← Session 33 — ii′-3b/3c crux + F-D finding; Lagarias letter package
thinking (3587 chars)
The stated certified bounds:
- `Bn_FAlt_80000_le : Bn FAlt 80000 ≤ 328000` (true old ≈ 322,309; new ≈ 323,391) — 328,000 still works with slack ~4,600.
- `Bn_FAlt_80000_ge : 274000 ≤ Bn FAlt 80000` (lower bound — still true, new value is bigger).
- `Bn_FAlt_100000_le : Bn FAlt 100000 ≤ 347000` (new ≈ 343,097) — works, slack ~3,900.
- `marginTilde_FAlt_80000_pos : 0 < marginTilde FAlt 80000` — via sInftyLower_80000_lo − Bn_FAlt_80000_le + 1 > 0: sInftyLower(8×10⁴) ≥ ~338,074 (sInftyLower_80000_lo's value — check) − 328,000 + 1 > 0 ✓ if the lo bound is ≥ 328,001-ish.
- `marginTilde_FAlt_80000_lt_100000` — via sInftyLower_80000_up, Bn_FAlt_80000_ge (lower), sInftyLower_100000_lo, Bn_FAlt_100000_le (upper).
So the strategy: KEEP the stated bound values (328000, 347000) — they still hold for the corrected Bn with slack ~4–5k. Only the PROOFS need updating:
1. In Bn_FAlt_80000_le's proof: replace hC0 with hZ2 (Z2hat ≤ 7.6705), update hbr to (2/3)·Z2hat ≤ 112.9637, hterm2 uses Z2hat, and the assembly handles 4·FAlt (≤ 4·273.82 = 1095.28) instead of 2·FAlt. New total: 1095.28 + 4·284·112.9637 + 53.41 + 2·55.6015·1763.52 + 2590.08 + 1 ≈ 1095.28 + 128,351 + 53.41 + 196,148 + 2590.08 + 1 ≈ 328,238?? — that EXCEEDS 328,000! Let me compute more carefully:
- 4·FAlt ≤ 4·273.82 = 1095.28
- 4T·hbr: 4·284·112.9637 = 1136·112.9637 = 128,326.7
- 40·1.3352 = 53.408
- 2(e⁴+1)(7Ĥ+3.3): 2·55.6015·(7·251.46+3.3) = 111.203·1763.52 = 196,109.3
- IIMaj ≤ 2590.08
- +1
Total: 1095.28+128,326.7+53.408+196,109.3+2590.08+1 = 328,175.8 > 328,000. The stated bound 328,000 FAILS for the corrected Bn! New true Bn(8×10⁴) ≈ 323,391; need a new certified upper bound — say 329,000 (slack ~5,600 over true, and the certified computation above gives 328,176 ≤ 329,000 ✓).
Check the downstream: marginTilde_FAlt_80000_pos needs sInftyLower_80000_lo > 329,000 − 1: what's sInftyLower_80000_lo's certified value? True sInftyLower(8×10⁴) ≈ 338,074 — certified lower bound probably ~337,000-ish. 337,000 − 329,000 + 1 = 8,001 > 0 ✓ (matches the designed ≥ 8001! The original certified margin ≥ 8001 came from exactly this: lo − up + 1 = 337,000 − 328,000 + 1 = 8,001. With the corrected Bn's new upper bound 329,000: 337,000 − 329,000 + 1 = 8,001 — SAME margin if I keep the lo bound and raise the up bound by exactly 1,000! Clever — the correction cost at 8×10⁴ is 1,082 ≈ 1,000; if I set the new certified upper bound at 329,000, margin ≥ 8,001 exactly as before... but the true margin dropped by 1,082: 15,764 − 1,082 = 14,682 — and the certified ≥ 8001 still holds ✓✓).
Hmm wait — let me double check: original: up = 328,000, lo = ? The margin ≥ 8001 was designed as: lo(8×10⁴) ≥ 336,000? Let me check sInftyLower_80000_lo's certified value. Let me grep.
2. Similarly Bn_FAlt_100000_le (347,000): new true ≈ 343,097 — recompute the certified total with Z2 and 4F at n=10⁵ (T ≤ 318): hFAlt ≤ ~292.5?, hZ2 ≤ 0.6·5.76+0.886·1.75+2.7 = 3.456+1.55+2.7 = 7.71, hbr ≤ (2/3)·7.71+10.5·5.76+48 = 5.14+60.48+48 = 113.62, 4T·hbr ≤ 4·318·113.62 = 144,525, 4·FAlt ≤ 4·292.5 = 1170, horizontal ≤ 2·55.6015·(7·~257+3.3) ≈ 111.2·1802 ≈ 200,400, IIMaj ≤ ~3180, +53.4+1. Total ≈ 1170+144,525+53.4+200,400+3180+1 = 349,329 > 347,000 — also fails. New bound: 350,000? Then margin(10⁵) ≥ lo(10⁵) − 350,000 + 1: lo(10⁵) ≈ 435,501-ish (certified ≥ 85,501 + 347,000 = 432,501? — the certified margin ≥ 85,501 came from lo − up + 1 = 432,501 − 347,000 + 1 = 85,501+1... hmm let me just read the actual certified lo values).
Let me look at the sInftyLower certified bounds and the margin proofs.
Checking the certified sInftyLower bounds that feed the margin proofs.
tool: Grep ok
input
{
"pattern": "theorem sInftyLower_80000_lo|theorem sInftyLower_80000_up|theorem sInftyLower_100000_lo|theorem sInftyLower_100000_up",
"path": "D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiBasePoints.lean",
"output_mode": "content",
"-n": true,
"-A": 2
}
result 2026-08-25 05:11:43Z → 2026-08-25 05:11:44Z
mode='content' num_files=0 filenames=[] content="197:theorem sInftyLower_80000_lo : (336000 : ℝ) ≤ sInftyLower 80000 := by\n198- rw [sInftyLower_eq']\n199- have h1 := log80000_ge\n--\n213:theorem sInftyLower_80000_up : sInftyLower 80000 ≤ (339000 : ℝ) := by\n214- rw [sInftyLower_eq']\n215- have h1 := log80000_le\n--\n229:theorem sInftyLower_100000_lo : (432500 : ℝ) ≤ sInftyLower 100000 := by\n230- rw [sInftyLower_eq']\n231- have h1 := log100000_ge" num_lines=11 num_matches=None applied_limit=None applied_offset=None
raw content_json (database row)
[{"type":"thinking","thinking":"The stated certified bounds:\n- `Bn_FAlt_80000_le : Bn FAlt 80000 ≤ 328000` (true old ≈ 322,309; new ≈ 323,391) — 328,000 still works with slack ~4,600.\n- `Bn_FAlt_80000_ge : 274000 ≤ Bn FAlt 80000` (lower bound — still true, new value is bigger).\n- `Bn_FAlt_100000_le : Bn FAlt 100000 ≤ 347000` (new ≈ 343,097) — works, slack ~3,900.\n- `marginTilde_FAlt_80000_pos : 0 < marginTilde FAlt 80000` — via sInftyLower_80000_lo − Bn_FAlt_80000_le + 1 > 0: sInftyLower(8×10⁴) ≥ ~338,074 (sInftyLower_80000_lo's value — check) − 328,000 + 1 > 0 ✓ if the lo bound is ≥ 328,001-ish.\n- `marginTilde_FAlt_80000_lt_100000` — via sInftyLower_80000_up, Bn_FAlt_80000_ge (lower), sInftyLower_100000_lo, Bn_FAlt_100000_le (upper).\n\nSo the strategy: KEEP the stated bound values (328000, 347000) — they still hold for the corrected Bn with slack ~4–5k. Only the PROOFS need updating:\n1. In Bn_FAlt_80000_le's proof: replace hC0 with hZ2 (Z2hat ≤ 7.6705), update hbr to (2/3)·Z2hat ≤ 112.9637, hterm2 uses Z2hat, and the assembly handles 4·FAlt (≤ 4·273.82 = 1095.28) instead of 2·FAlt. New total: 1095.28 + 4·284·112.9637 + 53.41 + 2·55.6015·1763.52 + 2590.08 + 1 ≈ 1095.28 + 128,351 + 53.41 + 196,148 + 2590.08 + 1 ≈ 328,238?? — that EXCEEDS 328,000! Let me compute more carefully:\n - 4·FAlt ≤ 4·273.82 = 1095.28\n - 4T·hbr: 4·284·112.9637 = 1136·112.9637 = 128,326.7\n - 40·1.3352 = 53.408\n - 2(e⁴+1)(7Ĥ+3.3): 2·55.6015·(7·251.46+3.3) = 111.203·1763.52 = 196,109.3\n - IIMaj ≤ 2590.08\n - +1\n Total: 1095.28+128,326.7+53.408+196,109.3+2590.08+1 = 328,175.8 > 328,000. The stated bound 328,000 FAILS for the corrected Bn! New true Bn(8×10⁴) ≈ 323,391; need a new certified upper bound — say 329,000 (slack ~5,600 over true, and the certified computation above gives 328,176 ≤ 329,000 ✓).\n\n Check the downstream: marginTilde_FAlt_80000_pos needs sInftyLower_80000_lo > 329,000 − 1: what's sInftyLower_80000_lo's certified value? True sInftyLower(8×10⁴) ≈ 338,074 — certified lower bound probably ~337,000-ish. 337,000 − 329,000 + 1 = 8,001 > 0 ✓ (matches the designed ≥ 8001! The original certified margin ≥ 8001 came from exactly this: lo − up + 1 = 337,000 − 328,000 + 1 = 8,001. With the corrected Bn's new upper bound 329,000: 337,000 − 329,000 + 1 = 8,001 — SAME margin if I keep the lo bound and raise the up bound by exactly 1,000! Clever — the correction cost at 8×10⁴ is 1,082 ≈ 1,000; if I set the new certified upper bound at 329,000, margin ≥ 8,001 exactly as before... but the true margin dropped by 1,082: 15,764 − 1,082 = 14,682 — and the certified ≥ 8001 still holds ✓✓).\n\n Hmm wait — let me double check: original: up = 328,000, lo = ? The margin ≥ 8001 was designed as: lo(8×10⁴) ≥ 336,000? Let me check sInftyLower_80000_lo's certified value. Let me grep.\n\n2. Similarly Bn_FAlt_100000_le (347,000): new true ≈ 343,097 — recompute the certified total with Z2 and 4F at n=10⁵ (T ≤ 318): hFAlt ≤ ~292.5?, hZ2 ≤ 0.6·5.76+0.886·1.75+2.7 = 3.456+1.55+2.7 = 7.71, hbr ≤ (2/3)·7.71+10.5·5.76+48 = 5.14+60.48+48 = 113.62, 4T·hbr ≤ 4·318·113.62 = 144,525, 4·FAlt ≤ 4·292.5 = 1170, horizontal ≤ 2·55.6015·(7·~257+3.3) ≈ 111.2·1802 ≈ 200,400, IIMaj ≤ ~3180, +53.4+1. Total ≈ 1170+144,525+53.4+200,400+3180+1 = 349,329 > 347,000 — also fails. New bound: 350,000? Then margin(10⁵) ≥ lo(10⁵) − 350,000 + 1: lo(10⁵) ≈ 435,501-ish (certified ≥ 85,501 + 347,000 = 432,501? — the certified margin ≥ 85,501 came from lo − up + 1 = 432,501 − 347,000 + 1 = 85,501+1... hmm let me just read the actual certified lo values).\n\nLet me look at the sInftyLower certified bounds and the margin proofs.","signature":""},{"type":"text","text":"Checking the certified sInftyLower bounds that feed the margin proofs."},{"type":"tool_use","id":"Grep_219","name":"Grep","input":{"pattern":"theorem sInftyLower_80000_lo|theorem sInftyLower_80000_up|theorem sInftyLower_100000_lo|theorem sInftyLower_100000_up","path":"D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiBasePoints.lean","output_mode":"content","-n":true,"-A":2},"extra_content":null}]