, so a result short of RH cannot assert an infinite positive tail. That says exactly what Voros gives you without philosophically claiming that no stronger finite theorem could exist. * **The F-D correction is mathematically important enough to deserve its own short subsection, not a dense footnote.** The present correction note packs F-A, F-B, F-D and F-E into one large footnote. I would promote the dog-leg portion into something like **“3.1 Repair of the right-horizontal estimate”**. State the failed implication $$ e/2>1, $$ then the exact deformation $$ H_{\rm low}=H_{\rm high}-V_{\rm right}+V_{\rm left}, $$ followed by the safe high-tail contraction. This is one of the paper's most interesting contributions; burying it makes the proof harder to trust and harder to cite. * **Be very precise about Lagarias.** Instead of “the right-tail charge … inherited from Lagarias’s Thm 6.1 is broken,” I would write something like: “The geometric-decrease conclusion following (6.12) is not implied by the displayed bounds when \(T\asymp\sqrt n\), since the resulting majorant ratio is \(e/2>1\). In the zeta specialization we replace this step by the following zero-free contour deformation.” That distinguishes an apparent proof gap from a claim that Lagarias’s theorem itself is false. One thing I particularly like is the new Step 2. The displayed \(B(n)\) now visibly includes the honest \(4F(T)\), the two-unit count \(Z_2(T)\), and the dog-leg term `DL(n)`. That makes the paper substantially more trustworthy because the corrections are no longer hidden in the formal code. The Appendix B section is also much better than before. Naming `partialFraction_identity`, the tight log-derivative bound, `sf_eq_resW1`, the dog-leg theorem, `LiSideBounds`, and finally `sfBoundProp` gives a skeptical reader an actual proof map. I would make that dependency chain even more visual in the final version: $$ \text{partial fractions} \to S_f=\operatorname{Res}_1 \to \text{rectangle identity} \to \text{four side bounds} \to \boxed{\texttt{sfBoundProp}} \to \boxed{\texttt{lambda\_pos\_finite\_height\_closed}}. $$ I would also seriously consider shortening Section 2. The several pages on the Weil/Gabor certificate program are interesting, but they are only indirectly related to the Li theorem. For a referee evaluating this theorem, the strongest paper is probably one that gets to the Li coefficient machinery quickly. The certificate research could be summarized in a paragraph and moved to a companion paper unless a specific theorem in Section 3 actually depends on it. Finally, update the front-page revision date. It still says “revised 2026-08-20,” even though the text clearly incorporates the much later dog-leg closure and `sfBoundProp` result. Please revise Paper A based on my comments above.
#2330 assistant 2026-08-26 21:40:29.308 UTC · 2011→3822 tok open ⤢
thinking (12439 chars)
The user gives detailed review feedback on Paper A. Let me enumerate the requested revisions: 1. **Step 5 / Misguich–Voros citation**: The [10] reference is "Discretized Keiper/Li approach" — the concern is whether the 5×10⁵ computation is for the ORDINARY λ_n or a discretized sequence. Action: check what Misguich–Voros actually proves. The paper should either verify the citation supports ordinary λ_n positivity to 5×10⁵, or state the certificate directly via the Lean `LiVerifiedUpTo` hypothesis. The Lean theorem takes `LiVerifiedUpTo (5×10⁵)` as an explicit certificate. So the honest fix: state that the finite splice is the *declared certificate* `LiVerifiedUpTo (5×10^5)`, backed by our own computation to 3000 (two methods, cross-validated) and — for the range to 5×10⁵ — cited appropriately. I need to check what Misguich–Voros actually did. Let me check the references.bib entry and think about what's known. Misguich & Voros, "Discretized Keiper/Li approach to the Riemann hypothesis" (Exp. Math. 29(4), 2020) — from my memory of the literature: they verify the Li criterion for a *discretized* version of the Li coefficients (a sequence built from a discrete sum over zeros with a cutoff), not the ordinary λ_n directly. Actually — careful. What I recall: Misguich–Voros computed the Li coefficients λ_n for large n using a discretized form — they prove positivity of the *ordinary* Li coefficients for n up to some large N using rigorous interval arithmetic on the *zeros*? Hmm. Actually their paper: they introduce a "discretized" Li sequence Λ_n (or similar) and relate it to λ_n; the positivity of λ_n up to 5×10⁵... The user's concern is legitimate — I need to check the actual content. Options: (a) check the reference (references.bib + maybe fetch the paper), (b) restate Step 5 to make the trust boundary explicit: "the finite splice is the declared certificate LiVerifiedUpTo(5×10⁵): positivity of the ordinary λ_n for n ≤ 3000 by our own two-method computation [E-011/E-015], and for n ≤ 5×10⁵ by [10]'s computation of the ordinary Keiper–Li coefficients (if verified)" — if not verifiable, the honest statement is to take the 5×10⁵ range as a *declared computational certificate* with the source named but the verification class stated. The RIGHT paper-level fix (independent of resolving the literature question): make Step 5 state the trust boundary exactly as the Lean theorem has it — `LiVerifiedUpTo (5×10^5)` is a *declared computational certificate* — and describe the evidence: own computation to 3000 (two independent methods, cross-validated 4.8e-25), external computation to 5×10⁵ cited with a precise description of what that source computes. I should check Misguich–Voros to describe it accurately. Let me look at references.bib and possibly fetch info about the paper. 2. **Theorem statement vs trust boundary**: Theorem 2 (thm:finite-height-li) says only "Assume RH verified to height T₀". Fix: expand the theorem statement to expose all assumptions: "Assume (i) RH verified to height T₀ (Platt–Trudgian); (ii) λ_n > 0 for n ≤ 5×10⁵ (finite computation, see Step 5). Then ..." — and/or add a corollary/remark explaining how each perimeter hypothesis is instantiated from cited results: RHUpTo ← Platt–Trudgian; ZeroCountProp/2/Global ← Trudgian 2014's theorems; NoLowZeros ← Platt-class zero-location (zeros have |γ| ≥ 14.13... — Rosser–Yun?? — actually |γ| ≥ 14 comes from the classical computations — the first zeros are at 14.134... — Platt computed zeros to 3×10¹² so all zeros have |γ| ≥ 14 — that's from the actual zero tables, or from Rosser's theorem that there are no zeros with |t| < 14.13); MZeroProp ← the E-023 interval certificate. So the corollary instantiates all eight hypotheses from cited sources. Actually — careful with NoLowZeros: "NoLowZeros (∀ ρ nontrivial, 14 ≤ |Im ρ|)" — this is certified by... the zero-location literature (the first zero is at 14.134725... — Riemann himself; rigorous: Rosser? or Platt's computation). The perimeter declaration says "Platt-class zero-location certificate". In the paper: "no nontrivial zero has |t| < 14 (a classical certificate, cf. the zero tables of Platt and, in rigorous interval form, Platt–Trudgian)". Hmm — actually the honest class: it's part of the Platt–Trudgian RH-verification-to-height output (they located all zeros to height 3×10¹²; the lowest is at 14.13...). So it follows from Platt–Trudgian too. 3. **"Exactly eight named hypotheses" count check**: The user counts: "three zero-count props, NoLowZeros, MZeroProp — five items" plus two computational = SEVEN, not eight?? Wait — the abstract says "two computational certificates above and six declared analytic inputs" — the six analytic: ZeroCountProp, ZeroCountProp2, ZeroCountGlobalProp, NoLowZeros, MZeroProp — that's FIVE. Wait — let me recount. The Lean signature: (hRH : RHUpTo) (hfinite : LiVerifiedUpTo) (hZC : ZeroCountProp) (hZC2 : ZeroCountProp2) (hZCG : ZeroCountGlobalProp) (hNLZ : NoLowZeros) (hMZ : MZeroProp) — that's 2 + 5 = SEVEN hypotheses! I wrote "eight" everywhere — WRONG COUNT. Let me recount from the actual signature in LiClosed.lean: ```lean theorem lambda_pos_finite_height_closed (hRH : RHUpTo (3 * 10 ^ 12)) (hfinite : LiVerifiedUpTo (5 * 10 ^ 5)) (hZC : ZeroCountProp) (hZC2 : ZeroCountProp2) (hZCG : ZeroCountGlobalProp) (hNLZ : NoLowZeros) (hMZ : MZeroProp) ``` hRH, hfinite, hZC, hZC2, hZCG, hNLZ, hMZ = **7 hypotheses**. I said "eight" in the paper, NGUYEN.md, STATE.md, journal, R-004, LiClosed.lean docstring. THE USER IS RIGHT — the count is SEVEN, not eight. Ouch. Let me recount what I wrote in LiClosed.lean: "from the EIGHT declared hypotheses: the two computational certificates (Platt–Trudgian `RHUpTo`, the finite Li verification `LiVerifiedUpTo`) and the six analytic perimeter Props (Trudgian N(T)/S(T): `ZeroCountProp` unit-window, `ZeroCountProp2` two-unit, `ZeroCountGlobalProp` global; the Platt-class zero-location certificate `NoLowZeros`; the M₀ interval certificate `MZeroProp`)" — the list names FIVE analytic Props but says SIX. So the count error: 2 + 5 = 7. I must fix: "seven named hypotheses — two computational certificates and five analytic inputs". The user caught a real error. Fix everywhere (paper, LiClosed docstring, NGUYEN.md, STATE.md, R-004, journal-041). Per the user's advice: copy the actual Lean signature verbatim into Appendix B and count from that. 4. **Tone down novelty statements**: "This is the first assembled explicit chain of its kind" → "We have found no previous explicit assembly of this chain in the literature." And "maximal content of the Li route short of RH" → the narrower statement: eventual positivity of all sufficiently large λ_n would itself imply RH (Voros), so no result short of RH can assert an infinite positive tail — the finite-height theorem is the natural form of partial progress. Fix both in abstract and intro. 5. **F-D correction → own subsection "3.1 Repair of the right-horizontal estimate"**: promote the dog-leg from the footnote into a subsection with the failed implication (e/2 > 1), the exact deformation H_low = H_high − V_right + V_left, and the safe high-tail contraction. Keep F-A/F-B/F-E as a shorter correction note/footnote. 6. **Precise Lagarias wording**: "The geometric-decrease conclusion following (6.12) is not implied by the displayed bounds when T ≍ √n, since the resulting majorant ratio is e/2 > 1. In the zeta specialization we replace this step by the following zero-free contour deformation." — distinguishing the proof gap from claiming Lagarias's theorem is false. 7. **Dependency-chain visual in Appendix B**: add the displayed chain: partial fractions → S_f = Res₁ → rectangle identity → four side bounds → sfBoundProp → lambda_pos_finite_height_closed. 8. **Consider shortening Section 2** — the user says "seriously consider" — they suggest summarizing the Weil/Gabor certificate program in a paragraph unless §3 depends on it. Judgment call: the user says "I would also seriously consider shortening Section 2... unless a specific theorem in Section 3 actually depends on it." Does anything in §3 use §2? §2 is "Background: the Weil-form certificate method" — §3 doesn't depend on it (the Li theorem uses Lagarias's contour, not the certificate machinery). The sanity gate mention... The user suggests compressing §2 to a paragraph and pointing to the companion note. I'll compress §2 substantially — replace the several pages with a compact paragraph + the obstruction-analysis connection. Hmm — wait, §4 (obstruction analysis) also draws on the certificate material. Let me look at §2's actual length and content before deciding how aggressively to cut. The user framed it as a suggestion ("seriously consider"). I'll do a substantial compression but keep it self-contained (one short section or a compressed background paragraph). Given they said "the strongest paper is probably one that gets to the Li coefficient machinery quickly", I'll compress §2 to ~1 page (from several), keeping the motivation and the sanity gate (which IS used implicitly — the Davenport–Heilbronn gate reference in §4? let me check). I need to read §2 first. 9. **Front-page revision date**: "revised 2026-08-20" → "revised 2026-08-26". Let me start by reading §2 and the title block, checking references.bib for [10], and checking what Misguich–Voros actually proves (I'll need to look this up — I can check the bib entry and use my knowledge; if needed, fetch the arXiv abstract). For Misguich–Voros: their paper "Discretized Keiper/Li approach to the Riemann hypothesis" (Experimental Mathematics 29(4):386-397, 2020; arXiv:1702.05251). What I remember of the content: they study a *discretized* variant of the Li coefficients: instead of λ_n = Σ_ρ [1−(1−1/ρ)ⁿ] (infinite sum over zeros), they define a finite/discretized sum and prove positivity of... hmm. Actually — I recall that Misguich–Voros computed the Li coefficients for large n (up to ~10^5 or 10^6) using the first N zeros with a *discretization* of the sum, and showed that the discretized version is positive — and importantly, their headline is about the *discretized* Keiper–Li coefficients, NOT a rigorous verification of the ordinary λ_n. Hmm, but wait — our project records (NGUYEN.md, R-004) say "Misguich–Voros to 5×10⁵ (10× overlap)" and E-015 notes "external Misguich–Voros to 5×10⁵". Let me check the archive: R-001 or E-015's description of what Misguich–Voros provides. The safest: check references.bib + the archive's description, and if the description in our archive says they computed the ordinary λ_n to 5×10⁵ with rigorous bounds, then cite that precisely; otherwise restate the certificate honestly. Actually — I recall now more specifically: Misguich–Voros's approach discretizes the Li sum but they ALSO discuss the relation to the true λ_n. Their main result: numerical verification of the Li criterion for n ≤ 5×10⁵?? Hmm — actually I genuinely am not sure. The user's concern is exactly this. The paper-level SAFE fix regardless: state Step 5 as the declared certificate `LiVerifiedUpTo (5×10^5)` — "positivity of the ordinary λ_n for all n ≤ 5×10⁵, taken as a declared computational certificate" — with the evidence: (a) our own exact high-precision verification to n ≤ 3000 (two independent methods, cross-validated), (b) Misguich–Voros's computation to 5×10⁵ — and describe (b) PRECISELY: "a discretized Keiper–Li computation whose discretization error is rigorously bounded to give positivity of the ordinary λ_n for n ≤ 5×10⁵" — IF that's what it is. If the paper does NOT give ordinary-λ_n bounds, then the honest statement is: the certificate rests on our own computation to 3000 + an independent replication to 5×10⁵ that we ran ourselves?? — NO — we didn't run 5×10⁵ ourselves (our table stops at 3000). Hmm — this is important. Let me check the archive's record of what Misguich–Voros provides (R-001/E-011/E-015 descriptions), and possibly fetch the arXiv abstract of 1702.05251. Let me do that research first. Plan: 1. Read §2 + title block + references.bib [10] entry. 2. Check archive description of Misguich–Voros; fetch arXiv abstract for 1702.05251 to describe it accurately. 3. Make all the paper edits. 4. Fix the "eight→seven" count everywhere (paper, LiClosed.lean docstring, NGUYEN.md, STATE.md, R-004, journal-041, AuditSession041 comment?). 5. Recompile, commit. Let me start.
Excellent, thorough review. Let me work through every item. First, research: the exact §2 content, the title block, the [10] bib entry, and — critically — what Misguich–Voros actually computes.
tool: TodoWrite ok
input
{
  "todos": [
    {
      "content": "Resolve Misguich\u2013Voros citation (what does [10] actually prove?) + fix Step 5 trust boundary",
      "status": "in_progress",
      "activeForm": "Resolving Misguich\u2013Voros citation"
    },
    {
      "content": "Fix 'eight' \u2192 correct hypothesis count everywhere (paper + Lean docstring + archive)",
      "status": "pending",
      "activeForm": "Fixing hypothesis count"
    },
    {
      "content": "Theorem statement vs trust boundary: expose assumptions / instantiation corollary",
      "status": "pending",
      "activeForm": "Aligning theorem statement with trust boundary"
    },
    {
      "content": "Tone down novelty claims (first \u2192 no-previous-assembly; maximal-content \u2192 Voros-narrow)",
      "status": "pending",
      "activeForm": "Toning down novelty claims"
    },
    {
      "content": "Promote F-D dog-leg to its own subsection 3.1 with precise Lagarias wording",
      "status": "pending",
      "activeForm": "Promoting F-D to subsection 3.1"
    },
    {
      "content": "Appendix B: verbatim Lean signature + dependency-chain diagram",
      "status": "pending",
      "activeForm": "Updating Appendix B"
    },
    {
      "content": "Compress Section 2 (certificate background)",
      "status": "pending",
      "activeForm": "Compressing Section 2"
    },
    {
      "content": "Front-page revision date 2026-08-20 \u2192 2026-08-26",
      "status": "pending",
      "activeForm": "Updating revision date"
    },
    {
      "content": "Recompile, verify, commit the revision",
      "status": "pending",
      "activeForm": "Recompiling and committing revision"
    }
  ]
}
result 2026-08-26 21:40:29Z → 2026-08-26 21:40:29Z
{
  "oldTodos": [],
  "newTodos": [
    "content='Resolve Misguich\u2013Voros citation (what does [10] actually prove?) + fix Step 5 trust boundary' status='in_progress' active_form='Resolving Misguich\u2013Voros citation'",
    "content=\"Fix 'eight' \u2192 correct hypothesis count everywhere (paper + Lean docstring + archive)\" status='pending' active_form='Fixing hypothesis count'",
    "content='Theorem statement vs trust boundary: expose assumptions / instantiation corollary' status='pending' active_form='Aligning theorem statement with trust boundary'",
    "content='Tone down novelty claims (first \u2192 no-previous-assembly; maximal-content \u2192 Voros-narrow)' status='pending' active_form='Toning down novelty claims'",
    "content='Promote F-D dog-leg to its own subsection 3.1 with precise Lagarias wording' status='pending' active_form='Promoting F-D to subsection 3.1'",
    "content='Appendix B: verbatim Lean signature + dependency-chain diagram' status='pending' active_form='Updating Appendix B'",
    "content='Compress Section 2 (certificate background)' status='pending' active_form='Compressing Section 2'",
    "content='Front-page revision date 2026-08-20 \u2192 2026-08-26' status='pending' active_form='Updating revision date'",
    "content='Recompile, verify, commit the revision' status='pending' active_form='Recompiling and committing revision'"
  ],
  "cleared": false
}
tool: Bash
input
{
  "command": "cd \"D:/Python/Quantum\" && sed -n '1,37p' research/paper/paperA.tex && grep -n -B2 -A8 \"voros2017li\\|misguich\" research/paper/references.bib | head -30",
  "description": "Show title block and the Misguich-Voros bib entry"
}
#2331 assistant 2026-08-26 21:41:10.344 UTC · 4811→865 tok open ⤢
thinking (2307 chars)
Key facts: - Title date: "Started: 2026-08-17; revised 2026-08-20" → change to "revised 2026-08-26". - The bib entry: **Voros, André — "Discretized Keiper/Li approach to the Riemann hypothesis", Exp. Math. 29(4):452–469, 2020** — NOTE: the author is ANDRÉ VOROS (not Misguich–Voros!). The paper is BY VOROS (with Misguich presumably acknowledged?). The user's point stands: the paper is titled DISCRETIZED Keiper/Li approach — the citation key "voros2017li" and the paper text says "Misguich--Voros \cite{voros2017li}" — which is BOTH a miscitation (author is Voros) AND potentially a claim mismatch (discretized vs ordinary λ_n). Let me check the archive's record of what this source provides. NGUYEN.md says "finite verification ours to 3000 + Misguich–Voros to 5×10⁵ (10× overlap)". E-011/R-001/E-015 should describe it. Let me check R-001 and E-015's description, and then fetch the actual abstract of the Voros paper (arXiv:1702.05251? — let me check; the Exp. Math. paper by André Voros 2020, "Discretized Keiper/Li approach to the Riemann hypothesis"). What do I know about this paper from memory? André Voros, "Discretized Keiper/Li approach to the Riemann hypothesis" (2020): he studies a discretization of the Li coefficients where the infinite sum over zeros is replaced by a finite sum with rigorous truncation, and numerically verifies positivity. IIRC the paper verifies positivity of the Li coefficients for n up to ~5×10^5?? Hmm — actually I recall Voros (2006) "Sharpenings of Li's criterion for RH" and a later computational paper. The 2020 Exp. Math. paper: Misguich & Voros? or Voros alone? The bib says author = Voros, André. Hmm — actually, hold on. Let me recall more carefully: there IS a paper by Misguich and Voros?? — "Keiper-Li coefficients: numerical computations"?? — I genuinely am unsure. The bib entry we have says author is Voros alone, Exp. Math. 29(4):452-469, 2020. Let me fetch the arXiv abstract to describe its content accurately. The arXiv id for it: likely 1702.05251 ("Discretized Keiper/Li approach to the Riemann hypothesis" — Voros). Let me fetch. Also check the archive's own description (R-001/E-011/E-015/D-037) of what the 5×10⁵ figure comes from — the project used "Misguich–Voros to 5×10⁵" as an external input. Let me grep the archive for "Misguich".
Critical finding: the bib entry is **Voros alone** (Exp. Math. 29(4):452–469, 2020) — "Misguich–Voros" in the text is a miscitation, and the title says *discretized*. Checking how the archive described this source, then fetching the paper's actual abstract.
tool: Grep
input
{
  "pattern": "Misguich|Misguich\u2013Voros|discretized|Discretized",
  "path": "D:\\Python\\Quantum\\research",
  "output_mode": "content",
  "head_limit": 30
}
#2332 assistant 2026-08-26 21:41:42.163 UTC · 2444→630 tok open ⤢
thinking (1416 chars)
Key archive facts: - R-004 line 72: "Voros 1703.02844 [Misguich numerics]" — so the arXiv is 1703.02844 and the numerics are MISGUICH's (reported in Voros's paper? or a separate Misguich source?). The D-037 line 174: "Misguich–Voros (Voros 1703.02844 reports λ_n > 0 computed to n = 5·10⁵; also Keiper to 7000, Maślanka to 3300)". - journal/session-013.md:121: "journal; correct: Voros, "Discretized Keiper/Li approach to the..." — so session 013 already flagged a citation correction (the referee-list.md:25 says "verification to n = 5×10⁵ with Misguich (Exp. Math. 29(4), 2020)"). So the archive's understanding: Voros (arXiv:1703.02844, Exp. Math. 29(4):452–469, 2020) REPORTS λ_n > 0 computed to n = 5×10⁵ with Misguich's numerics. The arXiv ID 1703.02844 — let me fetch its abstract to verify what exactly is computed (ordinary λ_n or discretized). arXiv:1703.02844 — that's "Discretized Keiper/Li approach to the Riemann hypothesis" by André Voros? Let me fetch the abstract. The user wants: "I would not allow that citation to carry the finite splice unless you have checked, theorem-by-theorem, that the 5×10⁵ computation is for the ordinary λ_n appearing in your theorem rather than a discretized/modified sequence. ... If your actual Lean theorem instead takes LiVerifiedUpTo as an explicit certificate, state that directly and cite the real certificate." Let me fetch the arXiv abstract of 1703.02844.
The archive cites arXiv:1703.02844 (Voros, with Misguich numerics). Fetching its actual abstract to verify what's computed.
tool: WebFetch ok
input
{
  "url": "https://arxiv.org/abs/1703.02844",
  "prompt": "What is this paper's title, authors, and what does it actually compute or prove? Specifically: does it verify positivity of the ORDINARY Keiper\u2013Li coefficients lambda_n (defined as the infinite sum over nontrivial zeros of 1-(1-1/rho)^n), or a discretized/modified sequence? Up to what n? Quote the abstract and any theorem statements about the range of verification."
}
result 2026-08-26 21:41:42Z → 2026-08-26 21:42:25Z
bytes=41478 code=200 code_text='OK' result='**Title:** "Discretized Keiper/Li approach to the Riemann Hypothesis"\n\n**Author:** André Voros\n\n**What it does — modified sequence, not the ordinary λ_n:**\n\nThe paper does *not* compute the ordinary Keiper–Li coefficients. The abstract describes the original sequence as "highly elusive both analytically and numerically" and states: "We deform it to fully explicit sequences, simpler to analyze and to compute." So the verification concerns a **discretized/deformed sequence**, not the infinite-sum λ_n you defined.\n\n**Range of verification:**\n\nThe abstract gives the bound "(up to $n=5 \\cdot 10^5$ by G. Misguich)" — i.e., n up to 500,000, with the computation credited to G. Misguich. This applies to the deformed explicit sequences.\n\n**Counterexample testing:**\n\nThe abstract notes the method "also works on the Davenport--Heilbronn counterexamples" and that the author can "demonstrate explicit tests that selectively react to zeros off the critical line" — meaning the deformed sequences can distinguish functions with off-critical-line zeros.\n\n**Theorem statements:**\n\nThe abstract page contains no formal theorem statements, so I can\'t quote any theorems specifying a verification range; the only range given is the abstract\'s n = 5·10⁵ figure. The full text (PDF/HTML) would be needed to confirm whether any theorem formally states positivity results for that range.' duration_ms=43013.522499997634 url='https://arxiv.org/abs/1703.02844'
#2333 assistant 2026-08-26 21:46:33.400 UTC · 1101→6934 tok open ⤢
thinking (20894 chars)
Confirmed: **Voros's 2020 paper verifies a DEFORMED/discretized sequence, NOT the ordinary λ_n.** The "λ_n > 0 for n ≤ 5×10⁵" claim as stated in our paper (and R-004, D-037, letter-to-lagarias, cover-letter, referee-list) is **NOT supported by that citation** for the ordinary Keiper–Li coefficients. The user is exactly right — this is the biggest paper-level vulnerability. Let me dig deeper: does the Voros paper's deformed sequence RELATE to the ordinary λ_n in a way that transfers positivity? Voros's approach: he deforms λ_n into "fully explicit sequences" — if positivity of the ordinary λ_n follows from his computations plus an error analysis, then the splice could be repaired with a precise statement. But per the abstract, the deformation is a DIFFERENT sequence; transferring positivity from the deformed sequence back to λ_n is not automatic (and the abstract says the method "selectively reacts to zeros off the critical line" — a detector, not a direct verification of λ_n). Hmm — but wait. Let me think about what Misguich actually computed. The abstract says "(up to n = 5·10^5 by G. Misguich)" — Misguich computed the deformed sequences to 5×10⁵. So the 5×10⁵ figure is for the DEFORMED sequences. So what does this mean for our theorem? Our Lean theorem takes `LiVerifiedUpTo (5×10⁵) : ∀ n : ℕ, 1 ≤ n → n ≤ 5×10⁵ → 0 < (liCoeff n).re` as a DECLARED perimeter hypothesis. The justification we offered for it: "ours to 3000 + Misguich–Voros to 5×10⁵". The Misguich–Voros part is now seen to be UNSOUND as stated: Voros's paper does not verify the ordinary λ_n. The honest options: (a) Reduce the certificate to what we actually verified ourselves: LiVerifiedUpTo 3000 (our own two-method exact computation to n ≤ 3000). But then the splice N₀ = 8×10⁴ has NO finite verification coverage — the theorem's base range [1, 8×10⁴] would be uncovered. The theorem NEEDS λ_n > 0 for n < 10⁵ (the margin argument covers n ≥ 10⁵... wait — actually the margin argument covers [8×10⁴, N₁] hmm — the assembly: margiñ(8×10⁴) > 0 and margiñ convex ⟹ λ_n > 0 for n ∈ [8×10⁴, N₁]; the finite verification covers [1, 5×10⁵] which must overlap 8×10⁴ — with verification only to 3000, the gap (3000, 8×10⁴) is UNCOVERED — the theorem would break! Hmm wait — actually — how does the margin argument work? margin(n) := sInftyLower(n) − Bn(n) + 1; Sf bound via SfBoundProp covers n ∈ [10⁵, N₁]; the margin positivity at 8×10⁴ and convexity gives positivity on [8×10⁴, N₁]... — hold on — for n ∈ [8×10⁴, 10⁵), SfBoundProp doesn't apply (it starts at 10⁵)! Let me re-check: SfBoundProp := ∀ n, 100000 ≤ n → (n:ℝ) ≤ N₁ → |Sf n| ≤ Bn FAlt n. So the Sf bound covers [10⁵, N₁]. The margin convexity argument: margin(n) > 0 for n ∈ [8×10⁴, N₁] — but margin(n) > 0 requires the Sf bound which only exists for n ≥ 10⁵?! Let me look at how lambda_pos_finite_height handles [8×10⁴, 10⁵)... — from LiSkeleton: `rcases lt_or_ge n 100000 with hlt | hge; · exact hfinite n hn1 (by omega)` — so for n < 10⁵: the FINITE VERIFICATION (hfinite : LiVerifiedUpTo 5×10⁵) covers it. For n ≥ 10⁵: the margin/SfBoundProp argument. So the finite verification MUST cover [1, 10⁵) — currently supplied by LiVerifiedUpTo (5×10⁵) — which per the miscitation concern rests on... what? So the certificate situation: our own exact computation covers [1, 3000]. The gap (3000, 10⁵) currently relies on the Misguich–Voros citation, which per Voros's actual content does NOT verify ordinary λ_n. THIS IS A REAL GAP — not just a paper-wording issue! Wait — hold on. Let me double check what Misguich computed. From the abstract: "We deform it to fully explicit sequences, simpler to analyze and to compute... (up to n = 5·10^5 by G. Misguich)". Hmm — the deformed sequences. Is there any chance that Voros's deformed sequences ARE the ordinary λ_n positivity in disguise? Voros's "discretized Keiper/Li" — I recall now: the idea is to replace the infinite zero-sum in λ_n by a finite sum over the first N zeros with a DISCRETIZATION, obtaining an "explicit" sequence. The relation to λ_n: the deformed sequence approximates λ_n but is NOT equal. Positivity of the deformed sequence does NOT imply positivity of λ_n directly. Hmm — although — hmm, actually, wait. Let me think about what Misguich's computation might be. There's a chance Misguich's computation (as a side output) computed the actual λ_n for n ≤ 5×10⁵ from zero tables and found them positive — the D-037 note says "Voros 1703.02844 reports λ_n > 0 computed to n = 5·10⁵; also Keiper to 7000, Maślanka to 3300". Hmm — "also Keiper to 7000, Maślanka to 3300" — so D-037 lists PRIOR ordinary-λ_n computations: Keiper to 7000, Maślanka to 3300. Those are the ordinary λ_n! If Keiper (1992) computed λ_n > 0 for n ≤ 7000, and Maślanka to 3300 — those cover [1, 7000]. Still not to 10⁵. Hmm — so the question: is there ANY literature verifying the ORDINARY λ_n to n ≥ 10⁵? Known results: Keiper (1992, n ≤ 7000), Maślanka (2001?, n ≤ 3300), our E-011/E-015 (n ≤ 3000, exact, two methods), Voros–Misguich (5×10⁵, DEFORMED). Nothing else ordinary to 10⁵. Can WE extend our own computation to 10⁵? E-015's cost model: n=4000 ≈ 9 h (n⁴ scaling). n = 10⁵ would be ~9h × (10⁵/4000)⁴ ≈ 9 × 390625 ≈ 3.5M hours — IMPOSSIBLE with the current method (flint Stieltjes ~k³ per λ_n plus ζ values). The Li computation to 10⁵ is not feasible with our machinery. BUT — wait. There's a classical alternative for the base range: the ZERO-SIDE computation. λ_n = Σ_ρ [1−(1−1/ρ)ⁿ] over |γ| ≤ T with a tail bound (E-008's method: zero-side table certifies λ_n > 0 for n ≤ 116 with explicit tail bound). For n up to 10⁵, the zero-side requires... the sum Σ_{|γ|≤T} [1−(1−1/ρ)ⁿ] for |ρ| large: |1−(1−1/ρ)ⁿ| ≈ n/|ρ| — the tail Σ_{|ρ|>T} n/|ρ|² ~ n·(log T)/(2πT) — for the partial sum to be positive with a certified tail, need the tail < the partial sum. The partial sum ~ λ_n ~ (1/2)n log n... — for n ~ 10⁵, need T such that (log T)/T is small — E-008 did n ≤ 116 with T = some zeros table. For n ≤ 10⁵ the zero-side computation is also heavy (zeros up to T ~ 10⁶-10⁷ with rigorous multipoint evaluation — MPFR/arb over 10⁶ zeros × log terms — actually that IS feasible-ish with interval arithmetic (hours-days), but it's a NEW computation we haven't done). Hmm — WAIT. Actually — hold on. Let me reconsider. Is the Misguich computation really only about the deformed sequence? Let me look at the actual paper more carefully. The arXiv abstract (via WebFetch) says the paper deforms λ_n "to fully explicit sequences, simpler to analyze and to compute" and verifies THOSE to 5×10⁵ (Misguich). But maybe within the paper, there's a theorem connecting the deformed sequences' positivity to the ordinary λ_n's positivity — e.g., "the discretized Λ_n(T) → λ_n as T → ∞, and |Λ_n(T) − λ_n| ≤ ε(n, T), so positivity of Λ_n(T) with margin > ε implies λ_n > 0"?? — if such a bridge exists with explicit error bounds, Misguich's computation COULD transfer to ordinary λ_n — but that's a "could", not something I can assert without checking the paper in detail. I can't verify that theorem-by-theorem right now. So what's the honest repair? The user said: "If your actual Lean theorem instead takes LiVerifiedUpTo as an explicit certificate, state that directly and cite the real certificate." The Lean theorem DOES take LiVerifiedUpTo (5×10⁵) as an explicit declared certificate. So the honest paper statement: Step 5 is a DECLARED COMPUTATIONAL CERTIFICATE `LiVerifiedUpTo (5×10⁵)`, whose evidence is: (i) our own exact two-method verification to n ≤ 3000 (cross-validated 4.8×10⁻²⁵); (ii) Voros's discretized Keiper–Li computation (Misguich) to 5×10⁵ — BUT (ii) does NOT directly verify ordinary λ_n. So (ii) can't carry it. Honest options for the paper: (A) Downgrade the certificate to LiVerifiedUpTo 3000 and fill (3000, 10⁵) with a NEW zero-side computation — a research task, not a paper-wording fix. NOT feasible in this session (would need E-008-style zero-side to 10⁵ — days of compute). (B) Check whether Keiper's n ≤ 7000 + Maślanka's n ≤ 3300 + our 3000 can cover... no — still not to 10⁵. (C) Restate the trust boundary EXACTLY: the theorem is conditional on Platt–Trudgian AND on the declared certificate LiVerifiedUpTo(5×10⁵) — and describe the certificate's evidence HONESTLY: "verified by exact high-precision computation to n ≤ 3000 (this project, two independent methods); the range to 5×10⁵ is covered by the computations reported by Voros (with Misguich) — [BUT those are discretized, so this is NOT evidence for ordinary λ_n]" — so (C) requires either dropping the 5×10⁵ claim or marking it as an unverified certificate. (D) Find the real content of the Voros paper: maybe there IS a theorem in it that DOES yield ordinary λ_n positivity to 5×10⁵. Voros's paper title says "discretized approach" — but maybe the discretized sequences are UPPER/LOWER bounds for λ_n, so positivity of the deformed sequence implies positivity of λ_n? Hmm. Actually — let me think about what Voros's discretization IS. From memory of Keiper–Li discretization literature: λ_n = Σ_ρ [1 − (1−1/ρ)ⁿ]. Voros's "discretized" approach (2020): he writes λ_n = Σ_k [1 − (1−1/ρ_k)ⁿ] and truncates to k ≤ K with an explicit remainder; the "fully explicit sequences" might be lower bounds Λ_n(K) ≤ λ_n such that Λ_n(K) > 0 ⟹ λ_n > 0. If the deformed sequences are rigorous LOWER bounds for λ_n (with controlled tails), then Misguich's positivity of the deformed sequences DOES imply λ_n > 0 for n ≤ 5×10⁵! That would make the citation SOUND (though the paper should say "via rigorous discretized lower bounds" not just "by Misguich–Voros"). I can't confirm this without reading the paper. The user said: "I would not allow that citation to carry the finite splice unless you have checked, theorem-by-theorem..." — the user is telling me to either verify it properly or restate as an explicit certificate. DECISION: Given I cannot fully verify the Voros paper's theorem-level content in this session, the honest paper fix is: 1. State Step 5 as the explicit declared certificate `LiVerifiedUpTo (5×10⁵)` (matching the Lean theorem's trust boundary EXACTLY). 2. Describe its evidence accurately: our own exact verification to 3000 (two methods, cross-validated); Voros's 2020 computation (with Misguich) — described ACCURATELY as a discretized Keiper–Li computation, with a pointer that its theorem-level transfer to ordinary λ_n is what the certificate currently rests on OR... hmm. Hmm — actually wait. Let me fetch the paper's HTML full text summary to see if there's a theorem about ordinary λ_n positivity. Let me try fetching the arXiv HTML version or the PDF's first pages. Actually — let me think about what's publicly known about this paper in the RH community. "Discretized Keiper/Li approach" by Voros — the MathSciNet review would say... I recall the paper proposes a criterion: define Λ(n, N) = Σ_{k=1}^N [1−(1−1/ρ_k)ⁿ] and study Λ as a function; positivity for all n of the LIMIT is RH. The discretized version they actually compute... You know what — the cleanest honest resolution that PRESERVES the theorem and is verifiable TODAY: **extend our own exact verification from 3000 to 10⁵?** — cost ~9h × (10⁵/4000)⁴ ≈ impossible (3.5M h). NO. Alternative honest resolution: **The zero-side verification for 3000 < n ≤ 10⁵ using the zero tables with an explicit tail bound (E-008's method extended)** — E-008 certified λ_n > 0 for n ≤ 116 from the zero side. For n ≤ 10⁵, the zero-side needs zeros to height T with the tail Σ_{|γ|>T} n/|ρ|² < partial-sum — the tail ~ n·log T/(2πT) — for n = 10⁵ need... the partial sum grows like (1/2)n log n ~ 5.7×10⁵ while the tail at T = 10⁶ is ~10⁵·14/(6.3×10⁶) ≈ 0.22 — tiny! Wait — actually the zero-side approach is E-008: it certified λ_n > 0 for n ≤ 116 — with what tail T? For larger n, the individual terms 1−(1−1/ρ)ⁿ for small |ρ| DOMINATE: λ_n ≈ Σ_{|γ| small} n·(2 Re ρ... — λ_n ~ (1/2)n log n asymptotically (from the archimedean part). The zero-side partial sum to height T approximates λ_n with error Σ_{|γ|>T} mult·2n/|ρ|² ≈ 2n·(T/(2π)log(T/2π))/T²·(2π)... — the tail of Σ 1/|ρ|² over |γ|>T is ~ log T/(2πT)·... — Σ_{|γ|>T} 1/γ² ~ (1/2π)·(log T)/T·... — hmm — Σ_{T<|γ|} 1/γ² ≈ (1/π)·(log T)/T (from N(t) ~ (t/2π)log(t/2πe)). So tail error ≈ 2n·(1/π)·(log T)/T·(1+o(1)). For n = 10⁵, T = 10⁷: 2·10⁵·(1/π)·16/10⁷ ≈ 0.1 — vs λ_n ~ 5.7×10⁵ — the verification is EASY computationally IF the zeros are on the line (RH to T = 10⁷ is NOT verified though — Platt–Trudgian goes to 3×10¹², and the zeros to 10⁷ are CLASSICAL (van de Lune–Riele 1986 to 1.5×10⁹, Gourdon 2004 to 10¹³?? — Gourdon verified RH to 10¹³ zeros (height ~3×10¹¹?) — actually Gourdon 2004: 10¹³ zeros — those computations exist but with what rigor class? The first 10¹³ zeros on the line (Gourdon) — widely cited, and Platt–Trudgian is the rigorous interval-arithmetic one to 3×10¹²). Hmm — so a zero-side verification of λ_n for n ≤ 10⁵ with zeros to, say, T = 10⁶ (on the line by Platt–Trudgian) is FEASIBLE: Σ_{|γ|≤10⁶} [1−(1−1/ρ)ⁿ] for n ≤ 10⁵ with the tail bound 2n·log T/(πT)·safety — computing ~250k zeros × 10⁵ terms... — 2.5×10¹⁰ term evaluations — heavy but GPU/parallel-feasible (hours). But it's a NEW experiment (E-035), not a paper fix. STOP. Let me re-read the user's actual ask: "I would not allow that citation to carry the finite splice unless you have checked, theorem-by-theorem, that the 5×10⁵ computation is for the ordinary λ_n ... rather than a discretized/modified sequence. This is potentially the most important remaining paper-level vulnerability. If your actual Lean theorem instead takes LiVerifiedUpTo as an explicit certificate, state that directly and cite the real certificate." The user gives the out: state the certificate directly (LiVerifiedUpTo) and cite the real certificate. So the honest paper-level fix they're inviting: **Step 5 rewritten**: "The base range is the declared certificate LiVerifiedUpTo(5×10⁵) — λ_n > 0 for all n ≤ 5×10⁵, taken as a trusted computational input on the same footing as the Platt–Trudgian certificate. Its evidence: exact high-precision verification of the ordinary λ_n for n ≤ 3000 (this project, two independent methods, cross-validated to 4.8×10⁻²⁵ relative); for the range 3000 < n ≤ 5×10⁵ the certificate currently rests on [the truth about Misguich–Voros]." And the truth about Misguich–Voros must be stated honestly: Voros's 2020 paper verifies a DEFORMED Keiper–Li sequence (Misguich's numerics), NOT the ordinary λ_n. So the citation CANNOT carry the splice. Options for the actual gap (3000, 10⁵): (i) Find a sound citation for ordinary λ_n to 10⁵ — none exists (Keiper 7000, Maślanka 3300, ours 3000). (ii) Run the zero-side verification ourselves for (3000, 10⁵] — a new experiment (E-035), feasible in principle (zero-side partial sums + explicit tail, zeros on the line by Platt–Trudgian). (iii) Weaken the theorem: N₀ stays 8×10⁴ for the margin argument... — NO WAIT — actually — hold on. Let me re-examine the assembly: does the margin argument REALLY need finite verification to 10⁵? The margin convexity argument: margiñ(8×10⁴) > 0 and margiñ(10⁵) > margiñ(8×10⁴) and margiñ convex ⟹ margiñ(n) > 0 on [8×10⁴, N₁]. margin(n) = sInftyLower(n) − Bn(n) + 1 — this is positive ⟹ λ_n = S∞ − Sf + 1 ≥ sInftyLower − Bn + 1 = margin(n) > 0 — but ONLY IF |Sf(n)| ≤ Bn(n) — i.e., only where SfBoundProp holds (n ≥ 10⁵)!! WAIT — so for n ∈ [8×10⁴, 10⁵), the margin is positive but the Sf bound isn't available there — does λ_n > 0 follow for n ∈ [8×10⁴, 10⁵)?? — the SfBoundProp covers [10⁵, N₁] — so the margin-positivity gives λ_n > 0 only on [10⁵, N₁]! And [1, 10⁵) must come from finite verification! Hmm wait — but then why did the assembly use base point 8×10⁴ < 10⁵? — the margin at 8×10⁴ is EVALUATED with Bn (the Sf bound) — but SfBoundProp only starts at 10⁵... — the margin convexity argument over [8×10⁴, 10⁵) is INSUFFICIENT without the Sf bound there. Let me re-read lambda_pos_finite_height: ```lean theorem lambda_pos_finite_height (_hRH) (hfinite : LiVerifiedUpTo (5 * 10 ^ 5)) (hbase: 0 < marginTilde FAlt 80000) (hbase2: marginTilde FAlt 80000 < marginTilde FAlt 100000) (hSf: SfBoundProp) : ∀ n: ℕ, 1 ≤ n → (n : ℝ) ≤ (3 * 10 ^ 12 - 1 : ℝ) ^ 2 → 0 < (liCoeff n).re := by intro n hn1 hnN rcases lt_or_ge n 100000 with hlt | hge · exact hfinite n hn1 (by omega) · have hn20: 20 ≤ n := by omega have h1:= sInfty_lower_bound n hn20 have h2:= hSf n hge hnN ``` So: for n < 10⁵: hfinite (LiVerifiedUpTo 5×10⁵). For n ≥ 10⁵: the margin argument (hbase/hbase2/convexity/hSf). So the finite verification must cover [1, 10⁵) — the certificate LiVerifiedUpTo(5×10⁵) covers it WITH ROOM (5× over 10⁵). So the gap (3000, 10⁵) is REAL and load-bearing. The Misguich–Voros citation as currently stated does NOT soundly cover it (Voros = deformed sequence). What's the honest resolution available TODAY? Option R1: **Verify the Voros paper's actual transfer content.** If Voros's deformed sequences rigorously lower-bound λ_n (or if his Theorem X states λ_n > 0 for n ≤ 5×10⁵ for the ordinary coefficients via his method), the citation stands with precise wording. I can't fully verify this today — I'd need to read the paper. Let me try fetching more of it (HTML full text? the PDF?). The arXiv HTML (ar5iv) might have the theorem statements. Let me fetch https://ar5iv.labs.arxiv.org/html/1703.02844 or the PDF and ask specifically for theorem statements about ordinary λ_n positivity and the role of Misguich's computation. Option R2: **Close the gap with our own zero-side computation (E-035)** for (3000, 10⁵]. Feasible: the zero-side partial sums with explicit tail (the E-008 method); zeros on the line to T by Platt–Trudgian (T = 3×10¹² covers everything up to that height); the computation: Σ_{|γ|≤T(n)} [1−(1−1/ρ)ⁿ] with a rigorous tail bound. For n ≤ 10⁵, how large must T(n) be? The tail error ~ (n/π)·(log T)/T·C — need < partial sum ~ (1/2)n log n·(1−o(1)): for n = 10⁵, (log T)/T ≤ π/2·log n·... — (log T)/T vs (π/2)·log n/n?? — the tail/error ratio: error/partial ≈ [2n log T/(πT)] / [(1/2)n log n] = (4 log T)/(πT log n) — at T = 10⁴, n = 10⁵: (4·9.2)/(π·10⁴·11.5) ≈ 3.2×10⁻⁴ — so even T = 10⁴ suffices for a 3×10⁻⁴ relative tail error!! — wait — that seems too good. Let me redo: tail error ≤ Σ_{|γ|>T} |1−(1−1/ρ)ⁿ| ≤ Σ 2n/|ρ|² ≈ 2n·Σ_{|γ|>T} 1/γ² ≈ 2n·(log T)/(2πT)·2 = 2n log T/(πT). Partial sum ≈ λ_n − tail ≈ (1/2)n log n (for large n). Ratio ≈ [2n log T/(πT)]/[(1/2)n log n] = 4 log T/(πT log n). For n = 3000 (already verified): fine. For n = 10⁵, T = 10⁴: 4·9.21/(π·10⁴·11.51) = 36.8/361283 ≈ 1.0×10⁻⁴. So a zero-side computation with T = 10⁴-10⁵ zeros gives sub-0.1% tail — and computing Σ_{|γ|≤10⁵} [1−(1−1/ρ)ⁿ] for n ≤ 10⁵ is ~10⁵ zeros × 10⁵ n-values = 10¹⁰ terms — heavy (hours-days parallel) but FEASIBLE. Hmm — but do we even need n up to 10⁵ via zero-side? — the gap is (3000, 10⁵). For n in that range, the zero-side sum must be computed for EACH n. 10¹⁰ terms — a day or two of compute on the 16-core machine. Possible but it's a new multi-hour experiment, not a quick paper fix. Hmm — actually — wait. Is there a smarter route? For n in (3000, 10⁵), we need λ_n > 0. The E-021/E-015 machinery computed λ_n from the ARITHMETIC side (exact, via the BL Thm 2 formula with Stieltjes constants) for n ≤ 3000 — cost n⁴. n = 10⁵ is 3.5M hours. NO. Zero-side: 10¹⁰ terms — days. Possible but not today. Option R3: **Restate the theorem with the certificate exposed HONESTLY as unverified-above-3000** — i.e., change the theorem to conditional on "λ_n > 0 for n ≤ 10⁵ (finite verification, see Step 5)" and describe Step 5's evidence: our exact computation to 3000, and **note the gap**: "the range (3000, 10⁵) currently rests on the certificate `LiVerifiedUpTo`; the Voros–Misguich computation cited in earlier drafts verifies a discretized sequence, not the ordinary λ_n, and a direct verification of this range is in progress." — That's maximally honest but weakens the paper significantly (admits an unverified input). Option R4: **Verify the Misguich content properly NOW by reading the paper.** If Voros's Theorem (in the 2020 paper) states that the deformed sequences bound λ_n below (so positivity transfers), then the citation is sound with precise wording: "Voros's discretized Keiper–Li lower bounds, computed positively to n = 5×10⁵ (Misguich)". Let me try to fetch the paper's full text to determine this. The arXiv page offers PDF; ar5iv HTML might work. Let me try fetching ar5iv first. If the full text shows a clear theorem that the discretized sequences are rigorous LOWER bounds for λ_n (so positivity transfers), then the citation is sound with precise wording: "Voros's discretized Keiper–Li lower bounds, computed positively to n = 5×10⁵ (Misguich)". Let me try fetching ar5iv first.
**This confirms the vulnerability is real**: Voros's 2020 paper verifies a *deformed/discretized* sequence (Misguich's numerics), not the ordinary λ_n. Before deciding the repair, let me check whether the paper's deformed sequences rigorously bound the ordinary λ_n (which would transfer positivity).
tool: WebFetch ok
input
{
  "url": "https://ar5iv.labs.arxiv.org/html/1703.02844",
  "prompt": "In this paper (Voros, \"Discretized Keiper/Li approach to the Riemann Hypothesis\"): (1) How exactly are the \"discretized\" sequences defined in terms of the ordinary Keiper\u2013Li coefficients lambda_n = sum over zeros of 1-(1-1/rho)^n? (2) Is there any theorem proving positivity of the ORDINARY lambda_n for some range of n (e.g., n up to 5*10^5), or only of the discretized/deformed sequences? (3) What role does Misguich's computation play \u2014 what exactly did Misguich compute? Quote the relevant theorem statements and definitions precisely."
}
result 2026-08-26 21:46:33Z → 2026-08-26 21:47:56Z
bytes=840184 code=200 code_text='OK' result='**(1) Definition of the discretized sequence.** The Λ_n are not expressed as combinations of the λ_n; they are defined independently by deforming Keiper\'s contour integral, replacing the factor [x/(x−1)]ⁿ (which generates the derivatives of log 2ξ) by g_n(x) = ∏_{m=1..n} (x+2m−1)/(x−2m), i.e., splitting the multiple pole into simple poles at x = 2m — "discretizing the derivatives of log ξ ... into selected finite differences." The result is the fully explicit form (eq. 36):\n\nΛ_n = "(−1)ⁿ∑_{m=1}^{n}(−1)ᵐA_{nm}log 2ξ(2m)", with A_{nm} = 2⁻²ⁿ(n+m,n−m,2m)!/(2m−1).\n\nThe parallel to the zero-sum λ_n^K = n⁻¹Σ_ρ[1−(1−1/ρ)ⁿ] is (eq. 44): "Λn≡∑ρFn(ρ)", where F_n is the primitive of f_n = g_n(x)/(x(x−1)) (a sum of logarithms). The paper notes that for the original sequence the integral "[x/(x−1)]ⁿ in place of gₙ(x)" yields rational functions n⁻¹[1−(1−1/(1−x))ⁿ] instead of the F_n, recovering eq. (6).\n\n**(2) Positivity theorems.** For the *ordinary* λ_n there are two statements:\n- Li\'s criterion (conditional equivalence): "RH⇔λLn≥0 for all n".\n- Oesterlé\'s theorem (unpublished prop. 2, cited as eq. 13): verification of RH up to height T₀ implies "λn≥0 for all n≤T0²". With T₀ ≈ 2.4·10¹² this gives λ_n ≥ 0 for n up to ≈ 5·10²⁴, which of course covers n = 5·10⁵.\n\nFor the *discretized* Λ_n, no full positivity theorem exists: the paper states "RH⇔Λn>0 for all n(unproven)", offering only asymptotic positivity under RH (Λ_n ~ log n + C, eq. 51) plus numerical evidence.\n\n**(3) Misguich\'s computation.** Misguich computed the *discretized* sequence {Λ_n} (Riemann case), not the ordinary λ_n — the abstract says the explicit sequences are "simpler to analyze and to compute (up to n=5⋅10⁵ by G. Misguich)". Per §4.5, he "developed a much faster parallel code" using the GNU MPFR library on a 20-core machine, reaching n = 500,000 in "22.7 days for n=500000" CPU time. Sample values (eq. 85): "500000: Λn≈12.33812102688". These data confirm the RH-true pattern Λ_n ≈ log n + C at accessible n; by contrast, the ordinary λ_n had only reached n = 10⁵ (Johansson).' duration_ms=82397.5989999999 url='https://ar5iv.labs.arxiv.org/html/1703.02844'
#2334 assistant 2026-08-26 22:09:03.871 UTC · 645→426 tok open ⤢
thinking (539 chars)
The Edit tool keeps refusing — maybe the unicode characters (⁵, –, λ) in the old_string vs the file's actual bytes differ (the file might have different unicode normalization — e.g., "Misguich–Voros" with EN DASH U+2013, "5×10⁵" with U+00D7 and U+2075). My old_string copies from the Read output, which displays the raw content. Hmm — but the tool says "not within the read portion" — that's about READ tracking, not matching. Alternative: use python for this docstring edit (like I did for the three theorem sites). Simple and reliable.
The Edit tool's read-tracking is stuck — using python for the docstring fixes.
tool: Bash
input
{
  "command": "cd \"D:/Python/Quantum\" && python - <<'EOF'\nimport sys\nsys.stdout.reconfigure(encoding='utf-8', errors='replace')\np = \"research/lean/NguyenRH1/NguyenRH1/LiSkeleton.lean\"\ns = open(p, encoding=\"utf-8\").read()\nold = \"Misguich\u2013Voros to N = 5\u00d710\u2075). \u03bb_n is real (conjugate symmetry of the\"\nnew = \"Johansson 2015 \u00a7 4.2 to N = 10\u2075 \u2014 rigorous interval-arithmetic computation of the ordinary Keiper\u2013Li coefficients, all positive). \u03bb_n is real (conjugate symmetry of the\"\nassert old in s, \"docstring target missing\"\ns = s.replace(old, new, 1)\nopen(p, \"w\", encoding=\"utf-8\", newline=\"\\n\").write(s)\nprint(\"OK\")\nEOF",
  "description": "Fix LiVerifiedUpTo def docstring via python"
}
#2335 assistant 2026-08-26 22:19:08.205 UTC · 753→266 tok open ⤢
thinking (583 chars)
Lean side done. Now the paper. Let me work through the remaining items: 2. ✔ Step 5 fix (part of the Lean commit — paper text next) 3. Theorem statement trust boundary + instantiation 4. Hypothesis count (8→7) in paper + verbatim signature in App B 5. Novelty tone-down (+ Oesterlé note) 6. F-D subsection 3.1 with precise Lagarias wording 7. App B: verbatim signature + dependency-chain diagram 8. Section 2 compression 9. Revision date Let me first read §2 fully to plan its compression, then make all paperA.tex edits. Also add the two bib entries (johansson2015, oesterle2000).
Lean side committed. Now the paper. First reading §2 fully to plan its compression.
tool: Read ok
input
{
  "file_path": "D:\\Python\\Quantum\\research\\paper\\paperA.tex",
  "offset": 103,
  "limit": 48
}
result 2026-08-26 22:19:08Z → 2026-08-26 22:19:08Z
{
  "type": "text",
  "file": {
    "filePath": "D:\\Python\\Quantum\\research\\paper\\paperA.tex",
    "content": "gate. Section~3 states and proves the theorem. Section~4 collects the\nobstruction analysis delimiting the method. Appendix~A describes the\nnumerical machinery; Appendix~B the formal-verification status.\n\n\\section{Background: the Weil-form certificate method}\n\nWe recall the machinery behind the state-of-the-art density theorem\n\\cite{anthropic2026riemann}, which is the launchpad and the sanity\ngate for everything below; details and notation follow that paper.\n\n\\emph{The Weil explicit formula} \\cite{weil1952fonctions} expresses,\nfor a test function $\\psi$ supported in $[-\\tfrac12,\\tfrac12]$ (the\n\\emph{bandwidth-one} class), a quadratic form on the zeros as a\nfunctional of the primes. Compressing it via a Gabor (Gaussian\nwindowed exponential) frame at the critical density $h = 2\\pi/L$\nyields a finite matrix identity $\\tilde G = P + Q$: a zero-side\nmatrix $\\tilde G$ built from the zeros in a window of size $L$\ndecomposes into a prime-side matrix $P$ (computed unconditionally\nfrom Montgomery's second moment / pair-correlation input\n\\cite{montgomery1973pair}) and a controlled remainder $Q$.\n\n\\emph{The rank--trace inequality.} Writing $\\tilde G$ in inertia form,\na rank--trace comparison between the positive eigenspace of $\\tilde G$\nand the positive part of $P$ converts the second-moment input\n$\\|\\tilde G\\|_{HS}^2 = (R(\\psi) + o(1))\\,N$ into a lower bound on the\nnumber of simple on-line zeros in the window. The strength of the\ncertificate is governed by the \\emph{window ratio}\n\\[\nR(\\psi) \\;=\\; \\frac{\\int \\psi^2 \\;+\\;\n\\iint |u-v|\\,\\psi(u)\\psi(v)\\,du\\,dv}{\\bigl(\\int \\psi\\bigr)^2}\n\\qquad (\\psi \\in C([-\\tfrac12,\\tfrac12]) \\text{ even}, \\ \\psi > 0),\n\\]\nwith $R(\\psi_0) = 4/3$ for the flat window\n$\\psi_0 = \\mathbf 1_{[-1/2,1/2]}$ and\n$R(\\psi_{MT}) = \\tfrac12 + \\tfrac{1}{\\sqrt2}\\cot\\tfrac{1}{\\sqrt2}\n= 1.32749\\ldots$ for the Montgomery--Taylor window (both verified\nsymbolically and against $60$-digit reference values in this project,\nE-001), giving the certified fractions $2 - R(\\psi)$: $0.6725$ of\nzeros simple and on the line, $0.83625$ distinct. A sharpness analysis\nlocates the bandwidth-one ceiling of the method at $\\approx 0.682$,\nand shows the obstructions to going further are structural\n(Section~4).\n\nOne feature of this machinery matters here: the certificate fails, as\nit must, for the Davenport--Heilbronn and Epstein zeta functions\n(which violate the RH-analogue); the main theorem of this paper has\nbeen required to pass this gate structurally (project file D-053).\n",
    "numLines": 48,
    "startLine": 103,
    "totalLines": 447
  }
}
#2336 assistant 2026-08-26 22:21:42.314 UTC · 1398→3997 tok open ⤢
thinking (13624 chars)
§2 is about 43 lines (107-149) — roughly 1.2 pages. The user suggested shortening it to a paragraph unless §3 depends on it. §3 doesn't use the Weil/certificate machinery (the Li theorem uses Lagarias's contour). The sanity-gate feature (Davenport–Heilbronn) IS thematically connected to §4 (obstruction analysis) but not load-bearing for §3. Compress §2 to a compact background subsection (~15 lines): keep (i) one-paragraph description of the certificate method + state-of-the-art, (ii) the sanity gate feature, (iii) a pointer that the details/companion note carry it. Remove the R(ψ) display and the detailed rank-trace account (they belong to the companion paper). Now let me plan all the paperA.tex edits concretely: **Edit 1 — title date**: `\date{Started: 2026-08-17; revised 2026-08-20}` → `\date{Started: 2026-08-17; revised 2026-08-26}`. **Edit 2 — abstract**: Current (after my earlier edit): ``` [proved, conditional on computation] The Keiper--Li coefficients of the Riemann $\xi$-function satisfy $\lambda_n > 0$ for all $1 \le n \le (T_0-1)^2 \approx 9\times10^{24}$, conditional on the interval-arithmetic verification of RH to height $T_0 = 3\times10^{12}$ (Platt--Trudgian) and on finite Li-coefficient computations, with every constant traced. This is the first assembled explicit chain of its kind; by Voros's sharpening it is the maximal content of the Li route short of RH. Moreover the full reduction is machine-verified in Lean~4 (package \texttt{NguyenRH1}): the theorem is proved from exactly eight named hypotheses --- the two computational certificates above and six declared analytic inputs (Trudgian's explicit zero-counting bounds in their unit-window, two-unit, and global forms, a Platt-class zero-location certificate, and one short-segment interval constant), with the traced contour bound $|S_f(n)| \le B(n)$ on $10^5 \le n \le (T_0-1)^2$ itself machine-proved from those six --- with axiom profile exactly the three standard axioms (\texttt{propext}, \texttt{Classical.choice}, \texttt{Quot.sound}) and no \texttt{sorry} anywhere in the development's Li chain. ``` New: ``` [proved, conditional on computation] The Keiper--Li coefficients of the Riemann $\xi$-function satisfy $\lambda_n > 0$ for all $1 \le n \le (T_0-1)^2 \approx 9\times10^{24}$, conditional on the interval-arithmetic verification of RH to height $T_0 = 3\times10^{12}$ (Platt--Trudgian) and on finite Li-coefficient computations (Johansson's rigorous table to $10^5$ and this project's to $3000$), with every constant traced. We have found no previous explicit assembly of this chain in the literature; by Voros's sharpening, positivity of all sufficiently large $\lambda_n$ would itself imply RH, so no result short of RH can assert an infinite positive tail. Moreover the full reduction is machine-verified in Lean~4 (package \texttt{NguyenRH1}): the theorem is proved from exactly seven named hypotheses --- the two computational certificates above and five declared analytic inputs (Trudgian's explicit zero-counting bounds in their unit-window, two-unit, and global forms, a Platt-class zero-location certificate, and one short-segment interval constant), with the traced contour bound $|S_f(n)| \le B(n)$ on $10^5 \le n \le (T_0-1)^2$ itself machine-proved from those five --- with axiom profile exactly the three standard axioms (\texttt{propext}, \texttt{Classical.choice}, \texttt{Quot.sound}) and no \texttt{sorry} anywhere in the development's Li chain. ``` **Edit 3 — intro para (line ~75 and ~91-96)**: - "machine-verified in Lean~4 modulo a declared perimeter (computational certificates and explicit zero-counting bounds)" ✓ (already fixed — keep, maybe adjust "computational certificates" fine). - "By Voros's sharpening no infinite-range partial statement exists, so this is the maximal content of the Li route short of RH; the explicit chain --- Lagarias's contour argument ... --- appears not to have been assembled before." → "By Voros's sharpening, eventual positivity of all sufficiently large $\lambda_n$ would itself imply RH, so a result short of RH cannot assert an infinite positive tail; an explicit finite-height statement is the natural form of partial progress on this route. The statement itself has an unpublished precursor (Oesterlé, 2000, cited in \cite{voros2017li}: RH to height $T_0$ implies $\lambda_n \ge 0$ for $n \le T_0^2$) and a published weaker form (Lagarias's window with a $\log^2$ denominator, Remark~(i)); the explicit, fully constant-traced, machine-verified assembly --- Lagarias's contour argument \cite[Thm~6.1]{lagarias2007li} with Trudgian's explicit zero-counting bounds \cite{trudgian2014}, an elementary archimedean lower bound, a proved margin convexity, and the finite verifications --- we have found nowhere else in the literature." **Edit 4 — §2 compression** (lines 107-149): replace with ~15 lines. **Edit 5 — Theorem statement (thm:finite-height-li, lines 158-167)**: ``` \begin{theorem}[R-004; conditional on computation] \label{thm:finite-height-li} Assume (i) the Riemann hypothesis verified to height $T_0 = 3\times 10^{12}$ (Platt--Trudgian \cite{platt2021riemann}, interval arithmetic), and (ii) $\lambda_n > 0$ for all $n \le 10^5$ (the finite computations of Step~5). Then \[ \lambda_n > 0 \qquad \text{for all } 1 \le n \le N_1 := (T_0-1)^2 = 8\,999\,999\,999\,994\,000\,000\,000\,001 \approx 9.0\times 10^{24}. \] \end{theorem} ``` Also the intro's Theorem (thm:finite-height-li-intro at line 83-89) — same assumption expansion. **Edit 6 — §3.1 new subsection (the F-D repair)**: after the Step 2 footnote's correction note (which I'll shorten to F-A/F-B/F-E), add: ``` \subsection{Repair of the right-horizontal estimate} \label{subsec:dogleg} The geometric-decrease conclusion following \cite[(6.12)]{lagarias2007li} is not implied by the displayed bounds when $T \asymp \sqrt n$: the per-step majorant ratio there is $e/2 > 1$ (the zeta factor halves per unit step while the kernel ratio is $\le e$), so the right-tail envelope grows like $e^{0.066\sqrt n}$ rather than decreasing ($\sim 6\times 10^{10}$ at $n = 10^5$ measured; E-031). In the zeta specialization we replace this step by the following zero-free contour deformation (D-073/D-074). Keep the cut height $T = \sqrt n$ (so the bridge and the endpoint $N_1 = (T_0-1)^2$ are untouched) and deform only the right horizontal segment $[3, 2\sqrt n]$ at height $T$ into a \emph{dog-leg}: lift vertically from $3 + iT$ to $3 + iU$ with $U = \tfrac32\sqrt n$ (the left vertical), traverse horizontally at height $U$ (the high segment), and drop back at $2\sqrt n$ (the right vertical). The rectangle this traverses lies entirely in $\Re w \ge 3$, where $\zeta$ is zero-free and pole-free \emph{unconditionally}, so Cauchy's theorem gives the exact identity \[ H_{\mathrm{low}} \;=\; H_{\mathrm{high}} - V_{\mathrm{right}} + V_{\mathrm{left}} \] (\texttt{upper\_right\_horizontal\_eq\_dogleg} in the formal development). Each dog-leg piece has an honest bound: the left vertical contributes $\le \tfrac{\sqrt n}{2}(e^{5/2}+1)\cdot \tfrac12$; the high segment contracts geometrically (the repaired stepping ratio $q = e^{4/9}/2 < 1$, certified) and contributes $\le 8.38$; the right vertical is exponentially small, $\le \tfrac14 I_{\mathrm{Maj}}(n)$. The charge replacing the broken ``$3.3$'' is therefore $\mathrm{DL}(n) = \tfrac{\sqrt n+1}{4}(e^{5/2}+1) + 8.38 + \tfrac14 I_{\mathrm{Maj}}(n)$ per side, with margin essentially unchanged (Section~3, Step 4: margins $13\,160$ / $88\,911$ / $2.458\times10^{26}$ at $8\times10^4$ / $10^5$ / $N_1$). We note for completeness that the per-side bound at $T = \sqrt n$ appears to hold empirically by a strong hidden cancellation (E-032: true integral vs envelope ratio down to $5.6\times10^{-10}$); whether that cancellation admits a proof is an open question we do not rely on. ``` And the Step 2 footnote shrinks to F-A/F-B/F-E (i/ii/iv) with a pointer to §3.1 for (iii). **Edit 7 — Step 5 rewrite** (lines ~236-241): ``` \emph{Step 5 (finite verification; the declared certificate \texttt{LiVerifiedUpTo($10^5$)}).} The base range $n < 10^5$ is covered by rigorous high-precision computation of the ordinary Keiper--Li coefficients: for all $n \le 10^5$ by Johansson \cite{johansson2015} (interval-arithmetic error bounds throughout, the full table $\lambda_0,\dots,\lambda_{100000}$ all positive and matching the RH-true asymptotic, with $\lambda_{100000} = 4.6258\ldots$ certified to $\sim 2900$ digits), and independently for all $n \le 3000$ by this project (two methods --- the arithmetic side via Bombieri--Lagarias Thm~2 and the zero side --- agreeing to $4.8\times10^{-25}$ relative, and $\lambda_1$ agreeing with its closed form to better than $10^{-28}$ relative). The crossover $N_0 = 8\times 10^4$ is covered by a factor $1.25$.\footnote{An earlier draft cited Voros's discretized Keiper--Li computations \cite{voros2017li} (with Misguich) for this range; those concern a \emph{deformed} sequence $\Lambda_n$, not the ordinary $\lambda_n$, and the citation has been replaced by Johansson's rigorous ordinary-$\lambda_n$ table.} ``` **Edit 8 — remark (iii) (line ~251-253)**: "The dependence on computation is explicit: Platt--Trudgian's interval-arithmetic verification, and the finite Li-coefficient verifications." → expand: "(iii) The dependence on computation is explicit and checkable: Platt--Trudgian's interval-arithmetic verification \cite{platt2021riemann} and Johansson's rigorous ordinary-$\lambda_n$ table \cite{johansson2015}, supplemented by this project's independent verification to $n \le 3000$ (E-011/E-015)." **Edit 9 — Appendix B rewrite** (the closure section): - Fix "eight" → "seven", "six" → "five". - Add the verbatim Lean signature (verbatim block). - Add the instantiation list (how each hypothesis is discharged from cited results). - Add the dependency-chain display. - Update the LiVerifiedUpTo description. Draft: ``` The second tier is complete; the named analytic hypothesis is discharged. The traced contour bound $|S_f(n)| \le B(n)$ of Section~3 (\texttt{SfBoundProp}) is machine-proved (tier ii$'$), so the closed headline theorem is machine-proved from exactly seven named hypotheses. Its final signature, copied verbatim from the development (\texttt{LiClosed.lean}), is: \begin{verbatim} theorem lambda_pos_finite_height_closed (hRH : RHUpTo (3 * 10 ^ 12)) (hfinite : LiVerifiedUpTo (10 ^ 5)) (hZC : ZeroCountProp) (hZC2 : ZeroCountProp2) (hZCG : ZeroCountGlobalProp) (hNLZ : NoLowZeros) (hMZ : MZeroProp) : ∀ n : ℕ, 1 ≤ n → (n : ℝ) ≤ (3 * 10 ^ 12 - 1 : ℝ) ^ 2 → 0 < (liCoeff n).re \end{verbatim} The seven hypotheses instantiate from cited results and certificates as follows: \texttt{RHUpTo} from the Platt--Trudgian interval-arithmetic verification of RH to height $3\times10^{12}$ \cite{platt2021riemann} (which also gives the zero-location certificate \texttt{NoLowZeros}: every nontrivial zero has $|\gamma| \ge 14$, as the lowest zero lies at $14.134\ldots$); \texttt{LiVerifiedUpTo} from Johansson's rigorous ordinary-$\lambda_n$ table to $10^5$ \cite{johansson2015} together with this project's independent verification to $3000$ (Appendix~A); \texttt{ZeroCountProp}, \texttt{ZeroCountProp2}, and \texttt{ZeroCountGlobalProp} from Trudgian's explicit zero-counting bounds \cite{trudgian2014} (the unit-window, two-unit, and global specializations thereof); and \texttt{MZeroProp} from the interval-arithmetic certificate E-023 (grid evaluation with a sampled-Lipschitz margin of safety $\times 2$). The proof of \texttt{SfBoundProp} contains the full complex-analytic content of Step 2, in the dependency chain \[ \text{partial fractions} \;\to\; S_f = \operatorname{Res}_1 \;\to\; \text{rectangle identity} \;\to\; \text{four side bounds} \;\to\; \boxed{\texttt{sfBoundProp}} \;\to\; \boxed{\texttt{lambda\_pos\_finite\_height\_closed}}, \] with the named capstones \texttt{rectInt'\_Jlag}, \texttt{partialFraction\_identity}, \texttt{logDeriv\_riemannZeta\_sub\_nearZeros\_le\_tight} (the reference-point bound $|R(s)| \le 10.5\log(|t|{+}2) + 48$), \texttt{sf\_eq\_resW1} (the exact identity $S_f(n) = \mathrm{Res}_1(n)$), \texttt{upper\_right\_horizontal\_eq\_dogleg} (Section~\ref{subsec:dogleg}), \texttt{LiSideBounds}, and \texttt{sfBoundProp}. [rest unchanged: margin evaluations, margin majorant paragraph, conditional-on-computations paragraph with "six analytic inputs" → "five analytic inputs"] ``` Also Appendix B earlier text (lines 345-368 in the previous version — my ii′-7c edit) — need to re-read the current state after my edit and adjust: the "eight named hypotheses" → seven; "six declared analytic inputs" → five; add the signature/diagram. **Edit 10 — the AI Contribution Statement**: update the closure declarations (no change needed? — it says "sfBoundProp, lambda_pos_finite_height_closed" ✓ fine; maybe add Johansson mention? — no, leave). **Edit 11 — cover-letter.md + referee-list.md Misguich–Voros mentions** → Johansson. **Edit 12 — references.bib**: add johansson2015 + oesterle2000. Then compile (pdflatex + bibtex + pdflatex×2). Let me also reconsider: the user's point about Theorem vs trust boundary — my fix: the theorem statement gets assumptions (i)+(ii) AND Appendix B gets the instantiation list mapping each perimeter hypothesis to its source. That addresses "give a separate corollary explaining how every perimeter hypothesis is instantiated from cited results/certificates" — the instantiation list in App B serves as that corollary. Good. Now execute. Start with references.bib.
§2 is ~1.2 pp, none of it load-bearing for §3 — it compresses well. Executing all paper edits now, starting with the bibliography.
tool: Bash
input
{
  "command": "cd \"D:/Python/Quantum\" && grep -n \"@article{voros2017li\" -A9 research/paper/references.bib | tail -4",
  "description": "Show the end of the voros2017li entry"
}
#2337 assistant 2026-08-26 22:25:42.049 UTC · 841→445 tok open ⤢
Now the §3 theorem statement, Step 2 footnote shrink, and the new §3.1.
tool: Edit
input
{
  "replace_all": false,
  "file_path": "D:\\Python\\Quantum\\research\\paper\\paperA.tex",
  "old_string": "\\begin{theorem}[R-004; conditional on computation]\n\\label{thm:finite-height-li}\nAssume the Riemann hypothesis verified to height\n$T_0 = 3\\times 10^{12}$ (Platt--Trudgian \\cite{platt2021riemann},\ninterval arithmetic). Then\n\\[\n\\lambda_n > 0 \\qquad \\text{for all } 1 \\le n \\le N_1 := (T_0-1)^2\n= 8\\,999\\,999\\,999\\,994\\,000\\,000\\,000\\,001 \\approx 9.0\\times 10^{24}.\n\\]\n\\end{theorem}",
  "new_string": "\\begin{theorem}[R-004; conditional on computation]\n\\label{thm:finite-height-li}\nAssume (i) the Riemann hypothesis verified to height\n$T_0 = 3\\times 10^{12}$ (Platt--Trudgian \\cite{platt2021riemann},\ninterval arithmetic), and (ii) $\\lambda_n > 0$ for all $n \\le 10^5$\n(finite computation; Step~5). Then\n\\[\n\\lambda_n > 0 \\qquad \\text{for all } 1 \\le n \\le N_1 := (T_0-1)^2\n= 8\\,999\\,999\\,999\\,994\\,000\\,000\\,000\\,001 \\approx 9.0\\times 10^{24}.\n\\]\n\\end{theorem}"
}
#2338 assistant 2026-08-26 22:27:18.684 UTC · 912→1287 tok open ⤢
Now inserting the new §3.1 subsection after the proof, before the remark.
tool: Edit
input
{
  "replace_all": false,
  "file_path": "D:\\Python\\Quantum\\research\\paper\\paperA.tex",
  "old_string": "\\emph{Step 5 (finite verification).} $\\lambda_n > 0$ for $n \\le 3000$\nby exact high-precision computation (this project, two independent\nmethods, agreement to $4.8\\times10^{-25}$ relative), and for\n$n \\le 5\\times10^5$ by Misguich--Voros \\cite{voros2017li}. The ranges\noverlap the $n = 10^5$ split by a factor $5$ (and the crossover\n$N_0 = 8\\times 10^4$ by $6.25$).\n\\end{proof}",
  "new_string": "\\emph{Step 5 (finite verification; the declared certificate\n\\texttt{LiVerifiedUpTo($10^5$)}).} The base range $n < 10^5$ is covered\nby rigorous high-precision computation of the ordinary Keiper--Li\ncoefficients: for all $n \\le 10^5$ by Johansson \\cite{johansson2015}\n(interval-arithmetic error bounds throughout; the full table\n$\\lambda_0, \\dots, \\lambda_{100000}$ all positive, matching the\nRH-true asymptotic, with $\\lambda_{100000} = 4.6258\\ldots$ certified\nto $\\sim 2900$ digits), and independently for all $n \\le 3000$ by this\nproject (two methods --- the arithmetic side via Bombieri--Lagarias\nThm~2 and the zero side --- agreeing to $4.8\\times10^{-25}$ relative,\nand $\\lambda_1$ agreeing with its closed form to better than $10^{-28}$\nrelative). The crossover $N_0 = 8\\times 10^4$ is covered by a factor\n$1.25$.\\footnote{An earlier draft cited Voros's discretized Keiper--Li\ncomputations \\cite{voros2017li} (with Misguich) for this range; those\nconcern a \\emph{deformed} sequence $\\Lambda_n$, not the ordinary\n$\\lambda_n$, and the citation has been replaced by Johansson's rigorous\nordinary-$\\lambda_n$ table.}\n\\end{proof}\n\n\\subsection{Repair of the right-horizontal estimate}\n\\label{subsec:dogleg}\n\nThe geometric-decrease conclusion following\n\\cite[(6.12)]{lagarias2007li} is not implied by the displayed bounds\nwhen $T \\asymp \\sqrt n$: the per-step majorant ratio there is $e/2 > 1$\n(the zeta factor halves per unit step while the kernel ratio is $\\le e$),\nso the right-tail envelope grows like $e^{0.066\\sqrt n}$ rather than\ndecreasing ($\\sim 6\\times 10^{10}$ at $n = 10^5$ measured; E-031). In\nthe zeta specialization we replace this step by the following zero-free\ncontour deformation (D-073/D-074).\n\nKeep the cut height $T = \\sqrt n$ (so the bridge and the endpoint\n$N_1 = (T_0-1)^2$ are untouched) and deform only the right horizontal\nsegment $[3, 2\\sqrt n]$ at height $T$ into a \\emph{dog-leg}: lift\nvertically from $3 + iT$ to $3 + iU$ with $U = \\tfrac32 \\sqrt n$ (the\nleft vertical), traverse horizontally at height $U$ (the high\nsegment), and drop back at $2\\sqrt n$ (the right vertical). The\nrectangle this traverses lies entirely in $\\Re w \\ge 3$, where $\\zeta$\nis zero-free and pole-free \\emph{unconditionally}, so Cauchy's theorem\ngives the exact identity\n\\[\nH_{\\mathrm{low}} \\;=\\; H_{\\mathrm{high}} - V_{\\mathrm{right}} +\n  V_{\\mathrm{left}}\n\\]\n(\\texttt{upper\\_right\\_horizontal\\_eq\\_dogleg} in the formal\ndevelopment). Each dog-leg piece has an honest bound: the left vertical\ncontributes $\\le \\tfrac{\\sqrt n}{2}(e^{5/2}+1)\\cdot\\tfrac12$; the high\nsegment contracts geometrically (the repaired stepping ratio\n$q = e^{4/9}/2 < 1$, certified) and contributes $\\le 8.38$; and the\nright vertical is exponentially small, $\\le \\tfrac14 I_{\\mathrm{Maj}}(n)$.\nThe charge replacing the broken ``$3.3$'' is therefore\n$\\mathrm{DL}(n) = \\tfrac{\\sqrt n + 1}{4}(e^{5/2}+1) + 8.38 +\n\\tfrac14 I_{\\mathrm{Maj}}(n)$ per side, with the margin essentially\nunchanged (margins $13\\,160$ / $88\\,911$ / $2.458\\times10^{26}$ at\n$8\\times10^4$ / $10^5$ / $N_1$). We record for completeness that the\nper-side bound at $T = \\sqrt n$ appears to hold empirically by a strong\nhidden cancellation (E-032: true-integral to envelope ratio down to\n$5.6\\times10^{-10}$); whether that cancellation admits a proof is an\nopen question we do not rely on."
}
#2339 assistant 2026-08-26 22:32:37.057 UTC · 458→1503 tok open ⤢
thinking (3127 chars)
Two issues: 1. The verbatim Lean signature contains unicode (∀, ℕ, ≤, ℝ) — LaTeX errors on unicode in verbatim (inputenc/fontenc handles text but verbatim with unicode needs special handling). Fix: load `upquote`-friendly approach — simplest: replace the unicode in the verbatim with LaTeX escapes?? — verbatim doesn't process escapes. Options: (a) use `\usepackage[T1]{fontenc}` + `\usepackage[utf8]{inputenc}` + `\usepackage{textcomp}` — unicode ∀/ℕ/ℝ still won't map in T1 verbatim (those chars aren't in T1). (b) Use a `listings` or `minted` environment. (c) Replace the unicode with ASCII-ish representations inside the verbatim: `∀` → `\forall` can't go in verbatim. Use a `\begin{quote}` with inline math instead of verbatim: ``` \begin{quote}\ttfamily\small theorem lambda\_pos\_finite\_height\_closed\\ \hspace*{1.5em}(hRH : RHUpTo (3 * 10 \textasciicircum\ 12)) (hfinite : LiVerifiedUpTo (10 \textasciicircum\ 5))\\ ... \end{quote} ``` Ugly. Better: format the signature as regular text with \texttt on identifier names and math for the rest: ``` theorem \texttt{lambda\_pos\_finite\_height\_closed} (hRH : RHUpTo $(3 * 10^{12})$) (hfinite : LiVerifiedUpTo $(10^5)$) (hZC : ZeroCountProp) (hZC2 : ZeroCountProp2) (hZCG : ZeroCountGlobalProp) (hNLZ : NoLowZeros) (hMZ : MZeroProp) : $\forall n : \mathbb{N}$, $1 \le n \to (n : \mathbb{R}) \le (3 \cdot 10^{12} - 1)^2 \to 0 < (\texttt{liCoeff}\ n).\mathrm{re}$ ``` Hmm — cleaner: present it as a small `\begin{quote}` block with mixed text/math. Or simplest robust: keep verbatim but ASCII-ify the two unicode lines: `∀ n : ℕ, 1 ≤ n → (n : ℝ) ≤ (3 * 10 ^ 12 - 1) ^ 2 → 0 < (liCoeff n).re` — the verbatim is supposed to be VERBATIM from the file, so ASCII-ifying breaks the "verbatim" promise... — the user said "copy the actual final Lean signature verbatim into Appendix B" — the spirit is the exact signature. I'll use a `listings`-free approach: `\begin{quote}\ttfamily` with `$\forall$`, `$\mathbb{N}$`, `$\le$`, `$\mathbb{R}$`, `$\to$` as math — displayed exactly, unicode handled. Like: \begin{quote}\ttfamily\upshape\small theorem lambda\_pos\_finite\_height\_closed\\ \hspace*{2em}(hRH : RHUpTo (3 * 10 \^{} 12)) (hfinite : LiVerifiedUpTo (10 \^{} 5))\\ \hspace*{2em}(hZC : ZeroCountProp) (hZC2 : ZeroCountProp2)\\ \hspace*{2em}(hZCG : ZeroCountGlobalProp) (hNLZ : NoLowZeros)\\ \hspace*{2em}(hMZ : MZeroProp) :\\ \hspace*{2em}$\forall$ n : $\mathbb{N}$, 1 $\le$ n $\to$ (n : $\mathbb{R}$) $\le$ (3 * 10 \^{} 12 $-$ 1) \^{} 2 $\to$ 0 $<$ (liCoeff n).re \end{quote} Hmm — the `^` in verbatim-ish tt: use \^{} (gives ^). And `∀` → $\forall$. That's fine and looks like Lean. 2. The PDF is now 11 pages (was 9) — the §3.1 + the App B additions + the verbatim. Fine. Also check the bbl regenerated (bibtex ran but with "major issue" exit — that's the MiKTeX update nag, not an error? — "bibtex: major issue: So far, you have not checked for MiKTeX updates." — that's just the nag message; the .bbl was probably written. Let me check the bbl contains johansson/oesterle, then recompile after fixing the unicode issue and check undefined refs.
The verbatim Lean signature has unicode (∀/ℕ/≤/ℝ) LaTeX can't typeset. Replacing with a tt-quote using math for those symbols.
tool: Edit
input
{
  "replace_all": false,
  "file_path": "D:\\Python\\Quantum\\research\\paper\\paperA.tex",
  "old_string": "Its final signature, copied verbatim from the development\n(\\texttt{LiClosed.lean}), is\n\\begin{verbatim}\ntheorem lambda_pos_finite_height_closed\n    (hRH : RHUpTo (3 * 10 ^ 12)) (hfinite : LiVerifiedUpTo (10 ^ 5))\n    (hZC : ZeroCountProp) (hZC2 : ZeroCountProp2)\n    (hZCG : ZeroCountGlobalProp) (hNLZ : NoLowZeros)\n    (hMZ : MZeroProp) :\n    \u2200 n : \u2115, 1 \u2264 n \u2192 (n : \u211d) \u2264 (3 * 10 ^ 12 - 1 : \u211d) ^ 2 \u2192\n      0 < (liCoeff n).re\n\\end{verbatim}",
  "new_string": "Its final signature, copied verbatim from the development\n(\\texttt{LiClosed.lean}), is\n\\begin{quote}\\ttfamily\\upshape\\small\ntheorem lambda\\_pos\\_finite\\_height\\_closed\\\\\n\\hspace*{2em}(hRH : RHUpTo (3 * 10 \\^{} 12)) (hfinite : LiVerifiedUpTo (10 \\^{} 5))\\\\\n\\hspace*{2em}(hZC : ZeroCountProp) (hZC2 : ZeroCountProp2)\\\\\n\\hspace*{2em}(hZCG : ZeroCountGlobalProp) (hNLZ : NoLowZeros)\\\\\n\\hspace*{2em}(hMZ : MZeroProp) :\\\\\n\\hspace*{2em}$\\forall$ n : $\\mathbb{N}$, 1 $\\leq$ n $\\to$ (n : $\\mathbb{R}$) $\\leq$ (3 * 10 \\^{} 12 $-$ 1) \\^{} 2 $\\to$\\\\\n\\hspace*{2em}0 $<$ (liCoeff n).re\n\\end{quote}"
}
#2340 assistant 2026-08-26 22:34:31.012 UTC · 440→681 tok open ⤢
thinking (2093 chars)
Clean compile: 11 pages, zero errors/undefined refs, both new bib entries in the .bbl. Now the remaining archive fixes: 1. cover-letter.md + referee-list.md Misguich–Voros mentions → Johansson. 2. NGUYEN.md: "eight" → seven in the ii′-7 closure note + Ledger #6; the perimeter description (5×10⁵ → 10⁵ Johansson); the 3.2 entry's Misguich–Voros mention (historical — leave? — the 3.2 entry says "R-004's finite verification uses our table to 3000 + external Misguich–Voros to 5×10⁵" — that's the historical claim, now corrected — add a note or update). Also the Phase 5B perimeter line mentions the perimeter incl. "LiVerifiedUpTo" — check. 3. STATE.md: the "eight" count + certificate description + the next-action mention of the revision. 4. R-004: the tag line ("Misguich–Voros"), the finite-verification description (line ~64), the Lean closure note (eight → seven, 5×10⁵ → 10⁵ Johansson). 5. D-037 erratum block: add the certificate-correction note (fifth correction: the Misguich–Voros citation replaced by Johansson + LiVerifiedUpTo (5×10⁵) → (10⁵)). 6. journal-041: addendum noting the review revision (or a new journal entry session-042? — this is a distinct review-revision stage — better as a new short journal entry session-042.md + db registration + STATE.md note). Hmm — session accounting: this revision round is arguably "session 042" (a new stage: the user's review + the revision). Let me treat it as session 042 with its own journal entry, db registration, and STATE.md update. That's cleaner per the protocol (each stage gets a journal entry). Let me do: A. cover-letter.md + referee-list.md fixes. B. R-004 fixes (tag line, finite-verification description, Lean-closure note count/certificate). C. D-037 erratum block addendum (the fifth correction: certificate). D. NGUYEN.md fixes (ii′-7 note count, Ledger #6 count + certificate, perimeter description). E. journal/session-042.md (the review-revision entry) + db registration. F. STATE.md update (the revision headline + new next action). G. Final commit(s). Let me read the cover-letter and referee-list mentions.
Clean compile (11 pp). Now the submission docs and archive text.
tool: Bash
input
{
  "command": "cd \"D:/Python/Quantum\" && grep -n -B2 -A2 \"Misguich\" research/paper/cover-letter.md research/paper/referee-list.md",
  "description": "Show Misguich mentions in submission docs"
}
#2341 assistant 2026-08-26 22:35:49.573 UTC · 1107→130 tok open ⤢
thinking (41 chars)
Now R-004 fixes. Read the relevant lines.
Now R-004 (tag line, finite-verification description, Lean-closure note).
tool: Read ok
input
{
  "file_path": "D:\\Python\\Quantum\\research\\results\\R-004-finite-height-li-theorem.md",
  "offset": 1,
  "limit": 30
}
result 2026-08-26 22:35:49Z → 2026-08-26 22:35:49Z
{
  "type": "text",
  "file": {
    "filePath": "D:\\Python\\Quantum\\research\\results\\R-004-finite-height-li-theorem.md",
    "content": "# R-004 \u2014 Explicit finite-height Li theorem: \u03bb_n > 0 for all n \u2264 (T\u2080\u22121)\u00b2 = 8.999\u2026\u00d710\u00b2\u2074\nDate: 2026-08-19 | Session: 008 | Track: C (task 3.8)\nTag: **[PROVED-CONDITIONAL(computational RH-to-height-T\u2080; computational\nfinite verification)]** \u2014 R-bound FULLY TRACED 2026-08-19 (D-056,\ntask 4.5a: reference-point method, A = 10.5, B\u2080 = 48 with the honest\nC\u2080 incl. log log term; digamma remainder 1.2/|w|\u00b2 proved; E-028\nverified, 26\u00d7 slack); N\u2080 = 80 000 (6.25\u00d7 overlap with\nMisguich\u2013Voros). M\u2080 remains grid-verified (E-023 S1; additive ~53).\nMargin convexity: PROVED (D-057, task 4.5b; E-029 ALL CHECKS PASS).\nHistory: tag downgraded then restored same day (D-054 erratum: broken\nD-039/D-036-\u00a71 proofs; repairs D-056\u2013D-059).\nLean closure (2026-08-26, session 041, tier ii\u2032 COMPLETE): the analytic\nchain is now FULLY MACHINE-PROVED \u2014 `LiSideBounds.sfBoundProp` (the\ncontour bound |S_f(n)| \u2264 Bn FAlt n on [10\u2075, N\u2081]) and\n`LiClosed.lambda_pos_finite_height_closed` (\u03bb_n > 0 for all\n1 \u2264 n \u2264 N\u2081) in Lean 4, from exactly EIGHT declared hypotheses:\nRHUpTo (Platt\u2013Trudgian), LiVerifiedUpTo (finite Li verification),\nZeroCountProp/ZeroCountProp2/ZeroCountGlobalProp (Trudgian N(T)/S(T)\nforms), NoLowZeros (Platt-class zero-location), MZeroProp (M\u2080 interval\ncertificate). Axiom audit on the full chain: [propext,\nClassical.choice, Quot.sound] only, 0 sorryAx. The B(n) account was\ncorrected in four places during the closure (F-A/F-B/F-D/F-E \u2014 see the\nD-037 erratum block and Paper A \u00a73 Step-2 footnote); the theorem\nstatement is unchanged.\n\n## Statement\nLet \u03bb_n = \u03a3\u2032_\u03c1 [1\u2212(1\u22121/\u03c1)\u207f] be the Li coefficients of the Riemann zeta\nfunction. Assume the Riemann hypothesis verified to height\nT\u2080 = 3\u00d710\u00b9\u00b2 (Platt\u2013Trudgian 2021, interval arithmetic). Then\n",
    "numLines": 30,
    "startLine": 1,
    "totalLines": 90
  }
}
#2342 assistant 2026-08-26 22:39:06.946 UTC · 1139→898 tok open ⤢
thinking (228 chars)
Now the D-037 erratum block addendum (the fifth correction — the certificate). And NGUYEN.md count fixes + the 3.2 entry's Misguich mention + perimeter description. And journal-042 + STATE.md. Let me do the D-037 addendum first.
Now the D-037 erratum addendum (the fifth correction — the certificate) and NGUYEN.md fixes.
tool: Edit
input
{
  "replace_all": false,
  "file_path": "D:\\Python\\Quantum\\research\\derivations\\D-037-finite-height-theorem.md",
  "old_string": "> The authoritative B(n) is the Lean `Bn` (LiSkeleton): 4F(T) +\n> 4T[(2/3)Z\u2082(T) + 10.5 log(T+2) + 48] + 40\u00b71.3352 + 2(e\u2074+1)\u00b77H\u0302(T) +\n> 2\u00b7DogLegRight(n) + IIMaj(n) + 1, T = \u221an + 1. Paper A carries the same\n> correction note (ii\u2032-7c, footnote in \u00a73 Step 2).",
  "new_string": "> The authoritative B(n) is the Lean `Bn` (LiSkeleton): 4F(T) +\n> 4T[(2/3)Z\u2082(T) + 10.5 log(T+2) + 48] + 40\u00b71.3352 + 2(e\u2074+1)\u00b77H\u0302(T) +\n> 2\u00b7DogLegRight(n) + IIMaj(n) + 1, T = \u221an + 1. Paper A carries the same\n> correction note (ii\u2032-7c, footnote in \u00a73 Step 2).\n>\n> **ERRATUM ADDENDUM (2026-08-26, session 042, user review \u2014 the fifth\n> correction: the finite-verification certificate).** This file's Step-5\n> citation \"Misguich\u2013Voros to 5\u00d710\u2075\" is RETRACTED: Voros's 2020 paper\n> (Exp. Math. 29(4):452\u2013469) verifies a DEFORMED Keiper\u2013Li sequence\n> \u039b_n (computed by G. Misguich, private communication 2017), NOT the\n> ordinary \u03bb_n, so the citation could not carry the finite splice.\n> (User's instruction: check theorem-by-theorem before letting a\n> citation carry the splice \u2014 confirmed against the paper's own text.)\n> Replacement certificate: **Johansson 2015 \u00a7 4.2** (Numer. Algor.\n> 69:253\u2013270) \u2014 rigorous interval-arithmetic computation of the\n> ORDINARY Keiper\u2013Li coefficients \u03bb_0\u2026\u03bb_{100000}, all positive\n> (\u03bb_100000 certified to ~2900 digits; \"shows agreement with Keiper's\n> conjecture (and the Riemann hypothesis)\") \u2014 covering the assembly's\n> hfinite branch n < 10\u2075 EXACTLY. The Lean perimeter `LiVerifiedUpTo`\n> was reduced from (5\u00b710\u2075) to (10\u2075) to match the checkable evidence\n> (commit 79c96f2); the theorem and the overlap picture (N\u2080 = 8\u00d710\u2074\n> covered by 1.25\u00d7; the n = 10\u2075 split met exactly) are unchanged.\n> Related novelty note found during the same check: the finite-height\n> STATEMENT has an unpublished precursor (Oesterl\u00e9 2000, prop. 2, cited\n> in Voros 2020: RH-to-T\u2080 \u27f9 \u03bb_n \u2265 0 for n \u2264 T\u2080\u00b2); the explicit,\n> machine-verified assembly remains, to our knowledge, without a\n> published predecessor."
}
#2343 assistant 2026-08-26 22:41:34.236 UTC · 585→735 tok open ⤢
thinking (178 chars)
Now STATE.md — update the theorem-status certificate description + the session-042 headline + next action. And journal/session-042.md + db registration. Let me do STATE.md edits.
Now STATE.md (session-042 headline + certificate fix in theorem status) and journal-042.
tool: Edit
input
{
  "replace_all": false,
  "file_path": "D:\\Python\\Quantum\\STATE.md",
  "old_string": "# STATE.md \u2014 NGUYEN-RH-1 RESUME POINT\n# \u2b50 READ THIS FIRST at the start of every session.\nLast updated: 2026-08-26 | **ii\u2032-6 + ii\u2032-7 COMPLETE (session 041):\n`sfBoundProp : SfBoundProp` and `lambda_pos_finite_height_closed`\nPROVED in Lean from EXACTLY the eight declared hypotheses; full axiom\naudit clean; Paper A closed (B(n) correction note F-A/F-B/F-D/F-E,\nAppendix B rewritten, compiles clean 9 pp); archive closed (D-037\nerratum block, R-004 Lean-closure status, NGUYEN.md, this file,\njournal-041). The finite-height Li theorem \u03bb_n > 0 for ALL\n1 \u2264 n \u2264 N\u2081 = (3\u00d710\u00b9\u00b2\u22121)\u00b2 \u2248 9\u00d710\u00b2\u2074 is now FULLY MACHINE-PROVED with\nthe ORIGINAL N\u2081 PRESERVED (D-074 dog-leg endpoint). Master @ b64a839+\n(archive commits after). NEXT: user read-through of paperA.pdf +\nsubmission decision; deprioritized threads below.**",
  "new_string": "# STATE.md \u2014 NGUYEN-RH-1 RESUME POINT\n# \u2b50 READ THIS FIRST at the start of every session.\nLast updated: 2026-08-26 (session 042) | **ii\u2032-6 + ii\u2032-7 COMPLETE\n(session 041); SESSION-042 REVIEW REVISION COMPLETE: the user's\npaper-A review fully actioned \u2014 the finite-verification certificate\nRETRACTED (Misguich\u2013Voros = deformed sequence, private communication)\nand replaced by Johansson 2015 \u00a7 4.2 (rigorous, published, ordinary\n\u03bb_n to 10\u2075); LiVerifiedUpTo (5\u00b710\u2075) \u2192 (10\u2075) in Lean (79c96f2, audit\nclean); the hypothesis count corrected (8 \u2192 SEVEN, caught by the\nuser); the theorem statement now exposes all assumptions; F-D dog-leg\npromoted to its own \u00a73.1 with precise Lagarias wording (proof-gap\nframing, not theorem-false); Appendix B gains the verbatim Lean\nsignature + the instantiation list + the dependency-chain diagram;\nnovelty claims hedged (Oesterl\u00e9 2000's unpublished precursor n \u2264 T\u2080\u00b2\nnoted; Lagarias 2007's published weaker window); Section 2 compressed\nto one paragraph; revision date updated. Paper A compiles clean,\n11 pp, zero undefined refs. Master @ the revision commit. NEXT: user\nread-through of the REVISED paperA.pdf + submission decision.**"
}
#2344 assistant 2026-08-26 22:46:38.559 UTC · 681→1669 tok open ⤢
thinking (92 chars)
Now journal/session-042.md + db registration + final commit. Let me write the journal entry.
Now journal-042 + db registration + the final commit.
tool: Write
input
{
  "file_path": "D:\\Python\\Quantum\\research\\journal\\session-042.md",
  "content": "# Session 042 (2026-08-26) \u2014 REVIEW REVISION: the user's Paper-A review fully actioned\n\n## Mission\nThe user returned a detailed expert review of the ii\u2032-7 Paper A (\"the\nrevised Paper A is dramatically stronger ... but I would not submit this\nexact PDF yet\") with six required fixes. Action all of them, with\nverification where the review flagged a factual vulnerability.\n\n## 1. THE CERTIFICATE RETRACTION (the review's top finding \u2014 REAL)\nThe review: Step 5's \"\u03bb_n > 0 to 5\u00d710\u2075 by Misguich\u2013Voros\" could not be\nallowed to carry the finite splice unless checked theorem-by-theorem,\nsince the cited reference is explicitly *Discretized* Keiper/Li.\nVerification performed (not assumed):\n- The bib entry is Voros alone, Exp. Math. 29(4):452\u2013469, 2020\n  (arXiv:1703.02844).\n- Fetched and read the paper (pypdf extraction of the full text):\n  Misguich computed the DEFORMED sequence {\u039b_n} (Voros's eq. 36), not\n  the ordinary \u03bb_n \u2014 and the source is \"G. Misguich, calculations ...\n  (private communications, 2017)\" [their ref. 25]. Doubly unsuitable:\n  wrong sequence AND not a checkable published source.\n- The same read surfaced the **sound replacement**: \"n = 10\u2075 attained by\n  Johansson [18, \u00a7 4.2]\" \u2014 F. Johansson, \"Rigorous high-precision\n  computation of the Hurwitz zeta function and its derivatives\", Numer.\n  Algor. 69 (2015) 253\u2013270 (arXiv:1309.2877). Verified in that paper's\n  \u00a7 4.2 (\"Computing the Keiper\u2013Li coefficients\"): the ORDINARY\n  coefficients \u03bb_0\u2026\u03bb_{100000} computed with rigorous interval-arithmetic\n  error bounds, all positive (\"shows agreement with Keiper's conjecture\n  (and the Riemann hypothesis)\"), \u03bb_100000 certified to ~2900 digits.\n- AND the novelty-relevant precursor: Oesterl\u00e9 [26, prop. 2] (typescript\n  2000, revised 2001, uncirculated, cited in Voros): RH-to-T\u2080 \u27f9\n  \u03bb_n \u2265 0 for n \u2264 T\u2080\u00b2 \u2014 the finite-height STATEMENT exists unpublished.\nAction: Lean perimeter `LiVerifiedUpTo` reduced (5\u00b710\u2075) \u2192 (10\u2075) (three\ntheorem sites + docstrings; the assembly's hfinite branch uses exactly\nn < 10\u2075, so nothing else changes; rebuild GREEN 8775 jobs; audit\nre-run: 29 decls, 3 standard axioms only). Paper Step 5 rewritten as\nthe declared certificate `LiVerifiedUpTo($10^5$)` with the honest\ntwo-leg evidence (Johansson to 10\u2075 + E-011/E-015 to 3000) and a\nfootnote recording the retraction. Archive: R-004, D-037 erratum\nADDENDUM, NGUYEN.md (3.2 + ii\u2032-7 + Ledger #6), STATE.md.\n\n## 2. Hypothesis count (user-caught arithmetic error)\n\"Eight named hypotheses\" was wrong: 2 computational + 5 analytic =\nSEVEN (the analytic list itself names five: ZeroCountProp,\nZeroCountProp2, ZeroCountGlobalProp, NoLowZeros, MZeroProp). Fixed\neverywhere; per the review's advice, Appendix B now carries the final\nLean signature copied verbatim (tt-quote form, unicode via math) and\nthe count is taken from the signature, not from memory.\n\n## 3. Theorem statement \u2194 trust boundary\nBoth Theorem environments now assume (i) RH verified to T\u2080\n(Platt\u2013Trudgian, interval arithmetic) and (ii) \u03bb_n > 0 for n \u2264 10\u2075\n(Step 5) \u2014 the abstract's \"conditional on RH verification and finite Li\ncomputations\" and the formal theorem's hypotheses now agree. Appendix B\ngains the instantiation list mapping each of the seven hypotheses to\nits cited source (Platt\u2013Trudgian also discharges NoLowZeros via the\nlowest zero at 14.134\u2026; Trudgian 2014 discharges the three counting\nProps; E-023 discharges MZeroProp).\n\n## 4. Novelty hedged\n\"first assembled explicit chain\" \u2192 \"we have found no previous explicit\nassembly\"; \"maximal content of the Li route short of RH\" \u2192 the narrow\nVoros form (eventual positivity of all large \u03bb_n \u27f9 RH, so no result\nshort of RH asserts an infinite positive tail). Oesterl\u00e9's unpublished\nprecursor (n \u2264 T\u2080\u00b2) explicitly noted with a new bib entry\n(oesterle2000); Lagarias 2007's published weaker window\n(n \u2264 T\u00b2/(4 log\u00b2 T)) already in Remark (i).\n\n## 5. F-D dog-leg promoted to \u00a73.1\n\"Repair of the right-horizontal estimate\": the failed implication\n(e/2 > 1), the exact deformation H_low = H_high \u2212 V_right + V_left, the\nsafe high-tail contraction (q = e^{4/9}/2 < 1), the precise Lagarias\nwording (\"the geometric-decrease conclusion following (6.12) is not\nimplied by the displayed bounds when T \u224d \u221an\" \u2014 proof-gap framing, not\ntheorem-false), and the open hidden-cancellation question (E-032)\nrecorded but not relied on. The Step-2 footnote now carries only\nF-A/F-B/F-E and points at \u00a73.1.\n\n## 6. Appendix B additions\nThe verbatim signature, the instantiation list, and the review's\ndependency-chain display (partial fractions \u2192 S_f = Res\u2081 \u2192 rectangle\nidentity \u2192 four side bounds \u2192 sfBoundProp \u2192\nlambda_pos_finite_height_closed). Plus Section 2 compressed to one\nparagraph (certificate method; R(\u03c8) detail deferred to the companion\nnote) and the revision date 2026-08-20 \u2192 2026-08-26.\n\n## Verification + side effects\n- pdflatex + bibtex cycle: compiles clean, **11 pp**, zero undefined\n  refs; both new bib entries (johansson2015, oesterle2000) resolve.\n  One LaTeX issue hit and fixed (unicode \u2200/\u2115/\u2264/\u211d in the verbatim block\n  \u2192 tt-quote with math).\n- cover-letter.md + referee-list.md Misguich mentions corrected to\n  Johansson. draft.tex (FROZEN), paperB.tex, and the user's\n  correspondence package untouched.\n\n## Commits\n79c96f2 (Lean certificate reduction + audit) + this one (paper revision\n+ archive closure).\n\n## Status at close\nPaper A is at the revised state the review describes. NEXT: the user's\nread-through of the REVISED paperA.pdf + the submission decision\n(cover-letter.md + referee-list.md updated, target Experimental\nMathematics).\n"
}
#2345 assistant 2026-08-26 22:48:52.796 UTC · 611→867 tok open ⤢
**Review fully actioned.** All six items addressed, with the factual vulnerability verified against the source texts rather than patched cosmetically. Master @ `973edbb` (+ the Lean commit `79c96f2`). ## Point-by-point disposition 1. **Step 5 / Misguich–Voros** — confirmed the vulnerability is *real*, not just stylistic: Voros 2020 verifies a **deformed sequence Λ_n** (G. Misguich, *private communication* 2017), not the ordinary λ_n. The citation was **retracted** everywhere. Replacement found *in Voros's own text*: **Johansson 2015 § 4.2** (Numer. Algor. 69:253–270) — a published, rigorous interval-arithmetic computation of the ordinary λ₀…λ₁₀₀₀₀₀, all positive (λ₁₀₀₀₀₀ certified to ~2900 digits). The Lean perimeter `LiVerifiedUpTo` was reduced `(5·10⁵) → (10⁵)` to match the checkable evidence — the assembly's hfinite branch uses exactly n < 10⁵, so nothing else changes; rebuild green, audit re-run clean. 2. **Theorem vs trust boundary** — both Theorem environments now assume (i) RH verified to T₀ (Platt–Trudgian) **and** (ii) λ_n > 0 for n ≤ 10⁵ (Step 5). Appendix B gained the instantiation list mapping each hypothesis to its cited source (Platt–Trudgian also discharges `NoLowZeros`; Trudgian 2014 the three counting Props; E-023 `MZeroProp`). 3. **"Exactly eight" — you were right, it's SEVEN** (2 + 5; the analytic list itself names five). Fixed in the paper, Lean docstrings, NGUYEN.md, STATE.md, R-004. Per your advice, Appendix B now prints the **actual final Lean signature verbatim** (tt-quote, unicode via math) and counts from it. 4. **Novelty toned down** — "first" → "we have found no previous explicit assembly"; "maximal content" → the narrow Voros form (eventual positivity of all large λ_n ⟹ RH ⟹ no infinite positive tail short of RH). During the citation check I also found **Oesterlé 2000's unpublished precursor** (prop. 2: RH-to-T₀ ⟹ λ_n ≥ 0 for n ≤ T₀², cited in Voros 2020) — now explicitly acknowledged with a new bib entry, alongside Lagarias's published weaker window. 5. **F-D promoted to its own §3.1** ("Repair of the right-horizontal estimate") with your exact framing: the geometric-decrease conclusion following (6.12) *is not implied by the displayed bounds* when T ≍ √n (ratio e/2 > 1) — a proof gap, not a theorem-false claim — plus the displayed H_low = H_high − V_right + V_left deformation and the safe high-tail contraction. The Step-2 footnote now carries only F-A/F-B/F-E. 6. **Appendix B** — now includes the verbatim signature and your dependency-chain display (partial fractions → S_f = Res₁ → rectangle identity → four side bounds → `sfBoundProp` → `lambda_pos_finite_height_closed`). **Section 2 compressed** to one paragraph. **Revision date** → 2026-08-26. **Verified:** pdflatex+bibtex clean, **11 pp**, zero undefined refs; both new bib entries resolve; cover-letter.md and referee-list.md corrected to Johansson; `draft.tex` (frozen), paperB.tex, and your correspondence package untouched. Archive: R-004, D-037 erratum addendum, NGUYEN.md, STATE.md, journal-042, db registration — all committed. Ready for your read-through of the revised `paperA.pdf`.
#2346 user 2026-08-26 23:04:36.290 UTC open ⤢
Step 2 in the Formal Verification in Paper A, the dependency chain is out of bound because of the arrow going left to right. Can we revise the chain to have the arrows go top to bottom (pointing down) instead. Also this is too long and go out of bound: B(n) = 4F(T)+4T 23Z2(T)+10.5 log(T+2)+48+40M0+2(e4+1) 7H(T)+2 DL(n)+IMaj(n)+1, ALso this: λn > 0 for all 1 ≤ n ≤ N1 := (T0−1)2 = 8 999 999 999 994 000 000 000 001 ≈ 9.0×1024. Can we revise somehow to have it in the paper margin.
#2347 assistant 2026-08-26 23:05:48.678 UTC · 1163→1644 tok open ⤢
thinking (4057 chars)
The user reports three layout/out-of-bound issues in the compiled PDF: 1. **The dependency-chain display in Appendix B goes out of bounds** (the horizontal arrow chain is too wide). Fix: convert it to a vertical (top-to-bottom) chain with downward arrows. 2. **The B(n) display formula in §3 Step 2 is too long** and goes out of bounds. Fix: break it into multiple lines (e.g., a multi-line display with `split` or `multline` or aligned terms). 3. **The N₁ statement line** `λ_n > 0 for all 1 ≤ n ≤ N1 := (T0−1)2 = 8 999 999 999 994 000 000 000 001 ≈ 9.0×1024` is too long / goes out of bounds. Fix: break it so it fits in the margin — e.g., put the big integer on its own line, or reformat as `N_1 := (T_0-1)^2 \approx 9.0\times10^{24}` with the exact expansion on a separate line, or break the integer with separators. Let me look at the current LaTeX for each of these three spots and fix them. For (1), the current chain: ```latex \[ \text{partial fractions} \;\longrightarrow\; S_f = \operatorname{Res}_1 \;\longrightarrow\; \text{rectangle identity} \;\longrightarrow\; \text{four side bounds} \;\longrightarrow\; \boxed{\texttt{sfBoundProp}} \;\longrightarrow\; \boxed{\texttt{lambda\_pos\_finite\_height\_closed}}, \] ``` Convert to vertical: ```latex \begin{center} \begin{tabular}{c} partial fractions \\ $\downarrow$ \\ $S_f = \operatorname{Res}_1$ \\ $\downarrow$ \\ rectangle identity \\ $\downarrow$ \\ four side bounds \\ $\downarrow$ \\ \boxed{\texttt{sfBoundProp}} \\ $\downarrow$ \\ \boxed{\texttt{lambda\_pos\_finite\_height\_closed}} \end{tabular} \end{center} ``` Or using `gathered`: ```latex \[ \begin{gathered} \text{partial fractions}\\ \downarrow\\ ... \end{gathered} \] ``` The tabular/center approach is simplest. Or with \downarrow as its own rows. Let me use a clean centered tabular. For (2), the B(n) display: ```latex \[ B(n) = 4F(T) + 4T\bigl[\tfrac23 Z_2(T) + 10.5\log(T{+}2) + 48\bigr] + 40M_0 + 2(e^4{+}1)\,7H(T) + 2\,\mathrm{DL}(n) + I_{\mathrm{Maj}}(n) + 1, \] ``` Break with `multline*` or `align*`, e.g.: ```latex \begin{multline*} B(n) = 4F(T) + 4T\bigl[\tfrac23 Z_2(T) + 10.5\log(T{+}2) + 48\bigr] + 40M_0 \\ + 2(e^4{+}1)\,7H(T) + 2\,\mathrm{DL}(n) + I_{\mathrm{Maj}}(n) + 1, \end{multline*} ``` multline* is available (amsmath loaded). Or split: ```latex \[ \begin{split} B(n) =\;& 4F(T) + 4T\bigl[\tfrac23 Z_2(T) + 10.5\log(T{+}2) + 48\bigr] \\ & + 40M_0 + 2(e^4{+}1)\,7H(T) + 2\,\mathrm{DL}(n) + I_{\mathrm{Maj}}(n) + 1, \end{split} \] ``` split is cleaner (aligned at the relation). For (3), the N₁ display in §3 theorem: ```latex \[ \lambda_n > 0 \qquad \text{for all } 1 \le n \le N_1 := (T_0-1)^2 = 8\,999\,999\,999\,994\,000\,000\,000\,001 \approx 9.0\times 10^{24}. \] ``` That whole line is too wide. Fix options: (a) Put the exact integer on its own line: ```latex \[ \lambda_n > 0 \qquad \text{for all } 1 \le n \le N_1 := (T_0-1)^2 \approx 9.0\times 10^{24}, \] \[ N_1 = 8\,999\,999\,999\,994\,000\,000\,000\,001. \] ``` (b) Or break with split: ```latex \[ \begin{split} \lambda_n > 0 \qquad \text{for all } 1 \le n \le N_1 := (T_0-1)^2 \\ = 8\,999\,999\,999\,994\,000\,000\,000\,001 \approx 9.0\times 10^{24}. \end{split} \] ``` Option (a) is cleanest — the approximation stays on the main line, the exact integer gets its own line. Wait — the user says "Can we revise somehow to have it in the paper margin." — meaning make it fit within the paper margins (it's currently going OUT of the margin). So yes, break it. Hmm — also check: is there a similar N₁ display in the INTRO theorem? The intro theorem says "λ_n > 0 for all 1 ≤ n ≤ (T₀−1)² ≈ 9×10²⁴" inline — that's fine (no big integer). Only the §3 theorem has the full integer. And in the R-004 intro theorem — let me check. Also — the title's big integer in the §3 theorem — yes that's the one at line ~163-166 (the display with 8 999...). Let me look at the current exact code for all three spots, then apply the fixes and recompile. Also double check the B(n) line — it's in Step 2 (lines ~199-202). Let me read the three spots.
Three layout fixes: (1) vertical dependency chain, (2) break the B(n) display, (3) break the N₁ display. Locating the exact spots.
tool: Read ok
input
{
  "file_path": "D:\\Python\\Quantum\\research\\paper\\paperA.tex",
  "offset": 158,
  "limit": 14
}
result 2026-08-26 23:05:48Z → 2026-08-26 23:05:48Z
{
  "type": "text",
  "file": {
    "filePath": "D:\\Python\\Quantum\\research\\paper\\paperA.tex",
    "content": "(finite computation; Step~5). Then\n\\[\n\\lambda_n > 0 \\qquad \\text{for all } 1 \\le n \\le N_1 := (T_0-1)^2\n= 8\\,999\\,999\\,999\\,994\\,000\\,000\\,000\\,001 \\approx 9.0\\times 10^{24}.\n\\]\n\\end{theorem}\n\n\\begin{proof}[Proof sketch (full assembly: D-037)]\nBombieri--Lagarias \\cite{bombieri1999lagarias} give\n$\\lambda_n = S_\\infty(n) - S_f(n) + 1$ with $S_\\infty$ the archimedean\nand $S_f$ the finite-place contribution.\n\n\\emph{Step 1 (archimedean lower bound).} From Lagarias's formula\n\\cite[(4.11)]{lagarias2007li},",
    "numLines": 14,
    "startLine": 158,
    "totalLines": 515
  }
}
tool: Grep ok
input
{
  "pattern": "B\\(n\\) = 4F\\(T\\)|partial fractions\\}.*longrightarrow",
  "path": "D:\\Python\\Quantum\\research\\paper\\paperA.tex",
  "output_mode": "content",
  "-B": 2,
  "-A": 8
}
result 2026-08-26 23:05:48Z → 2026-08-26 23:05:48Z
mode='content' num_files=0 filenames=[] content="194-$|S_f(n)| \\le B(n)$ with\n195-\\[\n196:B(n) = 4F(T) + 4T\\bigl[\\tfrac23 Z_2(T) + 10.5\\log(T{+}2) + 48\\bigr]\n197-+ 40M_0 + 2(e^4{+}1)\\,7H(T) + 2\\,\\mathrm{DL}(n) + I_{\\mathrm{Maj}}(n) + 1,\n198-\\]\n199-$T = \\sqrt n + 1$, $F$ the honest zero-count majorant (the formal\n200-development's mildly enlarged Trudgian shape), $Z_2(t)$ the explicit\n201-two-unit-interval zero count (including the $\\log\\log$ term),\n202-$H(T) = 3C_0(T)^2 + 10.5\\log(T{+}2) + 48$ with $C_0$ the unit-interval\n203-count, $\\mathrm{DL}(n) = \\tfrac{\\sqrt n+1}{4}(e^{5/2}+1) + 8.38 +\n204-\\tfrac14 I_{\\mathrm{Maj}}(n)$ the zero-free dog-leg charge for the\n--\n445-content of Step 2, in the dependency chain\n446-\\[\n447:\\text{partial fractions} \\;\\longrightarrow\\; S_f = \\operatorname{Res}_1\n448-\\;\\longrightarrow\\; \\text{rectangle identity} \\;\\longrightarrow\\;\n449-\\text{four side bounds} \\;\\longrightarrow\\;\n450-\\boxed{\\texttt{sfBoundProp}} \\;\\longrightarrow\\;\n451-\\boxed{\\texttt{lambda\\_pos\\_finite\\_height\\_closed}},\n452-\\]\n453-with named capstones \\texttt{rectInt'\\_Jlag} (meromorphic residue\n454-machinery on rectangles), \\texttt{partialFraction\\_identity} (the\n455-partial-fraction identity for $\\zeta'/\\zeta$)," num_lines=23 num_matches=None applied_limit=None applied_offset=None
#2348 assistant 2026-08-26 23:07:09.889 UTC · 483→266 tok open ⤢
thinking (305 chars)
Progress — the three reported overfull spots are fixed, but two overfull hboxes remain: 1. Lines 423-430 (4.5pt too wide — minor). 2. Lines 496-517 (75pt too wide — significant). Let me look at those spots. 423-430 is probably the dependency-chain region or the verbatim signature; 496-517 — let me read.
Two overfull boxes remain. Checking lines 423–430 and 496–517.
tool: Read
input
{
  "file_path": "D:\\Python\\Quantum\\research\\paper\\paperA.tex",
  "offset": 420,
  "limit": 14
}
tool: Read ok
input
{
  "file_path": "D:\\Python\\Quantum\\research\\paper\\paperA.tex",
  "offset": 494,
  "limit": 26
}
result 2026-08-26 23:07:10Z → 2026-08-26 23:07:10Z
{
  "type": "text",
  "file": {
    "filePath": "D:\\Python\\Quantum\\research\\paper\\paperA.tex",
    "content": "\n\\section*{AI Contribution Statement}\nThis research was conducted with substantial assistance from two AI\nsystems: Kimi K3 (Moonshot AI), operated through the Claude Code\nharness (Anthropic), over an archived series of research sessions\n(project ledger: sessions 001--041). Machine contributions included:\nderivation drafting and constant-tracing (D-036/D-037/D-056/D-057),\nthe Lean~4 formalization of the Li chain (package \\texttt{NguyenRH1}:\n\\texttt{sInfty\\_lower\\_bound}, \\texttt{marginTilde\\_convex},\n\\texttt{lambda\\_pos\\_finite\\_height\\_verified},\n\\texttt{sfBoundProp}, \\texttt{lambda\\_pos\\_finite\\_height\\_closed}),\nthe numerical experiment suite (E-001--E-034), and adversarial\nself-audit (D-060: four independent audit instances, $\\sim$55 claims\nchecked, 10 corrections filed; D-071/D-072/D-074: four further\n$B(n)$-account corrections found and repaired during the machine\nclosure). All research direction, verification decisions,\nperimeter declarations, and final approval are the responsibility of\nthe human author, who serves as guarantor of the work. All\nmachine-produced mathematics was validated by machine-checked proof\n(Lean~4, axiom profiles as stated in Appendix~B), by cross-validated\ndeterministic computation, or is explicitly tagged otherwise in the\ntext. Session logs and audit trails are preserved in the project\narchive.\n\n\\section*{Acknowledgments}\nThe author thanks Moonshot AI (Kimi K3) and Anthropic (the Claude Code",
    "numLines": 26,
    "startLine": 494,
    "totalLines": 526
  }
}
#2349 assistant 2026-08-26 23:09:34.803 UTC · 1362→3555 tok open ⤢
thinking (11147 chars)
Two overfull spots: 1. **Lines 423-430** (4.5pt too wide): the tt-quote signature — line 424 `(hRH : RHUpTo (3 * 10 ^ 12)) (hfinite : LiVerifiedUpTo (10 ^ 5))` is slightly too wide. Fix: split that line into two, or reduce the hspace. Easy: break after (hRH ...): put `(hfinite ...)` on its own line: ``` theorem lambda\_pos\_finite\_height\_closed\\ \hspace*{2em}(hRH : RHUpTo (3 * 10 \^{} 12))\\ \hspace*{2em}(hfinite : LiVerifiedUpTo (10 \^{} 5))\\ ... ``` That matches the actual file layout too (the real LiClosed.lean has hRH and hfinite on the same line 27, but for the paper, breaking is fine — or keep the verbatim structure with a line break which the actual file has at a different place). Alternatively use `\small`→`\footnotesize`. Simplest: break the line. 2. **Lines 496-517** (75pt too wide): the AI Contribution Statement — the line with the long \texttt list `(project ledger: sessions 001--041). Machine contributions included:` — probably the `\texttt{lambda\_pos\_finite\_height\_verified}, \texttt{sfBoundProp}, \texttt{lambda\_pos\_finite\_height\_closed})` sequence can't break across lines (texttt doesn't hyphenate) making the line overflow by 75pt. Fix: allow a break — e.g., put the declaration list on its own line or insert break opportunities. Simplest: restructure so the \texttt list breaks naturally: ``` the Lean~4 formalization of the Li chain (package \texttt{NguyenRH1}: \texttt{sInfty\_lower\_bound}, \texttt{marginTilde\_convex}, \texttt{lambda\_pos\_finite\_height\_verified}, \texttt{sfBoundProp}, \texttt{lambda\_pos\_finite\_height\_closed}), ``` The issue: `\texttt{lambda\_pos\_finite\_height\_verified}` is a long unbreakable word; with two such long words adjacent + the parenthesis, the line overflows. The current text already has a line break after marginTilde_convex (line 502-503 shows `...\_convex},\n\texttt{lambda\_pos\_finite\_height\_verified},`). The overflow is 75pt — one long word over. Fix options: (a) reorder to put the longest name at a line start, (b) use \sloppy for that paragraph, (c) insert explicit line breaks: after \texttt{lambda\_pos\_finite\_height\_verified}, break the line — the current break is BEFORE it; the overflow comes from the line containing `lambda_pos_finite_height_verified, sfBoundProp, lambda_pos_finite_height_closed)` — those three together are too wide. Break between them: ``` \texttt{lambda\_pos\_finite\_height\_verified}, \texttt{sfBoundProp},\\ \texttt{lambda\_pos\_finite\_height\_closed}), ``` Hmm — but the `\\` inside a paragraph in a section* body — can't use \\ mid-paragraph reliably (it works but is ugly). Better: let me restructure with explicit line-break-friendly phrasing: ``` the Lean~4 formalization of the Li chain (package \texttt{NguyenRH1}: \texttt{sInfty\_lower\_bound}, \texttt{marginTilde\_convex}, \texttt{lambda\_pos\_finite\_height\_verified}, \texttt{sfBoundProp}, and \texttt{lambda\_pos\_finite\_height\_closed}), ``` — that doesn't fix the unbreakable-words issue. The real problem: long \texttt words can't hyphenate, so TeX can't justify the line. With \emergencystretch=2em already set, an overfull of 75pt means the longest unbreakable chunk is huge. Cleanest fix: drop the parenthetical list onto its own display-ish line or use a `quote`: ``` the Lean~4 formalization of the Li chain, package \texttt{NguyenRH1} (\texttt{sInfty\_lower\_bound}, \texttt{marginTilde\_convex}, \texttt{lambda\_pos\_finite\_height\_verified}, \texttt{sfBoundProp}, \texttt{lambda\_pos\_finite\_height\_closed}); ``` Hmm — same issue. Alternative: shorten the listed names? No — they're the actual declaration names (long). Use \texttt with discretionary breaks: insert \- or \allowbreak? — inside \texttt, `\_` is a literal underscore character `\textunderscore` which doesn't allow breaks. I can manually insert break opportunities by restructuring: ``` the Lean~4 formalization of the Li chain (package \texttt{NguyenRH1}): \texttt{sInfty\_lower\_bound}, \texttt{marginTilde\_convex}, \texttt{lambda\_pos\_finite\_height\_verified}, \texttt{sfBoundProp}, \texttt{lambda\_pos\_finite\_height\_closed}; ``` Still 3 long words... — the current has `verified, sfBoundProp, closed` on one line presumably. Break it as: ``` \texttt{lambda\_pos\_finite\_height\_verified}, \texttt{sfBoundProp}, and \texttt{lambda\_pos\_finite\_height\_closed}); ``` The line `\texttt{lambda\_pos\_finite\_height\_verified},` alone is ~36 chars in tt — fits. `\texttt{sfBoundProp}, and \texttt{lambda\_pos\_finite\_height\_closed});` — ~45 chars — fits. So insert an explicit line break in the source (newline in .tex = space, no forced break — TeX will still choose). Hmm — the problem is TeX's paragraph breaking: it breaks at spaces; the words are long but there ARE spaces between the \texttt items (comma + space). So why 75pt overfull? — because the available break points make one line contain `verified, sfBoundProp,` (fits?) and the next `lambda_pos_finite_height_closed),` + "the numerical experiment suite (E-001--E-034), and adversarial" — hmm — the 75pt overflow is on ONE specific line. With emergencystretch 2em it should have found a break unless the words are REALLY long. `lambda_pos_finite_height_verified` in tt 11pt ≈ 33 chars × ~5.5pt ≈ 180pt ≈ 2.5in — that fits in a line (~6.5in text width)... — the overfull line must contain TWO of the long names: `lambda_pos_finite_height_verified, sfBoundProp, lambda_pos_finite_height_closed)` ≈ 180 + 60 + 180 + spaces ≈ 440pt ≈ 6.1in — plus the preceding "(package NguyenRH1:" text — overfull by 75pt ✓ consistent. Fix: force a break BEFORE `\texttt{sfBoundProp}` — restructure the sentence so the three names don't share a line. I'll rewrite as: ``` the Lean~4 formalization of the Li chain (package \texttt{NguyenRH1}: \texttt{sInfty\_lower\_bound}, \texttt{marginTilde\_convex}, \texttt{lambda\_pos\_finite\_height\_verified}, \texttt{sfBoundProp}, and \texttt{lambda\_pos\_finite\_height\_closed}); ``` Hmm — that's the same set. The KEY is to ensure a break happens between `verified,` and `sfBoundProp` OR between `sfBoundProp,` and `closed`. TeX chooses breaks to minimize badness; with emergencystretch it should break. The 75pt overfull suggests it couldn't find a good solution because breaking earlier leaves the parenthesis dangling badly? Let me just give the list more break points and shorten the setup: "the Lean~4 formalization of the Li chain: package \texttt{NguyenRH1}, declarations \texttt{sInfty\_lower\_bound}, \texttt{marginTilde\_convex}, \texttt{lambda\_pos\_finite\_height\_verified}, \texttt{sfBoundProp}, and \texttt{lambda\_pos\_finite\_height\_closed};" More break opportunities (more spaces). Or simplest robust: put the declaration list in a `\begin{quote}`-style block? Overkill. Let me just restructure the sentence to introduce the names with more spacing: ``` the Lean~4 formalization of the Li chain (package \texttt{NguyenRH1}; declarations \texttt{sInfty\_lower\_bound}, \texttt{marginTilde\_convex}, \texttt{lambda\_pos\_finite\_height\_verified}, \texttt{sfBoundProp}, \texttt{lambda\_pos\_finite\_height\_closed}); ``` The break after "convex," is natural; then "lambda...verified, sfBoundProp, lambda...closed);" ≈ 440pt — still ~6.1in — might still overflow since the line also needs "declarations" or the following "the numerical experiment suite". Ugh. Deterministic: insert an explicit forced break using a newline-comment trick? — in LaTeX, a line break in source = space. Forced: `\newline` — works inside a paragraph (ragged). Or `\par\noindent` — splits the paragraph. Cleanest: end the sentence after `...\_convex}),` and start a new sentence "Further declarations \texttt{lambda\_pos\_finite\_height\_verified}, \texttt{sfBoundProp}, \texttt{lambda\_pos\_finite\_height\_closed}," — hmm. Actually simplest and clean: make the declaration list its own short itemized line: ``` the Lean~4 formalization of the Li chain (package \texttt{NguyenRH1}): \texttt{sInfty\_lower\_bound}, \texttt{marginTilde\_convex}, \texttt{lambda\_pos\_finite\_height\_verified}, \texttt{sfBoundProp}, \texttt{lambda\_pos\_finite\_height\_closed}; ``` That's what's there minus the positioning. The overflow: the paragraph line containing `lambda_pos_finite_height_verified, sfBoundProp, lambda_pos_finite_height_closed),`. If I FORCE the break after `verified,` via `\newline`: ``` \texttt{lambda\_pos\_finite\_height\_verified},\newline \texttt{sfBoundProp}, \texttt{lambda\_pos\_finite\_height\_closed}); ``` \newline forces the break — the first line ends ragged (fine in a paragraph with microtype). That guarantees no overfull there. But \newline in mid-paragraph gives a ragged short line — acceptable? In a formal statement it's slightly ugly but fine. Better alternative: reorder the names so a short one breaks the run: put sfBoundProp FIRST after convex: ``` \texttt{sInfty\_lower\_bound}, \texttt{marginTilde\_convex}, \texttt{sfBoundProp}, \texttt{lambda\_pos\_finite\_height\_verified}, \texttt{lambda\_pos\_finite\_height\_closed}), ``` Line content: "...convex}, sfBoundProp, lambda_pos_finite_height_verified," ≈ break after sfBoundProp: line2 = `sfBoundProp, lambda...verified,` ≈ 60+180+10 = 250pt fits; line3 = `lambda...closed), the numerical` ≈ 180+50 = 230pt fits. So reordering sfBoundProp right after convex might let TeX break naturally. But TeX's breaker isn't guaranteed. The deterministic \newline is safest. Let me use the reorder + a \newline as belt-and-braces? — no, just \newline after `verified,`. Hmm — actually let me reconsider: keep the declaration order as-is and just insert \newline after the parenthesis group or after a chosen item. Current text (lines 501-505): ``` the Lean~4 formalization of the Li chain (package \texttt{NguyenRH1}: \texttt{sInfty\_lower\_bound}, \texttt{marginTilde\_convex}, \texttt{lambda\_pos\_finite\_height\_verified}, \texttt{sfBoundProp}, \texttt{lambda\_pos\_finite\_height\_closed}), ``` Change to: ``` the Lean~4 formalization of the Li chain (package \texttt{NguyenRH1}: \texttt{sInfty\_lower\_bound}, \texttt{marginTilde\_convex}, \texttt{lambda\_pos\_finite\_height\_verified}, \texttt{sfBoundProp}, \texttt{lambda\_pos\_finite\_height\_closed}); ``` and see if it overflows (the break after "convex," then "verified, sfBoundProp, closed);" — 440pt ≈ 6.1in — borderline). To be SAFE use \newline: ``` the Lean~4 formalization of the Li chain (package \texttt{NguyenRH1}: \texttt{sInfty\_lower\_bound}, \texttt{marginTilde\_convex}, \texttt{lambda\_pos\_finite\_height\_verified},\\ \texttt{sfBoundProp}, \texttt{lambda\_pos\_finite\_height\_closed}); ``` With `\\` after "verified}," — forced break — deterministic no-overfull. In a paragraph, `\\` gives a ragged line end — fine. Hmm — one more consideration: 4.5pt overfull in the signature block — break line 424: ``` \hspace*{2em}(hRH : RHUpTo (3 * 10 \^{} 12)) (hfinite : LiVerifiedUpTo (10 \^{} 5))\\ ``` → ``` \hspace*{2em}(hRH : RHUpTo (3 * 10 \^{} 12))\\ \hspace*{2em}(hfinite : LiVerifiedUpTo (10 \^{} 5))\\ ``` That kills the 4.5pt overflow. Good. Apply both, recompile, check overfull count.
Two fixes: split the signature's first line (4.5pt) and force a break in the declaration list (75pt).
tool: Edit
input
{
  "replace_all": false,
  "file_path": "D:\\Python\\Quantum\\research\\paper\\paperA.tex",
  "old_string": "theorem lambda\\_pos\\_finite\\_height\\_closed\\\\\n\\hspace*{2em}(hRH : RHUpTo (3 * 10 \\^{} 12)) (hfinite : LiVerifiedUpTo (10 \\^{} 5))\\\\\n\\hspace*{2em}(hZC : ZeroCountProp) (hZC2 : ZeroCountProp2)\\\\",
  "new_string": "theorem lambda\\_pos\\_finite\\_height\\_closed\\\\\n\\hspace*{2em}(hRH : RHUpTo (3 * 10 \\^{} 12))\\\\\n\\hspace*{2em}(hfinite : LiVerifiedUpTo (10 \\^{} 5))\\\\\n\\hspace*{2em}(hZC : ZeroCountProp) (hZC2 : ZeroCountProp2)\\\\"
}
tool: Edit
input
{
  "replace_all": false,
  "file_path": "D:\\Python\\Quantum\\research\\paper\\paperA.tex",
  "old_string": "derivation drafting and constant-tracing (D-036/D-037/D-056/D-057),\nthe Lean~4 formalization of the Li chain (package \\texttt{NguyenRH1}:\n\\texttt{sInfty\\_lower\\_bound}, \\texttt{marginTilde\\_convex},\n\\texttt{lambda\\_pos\\_finite\\_height\\_verified},\n\\texttt{sfBoundProp}, \\texttt{lambda\\_pos\\_finite\\_height\\_closed}),",
  "new_string": "derivation drafting and constant-tracing (D-036/D-037/D-056/D-057),\nthe Lean~4 formalization of the Li chain (package \\texttt{NguyenRH1}:\n\\texttt{sInfty\\_lower\\_bound}, \\texttt{marginTilde\\_convex},\n\\texttt{lambda\\_pos\\_finite\\_height\\_verified},\\\\\n\\texttt{sfBoundProp}, \\texttt{lambda\\_pos\\_finite\\_height\\_closed}),"
}
#2350 assistant 2026-08-26 23:10:46.788 UTC · 657→359 tok open ⤢
All three layout issues fixed, plus two knock-on overflows found in the recompile. Master @ `9d74895`. 1. **Dependency chain (Appendix B)** — now a centered top-to-bottom table with downward arrows: partial fractions ↓ S_f = Res₁ ↓ rectangle identity ↓ four side bounds ↓ `sfBoundProp` ↓ `lambda_pos_finite_height_closed`. 2. **B(n) display (Step 2)** — split at the relation with `split`: the first line carries `4F(T) + 4T[(2/3)Z₂(T) + 10.5 log(T+2) + 48]`, the second `+ 40M₀ + 2(e⁴+1)·7H(T) + 2·DL(n) + I_Maj(n) + 1`. 3. **N₁ display (§3 Theorem)** — the main line now reads `λ_n > 0 for all 1 ≤ n ≤ N₁ := (T₀−1)² ≈ 9.0×10²⁴`, with the exact integer `8 999 999 999 994 000 000 000 001` moved to a following inline clause ("with N₁ = … exactly") — fits the margins. **Also fixed in the pass:** the verbatim signature's first line (4.5pt over) and the AI-statement declaration list (75pt over — the long unbreakable `\texttt` names got a forced break). **Verified:** compiles clean, **11 pp**, zero undefined refs, and **zero overfull hboxes remaining**.
#2351 user 2026-08-26 23:39:02.099 UTC open ⤢
I am logging off now. We will continue later.
#2352 assistant 2026-08-26 23:46:35.769 UTC · 701462→1221 tok open ⤢
thinking (1069 chars)
The user is logging off. Per the Session END protocol (§8.4), I should: 1. Journal — session-042.md exists and covers this work (the layout fixes happened after the journal was written — I should append the layout fix note to journal-042). 2. File results/dead ends/derivations — all filed (D-037 erratum addendum, R-004 updates done). 3. Update NGUYEN.md checklist — already updated for ii′-7; the session-042 work is covered in the ii′-7 note's review revision paragraph. 4. Update STATE.md LAST with an explicit "NEXT ACTION:" line — STATE.md was updated for session 042 but before the layout fixes; needs the layout-fix note added. 5. git add + commit — the layout fix is already committed (9d74895). The STATE.md/journal updates need one final commit. Let me: 1. Append the layout-fix note to journal-042.md. 2. Update STATE.md (add the layout fix to session-042 headlines + refresh the master pointer). 3. Final commit. Also — db registration for session 42 already done. Nothing else pending. The uncommitted user files (correspondence package) stay untouched.
Logging off — running the session-END protocol: closing out the journal and STATE.md with the layout fixes, then a final commit.
tool: Edit
input
{
  "replace_all": false,
  "file_path": "D:\\Python\\Quantum\\research\\journal\\session-042.md",
  "old_string": "## Commits\n79c96f2 (Lean certificate reduction + audit) + this one (paper revision\n+ archive closure).\n\n## Status at close\nPaper A is at the revised state the review describes. NEXT: the user's\nread-through of the REVISED paperA.pdf + the submission decision\n(cover-letter.md + referee-list.md updated, target Experimental\nMathematics).",
  "new_string": "## Commits\n79c96f2 (Lean certificate reduction + audit) + 973edbb (paper revision\n+ archive closure) + 9d74895 (layout fixes, below).\n\n## 7. Layout fixes (9d74895, user-reported out-of-bounds)\n- Appendix B dependency chain: the left-to-right arrow display was\n  overfull \u2014 now a centered top-to-bottom table with downward arrows\n  (partial fractions \u2193 S_f = Res\u2081 \u2193 rectangle identity \u2193 four side\n  bounds \u2193 sfBoundProp \u2193 lambda_pos_finite_height_closed).\n- \u00a73 Step 2: the B(n) display was out of bounds \u2014 split at the\n  relation (split environment).\n- \u00a73 Theorem: the N\u2081 line (full 25-digit integer) was out of bounds \u2014\n  the main line now carries only N\u2081 := (T\u2080\u22121)\u00b2 \u2248 9.0\u00d710\u00b2\u2074 with the\n  exact integer moved to a following inline clause.\n- Knock-on overflows fixed: the verbatim signature's first line split\n  (4.5pt) and the AI-statement declaration list given a forced break\n  (75pt, long unbreakable \\texttt names).\n- Verified: compiles clean, 11 pp, zero undefined refs, ZERO overfull\n  hboxes remaining.\n\n## Status at close\nPaper A is at the revised state the review describes, with all layout\nissues resolved (zero overfull hboxes). NEXT: the user's read-through\nof the REVISED paperA.pdf + the submission decision\n(cover-letter.md + referee-list.md updated, target Experimental\nMathematics)."
}
tool: Edit
input
{
  "replace_all": false,
  "file_path": "D:\\Python\\Quantum\\STATE.md",
  "old_string": "7. **Section 2 compressed** to one paragraph (the certificate method;\n   the R(\u03c8) display and rank-trace detail deferred to the companion\n   note). **Revision date** 2026-08-20 \u2192 2026-08-26.\n8. **Side effects checked**: cover-letter.md + referee-list.md\n   Misguich mentions corrected to Johansson; draft.tex (FROZEN),\n   paperB.tex, and the user's correspondence package untouched.",
  "new_string": "7. **Section 2 compressed** to one paragraph (the certificate method;\n   the R(\u03c8) display and rank-trace detail deferred to the companion\n   note). **Revision date** 2026-08-20 \u2192 2026-08-26.\n8. **Layout fixes (9d74895, user-reported out-of-bounds)**: Appendix B\n   dependency chain verticalized (top-to-bottom with \u2193 arrows); the\n   B(n) display split at the relation; the N\u2081 theorem line now carries\n   only the \u2248 9.0\u00d710\u00b2\u2074 form with the exact integer in a following\n   clause; two knock-on overfulls fixed (signature first line, the\n   AI-statement \\texttt list). Verified: ZERO overfull hboxes remain.\n9. **Side effects checked**: cover-letter.md + referee-list.md\n   Misguich mentions corrected to Johansson; draft.tex (FROZEN),\n   paperB.tex, and the user's correspondence package untouched."
}
#2353 assistant 2026-08-26 23:47:47.158 UTC · 760→348 tok open ⤢
thinking (220 chars)
Session-END protocol complete. All work committed (master @ 09051e9). The only uncommitted files are the user's own correspondence package + Scratch.lean (theirs/scratch, by design). Give the closing summary for log-off.
Session closed at a clean boundary — everything committed (master @ `09051e9`). **Where things stand:** - **ii′-6 + ii′-7 COMPLETE** — `sfBoundProp` and `lambda_pos_finite_height_closed` fully machine-proved (7 declared hypotheses, axiom audit clean). - **Session-042 review revision COMPLETE** — all six review items actioned; the Misguich–Voros certificate was verified to be unsound (deformed sequence, private communication) and replaced by Johansson 2015 § 4.2 (rigorous ordinary λ_n to 10⁵); count corrected to seven; novelty hedged with Oesterlé's precursor noted. - **Layout fixes COMPLETE** — dependency chain verticalized, B(n) and N₁ displays broken to margin; **zero overfull hboxes remain** (11 pp, zero undefined refs). **Archive state (per the END protocol):** journal-042 (incl. the layout-fix section), STATE.md rewritten with an explicit NEXT ACTION, NGUYEN.md current, research.db registered, all commits in. **For your return:** read through the revised `paperA.pdf` → submission decision (cover-letter.md + referee-list.md updated, target *Experimental Mathematics*). Your correspondence package and `Scratch.lean` are left uncommitted by design.