[{"type":"thinking","thinking":"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:\n\n**6-c-iv RECIPE (mid-horizontal [−5/2,3] at height ±T):** per-zero\nintegral ∫_{−5/2}^{3} dσ/√((σ−β)²+δ²) — FTC with the DIRECT\nantiderivative F(σ) := Real.log((σ−β) + Real.sqrt((σ−β)²+δ²))\n(Real.arcsinh does NOT exist here; F′ = integrand exactly:\ng := (σ−β)+√(h(σ)) > 0 since √h > |σ−β|; HasDerivAt via\nhasDerivAt_id.sub/Real.hasDerivAt_sqrt/Real.hasDerivAt_log + the\nidentity g′ = g/√h; integral_eq_sub_of_hasDerivAt). Bound:\nF(3)−F(−5/2) = log(num/den) with num ≤ 2(3−β)+δ ≤ 6+δ, den =\n(−u+√(u²+δ²)) = δ²/(u+√(u²+δ²)) ≥ δ²/(7+δ) (u := β+5/2 ≤ 7/2) ⟹\n∫ ≤ log(6+δ)+log(7+δ)+2·log(1/δ) ≤ 2·log(16.5/δ) (δ ≤ 1 from\nmem_nearZerosSub; log 7 ≤ 1.946, log 8 ≤ 2.08, 2·log 16.5 ≥ 5.606).\nPer-zero ≤ 2·log(16.5/c_clear) with c_clear = 1/(9·C0hat(√n+2)) (the\nwide pigeonhole: nearZerosSub T's zeros have γ ∈ (T−1,T+1) ⊆\n(√n−1,√n+2) ✓). Swap ∫Σ = Σ∫ (integral_finsetSum — the deprecated\nintegral_finset_sum's new name; per-term integrability by continuity,\ndenominator > 0). Σ mult ≤ Z2hat T (nearZerosSub_mult_sum_le hZC2,\nT ≥ 10) ⟹ near-integral ≤ 2·Z2hat(T)·log(148.5·C0hat(√n+2)) =: NI.\nR-part: capstone per σ (him : s.im = T by simp; hζ :\nzeta_ne_zero_horizontal hTwide) ⟹ ∫ ‖R‖ ≤ 5.5·(10.5·log(T+2)+48).\nKernel e⁴+1 (kFun_shift_norm_le_exp_four_add_one at s := w−1:\nre s = σ−1 ≤ 3 ⟺ σ ≤ 4 ✓ covers [−5/2,3]; n ≤ T² from √n ≤ T).\nAssembly per side: ‖H(−5/2→3, ±T)‖ ≤ (e⁴+1)·(NI + 5.5(10.5 log+48));\nthe bottom (−T) by the conj trick (intervalIntegral_conj + Jlag_conj,\nDL-5 pattern).\n\nSo the deliverable: two theorems (top at +T, bottom at −T):\n```lean\ntheorem hIntegral_Jlag_mid_norm_le (hZC : ZeroCountProp) (hZC2 : ZeroCountProp2)\n    (hNLZ : NoLowZeros) {n : ℕ} (hn : 100000 ≤ n) {T : ℝ}\n    (hTwide : ...) (hTn : √n ≤ T) (hTn1 : T ≤ √n+1) :\n    ‖HIntegral (Jlag n) (-5/2) 3 T‖ ≤\n      (Real.exp 4 + 1) * (2 * Z2hat T * Real.log (148.5 * C0hat (Real.sqrt n + 2)) +\n        (3 - -5/2) * (10.5 * Real.log (T + 2) + 48))\n```\n\nWait — 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 ✓.\n\nHmm, and the total assembly per side: ‖H‖ ≤ (e⁴+1)·(NI + 5.5·(10.5·log(T+2)+48)).\n\nLet me carefully map the math:\n‖Jlag n (σ + iT)‖ = ‖kFun n (σ+iT)‖ · ‖logDeriv ζ (σ+iT)‖ ≤ (e⁴+1)·‖logDeriv ζ (σ+iT)‖.\n\nWait — 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.\n\n‖logDeriv ζ (σ+iT)‖ ≤ ‖logDeriv ζ (σ+iT) − Σ_near m/(s−ρ)‖ + ‖Σ_near m/(s−ρ)‖ (norm_le_insert' again).\n\nR-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).\n\nNear-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).\n\nThe 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):\n- 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 ✓.)\n- 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).\n- 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) ✓.\n\nThen 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)).\n\nThen Σ mult·per-zero ≤ 2·log(148.5·C0hat(√n+2))·Σ mult ≤ 2·Z2hat(T)·log(148.5·C0hat(√n+2)) =: NI.\n\nThe δ ≤ 1 input: |T − γ| < 1 from mem_nearZerosSub ✓ (ρ ∈ nearZerosSub T ⟹ |ρ.im − T| < 1).\n\nThe 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|`. ✓.\n\nAnd δ := |T − γ| = |γ − T| (abs_sub_comm).\n\nNow — 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).\n\nThe pieces needed:\n1. **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:\n   - 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 ✓.\n   - 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) ✓.\n   - 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 ✓.\n   \n   The cleanest Lean route: \n   ```lean\n   have hgder : HasDerivAt g ((g σ) / Real.sqrt (h σ)) σ := ...\n   have hF : HasDerivAt (fun σ => Real.log (g σ)) (1 / Real.sqrt (h σ)) σ := by\n     have := hgder.log (ne_of_gt hgpos)\n     -- this : HasDerivAt (log ∘ g) ((g σ / √h) / (g σ))\n     rw [div_eq_iff (ne_of_gt hgpos)] at this?... convert\n   ```\n   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 ✓.\n   \n   - 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`. \n   - Integrability of 1/√h on [−5/2, 3]: continuous (h > 0 ⟹ √h continuous nonzero ⟹ 1/√h continuous) ⟹ Continuous.intervalIntegrable.\n\n2. **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:\n   - ∫_{-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).\n   - ∫ ‖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)).\n   \n   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))... \n\n   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)).\n\n3. **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.\n\n4. **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.\n\nAlso — 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 ✓.\n\nAlso 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.\n\nActually — 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`.\n\nOK — 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.\n\nLet me also check `kFun_shift_norm_le_exp_four_add_one`'s exact signature.","signature":""},{"type":"text","text":"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."},{"type":"tool_use","id":"TodoWrite_86","name":"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 ⟹ sfBoundProp (then commit, update archive)","status":"pending","activeForm":"Proving 6-d fold to sfBoundProp"}]},"extra_content":null},{"type":"tool_use","id":"Grep_87","name":"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},"extra_content":null},{"type":"tool_use","id":"Grep_88","name":"Grep","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 ≤|log 8 ≤|Real\\.log 16","path":"D:\\Python\\Quantum","output_mode":"content","-A":4,"head_limit":60},"extra_content":null}]