[{"type":"thinking","thinking":"Now I can write both proof updates. For n = 10⁵:\n- hZ2 : Z2hat T ≤ 0.6·5.8 + 0.886·1.77 + 2.7 = 3.48 + 1.56822 + 2.7 = 7.74822. Bound: 7.7483.\n- hbr : (2/3)·Z2hat + 10.5·log(T+2) + 48 ≤ (2/3)·7.7483 + 10.5·5.8 + 48 = 5.1655 + 60.9 + 48 = 114.0655. Bound: 114.0655.\n- hterm2 : 4·318·114.0655 = 1272·114.0655 = 145,091.3.\n- Total: 4·311.45 = 1,245.8 + 145,091.3 + 53.408 + 2·55.6015·(7·255.05+3.3) + 2,900.16 + 1. The hterm4: 2·55.6015·1788.65 = 111.203·1788.65 = 198,903.9? Compute: 111.203·1788.65 ≈ 198,904. Total: 1,245.8+145,091.3+53.408+198,904+2,900.16+1 = 348,195.9 ≤ 350,000 ✓ slack 1,804.\n\nHmm wait — earlier estimate said 349,329; now 348,196 with the actual hbr 114.0655 — fine, ≤ 350,000 with slack ~1,800.\n\nFor n = 8×10⁴: computed 328,176 ≤ 329,000 ✓ slack 824.\n\nNow write the edits for both proofs. For 80000:\n- hZ2 : Z2hat (√80000+1) ≤ 7.6705: unfold Z2hat; linarith [hlogT2, hloglogT2] (0.6·5.7 + 0.886·1.75 + 2.7 = 3.42+1.5505+2.7 = 7.6705 ✓ exact).\n- hZ2nn : 0 ≤ Z2hat: linarith [hlogT2nn, hloglogT2nn].\n- hbr : (2/3)·Z2hat + 10.5·log(T+2) + 48 ≤ (2/3)·7.6705 + 59.85 + 48 = 5.11367+107.85 = 112.9637 ✓.\n- hbrnn.\n- hterm2 : 4·284·112.9637.\n- statement: 329000.\n\nLet me make the edits with python for precision.","signature":""},{"type":"text","text":"Writing both proof updates."},{"type":"tool_use","id":"Bash_222","name":"Bash","input":{"command":"cd D:/Python/Quantum && python - << 'EOF'\np = 'research/lean/NguyenRH1/NguyenRH1/LiBasePoints.lean'\nsrc = open(p, encoding='utf-8').read()\nsubs = [\n# === Bn_FAlt_80000_le: statement + Z2 block ===\n(\"\"\"theorem Bn_FAlt_80000_le : Bn FAlt 80000 ≤ (328000 : ℝ) := by\"\"\",\n \"\"\"theorem Bn_FAlt_80000_le : Bn FAlt 80000 ≤ (329000 : ℝ) := by\"\"\"),\n(\"\"\"  -- the big bracket: 0 ≤ · ≤ 112.4624\n  have hbr : (2 / 3 : ℝ) * C0hat (Real.sqrt 80000 + 1) +\n      10.5 * Real.log (Real.sqrt 80000 + 1 + 2) + 48 ≤ 112.4624 := by\n    have h1 := mul_le_mul_of_nonneg_left hC0 (by norm_num : (0 : ℝ) ≤ 2 / 3)\n    have h2 := mul_le_mul_of_nonneg_left hlogT2 (by norm_num : (0 : ℝ) ≤ 10.5)\n    linarith [h1, h2]\n  have hbrnn : (0 : ℝ) ≤ (2 / 3 : ℝ) * C0hat (Real.sqrt 80000 + 1) +\n      10.5 * Real.log (Real.sqrt 80000 + 1 + 2) + 48 := by\n    linarith [hC0pos, hlogT2nn]\"\"\",\n \"\"\"  -- Z₂ T ≤ 7.6705 and ≥ 0 (the corrected I_III count, D-071)\n  have hZ2 : Z2hat (Real.sqrt 80000 + 1) ≤ (7.6705 : ℝ) := by\n    unfold Z2hat\n    linarith [hlogT2, hloglogT2]\n  have hZ2nn : (0 : ℝ) ≤ Z2hat (Real.sqrt 80000 + 1) := by\n    unfold Z2hat\n    linarith [hlogT2nn, hloglogT2nn]\n  -- the big bracket: 0 ≤ · ≤ 112.9637\n  have hbr : (2 / 3 : ℝ) * Z2hat (Real.sqrt 80000 + 1) +\n      10.5 * Real.log (Real.sqrt 80000 + 1 + 2) + 48 ≤ 112.9637 := by\n    have h1 := mul_le_mul_of_nonneg_left hZ2 (by norm_num : (0 : ℝ) ≤ 2 / 3)\n    have h2 := mul_le_mul_of_nonneg_left hlogT2 (by norm_num : (0 : ℝ) ≤ 10.5)\n    linarith [h1, h2]\n  have hbrnn : (0 : ℝ) ≤ (2 / 3 : ℝ) * Z2hat (Real.sqrt 80000 + 1) +\n      10.5 * Real.log (Real.sqrt 80000 + 1 + 2) + 48 := by\n    linarith [hZ2nn, hlogT2nn]\"\"\"),\n(\"\"\"  have hterm2 : 4 * (Real.sqrt 80000 + 1) * ((2 / 3) * C0hat (Real.sqrt 80000 + 1) +\n      10.5 * Real.log (Real.sqrt 80000 + 1 + 2) + 48) ≤ 4 * 284 * 112.4624 := by\"\"\",\n \"\"\"  have hterm2 : 4 * (Real.sqrt 80000 + 1) * ((2 / 3) * Z2hat (Real.sqrt 80000 + 1) +\n      10.5 * Real.log (Real.sqrt 80000 + 1 + 2) + 48) ≤ 4 * 284 * 112.9637 := by\"\"\"),\n# === Bn_FAlt_100000_le: statement + Z2 block ===\n(\"\"\"theorem Bn_FAlt_100000_le : Bn FAlt 100000 ≤ (347000 : ℝ) := by\"\"\",\n \"\"\"theorem Bn_FAlt_100000_le : Bn FAlt 100000 ≤ (350000 : ℝ) := by\"\"\"),\n(\"\"\"  -- the big bracket: 0 ≤ · ≤ 113.5532\n  have hbr : (2 / 3 : ℝ) * C0hat (Real.sqrt 100000 + 1) +\n      10.5 * Real.log (Real.sqrt 100000 + 1 + 2) + 48 ≤ 113.5532 := by\n    have h1 := mul_le_mul_of_nonneg_left hC0 (by norm_num : (0 : ℝ) ≤ 2 / 3)\n    have h2 := mul_le_mul_of_nonneg_left hlogT2 (by norm_num : (0 : ℝ) ≤ 10.5)\n    linarith [h1, h2]\n  have hbrnn : (0 : ℝ) ≤ (2 / 3 : ℝ) * C0hat (Real.sqrt 100000 + 1) +\n      10.5 * Real.log (Real.sqrt 100000 + 1 + 2) + 48 := by\n    linarith [hC0pos, hlogT2nn]\"\"\",\n \"\"\"  -- Z₂ T ≤ 7.7483 and ≥ 0 (the corrected I_III count, D-071)\n  have hZ2 : Z2hat (Real.sqrt 100000 + 1) ≤ (7.7483 : ℝ) := by\n    unfold Z2hat\n    linarith [hlogT2, hloglogT2]\n  have hZ2nn : (0 : ℝ) ≤ Z2hat (Real.sqrt 100000 + 1) := by\n    unfold Z2hat\n    linarith [hlogT2nn, hloglogT2nn]\n  -- the big bracket: 0 ≤ · ≤ 114.0655\n  have hbr : (2 / 3 : ℝ) * Z2hat (Real.sqrt 100000 + 1) +\n      10.5 * Real.log (Real.sqrt 100000 + 1 + 2) + 48 ≤ 114.0655 := by\n    have h1 := mul_le_mul_of_nonneg_left hZ2 (by norm_num : (0 : ℝ) ≤ 2 / 3)\n    have h2 := mul_le_mul_of_nonneg_left hlogT2 (by norm_num : (0 : ℝ) ≤ 10.5)\n    linarith [h1, h2]\n  have hbrnn : (0 : ℝ) ≤ (2 / 3 : ℝ) * Z2hat (Real.sqrt 100000 + 1) +\n      10.5 * Real.log (Real.sqrt 100000 + 1 + 2) + 48 := by\n    linarith [hZ2nn, hlogT2nn]\"\"\"),\n(\"\"\"  have hterm2 : 4 * (Real.sqrt 100000 + 1) * ((2 / 3) * C0hat (Real.sqrt 100000 + 1) +\n      10.5 * Real.log (Real.sqrt 100000 + 1 + 2) + 48) ≤ 4 * 318 * 113.5532 := by\"\"\",\n \"\"\"  have hterm2 : 4 * (Real.sqrt 100000 + 1) * ((2 / 3) * Z2hat (Real.sqrt 100000 + 1) +\n      10.5 * Real.log (Real.sqrt 100000 + 1 + 2) + 48) ≤ 4 * 318 * 114.0655 := by\"\"\"),\n# === doc comments for the two margin base points ===\n(\"\"\"/-- [PROVED session 012] hbase: margiñ(8×10⁴) > 0. Certified:\nsInftyLower ≥ 336 000, Bn ≤ 328 000 ⟹ margin ≥ 8 001 (true 15 764). -/\"\"\",\n \"\"\"/-- [PROVED session 012; values updated session 038 for the D-071\ncorrection] hbase: margiñ(8×10⁴) > 0. Certified: sInftyLower ≥ 336 000,\nBn ≤ 329 000 ⟹ margin ≥ 7 001 (true 14 682 with the corrected Bn). -/\"\"\"),\n(\"\"\"/-- [PROVED session 012] hbase2: margiñ(8×10⁴) < margiñ(10⁵).\nCertified: margiñ(8×10⁴) ≤ 339 000 − 274 000 + 1 = 65 001 and\nmargiñ(10⁵) ≥ 432 500 − 347 000 + 1 = 85 501. -/\"\"\",\n \"\"\"/-- [PROVED session 012; values updated session 038 for the D-071\ncorrection] hbase2: margiñ(8×10⁴) < margiñ(10⁵).\nCertified: margiñ(8×10⁴) ≤ 339 000 − 274 000 + 1 = 65 001 and\nmargiñ(10⁵) ≥ 432 500 − 350 000 + 1 = 82 501. -/\"\"\"),\n]\ncount = 0\nfor old, new in subs:\n    if old in src:\n        src = src.replace(old, new)\n        count += 1\n    else:\n        print('MISS:', old[:70].replace('\\n',' | '))\nopen(p, 'w', encoding='utf-8').write(src)\nprint(f'{count}/{len(subs)} substitutions applied')\nEOF","description":"Update base-point proofs for corrected Bn"},"extra_content":null}]