[{"type":"thinking","thinking":"I see the structure of the Bn evaluations. This is a substantial rebuild. The Bn_FAlt_80000_le proof pattern:\n- hT: √80000+1 ≤ 284 (upper), and T-bounds\n- log T ≤ 5.7, log(T+2) ≤ 5.7, loglog ≤ 1.75\n- e⁴ ≤ 54.6015\n- FAlt T ≤ 273.82\n- Ĉ₀ T ≤ 6.9186, Ĥ T ≤ 251.46, Z2 T ≤ 7.6705\n- bracket ≤ 112.9637\n- IIMaj ≤ 2590.08\n- assemble: Bn ≤ 4·273.82 + 4·284·112.9637 + 53.408 + 2·55.6015·(7·251.46+3.3) + 2590.08 + 1 ≤ 329000\n\nAt c = 1.5 the new values (T = 1.5√n+1):\n- √80000 ∈ [282, 283]: T = 1.5√80000+1 ∈ [424, 425.5]\n- log T: log(424)... true log(425.26) = 6.0526. Certify log T ≤ 6.06 (e^{6.06} = e⁶·e^{0.06} = 403.43·1.0618 = 428.4 ≥ 425.5 ✓); log(T+2) ≤ 6.06 likewise (T+2 ≤ 427.5 ≤ 428.4 ✓). log T ≥ 6.04 (e^{6.04} = 403.43·e^{0.04} = 403.43·1.04081 = 419.9 ≤ 424 ✓).\n- loglog T: log(6.0526) = 1.8006. Certify ≤ 1.81 (e^{1.81} = e·e^{0.81} = 2.71828·2.2535 = 6.1257 ≥ 6.06 ✓); ≥ 1.79 (e^{1.79} = 2.71828·e^{0.79} = 2.71828·2.2034 = 5.9895 ≤ 6.04 ✓).\n- e^{16/9}+1 ≤ 6.93 (computed before: certify e^{16/9} ≤ 5.93 via e^{1.5}·e^{0.2778}: e^{1.5} ≤ 4.4818, e^{0.2778} ≤ 1.3240: product 5.9338 ≤ 5.934; so e^{16/9}+1 ≤ 6.934).\n- FAlt T: (1/2π)·T·log T + 0.137·log T + 0.443·loglog T + 2.463 ≤ (1/6)·425.5·6.06 + 0.137·6.06 + 0.443·1.81 + 2.463 = 429.755 + 0.830 + 0.802 + 2.463 = 433.85. Certify FAlt ≤ 434.\n- Ĉ₀ T ≤ 0.433·6.06 + 0.886·1.81 + 2.9 = 2.624 + 1.604 + 2.9 = 7.128 → ≤ 7.13.\n- Ĥ T ≤ 3·7.13² + 10.5·6.06 + 48 = 3·50.84 + 63.63 + 48 = 264.15 → ≤ 264.2.\n- Z2 T ≤ 0.6·6.06 + 0.886·1.81 + 2.7 = 3.636+1.604+2.7 = 7.94 → ≤ 7.94.\n- bracket: (2/3)·7.94 + 10.5·6.06 + 48 = 5.293 + 63.63 + 48 = 116.93 → ≤ 116.93.\n- term2: 4·425.5·116.93 = 1702·116.93 = 199,015.\n- term4: 2·6.934·(7·264.2+5.94) = 13.868·1855.34 = 25,727.\n- IIMaj: 1.5·4.56·(√80000+1)·2 ≤ 1.5·4.56·284·2 = 3,883.6 — wait, the Bn now has 1.5·IIMaj and IIMaj_le_two_mul gives IIMaj ≤ 4.56(√n+1)·2 = 4.56·284·2 = 2590.08·... = 5180.16/2... 4.56·284 = 1295.04; ×2 = 2590.08; ×1.5 = 3885.12.\n- bridge: 4·FAlt ≤ 4·434 = 1736.\n- Bn ≤ 1736 + 199015 + 53.41 + 25727 + 3885.12 + 1 = 230,417.5. \n\nCertify Bn_FAlt_80000_le : Bn ≤ 231,000. margin(8e4) ≥ 336000 − 231000 + 1 = 105,001. ✓ > 0.\n\nFor hbase2 need margin(8e4) upper: margin ≤ 339000 − Bn_lo + 1: Bn_lo: lower bound on Bn(8e4): the big terms only: term2 ≥ 4·424·((2/3)·Z2_lo + 10.5·log(T+2)_lo + 48): Z2_lo = 0.6·6.04+0.886·1.79+2.7 = 3.624+1.586+2.7 = 7.91... wait for a LOWER bound on the bracket I need lower bounds on the logs: log(T+2) ≥ 6.04, loglog ≥ 1.79: bracket_lo = (2/3)(0.6·6.04+0.886·1.79+2.7) + 10.5·6.04+48 = (2/3)·7.91+63.42+48 = 5.273+111.42 = 116.69. term2 ≥ 4·424·116.69 = 1696·116.69 = 197,906. Plus others ≥ 0 (bridge ≥ 0, term4 ≥ 0, IIMaj ≥ 0, 53.4+1): Bn ≥ 197,906 + 53.4 = 197,960. Certify Bn ≥ 197,000. margin(8e4) ≤ 339000 − 197000 + 1 = 142,001.\n\nAt 1e5: √100000 ∈ [316, 317]: T = 1.5√100000+1 ∈ [475, 476.5]:\n- log T ≤ 6.17 (e^{6.17} = e⁶·e^{0.17} = 403.43·1.1853 = 478.2 ≥ 476.5 ✓); log(T+2) ≤ 6.17 (T+2 ≤ 478.5?? 476.5+2 = 478.5 > 478.2 — tighten: e^{6.18} = 403.43·1.1972 = 483.0 ≥ 478.5 ✓ so log(T+2) ≤ 6.18).\n- Actually simplify: use log T ≤ 6.18 for both? No — for the upper bound on Bn use the worst: log T ≤ 6.17, log(T+2) ≤ 6.18. loglog T ≤ log(6.17) ≤ 1.822 (e^{1.822} = e·e^{0.822} = 2.71828·2.2752 = 6.185 ≥ 6.17 ✓; e^{1.82} = 6.172 ≥ 6.17 ✓ barely — use 1.83 for comfort: e^{1.83} = e·e^{0.83} = 2.71828·2.2933 = 6.2338 ≥ 6.18 ✓). Use loglog ≤ 1.83, loglog(T+2) ≤ 1.83.\n- lower: log T ≥ 6.15 (e^{6.15} = 403.43·e^{0.15} = 403.43·1.1618 = 468.7 ≤ 475 ✓); log(T+2) ≥ 6.15 ✓ (477 ≥ 468.7 ✓); loglog ≥ 1.81 (e^{1.81} = 6.1257 ≥ 6.15?? NO — need e^{1.81} ≤ log-argument 6.15: 6.1257 ≤ 6.15 ✓ so loglog T ≥ 1.81 ✓).\n- FAlt ≤ (1/6)·476.5·6.17 + 0.137·6.17 + 0.443·1.83 + 2.463 = 490.1+0.845+0.811+2.463 = 494.2 → 4·494.2 = 1977.\n- Ĉ₀ ≤ 0.433·6.18+0.886·1.83+2.9 = 2.676+1.621+2.9 = 7.197 → 7.20.\n- Ĥ ≤ 3·51.84+10.5·6.18+48 = 155.52+64.89+48 = 268.41 → 268.5.\n- Z2 ≤ 0.6·6.18+0.886·1.83+2.7 = 3.708+1.621+2.7 = 8.029 → 8.03.\n- bracket ≤ (2/3)·8.03+10.5·6.18+48 = 5.353+64.89+48 = 118.24.\n- term2 ≤ 4·476.5·118.24 = 1906·118.24 = 225,365.\n- term4 ≤ 2·6.934·(7·268.5+5.94) = 13.868·1885.44 = 26,146.\n- IIMaj: 1.5·4.56·318·2 = 4,354.6 (using √100000 ≤ 317 → √n+1 ≤ 318).\n- Bn ≤ 1977+225365+53.4+26146+4354.6+1 = 257,897. Certify ≤ 258,000. margin(1e5) ≥ 432500−258000+1 = 174,501.\n- Bn lower: bracket_lo = (2/3)(0.6·6.15+0.886·1.81+2.7)+10.5·6.15+48 = (2/3)·8.023+64.575+48 = 5.349+112.575 = 117.92; term2 ≥ 4·475·117.92 = 1900·117.92 = 224,048; Bn ≥ 224,100. margin(1e5) ≤ 339000?? — no upper needed on margin(1e5) for the two theorems (only margin(8e4) < margin(1e5): margin(8e4) ≤ 142,001 < 174,501 ≤ margin(1e5) ✓✓).\n\nGreat — targets:\n- Bn_FAlt_80000_le : Bn FAlt 80000 ≤ 231000\n- Bn_FAlt_80000_ge : 197000 ≤ Bn FAlt 80000\n- marginTilde_FAlt_80000_pos : 0 < marginTilde FAlt 80000 (via ≥ 105001)\n- Bn_FAlt_100000_le : Bn FAlt 100000 ≤ 258000\n- marginTilde_FAlt_80000_lt_100000 : marginTilde FAlt 80000 < marginTilde FAlt 100000 (via 142001 < 174501; needs Bn_FAlt_100000_ge : 224000 ≤ Bn FAlt 100000 and the sInftyLower bounds — existing: sInftyLower_100000_lo ≥ 432500 ✓; sInftyLower_80000_up ≤ 339000 ✓).\n\nWait — margin(1e5) lower = sInftyLower_100000_lo − Bn_up + 1 = 432500 − 258000 + 1 = 174501 ✓.\n\nNote: the old file ALSO has sInftyLower_100000_up? For hbase2's RHS I only need margin(1e5) LOWER (not upper) ✓ and margin(8e4) upper. The old file: does it have sInftyLower_100000_up? Lines 229+ has only _lo (margin(1e5) upper not needed). ✓.\n\nNow, these new evaluation proofs replace the old Bn_FAlt_*_le/ge (which will break with the new Bn anyway). I need to see the rest of the old file (402–720) to know the exact theorem inventory and the margin theorems + verified endpoint. Let me read the rest.","signature":""},{"type":"text","text":"Reading the rest of LiBasePoints (the lower bounds, the margin theorems, and the verified headline)."},{"type":"tool_use","id":"Read_77","name":"Read","input":{"file_path":"D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiBasePoints.lean","offset":402,"limit":290},"extra_content":null}]