[{"type":"thinking","thinking":"Good — `Ncount (T₁ T₂ : ℝ) : ℕ`, `IsNontrivialZero`, `zeroMult` all confirmed. Now let me update the todos and start the design + implementation.\n\nNow mark todo 1 complete, todo 2 (extract chain) complete (done via D-037/D-056/D-057 reads), todo 3 (design) in progress. Let me write down the design compactly, then implement.\n\n**DESIGN (12.1 + 12.2):**\n\nRefactor:\n1. `marginTilde_convex` + `continuousOn_marginExpanded` + `contDiffOn_marginExpanded` + `contDiffOn_mFblock`: change `hF : ContDiff ℝ 2 F` → `hF : ContDiffOn ℝ 2 F (Set.Ioi 1)`.\n   - contDiffOn_mFblock: `contDiffOn_const.mul (hF.comp contDiffOn_sqrt_shift mapsTo₁)` where mapsTo₁ : MapsTo (√·+1) (Ioi 20) (Ioi 1).\n   - vF: `chain_rule_sqrt_shift_on` needs Ioi(−1); add a generalized version with parameter c < 1. Actually simpler: prove `chain_rule_sqrt_shift_gt (φ) (hφ : ContDiffOn ℝ 2 φ (Set.Ioi c)) (hc : c < 1) ...` — generalize the existing one, update the ~9 call sites with `(c := -1) (hc := by norm_num)`. Hmm, wait — actually the existing chain_rule_sqrt_shift_on is used with blocks needing Ioi(−1). Let me generalize and fix call sites. The call sites: vTL1, vTL2, vId, vL1sq, vL1L2, vL2sq, vL1, vL2 (8 sites) each `chain_rule_sqrt_shift_on _ contDiffOn_X n hn0`. If I add c parameter with hc, each becomes `chain_rule_sqrt_shift_on _ contDiffOn_X (by norm_num) n hn0`. Mechanical.\n   - continuousOn_marginExpanded: `hF.continuous.continuousOn.comp hT (mapsTo_univ)` → `(hF.continuousOn).comp hT mapsTo₁ci` where mapsTo on Ici 20 → Ioi 1.\n   - marginTilde_convex signature: also keep hmaj (unused, documents intent; Ncount majorant — used by callers for the margin≥marginTilde comparison conceptually... hmm actually WAIT. Is the majorant property used ANYWHERE in the Lean assembly? marginTilde F = slower − Bn F + 1. The TRUE margin is slower − |S_f| + 1. SfBound gives |S_f| ≤ Bn F. So λ.re ≥ slower − Bn F + 1 = marginTilde F n DIRECTLY — no hmaj needed in the assembly either! hmaj (N ≤ F) is used only in the eventual SfBound proof (|Σ| ≤ 2N ≤ 2F inside Bn F's derivation). So hmaj is genuinely unused in ALL Lean proofs — it only documents what F is FOR. Hmm. Since marginTilde_convex already has it (compiles), leave it (no churn). Fine.\n\n2. Generalize `chain_rule_sqrt_shift_on` to parameter c.\n\nNew defs (after the margin section):\n3. `sInftyLower (n : ℝ) : ℝ := (1/2)*n*Real.log n + (lagariasC1 - Real.eulerMascheroniConstant/2)*n - 4`. Refactor sInfty_lower_bound statement to use it (statement change: `(1 / 2) * n * Real.log n + (lagariasC1 - ...)*n - 4 ≤ SInfty n` → `sInftyLower n ≤ SInfty n` — wait the binder is (n : ℕ) and the expression uses (n : ℝ) casts. sInftyLower takes ℝ; statement becomes `sInftyLower n ≤ SInfty n` with coercion. The proof's first lines establish the same expression — adjust via `show` or unfold. Mechanical.\n\n4. `FAlt (T : ℝ) : ℝ := (1/(2*Real.pi)) * T * Real.log T + 0.137 * Real.log T + 0.443 * Real.log (Real.log T) + (7/8 + 1.588)`.\n   Note: I'll write it in a form that makes the block decomposition obvious.\n\n5. `Bn (F : ℝ → ℝ) (n : ℝ) : ℝ := 2 * F (√n+1) + 4*(√n+1)*((2/3)*C0hat(√n+1) + 10.5*log(√n+1+2) + 48) + 40*1.3352 + 2*(exp 4 + 1)*(7*Hhat(√n+1) + 3.3) + IIMaj n + 1`.\n\n6. `marginTilde_eq_sub (F) (n : ℝ) : marginTilde F n = sInftyLower n - Bn F n + 1` — unfold + ring.\n\n7. `Sf (n : ℕ) : ℝ := SInfty n + 1 - (liCoeff n).re`.\n\n8. `SfBoundProp : Prop := ∀ n : ℕ, 80000 ≤ n → (n:ℝ) ≤ (3*10^12-1:ℝ)^2 → |Sf n| ≤ Bn FAlt n`.\n\n9. psiOf lemmas for the FAlt blocks:\n   - `psiOf_T_log_T`: ψ[T·log T] = 1/T − (log T + 1)/(T−1), < 0 for T > 1. Sign: ⟺ (T−1) < T(log T + 1) ⟸ −1 < T log T... for T > 1: T log T > 0 > −1 ✓. Proof: field_simp + the inequality (T−1) < T(log T+1) ⟺ −1 < T·log T; for T > 1, log T > 0 so T log T > 0 > −1 ✓.\n   - `psiOf_log`: ψ[log] = −1/T² − 1/(T(T−1)) < 0 for T > 1.\n   - `psiOf_log_log`: ψ[log log] = −(log T + 1)/(T log T)² − 1/(T log T (T−1)) < 0 for T > 1.\n   - `psiOf_const`: ψ[fun _ => K] = 0. (deriv const = 0; need deriv (deriv (fun _ => K)) T = 0 — deriv2_const_val exists! And deriv const = 0 (deriv_const). ψ = 0 − 0/(T−1) = 0.)\n   - `psiOf_add` (φ, ψ C²-ish at point): ψ[φ+ψ] = ψ[φ] + ψ[ψ] — needs deriv(deriv (φ+ψ)) = deriv deriv φ + deriv deriv ψ (deriv2_add) + deriv(φ+ψ) = deriv φ + deriv ψ (deriv_add) — field_simp assembly. Conditions: DifferentiableAt (deriv f) T etc. — for our blocks global differentiability on the relevant domain is available; state with explicit conditions like deriv2_add's.\n   - `psiOf_const_mul`: ψ[c·φ] = c·ψ[φ] (deriv2_const_mul + deriv_const_mul).\n   \n   Then: `psiOf_FAlt_neg : ∀ T ≥ √20+1, psiOf FAlt T < 0` — decompose FAlt = (1/2π)(T log T) + 0.137 log T + 0.443 log log T + (7/8+1.588) — need FAlt = that sum definitionally/by ring for deriv computation... Actually apply psiOf_const_mul/psiOf_add: ψ[FAlt] = (1/2π)ψ[TlogT] + 0.137 ψ[log] + 0.443 ψ[loglog] + ψ[const] = (1/2π)(neg) + 0.137(neg) + 0.443(neg) + 0 < 0. The decomposition: FAlt as written IS c₁·φ₁ + c₂·φ₂ + c₃·φ₃ + K — write FAlt := fun T => (1/(2π))*(T*log T) + 0.137*log T + 0.443*log(log T) + (7/8+1.588), then the psiOf computation via add/const_mul lemmas applied to literally that λ-expression. Doable — needs care with how the function is syntactically built (fun T => ... + ... + ... + ...). deriv lemmas operate on (fun T => c * φ T) etc. — the nested sums: ((c₁φ₁ + c₂φ₂) + c₃φ₃) + K. deriv2_add needs DifferentiableAt (deriv (left)) and (deriv (right)) at T and eventually-differentiable neighbors. For the blocks on T ≥ √20+1 > 1: T log T etc. all smooth on Ioi 0 / Ioi 1. The eventual-diff conditions: filter_upwards with Ioi 1 ∈ 𝓝 T (T > 1). This is the same pattern as marginTilde_convex's vF steps — I have the machinery patterns. It's maybe 150–250 lines.\n\n   Wait — do I even NEED psiOf additivity? Alternative: prove HasDerivAt FAlt (closed D1) and HasDerivAt D1 (closed D2) directly for the whole FAlt (like psiOf_T_mul_log_shift does for a block), then compute ψ[FAlt] in closed form and sign it directly:\n   ψ[FAlt](T) = (1/2π)[1/T − (log T + 1)/(T−1)] + 0.137[−1/T² − 1/(T(T−1))] + 0.443[−(log T+1)/(T log T)² − 1/(T log T (T−1))].\n   Sign: the 0.137 and 0.443 terms are negative for T > 1 ✓; the (1/2π) bracket: 1/T − (log T+1)/(T−1) < 0 ⟺ T log T > −1 ✓. So total < 0. The closed-form approach avoids general additivity lemmas but requires the full two-level HasDerivAt for FAlt — essentially the same work as blockwise + additivity. Blockwise + additivity is more modular and matches D-057's table style. I'll do blockwise psiOf lemmas for #1/#3/#4 + const, plus psiOf_add/psiOf_const_mul, then assemble. Actually hmm, psiOf_add needs deriv-level additivity conditions — deriv2_add already exists with the right conditions. psiOf_add: psiOf (fun x => f x + g x) T = psiOf f T + psiOf g T given (DifferentiableAt (deriv f) T), (DifferentiableAt (deriv g) T), (∀ᶠ m, DifferentiableAt f m), (∀ᶠ m, DifferentiableAt g m). Body: unfold psiOf; rw [deriv2_add, deriv_add (need DifferentiableAt f T, g T — from the ∀ᶠ self_of_nhds)]; ring. ✓ psiOf_const_mul similar with deriv2_const_mul + deriv_const_mul (needs DifferentiableAt f T). ✓\n\n10. hF for FAlt: ContDiffOn ℝ 2 FAlt (Set.Ioi 1): pieces: (contDiffOn_id.mul (contDiffOn_log? ...)). Mathlib: ContDiffOn.log — used in the file already: `(contDiffOn_id.log (fun x hx => ne_of_gt ...))`. For T log T: contDiffOn_id.mul (contDiffOn_id.log ...) on Ioi 1 ✓. log log: (contDiffOn_id.log ...).log (fun x hx => log x > 0 for x > 1 ✓ ne_of_gt (Real.log_pos ...)). Const mul/add: standard. ✓ ~15 lines.\n\n11. Base points (12.3): `marginTilde_FAlt_base : 0 < marginTilde FAlt 80000` and `0 < deriv (marginTilde FAlt) 80000`. margiñ(80000) ≈ 16020 (E-028/E-029): lower-bound each piece with crude rationals: need sInftyLower(80000) = (1/2)(80000)(log 80000) + (C₁−γ/2)(80000) − 4. log 80000 = log 8 + log 10⁴ = 3 log 2 + 4 log 10 ≈ 11.2898. (1/2)(80000)(11.2898) = 451 592. (C₁−γ/2) = −1.41894; ×80000 = −113 515. slower ≈ 338 073. Bn: 2F(T)+... with T = √80000+1 ≈ 283.84: F(T) ≈ (1/2π)(283.84)(5.6485) + ... ≈ 255.4 + 0.137·5.6485 + 0.443·1.7330 + 2.463 ≈ 258.9. 2F ≈ 518. 4T[(2/3)Ĉ₀ + 10.5 log(T+2) + 48]: Ĉ₀(283.84) = 0.433·log(285.84) + 0.886·log log(285.84) + 2.9 = 0.433·5.6555 + 0.886·1.7342 + 2.9 = 2.449 + 1.537 + 2.9 = 6.886. (2/3)(6.886) = 4.59; 10.5·log(285.84) = 59.38; bracket = 111.97. 4T = 1135.4 → 127 127?? Hmm that's big: 1135.4 × 111.97 ≈ 127 130. Plus 2(e⁴+1)(7Ĥ+3.3): Ĥ = 3Ĉ₀² + 10.5 log + 48 = 3·47.42 + 59.38 + 48 = 249.6. 7Ĥ + 3.3 = 1750.5. 2(e⁴+1) = 2·55.598 = 111.2. → 194 650. IIMaj tiny. Bn ≈ 518 + 127 130 + 53.4 + 194 650 + 1 ≈ 322 350. margin ≈ 338 073 − 322 350 + 1 ≈ 15 724 ✓ ≈ E-029's 16 020 (modulo my sloppy arithmetic — F here is F_alt not exact Trudgian; anyway positive with huge slack). Proving 0 < marginTilde FAlt 80000 in Lean: need lower bounds on log 80000, upper bounds on log 285.84, log log, e⁴, √80000 (upper), etc. All with generous slack (margin ~15 000, our errors can be ~1 000). This is a decent chunk of norm_num-style work: log 80000 = log(8×10⁴) = 3log2 + 4log10... need log 10 bounds: Mathlib has log 2 bounds (Real.log_two_gt_d9/lt_d9); log 10 = log 2 + log 5; log 5? Mathlib: Real.log five? There are d9 lemmas for log 2 only, I think. Can derive log 10 from log 2 + log 5, log 5 = log(10/2)... circular. log 10 via log(8) + log(1.25): log 10 = 3 log 2 + log 1.25; log 1.25 via series or bounds: log(5/4) ≤ 5/4 − 1 = 0.25 (log_le_sub_one) and ≥ ... Real.log_le_sub_one_of_pos gives upper; lower via −log(1−x)... Real.log_le_sub_one_of_pos (x := 4/5): log(4/5) ≤ −1/5 ⟹ log(5/4) ≥ 1/5. So 0.2 ≤ log 1.25 ≤ 0.25. log 10 ∈ [3·0.6931472 + 0.2, ...] ≈ [2.2794, 2.3294] — crude but FINE given our slack. √80000: upper bound via sqrt_le: √80000 ≤ 283? 283² = 80089 ≥ 80000 ✓ so √80000 ≤ 283 (need Real.sqrt_le' or le: Real.sqrt_le_sqrt/Real.sqrt_le' : √x ≤ y ⟺ x ≤ y² for y ≥ 0... `Real.sqrt_le' (hy : 0 < y) : √x ≤ y ↔ x ≤ y ^ 2`? something like that). T = √80000+1 ≤ 284. log(T+2) ≤ log 286: 286 ≤ e^5.66? e^5.66 ≥ ...: e^5 = 148.4, e^0.66 ≈ 1.935, → 287.1 ≥ 286 ✓ so log 286 ≤ 5.66. Provable: 286 ≤ exp(5.66) via sum bounds like iimaj_bracket_bound's approach (Real.sum_le_exp_of_nonneg). log log(286): log 5.66 ≤ ... e^1.74 = 5.70 ≥ 5.66 ✓ so ≤ 1.74. All doable with the patterns already in the file (Real.sum_le_exp_of_nonneg trick, Real.exp_one_lt_d9, etc.).\n\n    The deriv base point: margiñ′(80000) = 3.7. Need deriv (marginTilde FAlt) 80000 in closed form — deriv of each piece via the first-derivative HasDerivAt machinery (hasDerivAt_marginS gives S′ = (1/2)(log n + 1) + c; chain rule first-derivs for the T-blocks: d/dn φ(√n+1) = φ′(T)/(2√n) — available from the hg1 eventually-equal steps inside chain_rule_sqrt_shift_on... I'd need standalone first-derivative lemmas: deriv (fun n => φ(√n+1)) n = deriv φ (√n+1) / (2√n) — deriv_comp with Real.hasDerivAt_sqrt. Fine. And deriv IIMaj via psiTilde. The value: pieces ≈ S′ = (1/2)(12.2898) − 1.41894 = 4.726; minus B̃′ pieces ≈ 1.03 total. Sign window: need accuracy ~1. Crude bounds fine.\n\n    12.3 is maybe 200–400 lines of numeric grind. Might spill to next session — that's OK per sub-stage rule.\n\n12. Assembly theorem:\n```\ntheorem lambda_pos_finite_height\n    (hRH : RHUpTo (3 * 10 ^ 12)) (hfinite : LiVerifiedUpTo (5 * 10 ^ 5))\n    (hSf : SfBoundProp) :\n    ∀ n : ℕ, 1 ≤ n → (n : ℝ) ≤ (3 * 10 ^ 12 - 1 : ℝ) ^ 2 → 0 < (liCoeff n).re\n```\nproof: intro n hn1 hnN; rcases lt_or_ge n 80000 → \n  - n < 80000: hfinite n hn1 (n ≤ 79999 < 500000 via omega/Nat.cast) ✓\n  - 80000 ≤ n: \n    have h20 : (20:ℝ) ≤ n\n    have h1 := sInfty_lower_bound n (20 ≤ n as ℕ)\n    have h2 := hSf n h80000 hnN  : |Sf n| ≤ Bn FAlt n\n    have h3 : (liCoeff n).re = SInfty n - Sf n + 1 := by unfold Sf; ring\n    have h4 : sInftyLower n - Bn FAlt n + 1 ≤ (liCoeff n).re: from h1 (sInftyLower ≤ SInfty), h2 (Sf ≤ |Sf| ≤ Bn ⟹ −Bn ≤ −Sf), h3. linarith with abs: need Sf n ≤ |Sf n| (le_abs_self). \n    have h5 : marginTilde FAlt n = sInftyLower n - Bn FAlt n + 1 := marginTilde_eq_sub ...\n    have h6 : 0 < marginTilde FAlt n := pos_of_convex_base (marginTilde_convex FAlt hF hmaj?? hψ) ... \n\n    WAIT — marginTilde_convex needs hmaj : ∀ T, (Ncount 0 T : ℝ) ≤ F T. If I keep hmaj in marginTilde_convex's signature, the assembly must provide it → the Trudgian hypothesis enters the ASSEMBLY theorem as a hypothesis (making it 4 hypotheses: hRH, hfinite, hNTrudgian, hSf). Hmm! Since hmaj is UNUSED in the proof, better: REMOVE hmaj from marginTilde_convex (it's genuinely not part of the convexity statement). It's a small signature change — the theorem is about convexity of marginTilde F, full stop. The majorant property belongs to SfBound's eventual proof, not here. Removing = cleaner + honest. But churn: marginTilde_convex is referenced in journal/STATE as proved; signature change is fine (still proved). Also remove from convex-pos application. I'll remove hmaj from marginTilde_convex. (Check: any other file uses marginTilde_convex? Only LiSkeleton. PrintAxioms.lean references? Check later.)\n\n    Then: pos_of_convex_base hconv hfc hfd 80000 n (...) hbase hbase′ — with hbase/hbase′ PROVED (12.3) or as interim hypotheses.\n\nInterim structure for THIS session: I'll do the full thing in stages:\n- Stage A (today's first compile): everything except 12.3 base points → assembly theorem with hbase/hbase′ as explicit hypotheses:\n  Actually no — let me put hbase/hbase′ as hypotheses of an aux lemma, and state the MAIN theorem with them as hypotheses too until 12.3 discharges them. Hmm, the notes want the endpoint \"modulo SfBound + perimeter\". The base points are provable-in-Mathlib numerics — I'd rather not leave them as hypotheses in the final statement (they're not literature facts like Platt–Trudgian; they're elementary decidable inequalities). But for a first compiling milestone, hypotheses are fine, then discharge.\n\nPlan the file edits:\n1. Refactor chain_rule_sqrt_shift_on → parameterized c version (or add chain_rule_sqrt_shift_Ioi1). Decide: generalize with (c : ℝ) (hc : c < 1). Update 8 call sites.\n2. Refactor contDiffOn_mFblock, continuousOn_marginExpanded, contDiffOn_marginExpanded, marginTilde_convex: ContDiff ℝ 2 F → ContDiffOn ℝ 2 F (Set.Ioi 1). Add mapsTo_sqrt_shift_Ioi1 (MapsTo (√·+1) (Ioi 20) (Ioi 1)) and Ici version.\n3. Remove hmaj from marginTilde_convex. (Check Ncount import still used elsewhere — no, but `open Zeta23` stays.)\n4. Add sInftyLower def; refactor sInfty_lower_bound to conclude `sInftyLower n ≤ SInfty n` (proof: first line `show` the unfolded form? The current statement is the explicit expression; change statement to sInftyLower n and add `unfold sInftyLower` at proof start — wait the proof immediately does `have hmain : ∀ K, expr ≤ ... → expr ≤ SInfty n` where expr is the explicit form; simplest: keep statement explicit? For the assembly I need sInftyLower ≤ SInfty. If sInfty_lower_bound keeps explicit statement, the assembly can still use it directly (the explicit expression IS sInftyLower n by definition — `sInftyLower n` unfolds to it). Cleaner: change the theorem statement to `sInftyLower n ≤ SInfty n` and start proof with `unfold sInftyLower` — but `unfold` at the goal with the casted n... `show (1/2)*(n:ℝ)*... ≤ SInfty n` — the sInftyLower def on ℝ; sInftyLower (n : ℝ)... define sInftyLower (n : ℝ). Then statement `sInftyLower (n : ℝ) ≤ SInfty n`, proof starts `show (1/2)*(n:ℝ)*Real.log n + (lagariasC1 - Real.eulerMascheroniConstant/2)*n - 4 ≤ _` — needs the unfold to be definitional rfl: sInftyLower (n:ℝ) ≡ (1/2)*n*log n + ... — yes by rfl (with casts: (n : ℝ) inside — the def will have (n : ℝ) so sInftyLower n where n : ℕ needs ↑(n)... in the theorem `sInftyLower n` with n : ℕ and sInftyLower : ℝ → ℝ gives sInftyLower (↑n) ≡ (1/2)*↑n*log ↑n + ... The explicit statement has (1/2) * n * Real.log n with n : ℕ elaborated to ℝ — same ↑n. Should be rfl. Use `show` with the unfolded expression — if not syntactically rfl, tweak.)\n5. Add FAlt, Bn, Sf, SfBoundProp defs; marginTilde_eq_sub (ring).\n6. psiOf blocks: psiOf_T_log_T, psiOf_log, psiOf_log_log (formula + neg), psiOf_const_val, psiOf_add, psiOf_const_mul, psiOf_FAlt_neg.\n7. contDiffOn_FAlt.\n8. pos_of_convex_base lemma.\n9. lambda_pos_finite_height_aux (with hbase/hbase′ hypotheses) → then after 12.3, main theorem without them.\n10. Compile, PrintAxioms.\n\nBase points 12.3: two numeric theorems. This session if time permits.\n\nAlso: check whether `deriv (marginTilde FAlt) 80000` — the deriv_base needs the first-derivative closed forms. Defer to 12.3.\n\nLet me now check how PrintAxioms.lean looks and whether it references marginTilde_convex (signature change impact).","signature":""},{"type":"text","text":"All types confirmed (`Ncount T₁ T₂ : ℕ`, `IsNontrivialZero`, `zeroMult`). The vendored repo is at repo root. Checking PrintAxioms for signature dependencies before I refactor."},{"type":"tool_use","id":"Read_23","name":"Read","input":{"file_path":"D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\PrintAxioms.lean"},"extra_content":null}]