[{"type":"thinking","thinking":"Zero errors — DL-6-i compiles. Now DL-6-ii (the V_right quarter charge) and the DL-5 upgrade. Then the full build + audit + commit DL-6.\n\nDL-6-ii: `vIntegral_Jlag_right_norm_le_quarter` — clone of vIntegral_Jlag_right_norm_le but with norm_logDeriv_zeta_le_iimZeta (0.57·2^{−(2√n−2)}) instead of norm_logDeriv_zeta_le_two_pow (2^{−(2√n−2)}·dirichletC2). And the IIMaj/4 accounting: with U−T ≤ (√n+1)/2: V_right ≤ ((√n+1)/2)(e+1)·0.57·2^{−(2√n−2)} = IIMaj/4 EXACTLY (IIMaj/4 = (1/4)·2(√n+1)(e+1)·0.57·2^{−X} = ((√n+1)/2)(e+1)·0.57·2^{−X} ✓).\n\nWrite DL-6-ii as a clone with the accounting:\n\n```\n/-- **The far-right vertical drop, quarter charge** (DL-6-ii): for\nn ≥ 16, 0 < T ≤ U, T² ≥ n, ‖VIntegral (Jlag n) (2√n) T U‖ ≤\n(U−T)·(e^{√n/2}+1)·0.57·2^{−(2√n−2)} = the IIMaj/4 per-length form —\nvia the DIRECT dirichlet bound (norm_logDeriv_zeta_le_iimZeta, NO\ndirichletC2 ≤ 0.57 needed). -/\ntheorem vIntegral_Jlag_right_norm_le_quarter {n : ℕ} {T U : ℝ}\n    (hn : 16 ≤ n) (hT0 : (0 : ℝ) < T) (hTU : T ≤ U) (hTn : (n : ℝ) ≤ T ^ 2) :\n    ‖VIntegral (Jlag n) (2 * Real.sqrt (n : ℝ)) T U‖ ≤\n      (U - T) * (Real.exp (Real.sqrt (n : ℝ) / 2) + 1) *\n        ((0.57 : ℝ) * (2 : ℝ) ^ (-(2 * Real.sqrt (n : ℝ) - 2))) := by\n  have hnorm : ‖VIntegral (Jlag n) (2 * Real.sqrt (n : ℝ)) T U‖ =\n      ‖∫ t in T..U, Jlag n (((2 * Real.sqrt (n : ℝ)) : ℝ) + t * Complex.I)‖ := by\n    unfold VIntegral\n    rw [norm_smul, Complex.norm_I, one_mul]\n  rw [hnorm]\n  have hbound : ∀ t ∈ Set.Ioc T U,\n      ‖Jlag n (((2 * Real.sqrt (n : ℝ)) : ℝ) + t * Complex.I)‖ ≤\n        (Real.exp (Real.sqrt (n : ℝ) / 2) + 1) *\n          ((0.57 : ℝ) * (2 : ℝ) ^ (-(2 * Real.sqrt (n : ℝ) - 2))) := by\n    intro t ht\n    have htT : T < t := ht.1\n    have ht2 : (n : ℝ) ≤ t ^ 2 := by nlinarith [hTn, hT0]\n    have h1 := kFun_right_vertical_norm_le_exp_sqrt_half_add_one hn ht2\n    have h2 := norm_logDeriv_zeta_le_iimZeta (by omega : 5 ≤ n) (t := t)\n    have hnorm3 : ... (norm_mul, norm_neg as before)\n    rw [hnorm3]\n    exact mul_le_mul h1 h2 (norm_nonneg _)\n      (add_nonneg (Real.exp_nonneg _) (by norm_num))\n  have hint : ‖∫ t in T..U, Jlag n (((2 * Real.sqrt (n : ℝ)) : ℝ) + t * Complex.I)‖ ≤\n      ((Real.exp (Real.sqrt (n : ℝ) / 2) + 1) *\n        ((0.57 : ℝ) * (2 : ℝ) ^ (-(2 * Real.sqrt (n : ℝ) - 2)))) * |U - T| := by\n    apply intervalIntegral.norm_integral_le_of_norm_le_const\n    intro t ht\n    rw [Set.uIoc_of_le hTU] at ht\n    exact hbound t ht\n  have habs : |U - T| = U - T := abs_of_nonneg (by linarith)\n  rw [habs] at hint\n  calc ‖∫ t in T..U, Jlag n (((2 * Real.sqrt (n : ℝ)) : ℝ) + t * Complex.I)‖\n      ≤ ((Real.exp (Real.sqrt (n : ℝ) / 2) + 1) *\n          ((0.57 : ℝ) * (2 : ℝ) ^ (-(2 * Real.sqrt (n : ℝ) - 2)))) * (U - T) := hint\n    _ = (U - T) * (Real.exp (Real.sqrt (n : ℝ) / 2) + 1) *\n          ((0.57 : ℝ) * (2 : ℝ) ^ (-(2 * Real.sqrt (n : ℝ) - 2))) := by ring\n```\n\nnorm_logDeriv_zeta_le_iimZeta needs hn : 5 ≤ n — with hn : 16 ≤ n, `by omega : 5 ≤ n` ✓.\n\nAnd the IIMaj/4 accounting lemma:\n```\ntheorem vIntegral_Jlag_right_norm_le_IIMaj_quarter {n : ℕ} {T : ℝ}\n    (hn : 16 ≤ n) (hTn : Real.sqrt (n : ℝ) ≤ T) (hTn1 : T ≤ Real.sqrt (n : ℝ) + 1) :\n    ‖VIntegral (Jlag n) (2 * Real.sqrt (n : ℝ)) T ((3 / 2) * T)‖ ≤ IIMaj (n : ℝ) / 4 := by\n  ... (clone of DL-5's hVr2 with the quarter bound and the exact equality IIMaj/4 = ((√n+1)/2)(e+1)·0.57·2^{−X})\n```\n\nHmm — actually, for the DL-5 upgrade, the cleanest is to REPLACE hVr2's IIMaj/2 charge with IIMaj/4 via this new lemma. The DL-5's hVr2 currently proves V_right ≤ IIMaj/2 via hVr (vIntegral_Jlag_right_norm_le) + the dirichletC2 ≤ 1 accounting. With DL-6-ii I can prove V_right ≤ IIMaj/4 DIRECTLY:\n```\n  have hVr2 : ‖VIntegral (Jlag n) (2 * Real.sqrt (n : ℝ)) T ((3 / 2) * T)‖ ≤\n      IIMaj (n : ℝ) / 4 := by\n    have hVr' := vIntegral_Jlag_right_norm_le_quarter hn hT0 hTU hTn2 (n := n)\n    have hlen : (3 / 2) * T - T ≤ (Real.sqrt (n : ℝ) + 1) / 2 := by linarith [hTn1, hTn]\n    have hlen0 : (0 : ℝ) ≤ (3 / 2) * T - T := by linarith [hT0]\n    have e4 : IIMaj (n : ℝ) / 4 =\n        ((Real.sqrt (n : ℝ) + 1) / 2) * (Real.exp (Real.sqrt (n : ℝ) / 2) + 1) *\n          ((0.57 : ℝ) * (2 : ℝ) ^ (-(2 * Real.sqrt (n : ℝ) - 2))) := by\n      unfold IIMaj\n      ring\n    rw [e4]\n    refine hVr'.trans ?_\n    apply mul_le_mul hlen (le_refl _) (by positivity) hlen0\n```\ne4 : IIMaj/4 = ((√n+1)/2)·(e+1)·0.57·2^{−(2√n−2)}: IIMaj = 2(√n+1)(e+1)·0.57·2^{−X}: IIMaj/4 = (1/2)(√n+1)(e+1)·0.57·2^{−X} = ((√n+1)/2)(e+1)·0.57·2^{−X} ✓ — ring proves it? — IIMaj/4's unfolded form: `2 * (Real.sqrt n + 1) * (Real.exp (Real.sqrt n / 2) + 1) * 0.57 * 2 ^ (-(2 * Real.sqrt n - 2)) / 4` — vs `((Real.sqrt n + 1) / 2) * (Real.exp (Real.sqrt n / 2) + 1) * (0.57 * 2 ^ (-(2 * Real.sqrt n - 2)))` — ring over a field with the /4 and /2 divisions — ring handles divisions ✓ should close ✓.\n\nThen DL-5's dogLegUpper changes: 8.38 + IIMaj/4 (replacing IIMaj/2). The dogLegUpper lemmas' statements say IIMaj/2 — I must update the STATEMENTS too (they're the DL-5 deliverables) — edit them to IIMaj/4 + adjust the assembly arithmetic (the final `_ = VLeft + 8.38 + IIMaj/4 := by ring` ✓).\n\nWait — the I_I accounting: DL-5's dogLegUpper is per-side VLeft + 8.38 + VRight. With VRight = IIMaj/4 (new), dogLegUpper = VLeft + 8.38 + IIMaj/4. The E-034 fold's DogLegRight was VLeft + HHIGH + VRightBound with VRightBound = IIMaj/2 — update back to IIMaj/4 with the direct-route note + re-run.\n\nThen Bn's IIMaj coefficient: 1 (I_I) + 2·(1/4) = 1.5 ✓ convexity works with the 4/3-bracket.\n\nLet me write DL-6-ii + the DL-5 upgrade now. First DL-6-ii's two lemmas.","signature":""},{"type":"text","text":"Zero errors — DL-6-i compiles. Now DL-6-ii (the quarter-charge V_right lemma) + the DL-5 upgrade to IIMaj/4."},{"type":"tool_use","id":"Edit_493","name":"Edit","input":{"file_path":"D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiDogLeg.lean","old_string":"end NguyenRH1\n\nend","new_string":"/-! ## 9. The far-right vertical drop, quarter charge (DL-6-ii) -/\n\n/-- **The far-right vertical drop, quarter charge** (DL-6-ii): for\nn ≥ 16, 0 < T ≤ U, T² ≥ n, ‖VIntegral (Jlag n) (2√n) T U‖ ≤\n(U−T)·(e^{√n/2}+1)·0.57·2^{−(2√n−2)} — the IIMaj/4 per-length form,\nvia the DIRECT dirichlet bound (norm_logDeriv_zeta_le_iimZeta, NO\ndirichletC2 ≤ 0.57 needed). -/\ntheorem vIntegral_Jlag_right_norm_le_quarter {n : ℕ} {T U : ℝ}\n    (hn : 16 ≤ n) (hT0 : (0 : ℝ) < T) (hTU : T ≤ U) (hTn : (n : ℝ) ≤ T ^ 2) :\n    ‖VIntegral (Jlag n) (2 * Real.sqrt (n : ℝ)) T U‖ ≤\n      (U - T) * (Real.exp (Real.sqrt (n : ℝ) / 2) + 1) *\n        ((0.57 : ℝ) * (2 : ℝ) ^ (-(2 * Real.sqrt (n : ℝ) - 2))) := by\n  have hnorm : ‖VIntegral (Jlag n) (2 * Real.sqrt (n : ℝ)) T U‖ =\n      ‖∫ t in T..U, Jlag n (((2 * Real.sqrt (n : ℝ)) : ℝ) + t * Complex.I)‖ := by\n    unfold VIntegral\n    rw [norm_smul, Complex.norm_I, one_mul]\n  rw [hnorm]\n  have hbound : ∀ t ∈ Set.Ioc T U,\n      ‖Jlag n (((2 * Real.sqrt (n : ℝ)) : ℝ) + t * Complex.I)‖ ≤\n        (Real.exp (Real.sqrt (n : ℝ) / 2) + 1) *\n          ((0.57 : ℝ) * (2 : ℝ) ^ (-(2 * Real.sqrt (n : ℝ) - 2))) := by\n    intro t ht\n    have htT : T < t := ht.1\n    have ht2 : (n : ℝ) ≤ t ^ 2 := by nlinarith [hTn, hT0]\n    have h1 := kFun_right_vertical_norm_le_exp_sqrt_half_add_one hn ht2\n    have h2 := norm_logDeriv_zeta_le_iimZeta (by omega : 5 ≤ n) (t := t)\n    have hnorm3 : ‖Jlag n (((2 * Real.sqrt (n : ℝ)) : ℝ) + t * Complex.I)‖ =\n        ‖kFun n (((2 * Real.sqrt (n : ℝ)) : ℝ) + t * Complex.I)‖ *\n          ‖logDeriv riemannZeta (((2 * Real.sqrt (n : ℝ)) : ℝ) + t * Complex.I)‖ := by\n      show ‖kFun n (((2 * Real.sqrt (n : ℝ)) : ℝ) + t * Complex.I) *\n          (-logDeriv riemannZeta (((2 * Real.sqrt (n : ℝ)) : ℝ) + t * Complex.I))‖ = _\n      rw [norm_mul, norm_neg]\n    rw [hnorm3]\n    exact mul_le_mul h1 h2 (norm_nonneg _)\n      (add_nonneg (Real.exp_nonneg _) (by norm_num))\n  have hint : ‖∫ t in T..U, Jlag n (((2 * Real.sqrt (n : ℝ)) : ℝ) + t * Complex.I)‖ ≤\n      ((Real.exp (Real.sqrt (n : ℝ) / 2) + 1) *\n        ((0.57 : ℝ) * (2 : ℝ) ^ (-(2 * Real.sqrt (n : ℝ) - 2)))) * |U - T| := by\n    apply intervalIntegral.norm_integral_le_of_norm_le_const\n    intro t ht\n    rw [Set.uIoc_of_le hTU] at ht\n    exact hbound t ht\n  have habs : |U - T| = U - T := abs_of_nonneg (by linarith)\n  rw [habs] at hint\n  calc ‖∫ t in T..U, Jlag n (((2 * Real.sqrt (n : ℝ)) : ℝ) + t * Complex.I)‖\n      ≤ ((Real.exp (Real.sqrt (n : ℝ) / 2) + 1) *\n          ((0.57 : ℝ) * (2 : ℝ) ^ (-(2 * Real.sqrt (n : ℝ) - 2)))) * (U - T) := hint\n    _ = (U - T) * (Real.exp (Real.sqrt (n : ℝ) / 2) + 1) *\n          ((0.57 : ℝ) * (2 : ℝ) ^ (-(2 * Real.sqrt (n : ℝ) - 2))) := by ring\n\n/-- **V_right ≤ IIMaj/4** (DL-6-ii accounting): for n ≥ 16 and\nT ∈ [√n, √n+1], U = (3/2)·T, the far-right drop is ≤ IIMaj(n)/4\n(exactly: IIMaj/4 = ((√n+1)/2)·(e^{√n/2}+1)·0.57·2^{−(2√n−2)}). -/\ntheorem vIntegral_Jlag_right_le_IIMaj_quarter {n : ℕ} {T : ℝ}\n    (hn : 16 ≤ n) (hTn : Real.sqrt (n : ℝ) ≤ T) (hTn1 : T ≤ Real.sqrt (n : ℝ) + 1) :\n    ‖VIntegral (Jlag n) (2 * Real.sqrt (n : ℝ)) T ((3 / 2) * T)‖ ≤ IIMaj (n : ℝ) / 4 := by\n  have hT0 : (0 : ℝ) < T := by\n    have hnR : (16 : ℝ) ≤ (n : ℝ) := by exact_mod_cast hn\n    have hsqrt4 : (4 : ℝ) ≤ Real.sqrt (n : ℝ) := by\n      have h1 : Real.sqrt 16 ≤ Real.sqrt (n : ℝ) :=\n        Real.sqrt_le_sqrt (by exact_mod_cast hn : (16 : ℝ) ≤ (n : ℝ))\n      have h2 : Real.sqrt 16 = 4 := by\n        rw [Real.sqrt_eq_iff_eq_sq (by norm_num) (by norm_num)]; norm_num\n      rwa [h2] at h1\n    linarith\n  have hTn2 : (n : ℝ) ≤ T ^ 2 := by\n    have h1 : (Real.sqrt (n : ℝ)) ^ 2 ≤ T ^ 2 :=\n      pow_le_pow_left₀ (Real.sqrt_nonneg _) hTn 2\n    rwa [Real.sq_sqrt (by linarith [Real.sqrt_nonneg (n : ℝ)])] at h1\n  have hTU : T ≤ (3 / 2) * T := by linarith [hT0]\n  have hVr' := vIntegral_Jlag_right_norm_le_quarter hn hT0 hTU hTn2 (n := n)\n  have hlen : (3 / 2) * T - T ≤ (Real.sqrt (n : ℝ) + 1) / 2 := by linarith [hTn1, hTn]\n  have hlen0 : (0 : ℝ) ≤ (3 / 2) * T - T := by linarith [hT0]\n  have e4 : IIMaj (n : ℝ) / 4 =\n      ((Real.sqrt (n : ℝ) + 1) / 2) * (Real.exp (Real.sqrt (n : ℝ) / 2) + 1) *\n        ((0.57 : ℝ) * (2 : ℝ) ^ (-(2 * Real.sqrt (n : ℝ) - 2))) := by\n    unfold IIMaj\n    ring\n  rw [e4]\n  refine hVr'.trans ?_\n  apply mul_le_mul hlen (le_refl _) (by positivity) hlen0\n\nend NguyenRH1\n\nend"},"extra_content":null}]