{ "what": "The EinsteinArena / Together AI \"new state-of-the-art\" table (github.com/togethercomputer/EinsteinArena-new-SOTA, commit c388c6f7), every published construction re-decided in exact rational arithmetic from its own bytes read as the decimals they print. Per row: the exact value of the platform's objective, every constraint's exact slack, whether the printed digits are the exact value's (rounding, or ceiling for an upper bound), and a REPAIR where the bytes are a witness only up to the platform's tolerance.", "grammar": { "WITNESSED": "the bytes are an exact witness of the value stated", "REPAIRED": "a witness only up to the platform's tolerance; an exact witness built from the same bytes, its deficit printed", "UNWITNESSED": "the bytes fail exactly and no repair was built — a finding about the bytes, never a refutation of the bound" }, "scope": "constructions only; no upper-bound theorem, no optimality claim, and no search for better constructions. The \"improvement over the previous best\" rows are exact signs between two published constructions.", "provenance": { "repo": "github.com/togethercomputer/EinsteinArena-new-SOTA", "commit": "c388c6f7408c886311940896713339a1a70c2394", "fetched": "2026-09-08", "files": { "circles-rectangle/alphaevolve_2025.json": "ede505bcbca6dbcc420e646a25ec8af8190b6370ff3aa26e2a8e9524f92d7604", "circles-rectangle/ours_2026.json": "6060cc9f78f9d3ad2b035b6f1c15c3bfaac36f4e75532e9a94e1f804387d2307", "circles-rectangle/README.md": "ab45150d68d567e2596b62f2ac6c065fe4d6dede7b5cc07b6143da406e2cd54e", "circles-rectangle/verifier.py": "025424cd6ea2d979d9f0d53512e158c00579ba9ed9099c89a4ad543c89625cc9", "heilbronn-convex/alphaevolve_2025.json": "0006a49a12e12a4d5a941967aa0b65db79a74c879d44e3c5b66158b97422e8ae", "heilbronn-convex/ours_2026.json": "2b16263e299479b6a4c0b63991c73654456e3e2235d783b13942c8c6c3af46f7", "heilbronn-convex/README.md": "602bbf5a3254087236f6345c5c04889347065fc8e58d1db31ea5741a9d4ecd1d", 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"2.3658321334", "printedConvention": "rounding", "printedAgrees": true, "platformTolerance": "overlap allowed to 1e-9 in distance, box to 1e-9 in w + h", "ms": 0 }, { "id": "circles/ours_2026", "problem": "circles-rectangle", "claim": "n = 21 circles in a rectangle of perimeter 4 (bounding box w + h ≤ 2), no overlaps: Σ r = 2.3658323759", "claimant": "Together AI agents, this repository", "date": "2026-04", "file": "corpus/easota/circles-rectangle/ours_2026.json", "sha256": "6060cc9f78f9d3ad2b035b6f1c15c3bfaac36f4e75532e9a94e1f804387d2307", "verdict": "REPAIRED", "exact": "2.3658323758318174", "exactRational": "1182916187915908726997634896610790569/500000000000000000000000000000000000", "asPublished": { "sumR": "2.3658323759185154", "boxSlack": "-0.00000000000004204000", "overlappingPairs": 44, "worstPairSlackSquared": "-0.00000000000309437342", "worstPair": [ 3, 17 ] }, "repair": { "lambda": "0.999999999963354125", "sumR": "2.3658323758318174", "deficit": "0.000000000086697996", "how": "every radius scaled by λ, the largest rational with λ² ≤ d²/(rᵢ+rⱼ)² on every pair and the box inside w + h ≤ 2" }, "printed": "2.3658323759", "printedConvention": "rounding", "printedAgrees": false, "platformTolerance": "overlap allowed to 1e-9 in distance, box to 1e-9 in w + h", "ms": 0 }, { "id": "heilbronn/alphaevolve_2025", "problem": "heilbronn-convex", "claim": "n = 14 points, min triangle area / convex-hull area = 0.0278355715", "claimant": "AlphaEvolve V2 (arXiv:2511.02864)", "date": "2025-11", "file": "corpus/easota/heilbronn-convex/alphaevolve_2025.json", "sha256": "0006a49a12e12a4d5a941967aa0b65db79a74c879d44e3c5b66158b97422e8ae", "verdict": "WITNESSED", "exact": "0.0278355714584821", "exactRational": "26639300491813812188679511595248/957023660590097468704754145567911", "detail": { "hullVertices": 10, "hullArea": "0.4785118302950487", "minTriangle": [ 1, 6, 9 ], "minArea": "0.013319650245906906" }, "printed": "0.0278355715", "printedConvention": "rounding", "printedAgrees": true }, { "id": "heilbronn/ours_2026", "problem": "heilbronn-convex", "claim": "n = 14 points, min triangle area / convex-hull area = 0.0278355805", "claimant": "Together AI agents, this repository", "date": "2026-04", "file": "corpus/easota/heilbronn-convex/ours_2026.json", "sha256": "2b16263e299479b6a4c0b63991c73654456e3e2235d783b13942c8c6c3af46f7", "verdict": "WITNESSED", "exact": "0.0278355805175536", "exactRational": "1384426006618788439263350119322750000000000000000/49735840994792336683887689749880203672217096263811", "detail": { "hullVertices": 10, "hullArea": "1.9894336397916934", "minTriangle": [ 4, 7, 11 ], "minArea": "0.055377040264751537" }, "printed": "0.0278355805", "printedConvention": "rounding", "printedAgrees": true }, { "id": "mindist/alphaevolve_2025", "problem": "min-distance-ratio-2d", "claim": "n = 16 points, (max distance / min distance)² = 12.889266", "claimant": "AlphaEvolve (Novikov et al. — arXiv:2506.13131)", "date": "2025-06", "file": 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K. Haugland (arXiv:1609.08000)", "date": "2016", "file": "corpus/easota/erdos-minimum-overlap/haugland_2016.py", "sha256": "93bce13461a18cc1924512a685416359174cffa9a235f222e6d23703a3ab071a", "verdict": "WITNESSED", "exact": "0.3809268534330869", "exactRational": "64757565083624785688821235069923/170000000000000000000000000000000", "asPublished": { "steps": 51, "inRange": true, "sumMinusHalfN": "0.00000000000000000000", "bound": "0.3809268534330869", "lag": -2 }, "repair": null, "printed": "0.380927", "printedConvention": "ceiling", "printedAgrees": true, "platformTolerance": "Σh = n/2 within 1e-6" }, { "id": "overlap/alphaevolve_2025", "problem": "erdos-minimum-overlap", "claim": "a step function h with 95 steps, 0 ≤ h ≤ 1, Σh = n/2: upper bound 0.380924", "claimant": "AlphaEvolve (arXiv:2506.13131)", "date": "2025-06", "file": "corpus/easota/erdos-minimum-overlap/alphaevolve_2025.py", "sha256": "11f33879398ee8069125aecc588792cf3c049615c7af6d9613997374668b5301", "verdict": "REPAIRED", "exact": "0.3809230351084501", "exactRational": "5371609987271504817817482291257635482317212504/14101562500000001428177037500000036160702944129", "asPublished": { "steps": 95, "inRange": true, "sumMinusHalfN": "-0.00000000000000240535", "bound": "0.3809230351084501", "lag": -8 }, "repair": { "how": "1 − h scaled down to Σh = n/2", "bound": "0.3809230351084501", "delta": "0.00000000000000000353" }, "printed": "0.380924", "printedConvention": "ceiling", "printedAgrees": true, "platformTolerance": "Σh = n/2 within 1e-6" }, { "id": "overlap/ttt_discover_2026", "problem": "erdos-minimum-overlap", "claim": "a step function h with 600 steps, 0 ≤ h ≤ 1, Σh = n/2: upper bound 0.380876", "claimant": "TTT-Discover (Yuksekgonul et al. — arXiv:2601.16175)", "date": "2026-01", "file": "corpus/easota/erdos-minimum-overlap/ttt_discover_2026.py", "sha256": "586e77b49e07ace63ddf73e69137802d24d63c1231625448a77c9ef6fad7c82f", "verdict": "REPAIRED", "exact": "0.3808753232177187", "exactRational": 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"detail": { "area": "0.2124934484447972", "maxGap": "0.0500000429776642", "distinctRho": 100, "note": "a benchmark score, not a theorem: the area under the platform's slope-3 envelope plus 10 × the largest gap in edge density, recomputed exactly (its 1e-9 branches are exact comparisons here)" }, "printed": "-0.712494", "printedConvention": "rounding", "printedAgrees": true }, { "id": "edges/ours_2026", "problem": "edges-vs-triangles", "claim": "100 rows of 20 weights → (edge density, triangle density) points; the platform's envelope score = -0.712256", "claimant": "Together AI agents, this repository", "date": "2026-03", "file": "corpus/easota/edges-vs-triangles/ours_2026.py", "sha256": "350cbf63494f853979e9736916b6a1d40db279d9ae20461c66d0194c13abf0f3", "verdict": "WITNESSED", "exact": "-0.7122563326979318", "exactRational": 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"detail": { "area": "0.2122563326979293", "maxGap": "0.0500000000000002", "distinctRho": 99, "note": "a benchmark score, not a theorem: the area under the platform's slope-3 envelope plus 10 × the largest gap in edge density, recomputed exactly (its 1e-9 branches are exact comparisons here)" }, "printed": "-0.712256", "printedConvention": "rounding", "printedAgrees": true }, { "id": "autocorr/alphaevolve_2025", "problem": "first-autocorrelation", "claim": "f ≥ 0 as 600 values on [−1/4, 1/4]: C₁ ≤ 1.50529397", "claimant": "AlphaEvolve (arXiv:2506.13131)", "date": "2025-06", "file": "corpus/easota/first-autocorrelation/alphaevolve_2025.py", "sha256": "dea4517da3763e82c314606269718404c7b0f6d29c5522c0bc2033d3e894cb35", "verdict": "WITNESSED", "exact": "1.5052939684401608", "exactRational": "232137593339762332394128908075248066605881087348524259456000000000000/154214125750009859130178856148859502365769023151443785629054507630809", "detail": { "n": 600, "argmax": 769, "candidatesDecidedExactly": 1, "screenErrorBound": 5.329070518200751e-13, "decimalDigits": 32, "note": "for a step function the discrete maximum IS the supremum of f∗f; n² products screened in float64, every index within the forward-error bound of the float maximum decided exactly" }, "printed": "1.50529397", "printedConvention": "ceiling", "printedAgrees": true, "ms": 3 }, { "id": "autocorr/alphaevolve_v2_2025", "problem": "first-autocorrelation", "claim": "f ≥ 0 as 1319 values on [−1/4, 1/4]: C₁ ≤ (no value printed for this file)", "claimant": "AlphaEvolve V2 (arXiv:2511.02864)", "date": "2025-11", "file": "corpus/easota/first-autocorrelation/alphaevolve_v2_2025.json", "sha256": "92a023d37ea89c98b37a0778585dedafaf8acee540610b2afef922b39f51c430", "verdict": "WITNESSED", "exact": "1.5031635546815609", "exactRational": "925266108971129172015049350003168641445336690037048/615545863980944528771286708404495389967980829685049", "detail": { "n": 1319, "argmax": 1279, "candidatesDecidedExactly": 1, "screenErrorBound": 1.1715073355844652e-12, "decimalDigits": 23, "note": "for a step function the discrete maximum IS the supremum of f∗f; n² products screened in float64, every index within the forward-error bound of the float maximum decided exactly" }, "printed": null, "printedConvention": "ceiling", "printedAgrees": null, "ms": 4 }, { "id": "autocorr/ttt_discover_2026", "problem": "first-autocorrelation", "claim": "f ≥ 0 as 30000 values on [−1/4, 1/4]: C₁ ≤ 1.50286290", "claimant": "TTT-Discover (arXiv:2601.16175)", "date": "2026-01", "file": "corpus/easota/first-autocorrelation/ttt_discover_2026.json", "sha256": "4f24a0ef2515171617e3be99bc58e3ea8d350ff6b299775c029c5dc2971f413d", "verdict": "WITNESSED", "exact": "1.5028628982558265", "exactRational": "9017177389534960453073255343945818186531611658165103644880000/6000000000000000717612319644499700914825151864452703446981849", "detail": { "n": 30000, "argmax": 33235, "candidatesDecidedExactly": 1, "screenErrorBound": 2.6645352591003757e-11, "decimalDigits": 28, "note": "for a step function the discrete maximum IS the supremum of f∗f; n² products screened in float64, every index within the forward-error bound of the float maximum decided exactly" }, "printed": "1.50286290", "printedConvention": "ceiling", "printedAgrees": true, "ms": 920 }, { "id": "autocorr/ours_2026", "problem": "first-autocorrelation", "claim": "f ≥ 0 as 30000 values on [−1/4, 1/4]: C₁ ≤ 1.50286286", "claimant": "Together AI agents, this repository", "date": "2026-03", "file": "corpus/easota/first-autocorrelation/ours_2026.json", "sha256": "6cc34889cb40a4e2579c078b08c39d71fd29f07ab7f14f2417f770038e9c0caf", "verdict": "WITNESSED", "exact": "1.5028628587049711", "exactRational": "1502862858704110049736338328792230205046928615747268712036800/999999999999427044848111787787401518834592685139973858473881", "detail": { "n": 30000, "argmax": 28879, "candidatesDecidedExactly": 10, "screenErrorBound": 2.6645352591003757e-11, "decimalDigits": 30, "note": "for a step function the discrete maximum IS the supremum of f∗f; n² products screened in float64, every index within the forward-error bound of the float maximum decided exactly" }, "printed": "1.50286286", "printedConvention": "ceiling", "printedAgrees": true, "ms": 987 }, { "id": "flat/alphaevolve_2025", "problem": "flat-polynomials", "claim": "70 coefficients ±1: C⁺ = max_{|z|=1} |g(z)| / √71 = 1.340925", "claimant": "AlphaEvolve V2 (arXiv:2511.02864)", "date": "2025-11", "file": "corpus/easota/flat-polynomials/alphaevolve_2025.py", "sha256": "c403c792b1793cf618990e6b8e82fc174b1a001b8daa57f854a572f4ab26c07a", "verdict": "WITNESSED", "exact": "1.3409252804007615", "exactRational": null, "enclosure": { "lo": "1.34092528040076154302", "hi": "1.34092528040076161766", "width": "0.00000000000000007464", "maxSqLo": "127.6637231408681287", "maxSqHi": "127.6637231408681429" }, "detail": { "degree": 69, "gridScore": "1.3409252794557085", "gridShortfall": "0.0000000009450530", "argCosTheta": [ { "kind": "critical", "cosTheta": [ 0.6087949790870701, 0.6087949790870703 ] } ], "method": "chebyshev reduction → sturm isolation (grid+sturm-total) → interval-newton/bisection refinement → exact taylor value enclosures, outward-rounded", "counts": { "degree": 69, "sfDeg": 68, "nCrit": 39, "nCandidates": 41, "isolation": "grid+sturm-total" }, "ms": 368, "note": "the platform's score is a maximum over 1,000,000 grid points on the circle, a lower bound of the supremum; the enclosure is the supremum, certified (Sturm chain, interval Newton, exact Taylor bounds) — and the grid falls short of it by gridShortfall" }, "printed": "1.340925", "printedConvention": "rounding", "printedAgrees": true }, { "id": "flat/ours_2026", "problem": "flat-polynomials", "claim": "70 coefficients ±1: C⁺ = max_{|z|=1} |g(z)| / √71 = 1.280932", "claimant": "Together AI agents, this repository", "date": "2026-03", "file": "corpus/easota/flat-polynomials/ours_2026.py", "sha256": "d2a51b1c92cfb53de3b379e7599f00975e15e97876bb61d4b5ac9c485590d6a1", "verdict": "WITNESSED", "exact": "1.2809320528750416", "exactRational": null, "enclosure": { "lo": "1.28093205287504167568", "hi": "1.28093205287504175382", "width": "0.00000000000000007814", "maxSqLo": "116.4958716098694679", "maxSqHi": "116.4958716098694821" }, "detail": { "degree": 69, "gridScore": "1.2809320527987995", "gridShortfall": "0.0000000000762421", "argCosTheta": [ { "kind": "critical", "cosTheta": [ -0.7901917400320739, -0.7901917400320738 ] } ], "method": "chebyshev reduction → sturm isolation (grid+sturm-total) → interval-newton/bisection refinement → exact taylor value enclosures, outward-rounded", "counts": { "degree": 69, "sfDeg": 68, "nCrit": 43, "nCandidates": 44, "isolation": "grid+sturm-total" }, "ms": 403, "note": "the platform's score is a maximum over 1,000,000 grid points on the circle, a lower bound of the supremum; the enclosure is the supremum, certified (Sturm chain, interval Newton, exact Taylor bounds) — and the grid falls short of it by gridShortfall" }, "printed": "1.280932", "printedConvention": "rounding", "printedAgrees": true }, { "id": "hexagons/alphaevolve_2025", "problem": "hexagon-packing", "claim": "12 unit hexagons inside a hexagon of side 3.9419123, no overlaps, all inside: score = the outer side", "claimant": "AlphaEvolve V2 (arXiv:2511.02864)", "date": "2025-11", "file": "corpus/easota/hexagon-packing/alphaevolve_2025.json", "sha256": "daa887e9ba08b64bf405cf38a56cde2429ef07782db24f08c69bc91a2a250df0", "verdict": "WITNESSED", "exact": "3.9419123000000000", "exactRational": "39419123/10000000", "detail": { "pairs": { "SEPARATED": 66 }, "vertices": { "INSIDE": 72 }, "closestPair": { "i": 4, "j": 5, "gapAtLeast": 0.000006934900789179368 }, "tightestVertex": { "hexagon": 3, "vertex": 2, "crossAtLeast": 0.000002672621224775184 }, "enclosureWidth": 5.773159728050814e-15, "intersecting": [], "outside": [], "undecidedPairs": [], "undecidedVertices": [], "ms": 8, "note": "every vertex is a cosine and a sine of a published decimal, so the decision runs in outward-rounded interval arithmetic with a certified π and certified sin/cos (instruments/interval): a pair is SEPARATED when some edge normal has certainly disjoint projections, a vertex INSIDE when every edge cross product is certainly non-negative. gapAtLeast and crossAtLeast are certain lower bounds; the platform separates only past 1e-9 and admits a vertex outside by up to 1e-9" }, "printed": "3.9419123", "printedConvention": "exact", "printedAgrees": true, "platformTolerance": "a pair counts as intersecting unless separated by more than 1e-9; a vertex counts as inside unless outside by more than 1e-9" }, { "id": "hexagons/ours_2026", "problem": "hexagon-packing", "claim": "12 unit hexagons inside a hexagon of side 3.9416523, no overlaps, all inside: score = the outer side", "claimant": "Together AI agents, this repository", "date": "2026-04", "file": "corpus/easota/hexagon-packing/ours_2026.json", "sha256": "1d83e40b77862c0f46ec9891b3bbea16b4b303b644007207824acef9dbd6b693", "verdict": "WITNESSED", "exact": "3.9416523000000000", "exactRational": "39416523/10000000", "detail": { "pairs": { "SEPARATED": 66 }, "vertices": { "INSIDE": 72 }, "closestPair": { "i": 4, "j": 7, "gapAtLeast": 3.616782695559057e-8 }, "tightestVertex": { "hexagon": 1, "vertex": 4, "crossAtLeast": 4.059266311973175e-7 }, "enclosureWidth": 6.217248937900877e-15, "intersecting": [], "outside": [], "undecidedPairs": [], "undecidedVertices": [], "ms": 3, "note": "every vertex is a cosine and a sine of a published decimal, so the decision runs in outward-rounded interval arithmetic with a certified π and certified sin/cos (instruments/interval): a pair is SEPARATED when some edge normal has certainly disjoint projections, a vertex INSIDE when every edge cross product is certainly non-negative. gapAtLeast and crossAtLeast are certain lower bounds; the platform separates only past 1e-9 and admits a vertex outside by up to 1e-9" }, "printed": "3.9416523", "printedConvention": "exact", "printedAgrees": true, "platformTolerance": "a pair counts as intersecting unless separated by more than 1e-9; a vertex counts as inside unless outside by more than 1e-9" } ], "improvements": [ { "problem": "circles-rectangle", "direction": "maximise", "previous": "AlphaEvolve V2", "delta": "0.0000002424150543", "sign": 1, "note": "ours taken as its REPAIRED exact witness" }, { "problem": "heilbronn-convex", "direction": "maximise", "previous": "AlphaEvolve V2", "delta": "0.0000000090590715", "sign": 1 }, { "problem": "min-distance-ratio-2d", "direction": "minimise", "previous": "AlphaEvolve", "delta": "0.0000362043171149", "sign": 1 }, { "problem": "erdos-minimum-overlap", "direction": "minimise", "previous": "TTT-Discover", "delta": "0.0000050126314988", "sign": 1 }, { "problem": "edges-vs-triangles", "direction": "maximise", "previous": "AlphaEvolve V2", "delta": "0.0002375455235076", "sign": 1 }, { "problem": "first-autocorrelation", "direction": "minimise", "previous": "TTT-Discover", "delta": "0.0000000395508554", "sign": 1 }, { "problem": "flat-polynomials", "direction": "minimise", "previous": "AlphaEvolve V2", "delta": "0.0599932275257197", "sign": 1, "note": "a lower bound on the difference: the two certified enclosures are disjoint" }, { "problem": "hexagon-packing", "direction": "minimise", "previous": "AlphaEvolve V2", "delta": "0.0002600000000000", "sign": 1, "note": "both outer sides are exact decimals; both packings certified as witnesses" } ], "undecided": [ { "problem": "prime-number-theorem", "claim": "S(f) = 0.994179 (AlphaEvolve: 0.921292)", "status": "NOT DECIDABLE AS STATED", "why": "the platform's score is estimated from 10,000,000 random samples; a Monte Carlo score is not a mathematical claim about the construction and this ledger does not certify it." }, { "problem": "tammes / second & third autocorrelation / uncertainty (README rows)", "claim": "four further table rows", "status": "NEEDS DATA", "why": "the repository README lists them; no solution files are published in the repository at the pinned commit." } ] }