{
 "what": "The discoveries HorizonMath (arXiv 2603.15617v2) credits to frontier models, decided from what its Appendix A prints, by standard-library deciders that share no code with its validators.",
 "generated": "2026-09-29",
 "source": {
  "paper": "arXiv:2603.15617v2",
  "repo": "github.com/ewang26/HorizonMath",
  "commit": "4ef0b61a8a6d4c5ded661fab7c2e896af84abe6e",
  "metaSha256": "97ad6e3900cce882481d97905ea57a96929934895e07c72bb416e79d55aa218d"
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 "code": {
  "instruments/horizonmath/ramsey_region.py": "b2657578bd0b8cb7e4e7e261a634ee7ce8cdd8b0eb2a97cb526d88ace42fe420",
  "instruments/horizonmath/kakeya_area.py": "449530031a66bc0b4a5879141a2376dc8aa0d7293956b14a3d907e88a119159a"
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 "byVerdict": {
  "REFUTED": 1,
  "CERTIFIED": 1,
  "NEEDS DATA": 1
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   "id": "ramsey-asymptotic",
   "problem": "Asymptotic Upper Bound Constant for Diagonal Ramsey Numbers (solvability 1)",
   "claim": "R(k,k) <= 3.6960839126^(k+o(k)), improving 3.7992 (Gupta–Ndiaye–Norin–Wei), by a certificate that satisfies their Theorem 14",
   "claimant": "GPT-5.4 Pro, reproduced by GPT-5.6 Sol Max (HorizonMath, arXiv 2603.15617v2, Appendix A.3)",
   "verdict": "REFUTED",
   "kind": "checker-wider-than-definition",
   "scope": "the certificate is refuted, not the inequality: its pair (X(1), Y(1)) = (0.22745, 0.9988), at the point that sets c, lies outside R — Erdos's 1947 bound at e = 13/2000, p = 11981/500000 exceeds the pair's rate by 0.00130 — and 56 of its 201 decided points are outside R, 128 more not placed in R by the bound the problem names. The problem's own rule accepts every pair with x <= e^-U(1) = 0.26292, whatever y is: it asks one of the two inequalities membership needs",
   "decided": {
    "claim": "the certificate satisfies GNNW Theorem 14, so R(k,k) <= 3.6960839...^(k+o(k))",
    "verdict": "REFUTED",
    "checks": [
     {
      "name": "c = e^F(1) is the printed 3.6960839126...",
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      "detail": "[3.69608391263329, 3.69608391263329]"
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   ],
   "theirChecker": "valid: true — \"Valid split certificate; c = e^{F(1)} = 3.69608391263; polynomial_coeffs degree = 5; analytic tail endpoint = 0.00010846; worst small-lambda slack = 0.000272305; worst large R_0 slack = 0.000951234; worst large main slack = 0.000318787\"",
   "theirCheckerWithAnd": {
    "change": "validators/ramsey_asymptotic.py line 299, `return max(mp.mpf(0), min(bu, bs))` -> `max(bu, bs)`: the pair must clear BOTH orientations' thresholds, as membership in R requires",
    "on": "2026-09-29",
    "output": "valid: false — \"R_0 check failed somewhere on [0.001000000000000000020816681711721685132943093776702880859375, 0.00100138829520408798266616040823038247253862209618091583251953125]\" (the first interval; 0.2 s)"
   }
  },
  {
   "id": "keich-thin-triangles-128",
   "problem": "Thin-Triangle Kakeya (128 slopes): minimize union area (solvability 1)",
   "claim": "Area(E) = 0.1091479892... < 0.1148103258186177 (the AlphaEvolve baseline)",
   "claimant": "GPT-5.4 Pro, reproduced by GPT-5.6 Sol Max (HorizonMath, arXiv 2603.15617v2, Appendix A.2)",
   "verdict": "CERTIFIED",
   "kind": "none",
   "scope": "the union of the 128 printed triangles has area exactly 6008623/55050240 = 0.10914798918224…, 0.00566233 below the printed AlphaEvolve baseline; the printed 0.1091479892 is its rounding",
   "decided": {
    "claim": "Area(E) = 0.1091479892... < 0.1148103258186177 (the AlphaEvolve baseline)",
    "verdict": "CERTIFIED",
    "checks": [
     {
      "name": "the area is below the printed AlphaEvolve baseline 0.1148103258186177",
      "ok": true
     },
     {
      "name": "the printed 0.1091479892 is the exact area's truncation or rounding",
      "ok": true
     },
     {
      "name": "the area is at least the largest single triangle (1/256): the union is not empty or degenerate",
      "ok": true
     }
    ],
    "pieces": 4033,
    "printedIs": "its rounding",
    "area": {
     "num": "6008623",
     "den": "55050240",
     "decimal": "0.1091479891822451636904"
    },
    "improvement": "0.005662336636"
   },
   "theirChecker": "valid: true — \"Valid. Union area=0.109147989182245, baseline=0.119207265425702.\" (its baseline is Keich's construction, the problem statement's is AlphaEvolve's 0.1148103258186177)"
  },
  {
   "id": "gpt56-closed-forms",
   "problem": "Fifth Airy moment a5; equal-power TE+TM and non-resonant TM/TE spherical-mode quality factors (solvability 2)",
   "claim": "three closed forms that \"pass the compliance check and match the high-precision reference\"",
   "claimant": "GPT-5.6 Sol Max (HorizonMath, arXiv 2603.15617v2)",
   "verdict": "NEEDS DATA",
   "kind": "data-not-public",
   "scope": "the expressions are not published: the paper says all six discoveries \"are detailed in Appendix A\", which details three (A.1–A.3), none of these; agreement with a reference to 20 digits would not decide equality in any case"
  }
 ],
 "notDecided": [
  {
   "id": "spinor-norm-integral-i0",
   "why": "the closed form Gamma(1/4)^2/(8 sqrt(pi)) + Gamma(3/4)^2/sqrt(pi) comes with a printed derivation (Landen, then a Beltrami transform, to 2E(1/2) - K(1/2)/2); an enclosure of the integral to hundreds of digits, or a check of each step, is not built here yet"
  }
 ]
}