[{"type":"thinking","thinking":"All three key lemmas verified:\n\n1. **`Complex.dist_le_div_mul_dist_of_mapsTo_ball`** (Mathlib Schwarz.lean:190):\n```lean\ntheorem dist_le_div_mul_dist_of_mapsTo_ball (hd : DifferentiableOn ℂ f (ball c R₁))\n    (h_maps : MapsTo f (ball c R₁) (closedBall (f c) R₂)) (hz : z ∈ ball c R₁) :\n    dist (f z) (f c) ≤ R₂ / R₁ * dist z c\n```\n\n2. **`AnalyticOn.norm_le_of_norm_le_on_sphere`** (Zeta23 StrongPNTPrefix.lean:33):\n```lean\nlemma AnalyticOn.norm_le_of_norm_le_on_sphere {C r R : ℝ} {f : ℂ → ℂ} {w : ℂ}\n    (hyp_r : r ≤ R)\n    (analytic : AnalyticOn ℂ f (Metric.closedBall 0 R))\n    (cond : ∀ z ∈ Metric.sphere 0 r, ‖f z‖ ≤ C)\n    (wInS : w ∈ Metric.closedBall 0 r) :\n    ‖f w‖ ≤ C\n```\n\n3. **`Real.log_le_rpow_div {x ε : ℝ} (hx : 0 ≤ x) (hε : 0 < ε) : log x ≤ x ^ ε / ε`** — with ε = 1/4: log x ≤ 4·x^{1/4}.\n\nNow the plan for 5e-v-d. Design (STATE.md §NEXT ACTION):\n\n**Step 1** `gentire_circle_le`: for k ≥ 4000, R gap circle, ‖s‖ = R ⟹ ‖Gentire s‖ ≤ RB R where RB R := 2·lowZeroConst + dirichletC2 + 400000·A0count·C0hat (R+2)·(log R)². \n\nTo get this: Gentire s = Fexpr s needs ¬ badPoint s:\n- ¬ IsNontrivialZero s: from `not_isNontrivialZero_on_gapCircle hk hRlo hgap hs` (already used in Fexpr_circle_le).\n- s ≠ 0: ‖s‖ = R ≥ 16002 > 0.\n- s ≠ 1: ‖s‖ = R ≠ ‖1‖ = 1.\n- ¬ ∃ n ≥ 1, s = −2n: if s = −2·(n:ℂ), then s.re = −2n, so R/2 = ‖s‖/2 = |re s|... hmm wait. ‖s‖ = R and s = −2n gives R = ‖−2n‖ = 2n, so R/2 = n ∈ ℤ, contradicting hgapInt with m = n: |R/2 − n| = 0 < 1/4. Good — need hgapInt here.\n\n**Step 2** MapsTo: Gentire differentiable on ball 0 R. From gentire_analyticAt_and_eq.1: ∀ s, AnalyticAt ℂ Gentire s. So DifferentiableOn ℂ Gentire (ball 0 R) — via fun s hs => (gentire_analyticAt_and_eq.1 s).differentiableAt.differentiableWithinAt. Actually AnalyticAt.differentiableAt : AnalyticAt ℂ f s → DifferentiableAt ℂ f s. Then DifferentiableOn.\n\nThen: for w ∈ ball 0 R, ‖w‖ < R. Use the sphere lemma with r := ‖w‖... hmm wait. Actually the standard approach: we want MapsTo Gentire (ball 0 R) (closedBall (Gentire 0) (R₂+1)) where R₂ := RB R + ‖Gentire 0‖.\n\nWait, actually the sphere lemma gives ‖Gentire w‖ ≤ C for w ∈ closedBall 0 r where the bound C holds on sphere 0 r. For w with ‖w‖ < R: take r := max ‖w‖ something... hmm, simpler: for each w ∈ ball 0 R, choose r := (‖w‖ + R)/2 ∈ (‖w‖, R)? But then the circle bound on sphere 0 r requires r to be a GAP circle radius — which it isn't in general! The bound ‖Fexpr‖ ≤ RB only holds on the special gap circles.\n\nHmm wait. Let me re-think. The circle bound holds only on gap circles R_k. But R_k ∈ [4k+5/2, 4k+3] for each k. For a FIXED z₀, we pick k large enough that 4k+5/2 > ‖z₀‖, i.e., z₀ ∈ ball 0 R_k, and then:\n\n- Gentire analytic on closedBall 0 R_k (entire).\n- On sphere 0 R_k: ‖Gentire‖ ≤ RB R_k (step 1, using the gap-circle properties of R_k from exists_gapCircle_radius).\n- By sphere lemma (max modulus): ‖Gentire z₀‖ ≤ RB R_k as well.\n\nSo actually we don't need MapsTo on every ball — only on the gap-circle balls. Then:\n\ndist (Gentire z₀) (Gentire 0) ≤ ‖Gentire z₀‖ + ‖Gentire 0‖ ≤ RB R_k + ‖Gentire 0‖.\n\nThat's the naive triangle bound, which doesn't give → 0. We need the Schwarz lemma: apply Schwarz to f = Gentire on ball 0 R_k mapping into closedBall (Gentire 0) (RB R_k + ‖Gentire 0‖):\n\nMapsTo Gentire (ball 0 R_k) (closedBall (Gentire 0) (RB R_k + ‖Gentire 0‖)): for z ∈ ball 0 R_k, ‖Gentire z‖ ≤ RB R_k (max modulus on ball from sphere bound — using AnalyticOn.norm_le_of_norm_le_on_sphere with r = R_k, R = R_k, w ∈ closedBall 0 R_k) and dist (Gentire z) (Gentire 0) ≤ ‖Gentire z‖ + ‖Gentire 0‖ ≤ RB R_k + ‖Gentire 0‖. ✓.\n\nThen Schwarz: dist (Gentire z₀) (Gentire 0) ≤ (RB R_k + ‖Gentire 0‖)/R_k · dist z₀ 0 = (RB R_k + ‖Gentire 0‖)/R_k · ‖z₀‖.\n\n**Step 4**: (RB R_k + ‖Gentire 0‖)/R_k → 0 as k → ∞. RB R = 2·lowZeroConst + dirichletC2 + 400000·A₀·Ĉ₀(R+2)·(log R)².\n\nNeed: Ĉ₀(R+2) growth bound. What is C0hat? Let me check its definition. From STATE.md: \"Ĉ₀(R+2) ≤ 5·log R (log(R+4) ≤ 2·log R for R ≥ 3; loglog ≤ 1+log; L ≥ 9)\". Let me check the actual definition of C0hat.\n\nThen (RB R_k + ‖Gentire 0‖)/R_k ≤ (C + C'·(log R_k)³)/R_k with (log R)³ ≤ 64·R^{3/4} via Real.log_le_rpow_div with ε=1/4: log R ≤ R^{1/4}/(1/4) = 4·R^{1/4}, so (log R)³ ≤ 64 R^{3/4}. And R^{3/4}/R = R^{-1/4} → 0... but we don't need a limit — we do the direct argument: for fixed z₀ and ε > 0, choose k so large that (C + 64·C'·R_k^{3/4})/R_k < ε/‖z₀‖... Actually the cleanest: show dist (Gentire z₀) (Gentire 0) < ε for all ε > 0, hence = 0 (dist_eq_zero / eq_of_dist_eq_zero... the lemma: dist_lt_of... we use `dist_eq_zero` after showing ∀ ε>0, dist < ε — there's a lemma like `dist_le_zero`... The pattern: by_contra h; set ε := dist/2 > 0; get dist < ε contradiction. Or `eq_of_forall_dist_le` hmm. There's `dist_eq_zero : dist x y = 0 ↔ x = y`. And to show dist = 0: le_antisymm (le_of_forall_pos_le_add or le_of_forall_lt...) nonneg. There's the standard `le_of_forall_pos_le_add : (∀ ε > 0, a ≤ b + ε) → a ≤ b`? Hmm, or `le_of_forall_lt'`... Let me think: we show ∀ ε > 0, dist (Gentire z₀) (Gentire 0) ≤ ε. Then by `le_of_forall_le_of...`? Standard: `le_of_forall_pos_le_add` gives dist ≤ 0 from dist ≤ 0 + ε. Actually there's `le_of_forall_lt_imp_le_of_dense`... The cleanest known forms: \n- `le_of_forall_pos_le_add : (∀ ε : α, 0 < ε → a ≤ b + ε) → a ≤ b` (exists in ordered fields).\nSo show ∀ ε > 0, dist ≤ 0 + ε, i.e., dist ≤ ε.\n\nFor that: given ε > 0, we need k with (RB R_k + ‖Gentire 0‖)·‖z₀‖/R_k ≤ ε.\n\nBound: (RB R + G)/R ≤ (C₀' + C₁·(log R)³)/R where C₀' = 2·lowZeroConst + dirichletC2 + ‖Gentire 0‖ + ... wait plus we need Ĉ₀(R+2) ≤ 5 log R. Let me check C0hat's definition first.\n\nThen (C₀' + C₁·64·R^{3/4})/R = C₀'/R + 64·C₁·R^{-1/4}. To make this ≤ ε: choose R ≥ max((2C₀'/ε), (128·C₁/ε)⁴)... then C₀'/R ≤ ε/2 and 64 C₁ R^{-1/4} ≤ 64 C₁/(128 C₁/ε) = ε/2. ✓ (needs R^{-1/4} manipulation via rpow; R ≥ (128 C₁/ε)⁴ ⟹ R^{1/4} ≥ 128 C₁/ε ⟹ R^{-1/4} ≤ ε/(128 C₁). rpow arithmetic: (R^{1/4})... Let me handle with Real.rpow: R^{3/4}/R = R^{3/4 − 1} = R^{-1/4}. Hmm, dealing with negative rpow in Lean: Real.rpow_neg etc. Alternative cleaner: avoid rpow-negatives by writing: want (C₀' + 64·C₁·R^{3/4}) ≤ ε·R. Since R^{3/4} ≤ R... hmm.\n\nAlternative: use log R ≤ R^{1/4}·4 and demand: C₀' ≤ εR/2 AND 64·C₁·R^{3/4} ≤ εR/2. The second: 128·C₁·R^{3/4} ≤ ε·R ⟺ 128·C₁/ε ≤ R^{1/4} ⟺ (128 C₁/ε)⁴ ≤ R. With rpow: R^{3/4} = R^{(3/4:ℝ)} via rpow. Need (R⁴)^{1/4} = R for R ≥ 0: Real.rpow_nat_inv_pow... Actually `Real.rpow_natCast`, `Real.rpow_mul`... There's `Real.rpow_nat_inv_pow_nat (hx : 0 ≤ x) : (x ^ n) ^ ((n:ℝ)⁻¹) = x`? Something like Real.rpow_inv_natCast_pow? Let me just recall: `Real.rpow_natCast`, and the key identity: `Real.rpow_nat_inv_pow_nat {x : ℝ} (hx : 0 ≤ x) {n : ℕ} (hn : n ≠ 0)? : (x ^ n) ^ (n⁻¹ : ℝ) = x`. I believe there's `Real.rpow_inv_natCast_pow (hx : 0 ≤ x) (hn : n ≠ 0) : (x ^ (n:ℝ)) ^ (n:ℝ)⁻¹ = x`? I'll check in Mathlib when building. Alternatively avoid fourth roots: (log R)³ ≤ 64·R^{3/4} and R^{3/4}/R: note R^{3/4} ≤ R is wrong direction... \n\nCleaner approach avoiding rpow division gymnastics: log R ≤ 4·R^{1/4} gives log R ≤ 4·R^{1/4}. Cube: (log R)³ ≤ 64·(R^{1/4})³ = 64·R^{3/4}. Now I want: 64·C₁·(log R)³·.../R ≤ ε/2. Write R = (R^{1/4})⁴: then R^{3/4}/R = 1/R^{1/4}. So need 64·C₁/R^{1/4} ≤ ε/2, i.e., R^{1/4} ≥ 128·C₁/ε, i.e., R ≥ (128·C₁/ε)⁴.\n\nIn Lean: set Q := R^{(1/4 : ℝ)} (rpow). Then Q⁴ = R? `Real.rpow_nat_inv_pow_nat`: hmm. Let me instead set the choice directly in terms of a real M := max(‖z₀‖+1, (2C₀'/ε), (128 C₁/ε)⁴, 16002...) and pick k with 4k+5/2 ≥ M (exists by Archimedean: ∃ k:ℕ, M ≤ 4k — exists_nat_ge style). Then R_k ≥ 4k+5/2 ≥ M.\n\nThen:\n- C₀'/R ≤ C₀'/M ≤ ε/2 if M ≥ 2C₀'/ε (need C₀' ≥ 0 for the division monotonicity).\n- For the rpow part: R ≥ (128C₁/ε)⁴ ⟹ R^{1/4} ≥ ((128C₁/ε)⁴)^{1/4} = 128C₁/ε (using rpow monotonicity Real.rpow_le_rpow with 0 ≤ base and exponent 1/4 ≥ 0, and the identity (x⁴)^{1/4} = x). Then 64·C₁·R^{3/4}/R = 64·C₁·(R^{1/4})³/(R^{1/4})⁴ = 64·C₁/R^{1/4} ≤ 64·C₁·ε/(128·C₁) = ε/2. To do this cleanly in Lean I need R = (R^{1/4})⁴: `Real.rpow_nat_inv_pow_nat`? Let me search Mathlib for the exact lemma name later. Options: `Real.rpow_mul (hx : 0 ≤ x) (y z : ℝ) : x ^ (y * z) = (x ^ y) ^ z`. So R = R^1 = R^{(1/4)·4} = (R^{1/4})⁴ = ((R^{1/4}))^((4:ℕ):ℝ) = (R^{1/4})^4 via rpow_natCast. That works: R^(1/4 * 4) = R^1 = R. So set Q := R^(1/4:ℝ); R = Q^4. Then (log R)³ ≤ 64·Q³. And (C₀' + 64·C₁·Q³·... hmm wait I need to redo: RB R + ‖Gentire 0‖ ≤ C₀' + C₁·(log R)³ where C₁ = 400000·A₀·5 = 2·10⁶·A₀ (if Ĉ₀(R+2) ≤ 5 log R). Then numerator ≤ C₀' + C₁·64·Q³. Then numerator/R ≤ ε: suffices C₀' ≤ ε/2·Q⁴/... hmm — I set it up as: C₀'/R ≤ ε/2 and 64 C₁ Q³/Q⁴ ≤ ε/2.\n\nOK this is all standard real-number grinding. The nlinarith/linear tools won't handle rpow, so I'll convert: once Q ≥ max(128·C₁/ε, (2C₀'/ε)^{1/4}...)... Actually simpler to put both conditions in terms of Q: want C₀' ≤ (ε/2)·Q⁴ and 64·C₁ ≤ (ε/2)·Q. Two conditions: Q ≥ 128·C₁/ε and Q⁴ ≥ 2·C₀'/ε. Take Q ≥ max(128C₁/ε, (2C₀'/ε)^{1/4})... ugh, fourth root of C₀' — instead: Q ≥ max(128 C₁/ε, 2C₀'/ε) hmm no: if Q ≥ 2C₀'/ε and Q ≥ 1 then Q⁴ ≥ Q ≥ 2C₀'/ε ✓. So: Q ≥ max(1, 128·C₁/ε, 2·C₀'/ε). Then:\n- 64·C₁ ≤ (ε/2)·Q ⟹ 64·C₁·Q³ ≤ (ε/2)·Q⁴ ✓ (multiply by Q³ ≥ 0).\n- C₀' ≤ (ε/2)·Q ≤ (ε/2)·Q⁴ ✓ (Q ≥ 1 ⟹ Q ≤ Q⁴).\nSum: C₀' + 64·C₁·Q³ ≤ ε·Q⁴ = ε·R ✓.\n\nSo the condition is Q := R^{1/4} ≥ max(1, (128·C₁ + 2·C₀')/ε)-ish. And R = Q⁴ ≥ ((...)/ε)⁴. So choose M := max(‖z₀‖ + 1, 16002, ((128·C₁ + 2·C₀' + 1)/ε)⁴) — then R ≥ M gives Q = R^{1/4} ≥ M^{1/4} ≥ (128 C₁ + 2 C₀' + 1)/ε via rpow monotonicity + (x⁴)^{1/4} = x.\n\nThe rpow identity needed: (M^{1/4})⁴... I need M^{1/4} ≥ c where M ≥ c⁴: Real.rpow_le_rpow (0 ≤ c⁴... base monotonicity: c⁴ ≤ M, exponent 1/4 ≥ 0 ⟹ (c⁴)^{1/4} ≤ M^{1/4}; and (c⁴)^{1/4} = c for c ≥ 0: via Real.rpow_nat_inv_pow_nat or rpow_mul: (c⁴)^{1/4} = c^{4·(1/4)} = c^1 = c. rpow_mul needs 0 ≤ c: c^(4*1/4) = (c^4)^(1/4)... the direction: `Real.rpow_mul (hx : 0 ≤ x) (y z : ℝ) : x ^ (y * z) = (x ^ y) ^ z`. So (c^4)^((1/4):ℝ) hmm: c^((4:ℝ)·(1/4)) = (c^(4:ℝ))^(1/4). And c^(4:ℝ) = c^4 (rpow_natCast). 4·(1/4) = 1, c^1 = c (rpow_one). So (c^4)^(1/4:ℝ) = c after rewriting 4*(1/4)=1 — but rw inside exponent: `show c ^ (((4:ℝ) * (4:ℝ)⁻¹)) = ...`; the arithmetic 4 * (1/4) = 1 needs norm_num inside the exponent position — doable via `congr` or `rw [show (4:ℝ)*(1/4) = 1 by norm_num]`? The term is `c ^ ((4:ℝ) * (1/4))` — `rw [mul_inv_cancel₀]`? hmm (1/4) is 4⁻¹ defeq? (1/4 : ℝ) = 4⁻¹ is defeq (OneDiv). Might be easier to use exponent (0.25) no... Let me just use `Real.rpow_nat_inv_pow_nat`: searching memory — I believe Mathlib has `Real.rpow_nat_inv_pow_nat {x : ℝ} (hx : 0 ≤ x) {n : ℕ} (hn : n ≠ 0)? : (x ^ n) ^ ((n : ℝ)⁻¹) = x`? There's definitely `Real.rpow_inv_natCast_pow`? I'll grep Mathlib when writing the code. And (1/4 : ℝ) = (4:ℝ)⁻¹ by norm_num1 or `one_div`. Fine — details at build time.\n\nAlso need: (log R)³ ≤ 64·R^{3/4}: from log R ≤ R^{1/4}/(1/4) = 4·Q. Cube both sides (log R ≥ 9 ≥ 0, so pow monotone): (log R)³ ≤ (4Q)³ = 64 Q³. And R^{3/4} = Q³: (Q^4)^{3/4} = Q^3 — same rpow juggling: R^{(3/4:ℝ)} = (Q^4)^{3/4}... Alternatively skip R^{3/4} entirely: work wholly in Q: numerator bound C₀' + 64·C₁·Q³, denominator R = Q⁴. Never mention R^{3/4}. \n\nNow — the Ĉ₀(R+2) ≤ 5·log R bound. Need C0hat's definition. Let me grep. From the R-chain: C0hat is presumably defined in LiContour.lean or LiSkeleton as something like 3·log(x)·log(log x) or the Trudgian shape. STATE.md says: \"Ĉ₀(R+2) ≤ 5·log R (log(R+4) ≤ 2·log R for R ≥ 3; loglog ≤ 1+log; L ≥ 9)\". So C0hat(x) involves log·loglog presumably. Let me check.\n\nAlso `lowZeroConst`, `dirichletC2`, `A0count` are the constants. C₀' := 2·lowZeroConst + dirichletC2 + ‖Gentire 0‖ and C₁ := 400000·A0count·5 = 2·10⁶·A0count.\n\nWait, careful: the Schwarz application: h_maps : MapsTo Gentire (ball 0 R_k) (closedBall (Gentire 0) R₂) with R₂ := (RB R_k + ‖Gentire 0‖). For z ∈ ball: dist (Gentire z) (Gentire 0) ≤ ‖Gentire z − Gentire 0‖ ≤ ‖Gentire z‖ + ‖Gentire 0‖ ≤ RB R_k + ‖Gentire 0‖ where ‖Gentire z‖ ≤ RB R_k comes from the sphere lemma applied at radius... hmm — z ∈ ball 0 R_k, and the sphere bound is on sphere 0 R_k. AnalyticOn.norm_le_of_norm_le_on_sphere with r = R := R_k, analytic on closedBall 0 R_k, w ∈ closedBall 0 R_k (z ∈ ball ⊂ closedBall). Wait the signature: (hyp_r : r ≤ R) (analytic : AnalyticOn ℂ f (closedBall 0 R)) (cond : ∀ z ∈ sphere 0 r, ‖f z‖ ≤ C) (wInS : w ∈ closedBall 0 r). With r = R = R_k: analytic: AnalyticOn ℂ Gentire (closedBall 0 R_k) — from ∀ s, AnalyticAt ℂ Gentire s: AnalyticOnNhd, then .analyticOn. AnalyticOnNhd ℂ Gentire (closedBall 0 R_k) := fun s _ => gentire_analyticAt_and_eq.1 s. Then AnalyticOnNhd.analyticOn.\n\nHmm wait, but actually is the sphere 0 R_k in Metric vs the ‖s‖ = R hypothesis — sphere 0 R = {s : ‖s‖ = R}? Metric.sphere (0:ℂ) R = {s | dist s 0 = R} = {s | ‖s‖ = R} (dist_eq_norm, sub_zero). Fine — a mem-sphere to ‖·‖=R bridge: `Metric.mem_sphere`, dist_eq_norm, norm_sub... s - 0 = s (sub_zero). Standard.\n\nSo cond: ∀ z ∈ Metric.sphere 0 R_k, ‖Gentire z‖ ≤ RB R_k — from step 1 (gentire_circle_le) applied to each z on the sphere (using the gap properties of R_k from exists_gapCircle_radius hZC (k large)).\n\nThen the max-modulus gives: ∀ z ∈ closedBall 0 R_k, ‖Gentire z‖ ≤ RB R_k.\n\nThen MapsTo: z ∈ ball 0 R_k ⟹ dist (Gentire z) (Gentire 0) = ‖Gentire z − Gentire 0‖ ≤ ‖Gentire z‖ + ‖Gentire 0‖ ≤ RB R_k + ‖Gentire 0‖ =: R₂. mem_closedBall: dist z c ≤ R₂. Note dist (Gentire z) (Gentire 0) — closedBall (f c) R₂ membership is dist (Gentire z) (Gentire 0) ≤ R₂. ✓.\n\nSchwarz: hd : DifferentiableOn ℂ Gentire (ball 0 R_k). From analytic: (gentire_analyticAt_and_eq.1 s).differentiableAt for each s, then DifferentiableOn: fun z hz => ((...).differentiableAt).differentiableWithinAt. AnalyticAt.differentiableAt exists? There's `AnalyticAt.differentiableAt : AnalyticAt 𝕜 f z → DifferentiableAt 𝕜 f z`. Yes.\n\nhz : z₀ ∈ ball 0 R_k: dist z₀ 0 < R_k i.e. ‖z₀‖ < R_k. We pick R_k ≥ 4k+5/2 > ‖z₀‖.\n\nConclusion: dist (Gentire z₀) (Gentire 0) ≤ R₂/R_k · dist z₀ 0 = R₂/R_k·‖z₀‖.\n\nThen dist z₀ 0 = ‖z₀‖: dist_eq_norm + sub_zero.\n\nFinal bound: dist ≤ (RB R_k + ‖Gentire 0‖)/R_k·‖z₀‖ ≤ ε as computed. If z₀ = 0: dist = 0 ≤ anything; handle by cases (if ‖z₀‖ = 0 then trivial). Actually if z₀ = 0 then dist (Gentire 0) (Gentire 0) = 0 ≤ ε ✓ directly; or note the product is 0. The inequality R₂/R·0 = 0 ≤ ε fine — no case split needed since the Schwarz conclusion already gives dist ≤ R₂/R₁·dist z₀ 0 = R₂/R₁·‖z₀‖ and if ‖z₀‖ = 0 then RHS = 0... wait no, RHS = R₂/R₁ * 0 = 0, and we show RHS ≤ ε anyway. Actually our real-number argument: (C₀' + 64 C₁ Q³)/R · ‖z₀‖ ≤ ε — hmm, with ‖z₀‖ in the mix: we need (C₀' + 64·C₁·Q³)·‖z₀‖ ≤ ε·R. Since C₀' includes ‖Gentire 0‖... Let me redo: R₂ = RB + ‖Gentire 0‖ ≤ C₀' + 64·C₁·Q³ where C₀' := 2·lowZeroConst + dirichletC2 + ‖Gentire 0‖. So dist ≤ (C₀' + 64 C₁ Q³)·‖z₀‖/Q⁴. With Q ≥ max(1, 2·(C₀'+64·C₁)... hmm redo with ‖z₀‖ folded:\n\nWant: (C₀' + 64·C₁·Q³)·‖z₀‖ ≤ ε·Q⁴. Suffices: C₀'·‖z₀‖ ≤ (ε/2)·Q⁴ and 64·C₁·Q³·‖z₀‖ ≤ (ε/2)·Q⁴ ⟸ 64·C₁·‖z₀‖ ≤ (ε/2)·Q. So conditions: Q ≥ 128·C₁·‖z₀‖/ε and Q⁴ ≥ 2·C₀'·‖z₀‖/ε ⟸ (Q ≥ 1 ∧ Q ≥ 2·C₀'·‖z₀‖/ε). So Q ≥ max(1, (128·C₁·‖z₀‖ + 2·C₀'·‖z₀‖)/ε)-ish. Take c := (128·C₁ + 2·C₀')·‖z₀‖/ε + 1 hmm but ‖z₀‖ could be 0, fine, c ≥ 1 ensures Q ≥ 1.\n\nSo: c := 1 + (128·C₁ + 2·C₀')·‖z₀‖/ε; choose M := max(c⁴, ‖z₀‖ + 1); choose k with 4·k ≥ M (k ≥ 4000 automatic since M ≥ ... wait need k ≥ 4000 for Fexpr_circle_le; take M' := max(M, 16000) then k := ⌈M'/4⌉₊ ≥ 4000). Get R from exists_gapCircle_radius hZC (k := ...) (hk : 1 ≤ k): R ∈ [4k+5/2, 4k+3], gap properties. Then:\n- ‖z₀‖ < 4k+5/2 ≤ R (since M ≥ ‖z₀‖+1 and 4k ≥ M ⟹ 4k+5/2 ≥ M + 5/2 > ‖z₀‖) ⟹ z₀ ∈ ball 0 R.\n- R ≥ M ≥ c⁴ ⟹ Q := R^{1/4} ≥ c (rpow monotonicity + (c⁴)^{1/4} = c).\n- k ≥ 4000: 4k ≥ M ≥ 16000.\n\nWait — exists_gapCircle_radius needs (hk : 1 ≤ k); Fexpr_circle_le needs (hk : 4000 ≤ k). Choose k := max(4000, ⌈M/4⌉₊). Then 4k ≥ M ✓ (4·⌈M/4⌉₊ ≥ M by Nat.le_ceil... 4·ceil(M/4) ≥ M ✓) and k ≥ 4000 ✓.\n\nThen the estimate chain gives dist (Gentire z₀) (Gentire 0) ≤ ε. Since ε arbitrary: dist = 0 ⟹ Gentire z₀ = Gentire 0. ∀ z₀. Set B := Gentire 0.\n\n**Step 5 capstone**: for s good (ζ s ≠ 0, s ≠ 0, s ≠ 1, ∀ n ≥ 1, s ≠ −2n): ¬ badPoint s needs also ¬ IsNontrivialZero s. IsNontrivialZero ρ presumably means ζ ρ = 0 ∧ 0 < re ρ ∧ re ρ < 1? If ζ s ≠ 0 then s is not a nontrivial zero. Check the def of IsNontrivialZero. Then Gentire s = Fexpr s (gentire_eq_Fexpr) and Gentire s = B ⟹ Fexpr s = B ⟹ unfold Fexpr and rearrange: logDeriv ζ s = B + seriesFun s − 1/s − 1/(s−1) + (log π)/2 − ψ(s/2)/2. From Fexpr s = logDeriv ζ s − seriesFun s + 1/s + 1/(s−1) − (log π)/2 + ψ(s/2)/2 = B. Solve: logDeriv ζ s = B + seriesFun s − 1/s − 1/(s−1) + (log π)/2 − ψ(s/2)/2. linear_combination or field... just algebraic: linear_combination h (the equation Fexpr s = B) — after unfolding Fexpr it's a linear identity in the atoms. `linear_combination hF` should close it.\n\nNow, theorem statement (D-068 §5):\n```lean\ntheorem partialFraction_identity (hZC : ZeroCountProp) :\n    ∃ B : ℂ, ∀ s : ℂ, riemannZeta s ≠ 0 → s ≠ 0 → s ≠ 1 →\n      (∀ n : ℕ, 1 ≤ n → s ≠ -2 * (n : ℂ)) →\n      logDeriv riemannZeta s\n        = B + seriesFun s - 1 / s - 1 / (s - 1) + (Real.log Real.pi : ℂ) / 2\n          - Complex.digamma (s / 2) / 2\n```\n\nNow — where to put it? STATE.md says \"LiCircleBound.lean §13 or a new LiSqueeze.lean\". A new file LiSqueeze.lean is cleaner (imports LiCircleBound, LiFEntire, LiGapCircles). It also keeps the big file from getting bigger. Let me check LiCircleBound's imports to mirror them.\n\nAlso: is LiCircleBound root-imported? STATE says \"not yet root-imported\". I should root-import LiSqueeze (which imports LiCircleBound transitively) at the end — or root-import both. The closure commit should root-import LiSqueeze.lean.\n\nLet me now check:\n1. C0hat definition.\n2. IsNontrivialZero definition.\n3. LiCircleBound imports + namespace structure (to match).\n4. lowZeroConst, dirichletC2, A0count definitions/nonneg lemmas.\n5. AnalyticAt.differentiableAt name.\n6. The rpow lemma names.\n\nLet me grep these.","signature":""},{"type":"text","text":"All three key lemmas verified. Now checking the supporting definitions (C0hat, IsNontrivialZero, constants) and LiCircleBound's import header."},{"type":"tool_use","id":"Bash_18","name":"Bash","input":{"command":"cd D:/Python/Quantum/research/lean/NguyenRH1/NguyenRH1; sed -n '1,30p' LiCircleBound.lean; echo ===C0HAT===; grep -rn \"def C0hat\\|C0hat :=\" LiContour.lean LiSkeleton.lean LiDisc.lean 2>/dev/null | head -3; echo ===NTRIV===; grep -rn \"def IsNontrivialZero\" *.lean | head -2; echo ===CONSTS===; grep -rn \"def lowZeroConst\\|def dirichletC2\\|def A0count\\|theorem A0count_pos\\|theorem lowZeroConst_nonneg\\|theorem dirichletC2_nonneg\\|theorem A0count_ge_one\" *.lean | head -10","description":"Check imports, C0hat, IsNontrivialZero, constant definitions"},"extra_content":null}]