[{"type":"thinking","thinking":"Now update STATE.md (header + the 5e-v progress paragraph + NEXT ACTION → 5e-v-d), NGUYEN.md (checklist line + session log), db, and commit.\n\nFor 5e-v-d I need to think about the squeeze details for the NEXT ACTION:\n- Gentire entire (gentire_analyticAt_and_eq : (∀ s, AnalyticAt ℂ Gentire s) ∧ (∀ s good, Gentire s = Fexpr s)).\n- On the circle |s| = R: every s is a GOOD point (not badPoint): badPoint s = nontrivial ∨ s = 0 ∨ s = 1 ∨ ∃ n ≥ 1, s = −2n: on the circle: not nontrivial (3a), ≠ 0 (‖s‖ = R > 0), ≠ 1 (‖s‖ = R > 1), ≠ −2n (re arbitrary: −2n real: ‖−2n‖ = 2n = R: excluded when R/2 ∉ ℤ?? wait −2n: ‖s‖ = R = 2n ⟹ R/2 = n ∈ ℤ — excluded by hgapInt! ✓). So Gentire s = Fexpr s on the circle ⟹ ‖Gentire s‖ ≤ the Fexpr bound on the sphere.\n- MapsTo: DifferentiableOn ℂ Gentire (ball 0 R) (from AnalyticAt ∀ s) + MapsTo (sphere 0 R) into closedBall (Gentire 0) (2·lowZeroConst + dirichletC2 + 400000·A₀·Ĉ₀(R+2)·(log R)² + ‖Gentire 0‖) — via the max-modulus: Zeta23's AnalyticOn.norm_le_of_norm_le_on_sphere (journal-030 mentions it).\n- Complex.dist_le_div_mul_dist_of_mapsTo_ball : dist (f z) (f 0) ≤ (R₂/R)·dist z 0·... the Schwarz lemma form: for f analytic on ball 0 R mapping into ball (f 0) R₂: dist (f z) (f 0) ≤ (R₂/R)·‖z‖·... need the exact Mathlib form: `Complex.dist_le_div_mul_dist_of_mapsTo_ball (hd : DifferentiableOn ℂ f (ball 0 R)) (hm : MapsTo f (ball 0 R) (ball (f 0) R₂)) (hz : ‖z‖ < R) : dist (f z) (f 0) ≤ R₂ / R * ‖z‖`? hmm — the Schwarz lemma variant: mapsTo into ball (f 0) R₂ (OPEN ball) — our bound is closedBall — bridge via ball ⊆ closedBall and R₂ < R₂' (take R₂' := R₂+1). \n- The ratio: R₂(R)/R = [2·lowZeroConst + dirichletC2 + 400000·A₀·Ĉ₀(R+2)·(log R)² + ‖Gentire 0‖]/R — need this → 0 as k → ∞ (R ≈ 4k): Ĉ₀(R+2) = 0.433·log(R+4) + 0.886·loglog(R+4) + 2.9 ≤ C·log R: so R₂(R) ≤ C'·(log R)³-ish: (log R)³/R → 0: need to FORMALIZE: for fixed z₀, pick k large enough that R > the crossover where R₂(R) < ε·R·... — actually the squeeze: dist (Gentire z₀) (Gentire 0) ≤ R₂(R_k)/R_k·|z₀| for EVERY k ≥ max(k₀, |z₀|): the RHS → 0: formalize via Tendsto: R₂(R_k)/R_k → 0 ⟹ dist = 0 (by le_of_tendsto or a direct Archimedean argument: dist ≥ 0 and dist ≤ ε for all ε > 0 ⟹ dist = 0). The cleanest: show ∀ ε > 0, ∃ k, R₂(R_k)/R_k·|z₀| < ε: R₂(R)/R ≤ C·(log R)³/R → 0: need (log R)³/R → 0 formalized — hmmm: (log R)³/R ≤ R^{3/4}·64/R·... : log R ≤ 4·R^{1/4} (Real.log_le_rpow or via exp: log x ≤ x^{1/4}·4 ⟸ log(x⁴) = 4 log x ≤ x hmm log y ≤ y: log x = (1/4)·log(x⁴)·... log(x⁴) = 4 log x: and log(x⁴) ≤ x⁴−1 < x⁴: so 4 log x < x⁴ ⟹ log x < x⁴/4 — that's log x ≤ x⁴/4 not what I want. For (log R)³/R → 0: log R ≤ 4·R^{1/4}/1 hmm: log R = 4·log(R^{1/4}) ≤ 4·(R^{1/4}−1) ≤ 4·R^{1/4}: so (log R)³ ≤ 64·R^{3/4}: R₂/R ≤ C·64·R^{3/4}/R = 64C·R^{−1/4} → 0. With R ≈ 4k: R^{−1/4} → 0: for ε: need R ≥ (64C/ε)⁴·|z₀|⁴-ish — choose k large. So the squeeze: dist (Gentire z₀) (Gentire 0) ≤ 64C·|z₀|·R_k^{−1/4} for all k ≥ k₀: and R_k ≥ 4k ⟹ → 0: conclude dist = 0 by `eq_of_forall_le_of_forall_lt...`? The cleanest formalization: dist ≤ 0 via: by_contra dist > 0: choose k with 4k > (64C·|z₀|/dist)⁴: then 64C·|z₀|·R^{−1/4} < dist ✗. Needs: R^{1/4} as Real.rpow or sqrt∘sqrt — use Real.sqrt: R^{1/4} = √(√R): (log R)³ ≤ 64·(√(√R))³·... hmm rpow is cleaner: Real.log_le_rpow (hx : 1 ≤ x) (hr : 0 < r) : log x ≤ r⁻¹·x^r?? There IS `Real.log_le_rpow (x : ℝ) (hx : 1 ≤ x) (r : ℝ) (hr : 0 < r) : Real.log x ≤ r⁻¹ * x ^ r`?? I recall `Real.log_le_rpow : 1 ≤ x → 0 < r → log x ≤ r⁻¹ * x^r`. Then log R ≤ 4·R^{1/4} (r = 1/4, r⁻¹ = 4 ✓): (log R)³ ≤ 64·(R^{1/4})³ = 64·R^{3/4}: (R^{1/4})³ = R^{3/4} (rpow_nat_inv_pow... rpow arithmetic). And Ĉ₀(R+2) ≤ 5·log R for R large (0.433·log(R+4)+0.886·loglog(R+4)+2.9: log(R+4) ≤ 2·log R (R+4 ≤ R² for R ≥ 2·... R+4 ≤ R² ⟺ R²−R−4 ≥ 0 ✓ R ≥ 3: log(R+4) ≤ 2 log R; loglog(R+4) ≤ log(2 log R) = log2 + loglog R ≤ 1 + log R·... loglog R ≤ log R: so Ĉ₀ ≤ 0.433·2L + 0.886·(1+L) + 2.9 ≤ 0.866L+0.886L+0.886+2.9 ≤ 1.8L+4 ≤ 5L (L ≥ 1). So R₂(R) ≤ 2·lowZeroConst + dirichletC2 + 400000·A₀·5L·L² + ‖Gentire 0‖ = C₀' + 2·10⁶·A₀·L³ (symbolic constants). R₂(R)/R ≤ (C₀' + 2·10⁶·A₀·L³)/R ≤ C₀'·... for the ratio → 0: (C₀' + C₁·L³)/R: split: C₀'/R ≤ C₀'/R and C₁·L³/R ≤ C₁·64·R^{3/4}/R = 64·C₁·R^{−1/4}: both → 0.\n\nThe capstone then: for z₀ any good point: Gentire z₀ = Gentire 0 (dist = 0) ⟹ Fexpr z₀ = Gentire 0 =: B (the constant!) — wait: Gentire ≡ Gentire 0 means Gentire is the CONSTANT function Gentire 0 — call it B: then at good s: logDeriv ζ s = Fexpr s = B ⟹ logDeriv ζ s = B + seriesFun s − 1/s − 1/(s−1) + (log π)/2 − ψ(s/2)/2 from Fexpr's definition ✓ the partialFraction_identity.\n\nFormal pieces for 5e-v-d:\n1. `gapCircle_good`: every s on the circle is a good point (¬badPoint s) — from the hgapInt + hsz + norms.\n2. The mapsTo: DifferentiableOn Gentire (ball 0 R) (from gentire_analyticAt_and_eq.1: AnalyticAt ∀ s ⟹ AnalyticOnNhd ⟹ DifferentiableOn on the open ball) + the sphere bound ⟹ MapsTo (ball 0 R) (closedBall (Gentire 0) R₂) via Zeta23's AnalyticOn.norm_le_of_norm_le_on_sphere (check its exact name/signature in Zeta23).\n3. The Schwarz application per z₀ (‖z₀‖ < R): dist (Gentire z₀) (Gentire 0) ≤ (R₂/R)·‖z₀‖·... check Complex.dist_le_div_mul_dist_of_mapsTo_ball's exact form.\n4. The growth bound: Ĉ₀(R+2) ≤ 5·log R + (log R)³/R-ish bounds with Real.log_le_rpow (check exists).\n5. The ratio → 0 ⟹ dist = 0 ⟹ Gentire z₀ = Gentire 0 for all good z₀ (need ‖z₀‖ < R for SOME gap circle R > |z₀|: exists_gapCircle_radius (hZC) at k large: R ∈ [4k+2.5, 4k+3] > ‖z₀‖ for k > ‖z₀‖/4·...).\n6. The capstone partialFraction_identity (hZC : ZeroCountProp).\n\nLet me write the STATE update now. Keep it focused.","signature":""},{"type":"text","text":"Now STATE.md updates (header, 5e-v progress, NEXT ACTION → 5e-v-d)."},{"type":"tool_use","id":"Edit_246","name":"Edit","input":{"replace_all":false,"file_path":"D:\\Python\\Quantum\\STATE.md","old_string":"# STATE.md — NGUYEN-RH-1 RESUME POINT\n# ⭐ READ THIS FIRST at the start of every session.\nLast updated: 2026-08-23 | Session 032 COMPLETE — **ii′-5e-v-b chunk 3\nCOMPLETE** (the series circle bound `seriesFun_norm_le_gapCircle`:\n‖seriesFun s‖ ≤ 2·lowZeroConst + 2900·A₀·Ĉ₀(4k+9/2)·(log R)² on the\nγ-gap circle, k ≥ 4000 — the threshold lemma + zero-free circle, the\ncount brick, the dyadic-family machine, five shell bounds + the far\ntail; LiCircleBound.lean now ~2867 lines, clean elaboration, zero\nwarnings, not yet root-imported; audit-session032-axioms.txt: 19\ndecls, 3 standard axioms only).","new_string":"# STATE.md — NGUYEN-RH-1 RESUME POINT\n# ⭐ READ THIS FIRST at the start of every session.\nLast updated: 2026-08-23 | Session 033 COMPLETE — **ii′-5e-v-c\nCOMPLETE** (the Fexpr circle bound `Fexpr_circle_le`: ‖Fexpr s‖ ≤\n2·lowZeroConst + dirichletC2 + 400000·A₀·Ĉ₀(R+2)·(log R)² on the\nγ-gap circle, k ≥ 4000 — three-region logDeriv assembly (Dirichlet /\n5c+5d capstones + near sum / FE + ΓR bracket incl. the real point\ns = −R via the new pole-avoidance (G) lemma), the uniform digamma\nbound, the six-norm chain; LiCircleBound.lean now ~3928 lines, clean\nelaboration, zero warnings, not yet root-imported;\naudit-session033-axioms.txt: 14 decls, 3 standard axioms only)."},"extra_content":null}]