← Session 28 — ii′-5e: partial-fraction identity (zero series + F entire)
thinking (8922 chars)
`ENNReal.tsum_fiberwise (f : β → ℝ≥0∞) (g : β → γ) : ∑' x, ∑' b : g ⁻¹' {x}, f b = ∑' i, f i` — free (no summability side conditions) ✓. This gives the fiberwise decomposition in ℝ≥0∞.
The full tail bound proof sketch (in ℝ≥0∞ then back to ℝ):
- Let T := {ρ : zerosubtype // X < |ρ.1.im|} (the tail subtype-of-subtype). Hmm — actually cleaner: work directly on the zero subtype Z := {ρ : ℂ // IsNontrivialZero ρ} with the indicator... Let me define the tail as a subtype T of Z: T := {ρ : Z // X < |(ρ.1).im|}.
∑'_{ρ : T} (ofReal (m_ρ/(1+γ²))) ≤ ofReal (16 A₀ log X / X):
= ∑'_{j : ℕ} ∑'_{ρ : T, j(ρ) = j} ofReal(m/(1+γ²)) [ENNReal.tsum_fiberwise with g := j(·) : T → ℕ]
≤ ∑' j, ofReal((fiber count)·2/(4^j X²))
= ofReal(Σ' j, ...)...
the fiber count: {ρ : T, j(ρ) = j} — zeros with |γ| ∈ (2^j X, 2^{j+1} X]: Σ_{fiber} m ≤ 2A₀(2^jX+1)log(3·2^{j+1}X+5) — via Ncount_le_of_interval_gen ×2 (both sides) — the fiber is finite (bounded ordinates ✓ zerosIn_finite) — as a finset sum ≤ interval count ✓.
Then ∑'_{j : ℕ} 2A₀(2^jX+1)log(3·2^{j+1}X+5)·2/(4^jX²) ≤ (16A₀/X)·(10 log 2 + 2 log X)/... — the dyadic tsum over j : ℕ of an explicit ℝ≥0 sequence — summable (geometric×polynomial) — its sum: bound by the closed form 10 log 2 + 2 log X etc.
This is all doable but LONG (~250 lines for the tail alone).
PACING CHECK: This session has been very long already (5e-v-a + 5e-v-b chunk 1 committed). The remaining 5e-v work: tail bound (~250 lines) + F-finset shell bounds (~300) + assembly (~150) + squeeze + capstone (~200) ≈ 900 lines ≈ 1.5–2 sessions. Per the pacing directive and the natural breakpoint (helpers committed), the right move is to close the session cleanly now: journal-030 + STATE update + db + closure commit. The next session starts the tail bound with the full design in STATE.md.
Wait — actually, let me reconsider the tail ONE more time, because I want to double-check I'm not missing a 10× cheaper path...
CHEAP TAIL via Cauchy–Schwarz against zero_sum_inv_sq? Σ_{|γ|>X} m/(1+γ²) ≤ (Σ m/(1+γ²))^{1/2}·(Σ_{|γ|>X} m/(1+γ²))^{1/2} — circular.
Cheapest honest decay: the tail of a convergent series with a window-count input: is there really no simpler...
Actually — here's one: the tail Σ_{|γ|>X} m/(1+γ²) ≤ (1/X)·Σ_{|γ|>X} m·|γ|/(1+γ²)·... no: 1/(1+γ²) ≤ (1/X)·(|γ|/(1+γ²))·... for |γ| > X: |γ|/(1+γ²) ≥ |γ|/(2γ²) = 1/(2|γ|) — no.
How about: Σ_{|γ|>X} m/(1+γ²) ≤ Σ m·|γ|^{−2}·... the dyadic is really the way. And actually the fiberwise-in-ℝ≥0∞ route is not THAT bad. But it's for next session.
Actually — hold on, let me reconsider whether I even NEED the 1/(1+γ²) tail bound. What the series bound needs: ∑'_{‖ρ‖>2R} m·R/(‖ρ‖·‖s−ρ‖) ≤ ∑' m·2R/‖ρ‖² (‖s−ρ‖ ≥ ‖ρ‖/2) — and 1/‖ρ‖² ≤ 1/|γ|² (|γ| ≤ ‖ρ‖) — so ≤ ∑'_{‖ρ‖>2R} m·2R/|γ|² ≤ 2R·∑'_{|γ|>2R−1} m/γ² — the γ⁻² tail: Σ_{|γ|>X} m/γ² ≤ Σ_j (count_j)/(2^jX)² — same dyadic. OR the 1/(1+γ²) version: m/γ² ≤ 2m/(1+γ²) for |γ| ≥ 1 ✓. Same either way.
OK — one more idea to dodge the fiberwise decomposition: bound the tail by the WINDOW-INDEXED sum directly: the tail zeros = union over windows W_n (n : ℤ, |n| ≥ ⌈X⌉) of {γ ∈ (n, n+1]}: Σ_{tail} m/γ² ≤ Σ_{|n| ≥ ⌈X⌉} (count_n)·2/n² — the windows are INTEGER-indexed (n : ℤ): the assignment γ ↦ ⌊γ⌋ (or the `key` function from Zeta23!) — the fiberwise over ℤ with n ↦ window — Zeta23's key_lt/le_key_add_one machinery + one_add_sq_ge + summable_weight + totalWeight ALREADY prove ∑'_{n : ℤ} log(|n|+3)/(1+n²) < ∞!! And `weight_le`: (4/(1+n²))-style bounds. The Zeta23 `weight` function: weight n := log(|n|+3)/(1+n²)-ish (integer windows) with summable_weight : Summable weight and totalWeight := its tsum.
SO: Σ_{|γ|>X} m/(1+γ²) ≤ Σ_{n : ℤ, |n| ≥ X-ish} A₀·weight(n)·4-ish — the INTEGER-window version — and the tail of the (summable) weight series: Σ_{|n|≥N} log(|n|+3)/(1+n²) ≤ ∫-ish ≤ C·log N/N — an explicit tail bound on the weight series — provable by comparison with 1/n^{3/2} (log(n+3) ≤ n^{1/2} for n ≥ ...): weight(n) ≤ C/n^{3/2} for large n: Σ_{|n|≥N} ≤ 2C/√N·... — STILL a tail estimate but now on a 1D integer series with an explicit majorant — via `Summable.tail` bounds...
Hmm — the weight-tail: Σ_{|n| ≥ N} log(|n|+3)/(1+n²) ≤ Σ 2A₀/n^{3/2}-form: is there a packaged p-series tail bound: `Real.tsum_one_div_nat_rpow_le`? or `summable_one_div_nat_rpow` + tail via the integral test... The integral test in Lean: `Summable.of_integral_le`... the tail bound from the integral: there might be something like `Real.tsum_..._le_of_...`. OR the direct: Σ_{n ≥ N} 1/n^{3/2} ≤ 1/N^{3/2} + ∫_N^∞ x^{-3/2} = 1/N^{3/2} + 2/√N ≤ 3/√N: the integral comparison in Lean: `Summable.of_norm_bounded...` no. There IS the classic: for antitone f: Σ_{n ≥ N} f n ≤ f N + ∫...
OK. It's all ~the same length. This is next-session work. Let me close THIS session cleanly: journal-030 documenting 5e-v-a + 5e-v-b chunk 1 + the detailed remaining plan, STATE update, db, closure commit.
Actually — wait, let me reconsider the time/length budget once more. I've done 5e-v-a + 5e-v-b-helpers. The session isn't out of gas per se — but the tail bound + F-finset bounds + assembly + squeeze is easily another 600-900 lines and several compile cycles. The NGUYEN pacing rule: break into sub-stages with clean commits; also the BREAK PROMPT rule (approaching long work). Since everything is committed and the next sub-stage (the tail bound) is a clean chunk, let me continue — but efficiently. Let me build the tail bound now.
Actually no — let me be honest about the state: this turn has already produced 5e-v-a (3 commits) with substantial exploration. The 5e-v-b main lemma (series circle bound) is the single largest estimation lemma of the project and deserves a fresh sub-stage with full attention. The pacing directive is explicit: NO RUSH, correctness over speed, sub-stages ending in clean committed states. The clean state is now (5e-v-b chunk 1 committed). The remaining 5e-v-b (tail + main bound) is the next session's first item, with the full design recorded in STATE.md.
Hmm, but the user said "let's work on ii'-5e-v" — they want progress on 5e-v. I've made concrete progress (2 commits). A session-end now is legitimate (each sub-stage ends clean). But I could also push the tail bound now... Let me push ONE more meaningful chunk: the tail bound (the last genuinely-new analytic input). If it compiles, 5e-v-b finishes next session with the shell bounds + assembly.
Let me design the tail bound concretely and write it:
```lean
/-- The dyadic tail bound: for X ≥ 4, Σ over zeros with |γ| > X of
m/(1+γ²) ≤ 40·A₀·log X / X. Via the integer-window fiberwise
decomposition (ENNReal.tsum_fiberwise) + the window counts. -/
```
Hmm — actually, let me use a much simpler decomposition that avoids fiberwise: bound the tail tsum by a DIRECT comparison with the weight series restricted to high windows. For each zero with |γ| > X: its window n := ⌊γ⌋ (or the Zeta23 `key`): |n| ≥ ⌊X⌋... The map γ ↦ key γ : the zeros map to ℤ. Then:
∑'_{ρ : |γ|>X} m/(1+γ²) ≤ ∑'_{ρ : |γ|>X} m·(4/(1+(key γ)²)) [one_add_sq_ge: 1+γ² ≥ (1+n²)/4 for γ ∈ (n, n+1] ✓ banked in Zeta23!]
≤ (the fiberwise over key) = Σ_{n : ℤ} Σ_{ρ : |γ|>X, key ρ = n} m·4/(1+n²)
≤ Σ_{n : ℤ, |n| ≥ X-ish} 4A₀·log(|n|+3)/(1+n²) [window count]
≤ 4A₀·(tail of weight above ~X)
≤ 4A₀·(16·log X / X)?? — the weight tail bound.
The weight tail: Σ_{|n| ≥ N} log(|n|+3)/(1+n²) ≤ Σ_{|n|≥N} log(|n|+3)/n². For n ≥ N ≥ 16: log(n+3) ≤ log(2n) ≤ 2 log n·... log(n+3) ≤ log(2n) = log 2 + log n ≤ 1 + log n; and log n ≤ 2√n-ish... Let me bound: log(n+3)/n² ≤ n^{2/3}/n² = n^{-4/3}?? log(n+3) ≤ n^{2/3} for n ≥ 16: log(19) ≈ 2.94 ≤ 16^{2/3} ≈ 6.35 ✓ and grows — log(n+3) ≤ n^{2/3} for n ≥ 16 plausible (need a clean proof: log x ≤ x^{2/3}·... via log x ≤ 3(x^{1/3}−1)? hmm: log x ≤ x^r/r for r > 0 (Real.log_le_rpow_div?): with r = 1/2: log(n+3) ≤ 2√(n+3)... 2√(n+3)/n² ≤ 2√(2n)/n² = 2√2/n^{3/2} (n ≥ 3) — so weight(n) ≤ 2√2·log... no wait: log(|n|+3)/(1+n²) ≤ 2√2/|n|^{3/2} for |n| ≥ 3 ✓ (log(n+3) ≤ 2√(n+3) — from Real.log_le_sub_one? no — log y ≤ 2√y: log y = 2 log(√y) ≤ 2(√y − 1) ≤ 2√y using log z ≤ z−1 ✓ Real.log_le_sub_one_of_pos ✓ banked!). So weight(n) ≤ 2√2·|n|^{-3/2}: Σ_{|n| ≥ N} |n|^{-3/2} = 2·Σ_{n ≥ N} n^{-3/2}: the p-series tail with p = 3/2: Σ_{n≥N} n^{-3/2} ≤ N^{-3/2} + ∫_N^∞ x^{-3/2}dx = N^{-3/2} + 2N^{-1/2} ≤ 3N^{-1/2} (N ≥ 1). The integral-tail comparison in Lean — check: is there `Summable.tail_le_of_...` or the rpow series tail bound? There might be `Real.tsum_one_div_nat_rpow_le`? hmm.
The p-series tail estimate is ANOTHER estimation sub-fight. You know what — this whole tail is going to be ~200 lines no matter what. This is exactly where a fresh session should start. Let me close session 030 now with everything committed and the plan fully recorded.
DECISION: close session 030. Write journal-030, update STATE.md (NEXT ACTION: the tail bound with the full design), NGUYEN session log, db, closure commit.
The remaining 5e-v work (tail bound → shell bounds → assembly → squeeze) is a large analytic block that deserves a fresh sub-stage. Session 030 has two solid committed chunks. Closing cleanly with the full design recorded.
tool: Write ok
input
{
"file_path": "D:\\Python\\Quantum\\research\\journal\\session-030.md",
"content": "# Session 030 \u2014 2026-08-23 \u2014 ii\u2032-5e-v IN PROGRESS: count bridges + Dirichlet bound + split machinery (5e-v-a + 5e-v-b chunk 1)\n\n## Directive\nContinue the route per STATE.md: ii\u2032-5e-v (D-068 \u00a75) \u2014 the circle bound\n+ the squeeze \u27f9 the \u2203 B capstone. User: \"Let's work on ii'-5e-v.\"\n\n## Outcome (commits 5d24756, e6a654b; LiCircleBound.lean, ~340 lines so\nfar, clean elaboration, not yet root-imported)\n\n**5e-v-a COMPLETE** (\u00a70\u2013\u00a72):\n- `isNontrivialZero_conj` (the conjugate of a nontrivial zero is\n nontrivial \u2014 for the lower-half circle arcs; via Mathlib's\n unconditional `_root_.riemannZeta_conj`).\n- `A0count` + `Ncount_window_le`: Zeta23's RvM local count\n (`Zeta23.RvM.zeta_local_zero_count`: Ncount(t, t+1] \u2264 A\u2080\u00b7log(|t|+3),\n zeros WITH multiplicity, two-sided, ALL t \u2014 proved unconditionally in\n Zeta23; A\u2080 existential \u27f9 carried symbolically via Classical.choose).\n- `Ncount_add_of_le` (window additivity, disjoint finsum union) +\n `Ncount_chain_le` (N-step chain) + `Ncount_le_of_interval` +\n `Ncount_le_of_interval_gen` (dyadic interval count\n Ncount(a, b] \u2264 A\u2080\u00b7(b\u2212a+1)\u00b7log(3\u00b7max|a||b|+5), both signs).\n- `norm_logDeriv_zeta_le_dirichlet`: \u2016\u03b6\u2032/\u03b6 s\u2016 \u2264 dirichletC2 for\n re s \u2265 2 (\u2212\u03b6\u2032/\u03b6 = LSeries \u039b for re > 1 via Mathlib's\n `LSeries_vonMangoldt_eq_deriv_riemannZeta_div`; termwise \u2264 the s = 2\n majorant via `LSeries.norm_term_le_of_re_le_re`; the ZetaBounds.lean\n pattern mirrored).\n\n**5e-v-b chunk 1** (\u00a73 helpers):\n- `tsum_split_finset`: a summable real tsum = a finset sum + the\n if-zeroed complement tsum (via `tsum_eq_sum` + Summable.tsum_add +\n Summable.of_norm_bounded on the if-forms). [DecidableEq \u03b9] argument;\n the zero subtype discharges it via the (noncomputable) \u2102 instance.\n- `lowZeros` / `lowZeroConst`: the small-zero symbolic finite constant\n (\u2016\u03c1\u2016 \u2264 3; each \u2016\u03c1\u2016 > 0 since re > 0).\n\n## Design state for the remainder (all API verified this session)\nTarget: \u2016Fexpr s\u2016 \u2264 C\u00b7(log R)\u00b3-ish on the 5e-i \u03b3-gap circles\n(|s| = R_k, k \u2265 4000 so 13/R \u2264 1/(32\u00b7\u0108\u2080(4k+9/2)) \u2014 the numerics:\n\u0108\u2080(16006) \u2264 9.36 via log(16008) \u2264 10 \u27f8 16008 \u2264 e^10 \u27f8\n2.718281828^10 \u2265 22026 (norm_num chain) and log 10 \u2264 2.4 \u27f8\ne^{0.4} \u2265 1.4 = 1+0.4 (Real.add_one_le_exp) \u27f9 slack 4\u00d7).\n\n- **seriesFun circle bound** (the remaining analytic heart): ONE tsum\n split (tsum_split_finset) at the finset {\u2016\u03c1\u2016 \u2264 2R}; the finite part\n splits by Finset.filter into: low (\u2016\u03c1\u2016 \u2264 3: \u2264 lowZeroConst), inner\n (3 < \u2016\u03c1\u2016 \u2264 R\u22122: term \u2264 2/\u2016\u03c1\u2016 dyadic in \u2016\u03c1\u2016, and the R/2..R\u22122 band:\n term \u2264 2/(R\u2212\u2016\u03c1\u2016) dyadic in R\u2212\u2016\u03c1\u2016; counts via Ncount_le_of_interval_gen\n with |\u03b3| \u2265 \u2016\u03c1\u2016\u22121 [via Complex.norm_le_abs_re_add_abs_im]), critical\n (R\u22122 < \u2016\u03c1\u2016 < R+2: the \u03b3-gap \u2014 strip arcs |t\u2212\u03b3| \u2265 1/(32\u0108\u2080); near-axis\n arcs \u2016s\u2212\u03c1\u2016 \u2265 \u221a(R\u00b2\u2212100)\u22121 \u2265 R/2 \u2014 NO gap needed there), upper band\n (R+2 \u2264 \u2016\u03c1\u2016 \u2264 2R: term \u2264 1/(\u2016\u03c1\u2016\u2212R) dyadic). The tail (\u2016\u03c1\u2016 > 2R):\n term \u2264 (17R/6)\u00b7m/(1+\u03b3\u00b2) (5e-iii's (17/12) comparison) \u27f9 \u2264\n (17R/6)\u00b7(tail of \u03a3 m/(1+\u03b3\u00b2) over |\u03b3| > 2R\u22121).\n- **the tail bound** (the last new analytic input): \u03a3_{|\u03b3|>X} m/(1+\u03b3\u00b2)\n \u2264 16\u00b7A\u2080\u00b7log X/X for X \u2265 4\u00b710\u00b3-ish \u2014 via the \u2124-window fiberwise\n decomposition (ENNReal.tsum_fiberwise is FREE in \u211d\u22650\u221e \u2014 verified\n exists) with Zeta23's `key` machinery (key_lt, le_key_add_one,\n one_add_sq_ge: 1+\u03b3\u00b2 \u2265 (1+n\u00b2)/4 \u2014 all in Zeta23/Tail.lean +\n WeilEF/ZeroSummability.lean) + the window counts + the weight-series\n tail (weight(n) = log(|n|+3)/(1+n\u00b2) \u2264 2\u221a2\u00b7|n|^{-3/2} via\n log y \u2264 2(\u221ay\u22121) \u27f8 Real.log_le_sub_one_of_pos at \u221ay; p-series tail\n \u03a3_{n\u2265N} n^{-3/2} \u2264 3/\u221aN \u2014 the p-series tail-bound API to pin down at\n build time).\n- **logDeriv \u03b6 on the circle**: strip arcs (re \u2208 [\u22125/2, 7/2], |t| \u2265 10\n for R \u2265 11): 5c/5d capstones (100000/200000\u00b7(log|t|)\u00b2) + the near\n sum \u2264 32\u0108\u2080\u00b7(2\u00b7A\u2080\u00b7log(|t|+4)) [nearZeros \u2286 two Ncount windows] =\n O((log R)\u00b2); right near-axis (re \u2265 \u221a(R\u00b2\u2212100) \u2265 2): the Dirichlet\n bound (5e-v-a \u2714); left near-axis (re \u2264 \u2212\u221a(R\u00b2\u2212100)): 5d's\n logDeriv_zeta_fe pointwise + Dirichlet at 1\u2212s + the \u0393R brackets via\n 5e-ii (A)/(B) + the pointwise logDeriv \u0393R identity (5e-ii's (G) for\n im \u2260 0; the REAL point s = \u2212R via the closure/continuity extension \u2014\n the bound set is closed and the im \u2260 0 arc points are dense).\n- **elems**: 1/s, 1/(s\u22121) = O(1/R); \u03c8(s/2)/2 = O(log R): right arc\n via 5e-ii (B) (re \u2265 2, |im| \u2264 5); strip arcs via 5d's\n norm_digamma_sub_log_le_of_two_le_abs_im (|im| \u2265 5 there) + \u2016log w\u2016 \u2264\n log\u2016w\u2016 + \u03c0 (LiReflect:414); left arc via 5e-ii (A) with\n dist(s/2, \u2212\u2115) \u2265 1/8 (from dist(R/2, \u2124) \u2265 1/4 and |re s/2| \u2208\n [\u221a(R\u00b2\u2212100)/2, R/2] \u2286 [R/2 \u2212 26/R, R/2], R \u2265 208).\n- **the squeeze**: G(z) := Gentire entire; for fixed z\u2080, R_k > |z\u2080|:\n dist(Gentire z\u2080, Gentire 0) \u2264 (R\u2082(R_k)/R_k)\u00b7|z\u2080| with R\u2082(R) =\n C\u00b7(log R)\u00b3 + \u2016Gentire 0\u2016 \u2014 Mathlib's Schwarz\n `Complex.dist_le_div_mul_dist_of_mapsTo_ball` (DifferentiableOn on\n the ball + MapsTo into closedBall (Gentire 0) R\u2082 \u2014 the mapsTo from\n the sphere bound via Zeta23's\n AnalyticOn.norm_le_of_norm_le_on_sphere) \u27f9 dist \u2264 0 (the ratio \u2192 0:\n (log R)\u00b3/R \u2264 64\u00b7R^{3/4}/R via Real.log_le_rpow-ish) \u27f9 Gentire \u2261\n Gentire 0 =: B. Capstone (D-068 \u00a75):\n `partialFraction_identity (hZC : ZeroCountProp) : \u2203 B : \u2102, \u2200 s,\n riemannZeta s \u2260 0 \u2192 s \u2260 0 \u2192 s \u2260 1 \u2192 (\u2200 n \u2265 1, s \u2260 \u22122n) \u2192\n logDeriv riemannZeta s = B + seriesFun s \u2212 1/s \u2212 1/(s\u22121) +\n (Real.log Real.pi)/2 \u2212 Complex.digamma (s/2)/2` \u2014 from Gentire s =\n Fexpr s (5e-iv's gentire_eq_Fexpr at good points) + Gentire \u2261 B.\n\n## Banked gotchas (session 030 additions)\n- `le_or_lt` does NOT exist (banked session 026) \u2014 use `le_or_gt`.\n- finsum API: `finsum_mem_union (hst : Disjoint) (hs : s.Finite)\n (ht : t.Finite)` (to_additive of finprod_mem_union \u2014 plain grep for\n \"finsum_mem_union\" misses it), `finsum_mem_empty`, and\n `tsum_eq_sum (s := s) (h : \u2200 b \u2209 s, f b = 0) : \u2211' f = \u2211 b \u2208 s, f b`\n (NOT tsum_eq_sum_of_support_subset \u2014 that name does not exist).\n- `tsum_eq_sum`'s conclusion RHS keeps the SAME function: the if-form\n needs a closing Finset.sum_congr with if_pos.\n- **Bare `Nat.cast_nonneg N` as a term/have leaves the cast target a\n metavariable (IsOrderedRing stuck)** \u2014 always ascribe\n `(show (0:\u211d) \u2264 (N:\u211d) from Nat.cast_nonneg N)` (recurred 3\u00d7).\n- **AnalyticAt.comp / Ncount_add_of_le HOU**: pin the implicit\n endpoints (a := ..) (b := ..) \u2014 by-tactic proof args do NOT determine\n them (the elaborator assigns garbage from linarith internals).\n- A `calc ... := by` block nested inside a case/induction tactic block\n at deep indentation BROKE THE PARSER (mysterious \"unexpected token\n '/--'; expected ':='\" downstream + \"unknown identifier linarith\") \u2014\n restructure to avoid the nested-by calc in that position.\n- `Set.mem_union.mp` doesn't exist \u2014 union membership is defeq to Or:\n `rcases h with h1 | h2` directly.\n- positivity cannot see through `Classical.choose`-defined constants\n (A0count) \u2014 use mul_nonneg A0count_pos.le \u2026 explicitly.\n- `norm_num at hd` on `\u2016(1:\u2102) \u2212 0\u2016 < 1/2` leaked a metavariable side\n goal \u2014 prefer the explicit `rw [show \u2016..\u2016 = 1 by simp] at hd; linarith`.\n- Statement-level `if i \u2208 s` needs decidability BEFORE `classical` can\n run: use a [DecidableEq \u03b9] argument (the zero subtype inherits the\n noncomputable \u2102 instance); `open Classical in` before a doc-commented\n lemma FAILED to parse in this toolchain.\n- `riemannZeta_conj` is AMBIGUOUS (root Mathlib vs Zeta23's s \u2260 1\n version) \u2014 use `_root_.riemannZeta_conj` (unconditional).\n\n## Status / next\n5e-v-a \u2714 + 5e-v-b chunk 1 \u2714 committed. Next session: 5e-v-b chunk 2 \u2014\nthe tail bound (zero_tail_inv_sq_le via ENNReal.tsum_fiberwise +\nZeta23 key machinery + the weight-series tail) \u2192 the main seriesFun\ncircle bound (one tsum split + four finset shell bounds) \u2192 5e-v-c (the\nlogDeriv assembly + elems + Fexpr circle bound) \u2192 5e-v-d (Schwarz\nsqueeze + the \u2203 B capstone). Then ii\u2032-5f \u2192 ii\u2032-3b \u2192 ii\u2032-3c \u2192 ii\u2032-6 \u2192\nii\u2032-7.\n"
}
result 2026-08-23 18:29:26Z → 2026-08-23 18:29:26Z
{
"type": "create",
"filePath": "D:\\Python\\Quantum\\research\\journal\\session-030.md",
"content": "# Session 030 \u2014 2026-08-23 \u2014 ii\u2032-5e-v IN PROGRESS: count bridges + Dirichlet bound + split machinery (5e-v-a + 5e-v-b chunk 1)\n\n## Directive\nContinue the route per STATE.md: ii\u2032-5e-v (D-068 \u00a75) \u2014 the circle bound\n+ the squeeze \u27f9 the \u2203 B capstone. User: \"Let's work on ii'-5e-v.\"\n\n## Outcome (commits 5d24756, e6a654b; LiCircleBound.lean, ~340 lines so\nfar, clean elaboration, not yet root-imported)\n\n**5e-v-a COMPLETE** (\u00a70\u2013\u00a72):\n- `isNontrivialZero_conj` (the conjugate of a nontrivial zero is\n nontrivial \u2014 for the lower-half circle arcs; via Mathlib's\n unconditional `_root_.riemannZeta_conj`).\n- `A0count` + `Ncount_window_le`: Zeta23's RvM local count\n (`Zeta23.RvM.zeta_local_zero_count`: Ncount(t, t+1] \u2264 A\u2080\u00b7log(|t|+3),\n zeros WITH multiplicity, two-sided, ALL t \u2014 proved unconditionally in\n Zeta23; A\u2080 existential \u27f9 carried symbolically via Classical.choose).\n- `Ncount_add_of_le` (window additivity, disjoint finsum union) +\n `Ncount_chain_le` (N-step chain) + `Ncount_le_of_interval` +\n `Ncount_le_of_interval_gen` (dyadic interval count\n Ncount(a, b] \u2264 A\u2080\u00b7(b\u2212a+1)\u00b7log(3\u00b7max|a||b|+5), both signs).\n- `norm_logDeriv_zeta_le_dirichlet`: \u2016\u03b6\u2032/\u03b6 s\u2016 \u2264 dirichletC2 for\n re s \u2265 2 (\u2212\u03b6\u2032/\u03b6 = LSeries \u039b for re > 1 via Mathlib's\n `LSeries_vonMangoldt_eq_deriv_riemannZeta_div`; termwise \u2264 the s = 2\n majorant via `LSeries.norm_term_le_of_re_le_re`; the ZetaBounds.lean\n pattern mirrored).\n\n**5e-v-b chunk 1** (\u00a73 helpers):\n- `tsum_split_finset`: a summable real tsum = a finset sum + the\n if-zeroed complement tsum (via `tsum_eq_sum` + Summable.tsum_add +\n Summable.of_norm_bounded on the if-forms). [DecidableEq \u03b9] argument;\n the zero subtype discharges it via the (noncomputable) \u2102 instance.\n- `lowZeros` / `lowZeroConst`: the small-zero symbolic finite constant\n (\u2016\u03c1\u2016 \u2264 3; each \u2016\u03c1\u2016 > 0 since re > 0).\n\n## Design state for the remainder (all API verified this session)\nTarget: \u2016Fexpr s\u2016 \u2264 C\u00b7(log R)\u00b3-ish on the 5e-i \u03b3-gap circles\n(|s| = R_k, k \u2265 4000 so 13/R \u2264 1/(32\u00b7\u0108\u2080(4k+9/2)) \u2014 the numerics:\n\u0108\u2080(16006) \u2264 9.36 via log(16008) \u2264 10 \u27f8 16008 \u2264 e^10 \u27f8\n2.718281828^10 \u2265 22026 (norm_num chain) and log 10 \u2264 2.4 \u27f8\ne^{0.4} \u2265 1.4 = 1+0.4 (Real.add_one_le_exp) \u27f9 slack 4\u00d7).\n\n- **seriesFun circle bound** (the remaining analytic heart): ONE tsum\n split (tsum_split_finset) at the finset {\u2016\u03c1\u2016 \u2264 2R}; the finite part\n splits by Finset.filter into: low (\u2016\u03c1\u2016 \u2264 3: \u2264 lowZeroConst), inner\n (3 < \u2016\u03c1\u2016 \u2264 R\u22122: term \u2264 2/\u2016\u03c1\u2016 dyadic in \u2016\u03c1\u2016, and the R/2..R\u22122 band:\n term \u2264 2/(R\u2212\u2016\u03c1\u2016) dyadic in R\u2212\u2016\u03c1\u2016; counts via Ncount_le_of_interval_gen\n with |\u03b3| \u2265 \u2016\u03c1\u2016\u22121 [via Complex.norm_le_abs_re_add_abs_im]), critical\n (R\u22122 < \u2016\u03c1\u2016 < R+2: the \u03b3-gap \u2014 strip arcs |t\u2212\u03b3| \u2265 1/(32\u0108\u2080); near-axis\n arcs \u2016s\u2212\u03c1\u2016 \u2265 \u221a(R\u00b2\u2212100)\u22121 \u2265 R/2 \u2014 NO gap needed there), upper band\n (R+2 \u2264 \u2016\u03c1\u2016 \u2264 2R: term \u2264 1/(\u2016\u03c1\u2016\u2212R) dyadic). The tail (\u2016\u03c1\u2016 > 2R):\n term \u2264 (17R/6)\u00b7m/(1+\u03b3\u00b2) (5e-iii's (17/12) comparison) \u27f9 \u2264\n (17R/6)\u00b7(tail of \u03a3 m/(1+\u03b3\u00b2) over |\u03b3| > 2R\u22121).\n- **the tail bound** (the last new analytic input): \u03a3_{|\u03b3|>X} m/(1+\u03b3\u00b2)\n \u2264 16\u00b7A\u2080\u00b7log X/X for X \u2265 4\u00b710\u00b3-ish \u2014 via the \u2124-window fiberwise\n decomposition (ENNReal.tsum_fiberwise is FREE in \u211d\u22650\u221e \u2014 verified\n exists) with Zeta23's `key` machinery (key_lt, le_key_add_one,\n one_add_sq_ge: 1+\u03b3\u00b2 \u2265 (1+n\u00b2)/4 \u2014 all in Zeta23/Tail.lean +\n WeilEF/ZeroSummability.lean) + the window counts + the weight-series\n tail (weight(n) = log(|n|+3)/(1+n\u00b2) \u2264 2\u221a2\u00b7|n|^{-3/2} via\n log y \u2264 2(\u221ay\u22121) \u27f8 Real.log_le_sub_one_of_pos at \u221ay; p-series tail\n \u03a3_{n\u2265N} n^{-3/2} \u2264 3/\u221aN \u2014 the p-series tail-bound API to pin down at\n build time).\n- **logDeriv \u03b6 on the circle**: strip arcs (re \u2208 [\u22125/2, 7/2], |t| \u2265 10\n for R \u2265 11): 5c/5d capstones (100000/200000\u00b7(log|t|)\u00b2) + the near\n sum \u2264 32\u0108\u2080\u00b7(2\u00b7A\u2080\u00b7log(|t|+4)) [nearZeros \u2286 two Ncount windows] =\n O((log R)\u00b2); right near-axis (re \u2265 \u221a(R\u00b2\u2212100) \u2265 2): the Dirichlet\n bound (5e-v-a \u2714); left near-axis (re \u2264 \u2212\u221a(R\u00b2\u2212100)): 5d's\n logDeriv_zeta_fe pointwise + Dirichlet at 1\u2212s + the \u0393R brackets via\n 5e-ii (A)/(B) + the pointwise logDeriv \u0393R identity (5e-ii's (G) for\n im \u2260 0; the REAL point s = \u2212R via the closure/continuity extension \u2014\n the bound set is closed and the im \u2260 0 arc points are dense).\n- **elems**: 1/s, 1/(s\u22121) = O(1/R); \u03c8(s/2)/2 = O(log R): right arc\n via 5e-ii (B) (re \u2265 2, |im| \u2264 5); strip arcs via 5d's\n norm_digamma_sub_log_le_of_two_le_abs_im (|im| \u2265 5 there) + \u2016log w\u2016 \u2264\n log\u2016w\u2016 + \u03c0 (LiReflect:414); left arc via 5e-ii (A) with\n dist(s/2, \u2212\u2115) \u2265 1/8 (from dist(R/2, \u2124) \u2265 1/4 and |re s/2| \u2208\n [\u221a(R\u00b2\u2212100)/2, R/2] \u2286 [R/2 \u2212 26/R, R/2], R \u2265 208).\n- **the squeeze**: G(z) := Gentire entire; for fixed z\u2080, R_k > |z\u2080|:\n dist(Gentire z\u2080, Gentire 0) \u2264 (R\u2082(R_k)/R_k)\u00b7|z\u2080| with R\u2082(R) =\n C\u00b7(log R)\u00b3 + \u2016Gentire 0\u2016 \u2014 Mathlib's Schwarz\n `Complex.dist_le_div_mul_dist_of_mapsTo_ball` (DifferentiableOn on\n the ball + MapsTo into closedBall (Gentire 0) R\u2082 \u2014 the mapsTo from\n the sphere bound via Zeta23's\n AnalyticOn.norm_le_of_norm_le_on_sphere) \u27f9 dist \u2264 0 (the ratio \u2192 0:\n (log R)\u00b3/R \u2264 64\u00b7R^{3/4}/R via Real.log_le_rpow-ish) \u27f9 Gentire \u2261\n Gentire 0 =: B. Capstone (D-068 \u00a75):\n `partialFraction_identity (hZC : ZeroCountProp) : \u2203 B : \u2102, \u2200 s,\n riemannZeta s \u2260 0 \u2192 s \u2260 0 \u2192 s \u2260 1 \u2192 (\u2200 n \u2265 1, s \u2260 \u22122n) \u2192\n logDeriv riemannZeta s = B + seriesFun s \u2212 1/s \u2212 1/(s\u22121) +\n (Real.log Real.pi)/2 \u2212 Complex.digamma (s/2)/2` \u2014 from Gentire s =\n Fexpr s (5e-iv's gentire_eq_Fexpr at good points) + Gentire \u2261 B.\n\n## Banked gotchas (session 030 additions)\n- `le_or_lt` does NOT exist (banked session 026) \u2014 use `le_or_gt`.\n- finsum API: `finsum_mem_union (hst : Disjoint) (hs : s.Finite)\n (ht : t.Finite)` (to_additive of finprod_mem_union \u2014 plain grep for\n \"finsum_mem_union\" misses it), `finsum_mem_empty`, and\n `tsum_eq_sum (s := s) (h : \u2200 b \u2209 s, f b = 0) : \u2211' f = \u2211 b \u2208 s, f b`\n (NOT tsum_eq_sum_of_support_subset \u2014 that name does not exist).\n- `tsum_eq_sum`'s conclusion RHS keeps the SAME function: the if-form\n needs a closing Finset.sum_congr with if_pos.\n- **Bare `Nat.cast_nonneg N` as a term/have leaves the cast target a\n metavariable (IsOrderedRing stuck)** \u2014 always ascribe\n `(show (0:\u211d) \u2264 (N:\u211d) from Nat.cast_nonneg N)` (recurred 3\u00d7).\n- **AnalyticAt.comp / Ncount_add_of_le HOU**: pin the implicit\n endpoints (a := ..) (b := ..) \u2014 by-tactic proof args do NOT determine\n them (the elaborator assigns garbage from linarith internals).\n- A `calc ... := by` block nested inside a case/induction tactic block\n at deep indentation BROKE THE PARSER (mysterious \"unexpected token\n '/--'; expected ':='\" downstream + \"unknown identifier linarith\") \u2014\n restructure to avoid the nested-by calc in that position.\n- `Set.mem_union.mp` doesn't exist \u2014 union membership is defeq to Or:\n `rcases h with h1 | h2` directly.\n- positivity cannot see through `Classical.choose`-defined constants\n (A0count) \u2014 use mul_nonneg A0count_pos.le \u2026 explicitly.\n- `norm_num at hd` on `\u2016(1:\u2102) \u2212 0\u2016 < 1/2` leaked a metavariable side\n goal \u2014 prefer the explicit `rw [show \u2016..\u2016 = 1 by simp] at hd; linarith`.\n- Statement-level `if i \u2208 s` needs decidability BEFORE `classical` can\n run: use a [DecidableEq \u03b9] argument (the zero subtype inherits the\n noncomputable \u2102 instance); `open Classical in` before a doc-commented\n lemma FAILED to parse in this toolchain.\n- `riemannZeta_conj` is AMBIGUOUS (root Mathlib vs Zeta23's s \u2260 1\n version) \u2014 use `_root_.riemannZeta_conj` (unconditional).\n\n## Status / next\n5e-v-a \u2714 + 5e-v-b chunk 1 \u2714 committed. Next session: 5e-v-b chunk 2 \u2014\nthe tail bound (zero_tail_inv_sq_le via ENNReal.tsum_fiberwise +\nZeta23 key machinery + the weight-series tail) \u2192 the main seriesFun\ncircle bound (one tsum split + four finset shell bounds) \u2192 5e-v-c (the\nlogDeriv assembly + elems + Fexpr circle bound) \u2192 5e-v-d (Schwarz\nsqueeze + the \u2203 B capstone). Then ii\u2032-5f \u2192 ii\u2032-3b \u2192 ii\u2032-3c \u2192 ii\u2032-6 \u2192\nii\u2032-7.\n",
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[{"type":"thinking","thinking":"`ENNReal.tsum_fiberwise (f : β → ℝ≥0∞) (g : β → γ) : ∑' x, ∑' b : g ⁻¹' {x}, f b = ∑' i, f i` — free (no summability side conditions) ✓. This gives the fiberwise decomposition in ℝ≥0∞.\n\nThe full tail bound proof sketch (in ℝ≥0∞ then back to ℝ):\n- Let T := {ρ : zerosubtype // X < |ρ.1.im|} (the tail subtype-of-subtype). Hmm — actually cleaner: work directly on the zero subtype Z := {ρ : ℂ // IsNontrivialZero ρ} with the indicator... Let me define the tail as a subtype T of Z: T := {ρ : Z // X < |(ρ.1).im|}.\n\n∑'_{ρ : T} (ofReal (m_ρ/(1+γ²))) ≤ ofReal (16 A₀ log X / X):\n= ∑'_{j : ℕ} ∑'_{ρ : T, j(ρ) = j} ofReal(m/(1+γ²)) [ENNReal.tsum_fiberwise with g := j(·) : T → ℕ]\n≤ ∑' j, ofReal((fiber count)·2/(4^j X²))\n= ofReal(Σ' j, ...)... \n\nthe fiber count: {ρ : T, j(ρ) = j} — zeros with |γ| ∈ (2^j X, 2^{j+1} X]: Σ_{fiber} m ≤ 2A₀(2^jX+1)log(3·2^{j+1}X+5) — via Ncount_le_of_interval_gen ×2 (both sides) — the fiber is finite (bounded ordinates ✓ zerosIn_finite) — as a finset sum ≤ interval count ✓.\n\nThen ∑'_{j : ℕ} 2A₀(2^jX+1)log(3·2^{j+1}X+5)·2/(4^jX²) ≤ (16A₀/X)·(10 log 2 + 2 log X)/... — the dyadic tsum over j : ℕ of an explicit ℝ≥0 sequence — summable (geometric×polynomial) — its sum: bound by the closed form 10 log 2 + 2 log X etc.\n\nThis is all doable but LONG (~250 lines for the tail alone). \n\nPACING CHECK: This session has been very long already (5e-v-a + 5e-v-b chunk 1 committed). The remaining 5e-v work: tail bound (~250 lines) + F-finset shell bounds (~300) + assembly (~150) + squeeze + capstone (~200) ≈ 900 lines ≈ 1.5–2 sessions. Per the pacing directive and the natural breakpoint (helpers committed), the right move is to close the session cleanly now: journal-030 + STATE update + db + closure commit. The next session starts the tail bound with the full design in STATE.md.\n\nWait — actually, let me reconsider the tail ONE more time, because I want to double-check I'm not missing a 10× cheaper path...\n\nCHEAP TAIL via Cauchy–Schwarz against zero_sum_inv_sq? Σ_{|γ|>X} m/(1+γ²) ≤ (Σ m/(1+γ²))^{1/2}·(Σ_{|γ|>X} m/(1+γ²))^{1/2} — circular.\n\nCheapest honest decay: the tail of a convergent series with a window-count input: is there really no simpler... \n\nActually — here's one: the tail Σ_{|γ|>X} m/(1+γ²) ≤ (1/X)·Σ_{|γ|>X} m·|γ|/(1+γ²)·... no: 1/(1+γ²) ≤ (1/X)·(|γ|/(1+γ²))·... for |γ| > X: |γ|/(1+γ²) ≥ |γ|/(2γ²) = 1/(2|γ|) — no.\n\nHow about: Σ_{|γ|>X} m/(1+γ²) ≤ Σ m·|γ|^{−2}·... the dyadic is really the way. And actually the fiberwise-in-ℝ≥0∞ route is not THAT bad. But it's for next session.\n\nActually — hold on, let me reconsider whether I even NEED the 1/(1+γ²) tail bound. What the series bound needs: ∑'_{‖ρ‖>2R} m·R/(‖ρ‖·‖s−ρ‖) ≤ ∑' m·2R/‖ρ‖² (‖s−ρ‖ ≥ ‖ρ‖/2) — and 1/‖ρ‖² ≤ 1/|γ|² (|γ| ≤ ‖ρ‖) — so ≤ ∑'_{‖ρ‖>2R} m·2R/|γ|² ≤ 2R·∑'_{|γ|>2R−1} m/γ² — the γ⁻² tail: Σ_{|γ|>X} m/γ² ≤ Σ_j (count_j)/(2^jX)² — same dyadic. OR the 1/(1+γ²) version: m/γ² ≤ 2m/(1+γ²) for |γ| ≥ 1 ✓. Same either way.\n\nOK — one more idea to dodge the fiberwise decomposition: bound the tail by the WINDOW-INDEXED sum directly: the tail zeros = union over windows W_n (n : ℤ, |n| ≥ ⌈X⌉) of {γ ∈ (n, n+1]}: Σ_{tail} m/γ² ≤ Σ_{|n| ≥ ⌈X⌉} (count_n)·2/n² — the windows are INTEGER-indexed (n : ℤ): the assignment γ ↦ ⌊γ⌋ (or the `key` function from Zeta23!) — the fiberwise over ℤ with n ↦ window — Zeta23's key_lt/le_key_add_one machinery + one_add_sq_ge + summable_weight + totalWeight ALREADY prove ∑'_{n : ℤ} log(|n|+3)/(1+n²) < ∞!! And `weight_le`: (4/(1+n²))-style bounds. The Zeta23 `weight` function: weight n := log(|n|+3)/(1+n²)-ish (integer windows) with summable_weight : Summable weight and totalWeight := its tsum. \n\nSO: Σ_{|γ|>X} m/(1+γ²) ≤ Σ_{n : ℤ, |n| ≥ X-ish} A₀·weight(n)·4-ish — the INTEGER-window version — and the tail of the (summable) weight series: Σ_{|n|≥N} log(|n|+3)/(1+n²) ≤ ∫-ish ≤ C·log N/N — an explicit tail bound on the weight series — provable by comparison with 1/n^{3/2} (log(n+3) ≤ n^{1/2} for n ≥ ...): weight(n) ≤ C/n^{3/2} for large n: Σ_{|n|≥N} ≤ 2C/√N·... — STILL a tail estimate but now on a 1D integer series with an explicit majorant — via `Summable.tail` bounds... \n\nHmm — the weight-tail: Σ_{|n| ≥ N} log(|n|+3)/(1+n²) ≤ Σ 2A₀/n^{3/2}-form: is there a packaged p-series tail bound: `Real.tsum_one_div_nat_rpow_le`? or `summable_one_div_nat_rpow` + tail via the integral test... The integral test in Lean: `Summable.of_integral_le`... the tail bound from the integral: there might be something like `Real.tsum_..._le_of_...`. OR the direct: Σ_{n ≥ N} 1/n^{3/2} ≤ 1/N^{3/2} + ∫_N^∞ x^{-3/2} = 1/N^{3/2} + 2/√N ≤ 3/√N: the integral comparison in Lean: `Summable.of_norm_bounded...` no. There IS the classic: for antitone f: Σ_{n ≥ N} f n ≤ f N + ∫... \n\nOK. It's all ~the same length. This is next-session work. Let me close THIS session cleanly: journal-030 documenting 5e-v-a + 5e-v-b chunk 1 + the detailed remaining plan, STATE update, db, closure commit.\n\nActually — wait, let me reconsider the time/length budget once more. I've done 5e-v-a + 5e-v-b-helpers. The session isn't out of gas per se — but the tail bound + F-finset bounds + assembly + squeeze is easily another 600-900 lines and several compile cycles. The NGUYEN pacing rule: break into sub-stages with clean commits; also the BREAK PROMPT rule (approaching long work). Since everything is committed and the next sub-stage (the tail bound) is a clean chunk, let me continue — but efficiently. Let me build the tail bound now.\n\nActually no — let me be honest about the state: this turn has already produced 5e-v-a (3 commits) with substantial exploration. The 5e-v-b main lemma (series circle bound) is the single largest estimation lemma of the project and deserves a fresh sub-stage with full attention. The pacing directive is explicit: NO RUSH, correctness over speed, sub-stages ending in clean committed states. The clean state is now (5e-v-b chunk 1 committed). The remaining 5e-v-b (tail + main bound) is the next session's first item, with the full design recorded in STATE.md.\n\nHmm, but the user said \"let's work on ii'-5e-v\" — they want progress on 5e-v. I've made concrete progress (2 commits). A session-end now is legitimate (each sub-stage ends clean). But I could also push the tail bound now... Let me push ONE more meaningful chunk: the tail bound (the last genuinely-new analytic input). If it compiles, 5e-v-b finishes next session with the shell bounds + assembly.\n\nLet me design the tail bound concretely and write it:\n\n```lean\n/-- The dyadic tail bound: for X ≥ 4, Σ over zeros with |γ| > X of\nm/(1+γ²) ≤ 40·A₀·log X / X. Via the integer-window fiberwise\ndecomposition (ENNReal.tsum_fiberwise) + the window counts. -/\n```\n\nHmm — actually, let me use a much simpler decomposition that avoids fiberwise: bound the tail tsum by a DIRECT comparison with the weight series restricted to high windows. For each zero with |γ| > X: its window n := ⌊γ⌋ (or the Zeta23 `key`): |n| ≥ ⌊X⌋... The map γ ↦ key γ : the zeros map to ℤ. Then:\n\n∑'_{ρ : |γ|>X} m/(1+γ²) ≤ ∑'_{ρ : |γ|>X} m·(4/(1+(key γ)²)) [one_add_sq_ge: 1+γ² ≥ (1+n²)/4 for γ ∈ (n, n+1] ✓ banked in Zeta23!]\n≤ (the fiberwise over key) = Σ_{n : ℤ} Σ_{ρ : |γ|>X, key ρ = n} m·4/(1+n²)\n≤ Σ_{n : ℤ, |n| ≥ X-ish} 4A₀·log(|n|+3)/(1+n²) [window count]\n≤ 4A₀·(tail of weight above ~X)\n≤ 4A₀·(16·log X / X)?? — the weight tail bound.\n\nThe weight tail: Σ_{|n| ≥ N} log(|n|+3)/(1+n²) ≤ Σ_{|n|≥N} log(|n|+3)/n². For n ≥ N ≥ 16: log(n+3) ≤ log(2n) ≤ 2 log n·... log(n+3) ≤ log(2n) = log 2 + log n ≤ 1 + log n; and log n ≤ 2√n-ish... Let me bound: log(n+3)/n² ≤ n^{2/3}/n² = n^{-4/3}?? log(n+3) ≤ n^{2/3} for n ≥ 16: log(19) ≈ 2.94 ≤ 16^{2/3} ≈ 6.35 ✓ and grows — log(n+3) ≤ n^{2/3} for n ≥ 16 plausible (need a clean proof: log x ≤ x^{2/3}·... via log x ≤ 3(x^{1/3}−1)? hmm: log x ≤ x^r/r for r > 0 (Real.log_le_rpow_div?): with r = 1/2: log(n+3) ≤ 2√(n+3)... 2√(n+3)/n² ≤ 2√(2n)/n² = 2√2/n^{3/2} (n ≥ 3) — so weight(n) ≤ 2√2·log... no wait: log(|n|+3)/(1+n²) ≤ 2√2/|n|^{3/2} for |n| ≥ 3 ✓ (log(n+3) ≤ 2√(n+3) — from Real.log_le_sub_one? no — log y ≤ 2√y: log y = 2 log(√y) ≤ 2(√y − 1) ≤ 2√y using log z ≤ z−1 ✓ Real.log_le_sub_one_of_pos ✓ banked!). So weight(n) ≤ 2√2·|n|^{-3/2}: Σ_{|n| ≥ N} |n|^{-3/2} = 2·Σ_{n ≥ N} n^{-3/2}: the p-series tail with p = 3/2: Σ_{n≥N} n^{-3/2} ≤ N^{-3/2} + ∫_N^∞ x^{-3/2}dx = N^{-3/2} + 2N^{-1/2} ≤ 3N^{-1/2} (N ≥ 1). The integral-tail comparison in Lean — check: is there `Summable.tail_le_of_...` or the rpow series tail bound? There might be `Real.tsum_one_div_nat_rpow_le`? hmm.\n\nThe p-series tail estimate is ANOTHER estimation sub-fight. You know what — this whole tail is going to be ~200 lines no matter what. This is exactly where a fresh session should start. Let me close session 030 now with everything committed and the plan fully recorded.\n\nDECISION: close session 030. Write journal-030, update STATE.md (NEXT ACTION: the tail bound with the full design), NGUYEN session log, db, closure commit.","signature":""},{"type":"text","text":"The remaining 5e-v work (tail bound → shell bounds → assembly → squeeze) is a large analytic block that deserves a fresh sub-stage. Session 030 has two solid committed chunks. Closing cleanly with the full design recorded."},{"type":"tool_use","id":"Write_339","name":"Write","input":{"file_path":"D:\\Python\\Quantum\\research\\journal\\session-030.md","content":"# Session 030 — 2026-08-23 — ii′-5e-v IN PROGRESS: count bridges + Dirichlet bound + split machinery (5e-v-a + 5e-v-b chunk 1)\n\n## Directive\nContinue the route per STATE.md: ii′-5e-v (D-068 §5) — the circle bound\n+ the squeeze ⟹ the ∃ B capstone. User: \"Let's work on ii'-5e-v.\"\n\n## Outcome (commits 5d24756, e6a654b; LiCircleBound.lean, ~340 lines so\nfar, clean elaboration, not yet root-imported)\n\n**5e-v-a COMPLETE** (§0–§2):\n- `isNontrivialZero_conj` (the conjugate of a nontrivial zero is\n nontrivial — for the lower-half circle arcs; via Mathlib's\n unconditional `_root_.riemannZeta_conj`).\n- `A0count` + `Ncount_window_le`: Zeta23's RvM local count\n (`Zeta23.RvM.zeta_local_zero_count`: Ncount(t, t+1] ≤ A₀·log(|t|+3),\n zeros WITH multiplicity, two-sided, ALL t — proved unconditionally in\n Zeta23; A₀ existential ⟹ carried symbolically via Classical.choose).\n- `Ncount_add_of_le` (window additivity, disjoint finsum union) +\n `Ncount_chain_le` (N-step chain) + `Ncount_le_of_interval` +\n `Ncount_le_of_interval_gen` (dyadic interval count\n Ncount(a, b] ≤ A₀·(b−a+1)·log(3·max|a||b|+5), both signs).\n- `norm_logDeriv_zeta_le_dirichlet`: ‖ζ′/ζ s‖ ≤ dirichletC2 for\n re s ≥ 2 (−ζ′/ζ = LSeries Λ for re > 1 via Mathlib's\n `LSeries_vonMangoldt_eq_deriv_riemannZeta_div`; termwise ≤ the s = 2\n majorant via `LSeries.norm_term_le_of_re_le_re`; the ZetaBounds.lean\n pattern mirrored).\n\n**5e-v-b chunk 1** (§3 helpers):\n- `tsum_split_finset`: a summable real tsum = a finset sum + the\n if-zeroed complement tsum (via `tsum_eq_sum` + Summable.tsum_add +\n Summable.of_norm_bounded on the if-forms). [DecidableEq ι] argument;\n the zero subtype discharges it via the (noncomputable) ℂ instance.\n- `lowZeros` / `lowZeroConst`: the small-zero symbolic finite constant\n (‖ρ‖ ≤ 3; each ‖ρ‖ > 0 since re > 0).\n\n## Design state for the remainder (all API verified this session)\nTarget: ‖Fexpr s‖ ≤ C·(log R)³-ish on the 5e-i γ-gap circles\n(|s| = R_k, k ≥ 4000 so 13/R ≤ 1/(32·Ĉ₀(4k+9/2)) — the numerics:\nĈ₀(16006) ≤ 9.36 via log(16008) ≤ 10 ⟸ 16008 ≤ e^10 ⟸\n2.718281828^10 ≥ 22026 (norm_num chain) and log 10 ≤ 2.4 ⟸\ne^{0.4} ≥ 1.4 = 1+0.4 (Real.add_one_le_exp) ⟹ slack 4×).\n\n- **seriesFun circle bound** (the remaining analytic heart): ONE tsum\n split (tsum_split_finset) at the finset {‖ρ‖ ≤ 2R}; the finite part\n splits by Finset.filter into: low (‖ρ‖ ≤ 3: ≤ lowZeroConst), inner\n (3 < ‖ρ‖ ≤ R−2: term ≤ 2/‖ρ‖ dyadic in ‖ρ‖, and the R/2..R−2 band:\n term ≤ 2/(R−‖ρ‖) dyadic in R−‖ρ‖; counts via Ncount_le_of_interval_gen\n with |γ| ≥ ‖ρ‖−1 [via Complex.norm_le_abs_re_add_abs_im]), critical\n (R−2 < ‖ρ‖ < R+2: the γ-gap — strip arcs |t−γ| ≥ 1/(32Ĉ₀); near-axis\n arcs ‖s−ρ‖ ≥ √(R²−100)−1 ≥ R/2 — NO gap needed there), upper band\n (R+2 ≤ ‖ρ‖ ≤ 2R: term ≤ 1/(‖ρ‖−R) dyadic). The tail (‖ρ‖ > 2R):\n term ≤ (17R/6)·m/(1+γ²) (5e-iii's (17/12) comparison) ⟹ ≤\n (17R/6)·(tail of Σ m/(1+γ²) over |γ| > 2R−1).\n- **the tail bound** (the last new analytic input): Σ_{|γ|>X} m/(1+γ²)\n ≤ 16·A₀·log X/X for X ≥ 4·10³-ish — via the ℤ-window fiberwise\n decomposition (ENNReal.tsum_fiberwise is FREE in ℝ≥0∞ — verified\n exists) with Zeta23's `key` machinery (key_lt, le_key_add_one,\n one_add_sq_ge: 1+γ² ≥ (1+n²)/4 — all in Zeta23/Tail.lean +\n WeilEF/ZeroSummability.lean) + the window counts + the weight-series\n tail (weight(n) = log(|n|+3)/(1+n²) ≤ 2√2·|n|^{-3/2} via\n log y ≤ 2(√y−1) ⟸ Real.log_le_sub_one_of_pos at √y; p-series tail\n Σ_{n≥N} n^{-3/2} ≤ 3/√N — the p-series tail-bound API to pin down at\n build time).\n- **logDeriv ζ on the circle**: strip arcs (re ∈ [−5/2, 7/2], |t| ≥ 10\n for R ≥ 11): 5c/5d capstones (100000/200000·(log|t|)²) + the near\n sum ≤ 32Ĉ₀·(2·A₀·log(|t|+4)) [nearZeros ⊆ two Ncount windows] =\n O((log R)²); right near-axis (re ≥ √(R²−100) ≥ 2): the Dirichlet\n bound (5e-v-a ✔); left near-axis (re ≤ −√(R²−100)): 5d's\n logDeriv_zeta_fe pointwise + Dirichlet at 1−s + the ΓR brackets via\n 5e-ii (A)/(B) + the pointwise logDeriv ΓR identity (5e-ii's (G) for\n im ≠ 0; the REAL point s = −R via the closure/continuity extension —\n the bound set is closed and the im ≠ 0 arc points are dense).\n- **elems**: 1/s, 1/(s−1) = O(1/R); ψ(s/2)/2 = O(log R): right arc\n via 5e-ii (B) (re ≥ 2, |im| ≤ 5); strip arcs via 5d's\n norm_digamma_sub_log_le_of_two_le_abs_im (|im| ≥ 5 there) + ‖log w‖ ≤\n log‖w‖ + π (LiReflect:414); left arc via 5e-ii (A) with\n dist(s/2, −ℕ) ≥ 1/8 (from dist(R/2, ℤ) ≥ 1/4 and |re s/2| ∈\n [√(R²−100)/2, R/2] ⊆ [R/2 − 26/R, R/2], R ≥ 208).\n- **the squeeze**: G(z) := Gentire entire; for fixed z₀, R_k > |z₀|:\n dist(Gentire z₀, Gentire 0) ≤ (R₂(R_k)/R_k)·|z₀| with R₂(R) =\n C·(log R)³ + ‖Gentire 0‖ — Mathlib's Schwarz\n `Complex.dist_le_div_mul_dist_of_mapsTo_ball` (DifferentiableOn on\n the ball + MapsTo into closedBall (Gentire 0) R₂ — the mapsTo from\n the sphere bound via Zeta23's\n AnalyticOn.norm_le_of_norm_le_on_sphere) ⟹ dist ≤ 0 (the ratio → 0:\n (log R)³/R ≤ 64·R^{3/4}/R via Real.log_le_rpow-ish) ⟹ Gentire ≡\n Gentire 0 =: B. Capstone (D-068 §5):\n `partialFraction_identity (hZC : ZeroCountProp) : ∃ B : ℂ, ∀ s,\n riemannZeta s ≠ 0 → s ≠ 0 → s ≠ 1 → (∀ n ≥ 1, s ≠ −2n) →\n logDeriv riemannZeta s = B + seriesFun s − 1/s − 1/(s−1) +\n (Real.log Real.pi)/2 − Complex.digamma (s/2)/2` — from Gentire s =\n Fexpr s (5e-iv's gentire_eq_Fexpr at good points) + Gentire ≡ B.\n\n## Banked gotchas (session 030 additions)\n- `le_or_lt` does NOT exist (banked session 026) — use `le_or_gt`.\n- finsum API: `finsum_mem_union (hst : Disjoint) (hs : s.Finite)\n (ht : t.Finite)` (to_additive of finprod_mem_union — plain grep for\n \"finsum_mem_union\" misses it), `finsum_mem_empty`, and\n `tsum_eq_sum (s := s) (h : ∀ b ∉ s, f b = 0) : ∑' f = ∑ b ∈ s, f b`\n (NOT tsum_eq_sum_of_support_subset — that name does not exist).\n- `tsum_eq_sum`'s conclusion RHS keeps the SAME function: the if-form\n needs a closing Finset.sum_congr with if_pos.\n- **Bare `Nat.cast_nonneg N` as a term/have leaves the cast target a\n metavariable (IsOrderedRing stuck)** — always ascribe\n `(show (0:ℝ) ≤ (N:ℝ) from Nat.cast_nonneg N)` (recurred 3×).\n- **AnalyticAt.comp / Ncount_add_of_le HOU**: pin the implicit\n endpoints (a := ..) (b := ..) — by-tactic proof args do NOT determine\n them (the elaborator assigns garbage from linarith internals).\n- A `calc ... := by` block nested inside a case/induction tactic block\n at deep indentation BROKE THE PARSER (mysterious \"unexpected token\n '/--'; expected ':='\" downstream + \"unknown identifier linarith\") —\n restructure to avoid the nested-by calc in that position.\n- `Set.mem_union.mp` doesn't exist — union membership is defeq to Or:\n `rcases h with h1 | h2` directly.\n- positivity cannot see through `Classical.choose`-defined constants\n (A0count) — use mul_nonneg A0count_pos.le … explicitly.\n- `norm_num at hd` on `‖(1:ℂ) − 0‖ < 1/2` leaked a metavariable side\n goal — prefer the explicit `rw [show ‖..‖ = 1 by simp] at hd; linarith`.\n- Statement-level `if i ∈ s` needs decidability BEFORE `classical` can\n run: use a [DecidableEq ι] argument (the zero subtype inherits the\n noncomputable ℂ instance); `open Classical in` before a doc-commented\n lemma FAILED to parse in this toolchain.\n- `riemannZeta_conj` is AMBIGUOUS (root Mathlib vs Zeta23's s ≠ 1\n version) — use `_root_.riemannZeta_conj` (unconditional).\n\n## Status / next\n5e-v-a ✔ + 5e-v-b chunk 1 ✔ committed. Next session: 5e-v-b chunk 2 —\nthe tail bound (zero_tail_inv_sq_le via ENNReal.tsum_fiberwise +\nZeta23 key machinery + the weight-series tail) → the main seriesFun\ncircle bound (one tsum split + four finset shell bounds) → 5e-v-c (the\nlogDeriv assembly + elems + Fexpr circle bound) → 5e-v-d (Schwarz\nsqueeze + the ∃ B capstone). Then ii′-5f → ii′-3b → ii′-3c → ii′-6 →\nii′-7.\n"},"extra_content":null}]