{
 "what": "The registry's asterisked upper bound C84b <= 1.999281, decided from the chain of the note it cites: the note's own constant 0.0007 holds (C84b <= 1.9993, given BSSZ2026 §5, the note's lattice-doubling lemma and the regulator bound), and no choice of the chain's parameters reaches the 0.000719 the registry quotes.",
 "generated": "2026-09-29",
 "registry": {
  "commit": "2c1968cd520b60f1cf3cf50749f7285e800e4e15",
  "row": "| $1.999281$ | [Al26], [EPF52] | Althoefer (28 May 2026), posted on the Erdős Problems forum [EPF52]; a ChatGPT 5.5 long-thinking optimization of the explicit constant of [BSSZ2026, §5] giving $c \\ge 0.000719$.  Unverified. |"
 },
 "note": {
  "title": "Improved constant for [BSSZ2026]",
  "author": "Ingo Althoefer",
  "date": "2026-05-28",
  "tex": "https://althofer.de/improved_constant_052.tex",
  "texSha256": "2f6bbdcbf38c5430e79709693528219ae793451f86f400c820ada767cbe05568",
  "pdf": "https://althofer.de/improved_constant_052.pdf",
  "pdfSha256": "ff30e04ed76103fda961b04b6b0d72ac3029df5b2a45c4b67c3ba1b15d8caf16",
  "stored": false,
  "theorem": "\"there are arbitrarily large finite sets A ⊂ R such that max(|A+A|,|AA|) ≤ |A|^{2-0.0007}. The optimized value suggested by the same calculation is about 0.000719, but 0.0007 is the safer quoted constant.\""
 },
 "code": {
  "instruments/sumproduct/c84b.py": "6546ed1a6151cd1836d3f92186d3195400e1646864cd19c1d8c7dbac6c391778"
 },
 "rows": [
  {
   "id": "84b",
   "claim": "C84b <= 1.999281 (c >= 0.000719)",
   "claimant": "I. Althoefer with ChatGPT 5.5 (the note, 28 May 2026), as the registry quotes it",
   "verdict": "REPAIRED",
   "repairedTo": "C84b <= 1.9993 (c >= 0.0007, the note's own theorem)",
   "ceiling": "0.0007150507",
   "cAtNoteChoice": [
    "0.0007062455",
    "0.0007062455"
   ],
   "checks": [
    {
     "name": "f(1.371966384) <= 103 (the note: about 101.956)",
     "ok": true,
     "detail": "[101.9560758327, 101.9560758327]"
    },
    {
     "name": "515/Y = 515/1415 < 0.364",
     "ok": true
    },
    {
     "name": "M1 e^Y/(X Y^2) + M2/(X^2 Y^2) < 0.136 (and so < 0.364)",
     "ok": true,
     "detail": "0.1352394769"
    },
    {
     "name": "log|A|/d < L < 1432",
     "ok": true,
     "detail": "1431.1139757"
    },
    {
     "name": "-log(0.364) > 0.0007 * 1432 = 1.0024: c = 0.0007 holds",
     "ok": true,
     "detail": "1.01060141"
    },
    {
     "name": "inf over s > 1 of f(s) >= 101.9 (a sweep of [1.01, 2.9] by 0.001, of [1.2, 1.6] by 0.0001; below, zeta > 100; above, the power dominates)",
     "ok": true,
     "detail": "smallest endpoint bound 101.927484 at s = 1.372"
    },
    {
     "name": "for every s > 1, X and Y, the chain gives c <= 0.0007150 < 0.000719",
     "ok": true,
     "detail": "the maximum near Y = 1399.0"
    }
   ]
  }
 ]
}