{
 "what": "The Tao #179 sublevel campaign (Erdős #1038, supremum side). Rational-weight discrete measures on [-1,1] with denominator N are monic degree-N polynomials with roots in [-1,1]; this record holds certified sublevel measures |{|q|<1}| (lower-bound witnesses for sup = 2*sqrt(2), a theorem of Tao, December 2025), certified grid champions, and per-degree branch-and-bound THEOREMS — independent re-decisions of that known theorem: odd degrees 3, 5, 7 fall strictly below 2.82 < 2*sqrt(2); even degrees 4, 6, 8 have supremum in [2*sqrt(2), 2.82845], the left end attained by (x^2-1)^{N/2}. A degree marked failed was ATTEMPTED and refused on its budget, not silently dropped. Every measure is an exact outward enclosure from BigInt Sturm root isolation; every branch-and-bound pruning uses the certified min-product box bound (calibrated: thin boxes reproduce measures to all printed digits).",
 "provenance": "Framing: T. Tao, teorth/erdosproblems#179 (16 Dec 2025), on erdosproblems.com #1038. The SUPREMUM is settled: Tao proved sup = 2*sqrt(2), the two-atom measure the only case of equality (forum thread 1038, 21 Dec 2025; notes of 22 and 27 Dec 2025, Theorem 2.1; formal-conjectures erdos_1038.parts.ii \"research solved\"), before this campaign ran. The INFIMUM is open on erdosproblems.com (parts.i \"research open\"); Darvas-Peng-Tao (July 2026) is one of three AI-assisted proof claims for it. This campaign addresses the supremum side only. Sources pinned in corpus/sources/erdos1038. (Corrected 2026-10-05: this field said the infimum side was resolved by Darvas-Peng-Tao.)",
 "witness": {
  "config": "(x^2-1): uniform measure on {-1,+1}",
  "measure": {
   "lo": "7774721279937/2748779069440",
   "hi": "388736063997/137438953472",
   "loD": 2.828427124745576,
   "hiD": 2.8284271247466677
  },
  "equals": "2*sqrt(2), certified"
 },
 "sweeps": {
  "n=3 d=8": {
   "configs": 505,
   "champion": {
    "lo": "518676742335/188978561024",
    "hi": "8298827877363/3023656976384",
    "loD": 2.7446327219579625,
    "hiD": 2.7446327219589546,
    "roots": "-8,6,8",
    "d": 8
   }
  },
  "n=4 d=4": {
   "configs": 264,
   "champion": {
    "lo": "49758216191595/17592186044416",
    "hi": "49758216191613/17592186044416",
    "loD": 2.828427124745474,
    "hiD": 2.828427124746497,
    "roots": "-4,-4,4,4",
    "d": 4
   }
  },
  "n=5 d=2": {
   "configs": 69,
   "champion": {
    "lo": "9817124967843/3573412790272",
    "hi": "9817124967849/3573412790272",
    "loD": 2.7472686599679808,
    "hiD": 2.74726865996966,
    "roots": "-2,-2,1,2,2",
    "d": 2
   }
  }
 },
 "cubicChampion": {
  "config": "(-1, 201/256, 1)",
  "measure": {
   "lo": "757061440731/274877906944",
   "hi": "2081918962011/755914244096",
   "loD": 2.754173477045697,
   "hiD": 2.754173477046689
  },
  "note": "interior critical root — the best certified degree-3 value in the record is at r = 201/256, not at a lattice point, near 2.7542: a lower bound on the degree-3 supremum (that it is the maximum is not certified)"
 },
 "quinticPeak": {
  "config": "(-1 dbl, 905/1024, 1 dbl)",
  "measure": {
   "lo": "10009453274475/3573412790272",
   "hi": "10009453274481/3573412790272",
   "loD": 2.8010906833165232,
   "hiD": 2.8010906833182023
  },
  "note": "the family (x^2-1)^2(x-r) peaks near r = 905/1024 while its sublevel set is one interval, then descends steeply but continuously as a gap opens and the set splits in two (corrected 2026-10-05: this note said it drops discontinuously; for fixed degree the measure is continuous in the roots)"
 },
 "theorems": {
  "deg3 < 2.82": {
   "n": 3,
   "T": "141/50",
   "TD": 2.82,
   "leaves": 45,
   "explored": 127,
   "maxDepth": 16,
   "ms": 12
  },
  "deg5 < 2.82": {
   "n": 5,
   "T": "141/50",
   "TD": 2.82,
   "leaves": 522,
   "explored": 1583,
   "maxDepth": 37,
   "ms": 212
  },
  "deg7 < 2.82": {
   "n": 7,
   "T": "141/50",
   "TD": 2.82,
   "leaves": 9472,
   "explored": 28773,
   "maxDepth": 63,
   "ms": 8771
  },
  "deg4 sup in [2sqrt2, 2.82845]": {
   "n": 4,
   "T": "56569/20000",
   "TD": 2.82845,
   "leaves": 213,
   "explored": 549,
   "maxDepth": 59,
   "ms": 57
  },
  "deg6 sup in [2sqrt2, 2.82845]": {
   "n": 6,
   "T": "56569/20000",
   "TD": 2.82845,
   "leaves": 1370,
   "explored": 4219,
   "maxDepth": 89,
   "ms": 645
  },
  "deg8 sup in [2sqrt2, 2.82845]": {
   "n": 8,
   "T": "56569/20000",
   "TD": 2.82845,
   "leaves": 9330,
   "explored": 31425,
   "maxDepth": 118,
   "ms": 5671
  },
  "deg9": {
   "n": 9,
   "failed": "bnb: box budget exhausted",
   "ms": 2871488
  }
 },
 "meta": {
  "date": "2026-09-01",
  "git": "cd057d7",
  "ms": 2887031
 }
}