[{"type":"thinking","thinking":"Wait — I removed the doc comment of chain_rule_sqrt by placing the new lemma before it. Let me check: the old_string was the doc comment of chain_rule_sqrt; my new_string ends with the same doc comment — so chain_rule_sqrt keeps its doc ✓ (I inserted the new theorem BEFORE chain_rule_sqrt's doc). ✓ Good.\n\nNow edit marginTilde (766), the blocks (1815–1832), contDiffOn_m* (1857–1884), continuousOn (1901), and the convexity proof. Note: line numbers have shifted due to my inserts (+~150 lines). Use content-based matching.\n\nmarginTilde current:\n```\n/-- The margin majorant chain (D-057, corrected per D-071): ...\ndef marginTilde (F : ℝ → ℝ) (n : ℝ) : ℝ :=\n  let T := Real.sqrt n + 1\n  ((1 / 2) * n * Real.log n +\n      (lagariasC1 - Real.eulerMascheroniConstant / 2) * n - 4) -\n    (4 * F T + 4 * T * ((2 / 3) * Z2hat T + 10.5 * Real.log (T + 2) + 48) +\n      40 * 1.3352 + 2 * (Real.exp 4 + 1) * (7 * Hhat T + 3.3) +\n      IIMaj n + 1) + 1\n```\nNew:\n```\ndef marginTilde (F : ℝ → ℝ) (n : ℝ) : ℝ :=\n  let T := 1.5 * Real.sqrt n + 1\n  ((1 / 2) * n * Real.log n +\n      (lagariasC1 - Real.eulerMascheroniConstant / 2) * n - 4) -\n    (4 * F T + 4 * T * ((2 / 3) * Z2hat T + 10.5 * Real.log (T + 2) + 48) +\n      40 * 1.3352 + 2 * (Real.exp (16 / 9) + 1) * (7 * Hhat T + 5.94) +\n      1.5 * IIMaj n + 1) + 1\n```\nWith doc comment update (protocol step 2: D-073 correction comment).","signature":""},{"type":"text","text":"Now the core redefinitions — `marginTilde`, the blocks, `mK`, and `Bn`."},{"type":"tool_use","id":"Edit_84","name":"Edit","input":{"file_path":"D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiSkeleton.lean","old_string":"/-- The margin majorant chain (D-057, corrected per D-071): margiñ(n) =\nS(n) − B̃(n) + 1 with the counting function majorized by a smooth\nfunction F. M₀ = 1.3352 (E-023, grid-verified — see D-060 §4).\nCORRECTION (D-071, user-approved session 038): the bridge is 4F (the\nresidue sum λ_n(T) runs over BOTH signs of γ — D-036/D-037's 2N(T)\nwas a factor-2 accounting slip, R-004 unaffected) and the I_III\nnear-sum uses the two-unit count (2/3)·Z2hat (D-069 Finding 2 made\nexact). See D-071 + E-030. -/\ndef marginTilde (F : ℝ → ℝ) (n : ℝ) : ℝ :=\n  let T := Real.sqrt n + 1\n  ((1 / 2) * n * Real.log n +\n      (lagariasC1 - Real.eulerMascheroniConstant / 2) * n - 4) -\n    (4 * F T + 4 * T * ((2 / 3) * Z2hat T + 10.5 * Real.log (T + 2) + 48) +\n      40 * 1.3352 + 2 * (Real.exp 4 + 1) * (7 * Hhat T + 3.3) +\n      IIMaj n + 1) + 1","new_string":"/-- The margin majorant chain (D-057, corrected per D-071, repaired\nper D-073): margiñ(n) = S(n) − B̃(n) + 1 with the counting function\nmajorized by a smooth function F. M₀ = 1.3352 (E-023, grid-verified —\nsee D-060 §4).\nCORRECTION 1 (D-071, user-approved session 038): the bridge is 4F\n(the residue sum λ_n(T) runs over BOTH signs of γ — D-036/D-037's\n2N(T) was a factor-2 accounting slip, R-004 unaffected) and the I_III\nnear-sum uses the two-unit count (2/3)·Z2hat (D-069 Finding 2 made\nexact). See D-071 + E-030.\nCORRECTION 2 (D-073, the F-D repair, session 040): the horizontal\nheight moves T = √n+1 → T = 1.5·√n+1 (the (6.12) stepping is broken\nat T = √n — Finding F-D/D-072; the repaired stepping needs kernel\nratio e^{1/c²} < 2, i.e. c > 1/√log 2). Constants: kernel charge\ne⁴+1 → e^{16/9}+1 (T² = 2.25n); the broken right-tail charge 3.3 →\nR(1.5) = 5.94 (D-073 Cor 4.2, explicit repaired bound); I_I length\n2(√n+1) → 2(1.5√n+1), charged 1.5·IIMaj (1.5√n+1 ≤ 1.5(√n+1));\nN₁ → ((2/3)(3×10¹²−1))² = 4×10²⁴. Recommended c = 1.5 (D-073 §5,\nE-033 cross-validation). See D-073 + E-033. -/\ndef marginTilde (F : ℝ → ℝ) (n : ℝ) : ℝ :=\n  let T := 1.5 * Real.sqrt n + 1\n  ((1 / 2) * n * Real.log n +\n      (lagariasC1 - Real.eulerMascheroniConstant / 2) * n - 4) -\n    (4 * F T + 4 * T * ((2 / 3) * Z2hat T + 10.5 * Real.log (T + 2) + 48) +\n      40 * 1.3352 + 2 * (Real.exp (16 / 9) + 1) * (7 * Hhat T + 5.94) +\n      1.5 * IIMaj n + 1) + 1"},"extra_content":null},{"type":"tool_use","id":"Edit_85","name":"Edit","input":{"file_path":"D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiSkeleton.lean","old_string":"/-- The margin pieces as block compositions (for chain-rule application).\nConstants updated for the D-071 correction (4F bridge; (2/3)Z2 I_III). -/\nprivate def mFblock (F : ℝ → ℝ) : ℝ → ℝ := fun m => (-4) * F (Real.sqrt m + 1)\nprivate def mTL1b : ℝ → ℝ := fun m => (-((8/3)*0.6 + 42)) *\n  ((fun t => t * Real.log (t + 2)) (Real.sqrt m + 1))\nprivate def mTL2b : ℝ → ℝ := fun m => (-((8/3)*0.886)) *\n  ((fun t => t * Real.log (Real.log (t + 2))) (Real.sqrt m + 1))\nprivate def mIdb : ℝ → ℝ := fun m => (-((8/3)*2.7 + 192)) * (Real.sqrt m + 1)\nprivate def mL1sqb : ℝ → ℝ := fun m => (-(42*0.433^2*(Real.exp 4 + 1))) *\n  ((fun t => Real.log (t + 2) ^ 2) (Real.sqrt m + 1))\nprivate def mL1L2b : ℝ → ℝ := fun m => (-(42*(2*0.433*0.886)*(Real.exp 4 + 1))) *\n  ((fun t => Real.log (t + 2) * Real.log (Real.log (t + 2))) (Real.sqrt m + 1))\nprivate def mL2sqb : ℝ → ℝ := fun m => (-(42*0.886^2*(Real.exp 4 + 1))) *\n  ((fun t => Real.log (Real.log (t + 2)) ^ 2) (Real.sqrt m + 1))\nprivate def mL1b : ℝ → ℝ := fun m => (-((42*(2*0.433*2.9) + 147)*(Real.exp 4 + 1))) *\n  ((fun t => Real.log (t + 2)) (Real.sqrt m + 1))\nprivate def mL2b : ℝ → ℝ := fun m => (-(42*(2*0.886*2.9)*(Real.exp 4 + 1))) *\n  ((fun t => Real.log (Real.log (t + 2))) (Real.sqrt m + 1))\nprivate def mIIM : ℝ → ℝ := fun m => (-1) * IIMaj m\nprivate def mK : ℝ := -40*1.3352 - 2*(Real.exp 4 + 1)*(7*(3*2.9^2 + 48) + 3.3) - 1 + 1","new_string":"/-- The margin pieces as block compositions (for chain-rule application).\nConstants updated for the D-071 correction (4F bridge; (2/3)Z2 I_III)\nand the D-073 F-D repair (T = 1.5·√m+1; kernel charge e^{16/9}+1;\nright-tail 5.94; I_I charged 1.5·IIMaj). -/\nprivate def mFblock (F : ℝ → ℝ) : ℝ → ℝ := fun m => (-4) * F (1.5 * Real.sqrt m + 1)\nprivate def mTL1b : ℝ → ℝ := fun m => (-((8/3)*0.6 + 42)) *\n  ((fun t => t * Real.log (t + 2)) (1.5 * Real.sqrt m + 1))\nprivate def mTL2b : ℝ → ℝ := fun m => (-((8/3)*0.886)) *\n  ((fun t => t * Real.log (Real.log (t + 2))) (1.5 * Real.sqrt m + 1))\nprivate def mIdb : ℝ → ℝ := fun m => (-((8/3)*2.7 + 192)) * (1.5 * Real.sqrt m + 1)\nprivate def mL1sqb : ℝ → ℝ := fun m => (-(42*0.433^2*(Real.exp (16/9) + 1))) *\n  ((fun t => Real.log (t + 2) ^ 2) (1.5 * Real.sqrt m + 1))\nprivate def mL1L2b : ℝ → ℝ := fun m => (-(42*(2*0.433*0.886)*(Real.exp (16/9) + 1))) *\n  ((fun t => Real.log (t + 2) * Real.log (Real.log (t + 2))) (1.5 * Real.sqrt m + 1))\nprivate def mL2sqb : ℝ → ℝ := fun m => (-(42*0.886^2*(Real.exp (16/9) + 1))) *\n  ((fun t => Real.log (Real.log (t + 2)) ^ 2) (1.5 * Real.sqrt m + 1))\nprivate def mL1b : ℝ → ℝ := fun m => (-((42*(2*0.433*2.9) + 147)*(Real.exp (16/9) + 1))) *\n  ((fun t => Real.log (t + 2)) (1.5 * Real.sqrt m + 1))\nprivate def mL2b : ℝ → ℝ := fun m => (-(42*(2*0.886*2.9)*(Real.exp (16/9) + 1))) *\n  ((fun t => Real.log (Real.log (t + 2))) (1.5 * Real.sqrt m + 1))\nprivate def mIIM : ℝ → ℝ := fun m => (-1.5) * IIMaj m\nprivate def mK : ℝ := -40*1.3352 - 2*(Real.exp (16/9) + 1)*(7*(3*2.9^2 + 48) + 5.94) - 1 + 1"},"extra_content":null}]