[{"type":"thinking","thinking":"Cherry-pick committed (e11ad95). Now ii′-6-a2-i: the Bn/marginTilde/marginExpanded/mK dog-leg redefinition.\n\nThe new Bn (D-074's DogLegRight at the DL-5 quarter-charge constants):\n```\nBn(F, n) := 4F(T) + 4T[(2/3)Z2(T) + 10.5 log(T+2) + 48] + 40·1.3352\n          + 2(e⁴+1)(7Ĥ(T)) + 2·DogLegRight(n) + IIMaj(n) + 1,   T = √n+1\nDogLegRight(n) = ((√n+1)/2)(e^{5/2}+1)/2 + 8.38 + IIMaj(n)/4\n```\nExpanded: 2·DogLegRight = (√n+1)(e^{5/2}+1)/2 + 16.76 + IIMaj(n)/2.\nTotal IIMaj: 1 (I_I) + 1/2 (dog-leg) = 1.5·IIMaj ✓.\n\nvs session-038 honest Bn: `... + 2(e⁴+1)(7Ĥ + 3.3) + IIMaj + 1`. Changes:\n- horizontal share: 2(e⁴+1)(7Ĥ + 3.3) → 2(e⁴+1)(7Ĥ) + 2·DogLegRight.\n- So Bn_new = Bn_old − 2(e⁴+1)·3.3 + (√n+1)(e^{5/2}+1)/2 + 16.76 + IIMaj(n)/2.\n\nWait — 2·DogLegRight = (√n+1)(e^{5/2}+1)/2 + 16.76 + IIMaj/2, and the I_I piece is IIMaj. So:\nBn_new = 4F + 4T[...] + 40·1.3352 + 2(e⁴+1)(7Ĥ) + [(√n+1)(e^{5/2}+1)/2 + 16.76 + IIMaj/2] + IIMaj + 1.\n\nTotal IIMaj coefficient: 1/2 + 1 = 1.5 ✓.\n\nFor the margin blocks (marginExpanded):\n- The √n-linear part (√n+1)(e^{5/2}+1)/2: merges with mIdb: coefficient −((8/3)·2.7+192) = −199.2 → −[199.2 + (e^{5/2}+1)/2] = −[199.2 + 6.591] = −205.791. Note: (√n+1)(e^{5/2}+1)/2 = (√n+1)·(e^{5/2}+1)/2 — as a function of m: ((√m+1)·(e^{5/2}+1)/2) — the mIdb block is (−coef)·(√m+1) with coef = 199.2: new coef = 199.2 + (e^{5/2}+1)/2.\n- The 16.76: merges with mK: mK := −40·1.3352 − 2(e⁴+1)(7(3·2.9²+48) + 3.3) − 1 + 1 → −40·1.3352 − 2(e⁴+1)(7(3·2.9²+48)) − 16.76 − 1 + 1. (The 3.3 inside the (e⁴+1) bracket is REMOVED and the 16.76 appears raw.)\n- mIIM: (−1)·IIMaj → (−1.5)·IIMaj.\n- Everything else unchanged (the Ĥ log-blocks mL1sqb/mL1L2b/mL2sqb/mL1b/mL2b with the 2(e⁴+1)·7Ĥ expansion — the Ĥ inside is UNCHANGED: Hhat = 3Ĉ₀² + 10.5 log + 48 ✓ so the log-block coefficients 42·0.433²·(e⁴+1) etc. are UNCHANGED ✓).\n\nWait — let me double check the Ĥ-block expansion: 2(e⁴+1)·7Ĥ = 2(e⁴+1)·7·(3Ĉ₀² + 10.5 log(T+2) + 48) = 42(e⁴+1)Ĉ₀² + 147(e⁴+1)log(T+2) + 336(e⁴+1). The blocks:\n- mL1sqb: −(42·0.433²·(e⁴+1))·log²(T+2) ✓ unchanged\n- mL1L2b: −(42·2·0.433·0.886·(e⁴+1))·log·loglog ✓ unchanged\n- mL2sqb: −(42·0.886²·(e⁴+1))·loglog² ✓ unchanged\n- mL1b: −((42·2·0.433·2.9 + 147)·(e⁴+1))·log ✓ unchanged\n- mL2b: −(42·2·0.886·2.9·(e⁴+1))·loglog ✓ unchanged\n- mK: the constants −40·1.3352 − 336(e⁴+1) − 2(e⁴+1)·7·3·2.9²·... — wait, the old mK = −40·1.3352 − 2(e⁴+1)(7(3·2.9² + 48) + 3.3) − 1 + 1 — that has the Ĉ₀ constant part 3·2.9² = 25.23·... and the 3.3. New mK = −40·1.3352 − 2(e⁴+1)(7(3·2.9² + 48)) − 16.76 − 1 + 1 — the (7(3·2.9²+48)) = 7·(25.23+48) = 7·73.23 = 512.61 — that's the constant part of 2(e⁴+1)·7Ĥ's Ĉ₀-constant + 48 ✓ — and NO 3.3 (moved to the dog-leg R form) + 16.76 (the 2·8.38).\n\nNow the Bn def itself:\n```\ndef Bn (F : ℝ → ℝ) (n : ℝ) : ℝ :=\n  let T := Real.sqrt n + 1\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) +\n    2 * DogLegRight n + IIMaj n + 1\n```\nwith a new def DogLegRight:\n```\ndef DogLegRight (n : ℝ) : ℝ :=\n  ((Real.sqrt n + 1) / 2) * (Real.exp (5 / 2) + 1) / 2 + 8.38 + IIMaj n / 4\n```\nAnd 2·DogLegRight = (√n+1)(e^{5/2}+1)/2 + 16.76 + IIMaj/2.\n\nHmm — for the marginExpanded bookkeeping (marginTilde_eq_expanded's ring check), the blocks must sum to exactly marginTilde = sInfty − Bn + 1. The mIdb new coefficient (199.2 + (e^{5/2}+1)/2) and mIIM (−1.5) and mK (−16.76, no 3.3). Let me write the new forms:\n\n```\nprivate def mIdb : ℝ → ℝ := fun m => (-(((8/3)*2.7 + 192) + (Real.exp (5/2) + 1) / 2)) *\n  ((1.5 * Real.sqrt m + 1) ... \n```\nWAIT — the mIdb block's function: (−coef)·(√m+1) — the dog-leg's √n-linear part is (√n+1)(e^{5/2}+1)/2 — and mIdb's (−coef)·id(√m+1) — the id block: (−((8/3)·2.7+192))·(Real.sqrt m + 1) — the dog-leg part is ((√m+1)·(e^{5/2}+1)/2) = (e^{5/2}+1)/2·(√m+1) — so the TOTAL coefficient of (√m+1) in margiñ is −((8/3)·2.7+192) [from I_III] − (e^{5/2}+1)/2 [from dog-leg] ✓ — so mIdb_new = (−(((8/3)*2.7 + 192) + (Real.exp (5/2) + 1) / 2))·(Real.sqrt m + 1) ✓ — NOT the 1.5-scaled form (the dog-leg keeps T = √n+1!! — no k-shift — the inner function stays √m+1 ✓✓).\n\nAnd the I_I majorant IIMaj: the Bn's I_I piece is IIMaj(n) [coefficient 1] + dog-leg IIMaj/4 ×2 = IIMaj/2: total 1.5. mIIM = (−1.5)·IIMaj m ✓.\n\nAnd the DogLegRight def + the Bn def. Also marginTilde's def (lockstep with Bn via marginTilde_eq_sub's simp only).\n\nLet me also double-check marginTilde_eq_sub: marginTilde F n = sInftyLower n − Bn F n + 1 — marginTilde's def is literally (sInfty-expression) − (Bn-expression) + 1 — so if I change BOTH consistently, the simp only [marginTilde, Bn, sInftyLower] proof stays ✓.\n\nAnd marginTilde_eq_expanded: the blocks sum to the marginTilde form via ring — the mIdb/mIIM/mK changes are consistent on both sides ✓ ring verifies.\n\nAnd contDiffOn_mIdb: `contDiffOn_const.mul contDiffOn_sqrt_shift` — the block's constant changed (now a sum-of-constants including e^{5/2}) — contDiffOn_const.mul works for any constant ✓ no change needed.\n\nAnd contDiffOn_mIIM: mIIM = (−1.5)·IIMaj — the unfold + contDiffOn_const.mul ✓ unchanged (constant-agnostic).\n\nAnd marginTilde_convex's v-blocks: vId (mIdb's deriv) — the coefficient changed: vId's value = (−new-coef)·(psiOf id (√n+1)/(4n)) — the chain_rule_sqrt_shift_on form ✓ unchanged (constant-agnostic) — just the literal updates. vIIM: (−1.5)·IIMaj″ — the final assembly: 1/(2n) + nonnegs + (−1.5)·IIMaj″ ≥ 1/(2n) − 1.5·IIMaj″ > 0 ⟸ IIMaj″ < 1/(3n) ✓ lt_three + the bridge 1.5·(1/(3n)) = 1/(2n) (field_simp — hmm, the bridge `1.5·(1/(3n)) = 1/(2n)` — field_simp [hn0.ne'] closed it earlier ✓ or div_eq_div_iff).\n\nAnd the mK constant block deriv2_const_val ✓ unchanged (constant).\n\nSo the a2-i edits:\n1. `DogLegRight` def (new).\n2. `Bn` def: horizontal share → 2(e⁴+1)(7Ĥ) + 2·DogLegRight n.\n3. `marginTilde` def: lockstep.\n4. `marginExpanded` blocks: mIdb (new coefficient), mIIM (−1.5), mK (new).\n5. marginTilde_eq_expanded/eq_sub (should be ring/simp-clean).\n6. contDiffOn_mIdb/mIIM (probably fine as-is).\n7. marginTilde_convex's vId/vIIM literals + final linarith (the 1.5 bridge).\n\nThen a2-ii: base points (LiBasePoints rebuild at the new Bn — targets margin(8e4) ≈ 13 160, margin(10⁵) ≈ 88 911 — vs E-034's margins computed with Bn_DL: 13 160.089 / 88 910.825 — hmm — E-034's Bn_DL used 2·DogLegRight with VRightBound = IIMaj/4 ✓ same as the new Bn ✓ margins match ✓) + endpoint margins.\n\nWait — one nuance: E-034's Bn_DL has `2 * (e^4 + 1) * (7 * Hhat(T))` (no +3.3, replaced by DogLegRight) ✓ matches the new Bn's structure ✓. And E-034's margins (13 160.089 / 88 910.825 / 2.458e26) are computed with the certified-constant form (e^{5/2}+1 ≤ 13.19 charged as E52P1 = 13.19 ✓ same as Lean's (Real.exp (5/2)+1)/2 with the true exp value ✓ — hmm — the Lean Bn uses Real.exp (5/2) (the TRUE exp) while E-034 charges 13.19 ≥ e^{5/2}+1 = 13.182 — the Lean Bn uses the TRUE exp (5/2)+1 = 13.182+1 = 14.182?? — wait: Real.exp (5/2) = 12.1825, +1 = 13.1825 — E-034's E52P1 = 13.19 (charged upper) — so E-034's Bn_DL is SLIGHTLY bigger (13.19 vs 13.1825) than the Lean Bn (true 13.1825) — the margins differ by ~2·(13.19−13.1825)/2·(√n+1)·... ≈ 2·0.00375·(√n+1) ≈ 2 units at 8e4 — negligible ✓. So E-034's margins ≈ the Lean margins to within a few units ✓ targets fine.\n\nLet me start the a2-i edits. First the DogLegRight + Bn defs.","signature":""},{"type":"text","text":"Cherry-pick committed (`e11ad95`). Now **ii′-6-a2-i**: the Bn dog-leg redefinition. Starting with the `DogLegRight` def and the Bn/marginTilde/block updates."},{"type":"tool_use","id":"TodoWrite_591","name":"TodoWrite","input":{"todos":[{"content":"Cherry-pick bracket < 4/3 + IIMaj_deriv2_lt_three onto master (committed e11ad95)","status":"completed","activeForm":"Cherry-picking the bracket strengthening"},{"content":"ii′-6-a2-i: DogLegRight def + Bn/marginTilde/marginExpanded/mK dog-leg redefinition + eq_expanded/eq_sub + contDiffOn + convexity literals","status":"in_progress","activeForm":"Redefining Bn at the dog-leg constants"},{"content":"ii′-6-a2-ii: base-point re-certification (LiBasePoints at the new Bn) + endpoint margins","status":"pending","activeForm":"Re-certifying base points"},{"content":"Full build + axiom audit + commit ii′-6-a2; then ii′-6-c per the return map","status":"pending","activeForm":"Closing ii′-6-a2"}]},"extra_content":null},{"type":"tool_use","id":"Read_592","name":"Read","input":{"file_path":"D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiSkeleton.lean","offset":2421,"limit":45},"extra_content":null}]