[{"type":"thinking","thinking":"I see the structure: `4.56 = 2·2·0.57·4` (the 4.56 constant in the psiTilde rewriting: IIMaj = 4.56·(√n+1)·(exp-stuff)·... where 4.56 = 2×0.57×4 with 4 = 2^2 from the 2^{−(2√n−2)} = 4·2^{−2√n} rewrite). So the literal 0.57 → 4.56 propagation: the psiTilde machine has 4.56 baked in.\n\nDownstream impact of changing 0.57 → 0.8: 4.56 → 2×0.8×4 = 6.4. Sites:\n1. IIMaj_eq_psiTilde_sqrt (the `show` with 4.56): literal → 6.4.\n2. IIMaj_le_two_mul: 4.56 literal → 6.4 (statement AND proof — statement itself has 4.56!). Consumers: LiBasePoints hII : IIMaj 80000 ≤ 2590.08 via IIMaj_le_two_mul 80000: 4.56·(√80000+1)·2 = 4.56·283.84·2 = 2590.08 ✓. With 6.4: 6.4·283.84·2 = 3633.2 — the hII bound 2590.08 must change to e.g. 3633.2. Then hDL (2·DogLegRight ≤ 3184) uses IIMaj/4 ≤ 2590.08/4 = 647.52 — with the new constant: 3633.2/4 = 908.3. Then hDL's total: 2·DogLegRight = (√n+1)(e^{5/2}+1)/2 + 16.76 + IIMaj/2 ≤ 283.84·13.183/2·... let me recompute: hDL : 2·DogLegRight 80000 ≤ 3184: DogLegRight = ((√n+1)/2)(e^{5/2}+1)/2 + 8.38 + IIMaj/4. 2·DogLegRight = ((√n+1)/2)(e^{5/2}+1) + 16.76 + IIMaj/2 ≤ (283.84/2)·13.183 + 16.76 + 908.3·... hmm wait IIMaj/2 ≤ 3633.2/2 = 1816.6. Total: 141.92·13.183 = 1871.1 + 16.76 + 1816.6 = 3704.5 vs old 3184. The Bn upper bound 332000 has slack? Bn(8e4) ≤ 332000 certified with the old IIMaj/4 ≤ 647.52 contribution... the increase is (1816.6 − 647.5)·2... no wait: hDL's increase = new IIMaj/2 − old IIMaj/2 = 1816.6 − 647.5 = 1169. So the new 2·DogLegRight ≤ 3184 + 1169 = 4353, and Bn(8e4) upper bound increases by 1169: 332000 → 333169. Then margin(8e4) = sInftyLower − Bn + 1 ≥ [certified sInftyLower lower bound] − 333169 + 1. The OLD margin was ≥ 4001 (from summary: margin(8e4) ≥ 4,001). Wait — actually from the a2 work: \"Bn(8e4) ∈ [273,000, 332,000], Bn(10⁵) ≤ 352,000 ⟹ margin(8e4) ≥ 4,001 > 0, margin(8e4) ≤ 66,001 < margin(10⁵) ≥ 80,501\". So margin(8e4) ≥ 4001 with Bn ≤ 332000. With Bn ≤ 333169: margin ≥ 4001 − 1169 = 2832 > 0 ✓ STILL POSITIVE. And margin(8e4) ≤ 66001 (from Bn ≥ 273000 — the LOWER bound uses hDLnn : 0 ≤ 2·DogLegRight — nonnegativity only ✓ constant-change-immune). margin(10⁵) ≥ 80501 with Bn(10⁵) ≤ 352000: increase by IIMaj/2(10⁵): old hII4 : IIMaj/4 ≤ 2900.16/4 (2900.16 = 4.56·(√10⁵+1)·2 = 4.56·317.23·2) → new 6.4·317.23·2 = 4060.6; IIMaj/2: 2030.3 vs old 1450.1: increase 580. Bn(10⁵) ≤ 352580: margin(10⁵) ≥ 80501 − 580 = 79921 > 66001 ✓ hbase2 SURVIVES (66001 ≤ 79921). Phew — both base points survive the constant change with margin to spare.\n\nBut the scripts need literal updates (2590.08 → 3633.2-ish etc.). Mechanical.\n\n3. iimaj_bracket_bound (LiSkeleton:1511): need to check whether 0.57/4.56 appears in its statement/proof. The statement: let me read it. It says {s} (hs : √20 ≤ s) — the bracket is |psiTilde''(s)|·(4s²).../psiTilde(s) < 4/3? As a RATIO the 4.56 cancels — but the certified product 4.56×1.2237×0.2124×1.1099 HAS 4.56 as a factor!! So the bracket proof literally uses 4.56 — meaning the ratio does NOT cancel in the proof?! Let me read it.\n\nActually wait — 4.56×1.2237×0.2124×1.1099 = 1.3155. If the bracket is deriv2-related over IIMaj, 4.56 in the numerator AND denominator... but the product shows 4.56 only once. So either the bracket bound has 4.56 in it linearly (i.e., NOT a pure ratio) or... Let me read the statement.","signature":""},{"type":"text","text":"The `0.57` propagates as `4.56 = 2·2·0.57·4` into the psiTilde machine. Need to check how `iimaj_bracket_bound` uses it — the certified product `4.56×1.2237×0.2124×1.1099` has 4.56 as a linear factor, which would break under a constant change."},{"type":"tool_use","id":"Read_33","name":"Read","input":{"file_path":"D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiSkeleton.lean","offset":1495,"limit":120},"extra_content":null}]