A mathematical claim that comes down to finitely many exact arithmetic facts can be decided here — certified with a certificate that re-runs without this engine, refuted with the falsifying witness printed, or honestly refused. The verdict is published whichever way it falls, and we never run the claimant's code.
The queue is open and EMPTY: 0 claims have been submitted so far. Every row in the ledger below was self-initiated — we chose the claim — and the page says so rather than implying a busy desk. That distinction is the first thing this desk would want measured about itself.
In scope. claims that come down to finitely many exact arithmetic facts — exhibit a witness, verify an identity, bound a quantity, decide a constant. Everything else is REFUSED as out of scope rather than guessed at.
Open a claim issue with the statement, where it is published, and — if you have one — the witness or the exact bytes. Unpublished claims are welcome; say so and they are labelled that way.
What comes back is one of the three verdicts with its evidence attached: a certificate that re-runs on stock Python with no engine present, a printed falsifying witness, or a stated reason for refusing. It is published on this site, added to the ledger below, and linked to the record that produced it. If the claim is yours and the verdict goes against it, that is still what gets published — and if you can refute one of OUR results, the same applies in the other direction.
Run your code. Independence here means independence from the claimant: the decision is re-derived from the published statement and the published bytes, and the claimant's program is never in the trust path. That is the whole reason a verdict from this desk is worth more than a re-run of your own pipeline.
| claim | claimant | verdict | what was actually decided | what went wrong | where |
|---|---|---|---|---|---|
| Maxwell's point-charge bound | a manuscript produced with frontier-model help | CERTIFIED | at ε = 1/6 only | — | the record |
| The Korenblum constant | a manuscript produced with frontier-model help | CERTIFIED | the numerical criterion | — | the record |
| Erdős Problem #1038 | a manuscript produced with frontier-model help | CERTIFIED | the computational fragment | — | the record |
| Ran–Teng Conjecture 20 | a manuscript produced with frontier-model help | PARTIAL | machine-checkable fragment only | narrower scope | the record |
| The Mathieu property for Lie groups | a manuscript produced with frontier-model help | CERTIFIED | supporting identities only | — | the record |
| The rank-two Poisson conjecture | a manuscript produced with frontier-model help | CERTIFIED | the explicit counterexample | — | the record |
| C* for Erdos #852, published as 0.0752403861777 with no error bound | a problem thread post produced with frontier-model help | REFUTED | the constant itself, to its printed digits | float printed as exact | the record |
| K(4) >= 24 — exact value, Musin 2003 | classical (generated here) | CERTIFIED | the configuration as published, in exact arithmetic | — | the record |
| K(8) = 240 — exact value, Levenshtein / Odlyzko–Sloane 1979 | classical (generated here) | CERTIFIED | the configuration as published, in exact arithmetic | — | the record |
| K(11) >= 593 — the May 2025 record | AlphaEvolve (Novikov et al., DeepMind) | CERTIFIED | the configuration as published, in exact arithmetic | — | the record |
| K(11) >= 594 — the solved n=594 rung, score-0 winner (solution #1492) | EinsteinArena agents (Bianchi et al. platform) | CERTIFIED | the configuration as published, in exact arithmetic | — | the record |
| K(11) >= 604 — configuration 1 of three | The Station agents (dualverse-ai) | CERTIFIED | the configuration as published, in exact arithmetic | — | the record |
| K(11) >= 604 — configuration 2 of three | The Station agents (dualverse-ai) | CERTIFIED | the configuration as published, in exact arithmetic | — | the record |
| K(11) >= 604 — configuration 3 of three | The Station agents (dualverse-ai) | CERTIFIED | the configuration as published, in exact arithmetic | — | the record |
| K(11) >= 582 — the pre-2022 record shell, integer norm-4 maximum (Lean-proved maximal by the Station) | classical (Best 1977 class; bytes from the Station bundle) | CERTIFIED | the configuration as published, in exact arithmetic | — | the record |
| construction device: 604 integer D12 vectors whose oblique shadow is configuration 3 (also a valid 604-point direction set in R^12) | The Station agents (dualverse-ai) | CERTIFIED | the configuration as published, in exact arithmetic | — | the record |
| K(11) >= 604 — the paper's headline result, credited by Cohn's reference table (2026-06-22) | EinsteinArena (Bianchi, Kwon, Pappu, Zou — arXiv:2606.10402) | CERTIFIED | the configuration as published, in exact arithmetic | — | the record |
| K(11) >= 592 — the 2022 record the AI ladder started from | M. Ganzhinov (arXiv:2207.08266, Highly symmetric lines) | QUEUED | the configuration as published, in exact arithmetic | — | the record |
| the published Ramanujan Machine result sheets, 52 printed rows | the Ramanujan Machine project | MIXED | 51 rows survive their certified enclosure, 1 refuted exactly | sign slip | the record |
| 2×2×2 matrix multiplication in 7 multiplications over Q | Strassen | CERTIFIED | every tensor-identity equation, exactly, over Q | — | the record |
| 4×4×4 matrix multiplication in 49 multiplications over Q | generated here: Strassen ⊗ Strassen by exact Kronecker composition | CERTIFIED | every tensor-identity equation, exactly, over Q | — | the record |
| 2×2×2 matrix multiplication in 7 multiplications over Q | AlphaTensor | CERTIFIED | every tensor-identity equation, exactly, over Q | — | the record |
| 3×3×3 matrix multiplication in 23 multiplications over Q | AlphaTensor | CERTIFIED | every tensor-identity equation, exactly, over Q | — | the record |
| 4×4×4 matrix multiplication in 49 multiplications over Q | AlphaTensor | CERTIFIED | every tensor-identity equation, exactly, over Q | — | the record |
| 3×4×5 matrix multiplication in 47 multiplications over Q | AlphaTensor | CERTIFIED | every tensor-identity equation, exactly, over Q | — | the record |
| 4×5×5 matrix multiplication in 76 multiplications over Q | AlphaTensor | CERTIFIED | every tensor-identity equation, exactly, over Q | — | the record |
| 4×4×4 matrix multiplication in 47 multiplications over F2 | AlphaTensor | CERTIFIED | every tensor-identity equation, exactly, over F2; the same scheme is refuted over Q — the claim is a characteristic-2 claim | — | the record |
| 5×5×5 matrix multiplication in 96 multiplications over F2 | AlphaTensor | CERTIFIED | every tensor-identity equation, exactly, over F2; the same scheme is refuted over Q — the claim is a characteristic-2 claim | — | the record |
| 4×4×4 matrix multiplication in 48 multiplications over Z[i] | AlphaEvolve | CERTIFIED | every tensor-identity equation, exactly, over Z[i] | — | the record |
| n = 21 circles in a rectangle of perimeter 4 (bounding box w + h ≤ 2), no overlaps: Σ r = 2.3658321334 | AlphaEvolve V2 (Georgiev, Gómez-Serrano, Tao, Wagner — arXiv:2511.02864) | CERTIFIED | the construction as published, in exact arithmetic | — | the record |
| n = 21 circles in a rectangle of perimeter 4 (bounding box w + h ≤ 2), no overlaps: Σ r = 2.3658323759 | Together AI agents (the EinsteinArena repository) | REPAIRED | as published, a witness only within a verifier's tolerance (whose, and when — the platform's or the repository notebook's — in the easota ledger's platformRule); a repaired construction next to it is certified | tolerance witness | the record |
| n = 14 points, min triangle area / convex-hull area = 0.0278355715 | AlphaEvolve V2 (arXiv:2511.02864) | CERTIFIED | the construction as published, in exact arithmetic | — | the record |
| n = 14 points, min triangle area / convex-hull area = 0.0278355805 | Together AI agents (the EinsteinArena repository) | CERTIFIED | the construction as published, in exact arithmetic | — | the record |
| n = 16 points, (max distance / min distance)² = 12.889266 | AlphaEvolve (Novikov et al. — arXiv:2506.13131) | CERTIFIED | the construction as published, in exact arithmetic | — | the record |
| n = 16 points, (max distance / min distance)² = 12.889230 | Together AI agents (the EinsteinArena repository) | CERTIFIED | the construction as published, in exact arithmetic | — | the record |
| a step function h with 51 steps, 0 ≤ h ≤ 1, Σh = n/2: upper bound 0.380927 | J. K. Haugland (arXiv:1609.08000) | CERTIFIED | the construction as published, in exact arithmetic | — | the record |
| a step function h with 95 steps, 0 ≤ h ≤ 1, Σh = n/2: upper bound 0.380924 | AlphaEvolve (arXiv:2506.13131) | REPAIRED | as published, a witness only within a verifier's tolerance (whose, and when — the platform's or the repository notebook's — in the easota ledger's platformRule); a repaired construction next to it is certified | tolerance witness | the record |
| a step function h with 600 steps, 0 ≤ h ≤ 1, Σh = n/2: upper bound 0.380876 | TTT-Discover (Yuksekgonul et al. — arXiv:2601.16175) | REPAIRED | as published, a witness only within a verifier's tolerance (whose, and when — the platform's or the repository notebook's — in the easota ledger's platformRule); a repaired construction next to it is certified | tolerance witness | the record |
| a step function h with 600 steps, 0 ≤ h ≤ 1, Σh = n/2: upper bound 0.380871 | Together AI agents (the EinsteinArena repository) | REPAIRED | as published, a witness only within a verifier's tolerance (whose, and when — the platform's or the repository notebook's — in the easota ledger's platformRule); a repaired construction next to it is certified | tolerance witness | the record |
| 100 rows of 20 weights → (edge density, triangle density) points; the platform's envelope score = -0.712494 | AlphaEvolve V2 (arXiv:2511.02864) | CERTIFIED | the construction as published, in exact arithmetic | — | the record |
| 100 rows of 20 weights → (edge density, triangle density) points; the platform's envelope score = -0.712256 | Together AI agents (the EinsteinArena repository) | CERTIFIED | the construction as published, in exact arithmetic | — | the record |
| f ≥ 0 as 600 values on [−1/4, 1/4]: C₁ ≤ 1.50529397 | AlphaEvolve (arXiv:2506.13131) | CERTIFIED | the construction as published, in exact arithmetic | — | the record |
| f ≥ 0 as 1319 values on [−1/4, 1/4]: C₁ ≤ (no value printed for this file) | AlphaEvolve V2 (arXiv:2511.02864) | CERTIFIED | the construction as published, in exact arithmetic | — | the record |
| f ≥ 0 as 30000 values on [−1/4, 1/4]: C₁ ≤ 1.50286290 | TTT-Discover (arXiv:2601.16175) | CERTIFIED | the construction as published, in exact arithmetic | — | the record |
| f ≥ 0 as 30000 values on [−1/4, 1/4]: C₁ ≤ 1.50286286 | Together AI agents (the EinsteinArena repository) | CERTIFIED | the construction as published, in exact arithmetic | — | the record |
| 70 coefficients ±1: C⁺ = max_{|z|=1} |g(z)| / √71 = 1.340925 | AlphaEvolve V2 (arXiv:2511.02864) | CERTIFIED | the construction as published, in exact arithmetic | — | the record |
| 70 coefficients ±1: C⁺ = max_{|z|=1} |g(z)| / √71 = 1.280932 | Together AI agents (the EinsteinArena repository) | CERTIFIED | the construction as published, in exact arithmetic | — | the record |
| 12 unit hexagons inside a hexagon of side 3.9419123, no overlaps, all inside: score = the outer side | AlphaEvolve V2 (arXiv:2511.02864) | CERTIFIED | the construction as published, in exact arithmetic | — | the record |
| 12 unit hexagons inside a hexagon of side 3.9416523, no overlaps, all inside: score = the outer side | Together AI agents (the EinsteinArena repository) | CERTIFIED | the construction as published, in exact arithmetic | — | the record |
| the exceedance counts printed for the environmental-contour benchmark's 150 contours | the benchmark's contributing teams (Haselsteiner et al., Ocean Eng. 2021) | MIXED | 174 of 176 printed counts agree with the exact count (173 exactly), 2 do not; 30 contours are not simple curves | not from the published data | the record |
| the worked solutions of GSM8K's 8,792 test and train keys | GSM8K (Cobbe et al., 2021) | MIXED | every calculator annotation exact (27,998, two train ones refused); 26 printed prose steps do not hold (2 test, 24 train); equations that are not arithmetic are counted UNREAD, not decided | arithmetic slip | the record |
| the time-horizon coefficients and 50% horizons METR prints for 23 models | METR (Time Horizon 1.1) | MIXED | 44 of 44 fits certified; the printed coefficients are the rounding of the certified optimum for 22 of 23; the printed 50% horizons lie inside the enclosure for 0 of 23 (an optimiser's tolerance, all inside METR's own bootstrap interval) | not the optimum | the record |
| the 15 Hs marginal fits the benchmark teams printed for buoys A, B and C | the environmental-contour benchmark's contributing teams | MIXED | 5 reproduce a certified maximum-likelihood fit; 1 printed as Hs reproduce the certified fit of Tz; 0 stated as maximum likelihood are off its maximum; 3 use another estimator (least squares) and are not decided here; 6 are not decidable at the printed precision | wrong quantity | the record |
| the annual-maximum Hs fits of 20 printed rows for five South Atlantic lease areas | Bhaskaran et al. (Energies 16, 6935, 2023) | NEEDS DATA | the fits are to a commercial hindcast nobody outside can re-run; against the nearest cells of the public one they are 19 OFF THE MAXIMUM, 1 OUTSIDE ITS SUPPORT — the distance between two records, not an error | data not public | the record |
| finite-time blowup for the 3D Navier–Stokes equations with finite energy (Clay alternatives C and D) | OpenAI (2026-09-08; the Lean amended 2026-09-10) | CERTIFIED | at f9e8bc5b (2026-09-10) the Lean proves the paper's Theorem 1.1 as stated, finite energy included (theorem_1_1, its statement read here clause by clause), and derives Clay's (C) and (D) from it — 486 changed or dependent modules rebuilt here over the pin's build, only the standard axioms for all 7 declarations asked; at the announced commit 8937a8f4 the finite-energy clause was not formal (PARTIAL, kept as history). Comparator judged (C) and (D) at 8937a8f4 and was not re-run at f9e8bc5b; Theorem 1.1's statement is OpenAI's own, not Comparator-judged | — | the record |
| C3b >= 1.77898884 (the registry's asterisked lower bound) | Mosaic Intelligence (2026) | CERTIFIED | the entropy ratio of the cited 13-point certificate, enclosed to 40 digits: 1.77898884142069354980… | — | the record |
| C3c >= 1.6747338950414058 (the registry's asterisked lower bound) | Y. Lin (2026) | CERTIFIED | the entropy ratio of the cited 147-point certificate, enclosed to 40 digits: 1.67473389504140587006… | — | the record |
| C3c >= 1.6747338950208249 (superseded in the registry) | Mosaic Intelligence (2026) | CERTIFIED | the entropy ratio of the cited 95-point certificate, enclosed to 40 digits: 1.67473389502082499337… | — | the record |
| C71 > 6.521845710923046575 (the registry's asterisked lower bound) | Numaro (numaro.tech, "AI Autoresearch"), Zenodo 10.5281/zenodo.21497769 | CERTIFIED | H/(I-1) for the balanced n = 18 function, I = 261/128 exact and H in outward-rounded intervals: 6.52184571092304657565…; the amplification rule (O'Donnell–Tan, Hod) is cited, not re-proved | — | the record |
| C84b <= 1.999281 (c >= 0.000719) (the registry's asterisked upper bound) | I. Althoefer with ChatGPT 5.5 (the note, 28 May 2026), as the registry quotes it | REPAIRED | holds as C84b <= 1.9993 (c >= 0.0007, the note's own theorem); for every choice of the chain's parameters c <= 0.0007150507 < 0.000719, so the quoted constant is out of reach of the calculation cited | arithmetic slip | the record |
| C42 <= 0.6906538 (the registry's asterisked upper bound) | S. Griego (github.com/sebastian-griego/turan-c42-certificate v1.0.0) | PARTIAL | the limiting certificate decided (|Y|/D <= 0.690653695151… < 0.6906538, |1-alpha| < C, |eta| < C, tau > 1/3); the asymptotic reduction to it, with no finite threshold, is the note's prose | narrower scope | the record |
| C42 <= 0.688983 (an open pull request to the registry) | A. Röhrig with Codex (teorth/optimizationproblems PR #184, 9 Sep 2026) | PARTIAL | the eight-block limiting certificate decided (|Y|/D <= 0.688982098504… < 0.688983, every block inside C, gap 9.7997e-07); the asymptotic reduction is the note's prose | narrower scope | the record |
| Borcea-Branden AIM problems | GPT-5 (Pro), GPT-5.6 (Pro), Opus 5 (S. Sra's counterexample library) | PARTIAL | Problems 36, 37 and 38 certified; Problem 35's nonexistence over all positive semidefinite A rests on universal theorems the case cites (Marcus 1963, Lieb 1966, via Wanless 2022) — the witness's own violations of both are certified, the universal step is not re-derived | narrower scope | the record |
| Courtade's volume conjecture for Minkowski sums is false | Suvrit Sra, by hand (S. Sra's counterexample library) | CERTIFIED | every fact the counterexample needs, re-derived from the published artifact | — | the record |
| Feasible Picard steps for DPP likelihood | GPT-5.6 (S. Sra's counterexample library) | PARTIAL | CERTIFIED when "feasible" means the step keeps the iterate positive definite (the case's context paragraph): a = 5 does, and the likelihood falls. The case's statement block reads "feasibility is the bound a ≤ 1/(1 − γ) of Prop. A.1", and for this L0 that bound is about 1.90, which a = 5 exceeds; under that reading the witness refutes nothing, and the steps tried inside the bound (a = 1, 3/2, 9/5, 1899/1000) all ascend | depends on reading | the record |
| Failure of the proposed Hamiltonian NEPv Rayleigh identity | OpenAI Codex (S. Sra's counterexample library) | CERTIFIED | every fact the counterexample needs, re-derived from the published artifact | — | the record |
| Log-volume midpoint gap and its Lorentzian generalization | GPT-5.6 Pro (S. Sra's counterexample library) | PARTIAL | the Lorentzian generalization certified (G strictly Lorentzian, the triangle violation enclosed to 60 digits); the log-volume-distance result needs convex bodies that exist only through a cited realization theorem (Shephard), none exhibited | narrower scope | the record |
| Macdonald lattice Schur-convexity | GPT-5.6 (Pro) (S. Sra's counterexample library) | CERTIFIED | every fact the counterexample needs, re-derived from the published artifact | — | the record |
| No dimension-free constant in O'Donnell's matrix conjecture | GPT-5.6 Pro (S. Sra's counterexample library) | PARTIAL | every family member to m = 11 certified (unit trace, PSD, ratio above m/32; built exactly, densely at n = 512); unboundedness over all m, which refuting a universal constant needs, is argued in prose (a trace-norm duality bound and dyadic sums past m = 14) | narrower scope | the record |
| An oblivious subspace injection need not give relative-error sketch-and-solve | OpenAI Codex (S. Sra's counterexample library) | CERTIFIED | every fact the counterexample needs, re-derived from the published artifact | — | the record |
| Exact QRCP can miss the orthonormal-row conditioning bound | OpenAI Codex (S. Sra's counterexample library) | CERTIFIED | every fact the counterexample needs, re-derived from the published artifact | — | the record |
| Quantum coupon collection: positivity of an alternating sum of inverses | GPT Pro (S. Sra's counterexample library) | CERTIFIED | every fact the counterexample needs, re-derived from the published artifact | — | the record |
| A mixed-norm Cauchy-Schwarz question of Sah, Sawhney, Stoner and Zhao | GPT-5.6 Sol (Pro), Opus 5 (S. Sra's counterexample library) | CERTIFIED | every fact the counterexample needs, re-derived from the published artifact | — | the record |
| Strict diagonal dominance does not ensure diminishing Nyström error reductions | OpenAI Codex (S. Sra's counterexample library) | CERTIFIED | every fact the counterexample needs, re-derived from the published artifact | — | the record |
| Derivatives of the Jacobi-theta kernel | GPT-5.5 (Pro) (S. Sra's counterexample library) | CERTIFIED | every fact the counterexample needs, re-derived from the published artifact | — | the record |
| Variance-sensitive Matrix Spencer | GPT-5.5 (Pro) (S. Sra's counterexample library) | PARTIAL | m = 2..10 certified exactly (discrepancy by exhaustion, ratio squared up to 81/11 at m = 10); unboundedness over all m, which refuting a universal constant needs, is a three-line argument in the case's prose, checked by hand and not by code | narrower scope | the record |
| R(k,k) <= 3.6960839126^(k+o(k)), improving 3.7992 (Gupta–Ndiaye–Norin–Wei), by a certificate that satisfies their Theorem 14 | GPT-5.4 Pro, reproduced by GPT-5.6 Sol Max (HorizonMath, arXiv 2603.15617v2, Appendix A.3) | REFUTED | 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 | checker wider than definition | the record |
| Area(E) = 0.1091479892... < 0.1148103258186177 (the AlphaEvolve baseline) | GPT-5.4 Pro, reproduced by GPT-5.6 Sol Max (HorizonMath, arXiv 2603.15617v2, Appendix A.2) | CERTIFIED | 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 | — | the record |
| three closed forms that "pass the compliance check and match the high-precision reference" | GPT-5.6 Sol Max (HorizonMath, arXiv 2603.15617v2) | NEEDS DATA | 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 | data not public | the record |
| R(k,k) <= 3.78233^(k+o(k)): Theorem 1 of Gupta–Ndiaye–Norin–Wei holds with G_AI (one more iteration of Theorem 14) | ChatGPT 5.6 Sol, printed by Gupta, Ndiaye, Norin and Wei as preliminary and unverified (arXiv 2407.19026v2) | CERTIFIED | Theorem 14's inequality decided on all of (0, 1] in interval arithmetic for F = h + G_AI, with a continuous M chosen here and Y from Lemma 15 for the proved F_0.03, by two independent programs; the theorem, the lemma and Theorem 1 are the paper's. Base e^F(1) = 3.78232877553731…; the printed 3.78233 is its rounding | — | the record |
| G: a polynomial map of C^3 with det J = 2 that is not injective (a counterexample to the Jacobian conjecture), printed in full | S. Gao, with Claude Fable 5 assisting (arXiv 2608.00222) | CERTIFIED | every fact the claim needs, as an exact identity or evaluation | — | the record |
| F4: a polynomial map of C^4 with det J = -44/9 that is not injective (a counterexample to the Jacobian conjecture), printed in full | S. Gao, with Claude Fable 5 assisting (arXiv 2608.00222) | CERTIFIED | every fact the claim needs, as an exact identity or evaluation | — | the record |
| F5: a polynomial map of C^4 with det J = 160/29 that is not injective (a counterexample to the Jacobian conjecture), printed in full | S. Gao, with Claude Fable 5 assisting (arXiv 2608.00222) | CERTIFIED | every fact the claim needs, as an exact identity or evaluation | — | the record |
| F6: the map the printed construction defines (n = 5), det J = -290, not injective | S. Gao, with Claude Fable 5 assisting (arXiv 2608.00222) | CERTIFIED | every fact the claim needs, as an exact identity or evaluation (the full map is not printed; rebuilt from the printed construction) | — | the record |
| F7: the map the printed construction defines (n = 5), det J = 119377, not injective | S. Gao, with Claude Fable 5 assisting (arXiv 2608.00222) | PARTIAL | non-injective and the factors' determinants decided; the constant determinant of the whole rests on the paper's factorisation lemma (the full map is not printed; rebuilt from the printed construction) | narrower scope | the record |
| X on R^14 (degree 7): JX has spectrum {-1} at every point and X has three zeros, so the weak Markus–Yamabe conjecture fails in dimension 14 | Castañeda, Honorato, Valenzuela-Henríquez (arXiv 2608.05392) | CERTIFIED | every fact the claim needs, as an exact identity or evaluation | — | the record |
| Phi, a cubic map of C^11 with det J = -2 that is not injective (the gist that first published it says ChatGPT generated it) | Castañeda, Honorato, Valenzuela-Henríquez (arXiv 2608.05392); the map first published in a gist that says ChatGPT generated it | CERTIFIED | every fact the claim needs, as an exact identity or evaluation | — | the record |
| X-hat on R^18 (degree 3): spectrum {-1} everywhere, three zeros — decided as a symbolic identity where the paper's appendix checks one sample point | Castañeda, Honorato, Valenzuela-Henríquez (arXiv 2608.05392) | CERTIFIED | every fact the claim needs, as an exact identity or evaluation | — | the record |
| the notebook's "Rank-32 decomposition of <2,4,5> over 0.5*Z" is a decomposition of that tensor of rank 32 over that ring | AlphaEvolve (DeepMind, 2025; arXiv:2506.13131) | CERTIFIED | every one of the 1,600 tensor-identity equations, exactly, over (1/2)Z, decided as the integer identity 8·T after the factors were scaled by 2·2·2 (so the scheme holds over every commutative ring in which 2 is invertible) — C-layout CA; rank 32 < 40 naive | — | the record |
| the notebook's "Rank-45 decomposition of <2,4,7> over Z" is a decomposition of that tensor of rank 45 over that ring | AlphaEvolve (DeepMind, 2025; arXiv:2506.13131) | CERTIFIED | every one of the 3,136 tensor-identity equations, exactly, over Z (an integer identity, so the scheme holds over every commutative ring) — C-layout CA; rank 45 < 56 naive | — | the record |
| the notebook's "Rank-51 decomposition of <2,4,8> over Z" is a decomposition of that tensor of rank 51 over that ring | AlphaEvolve (DeepMind, 2025; arXiv:2506.13131) | CERTIFIED | every one of the 4,096 tensor-identity equations, exactly, over Z (an integer identity, so the scheme holds over every commutative ring) — C-layout CA; rank 51 < 64 naive | — | the record |
| the notebook's "Rank-47 decomposition of <2,5,6> over Z" is a decomposition of that tensor of rank 47 over that ring | AlphaEvolve (DeepMind, 2025; arXiv:2506.13131) | CERTIFIED | every one of the 3,600 tensor-identity equations, exactly, over Z (an integer identity, so the scheme holds over every commutative ring) — C-layout CA; rank 47 < 60 naive | — | the record |
| the notebook's "Rank-23 decomposition of <3,3,3> over Z" is a decomposition of that tensor of rank 23 over that ring | AlphaEvolve (DeepMind, 2025; arXiv:2506.13131) | CERTIFIED | every one of the 729 tensor-identity equations, exactly, over Z (an integer identity, so the scheme holds over every commutative ring) — C-layout CA; rank 23 < 27 naive | — | the record |
| the notebook's "Rank-54 decomposition of <3,4,6> over 0.5*Z" is a decomposition of that tensor of rank 54 over that ring | AlphaEvolve (DeepMind, 2025; arXiv:2506.13131) | CERTIFIED | every one of the 5,184 tensor-identity equations, exactly, over (1/2)Z, decided as the integer identity 8·T after the factors were scaled by 2·2·2 (so the scheme holds over every commutative ring in which 2 is invertible) — C-layout CA; rank 54 < 72 naive | — | the record |
| the notebook's "Rank-63 decomposition of <3,4,7> over 0.5*C" is a decomposition of that tensor of rank 63 over that ring | AlphaEvolve (DeepMind, 2025; arXiv:2506.13131) | CERTIFIED | every one of the 7,056 tensor-identity equations, exactly, over (1/2)Z[i], decided as the Gaussian-integer identity 8·T after the factors were scaled by 2·2·2, imaginary part exactly 0 — C-layout CA; rank 63 < 84 naive | — | the record |
| the notebook's "Rank-74 decomposition of <3,4,8> over Z" is a decomposition of that tensor of rank 74 over that ring | AlphaEvolve (DeepMind, 2025; arXiv:2506.13131) | CERTIFIED | every one of the 9,216 tensor-identity equations, exactly, over Z (an integer identity, so the scheme holds over every commutative ring) — C-layout CA; rank 74 < 96 naive | — | the record |
| the notebook's "Rank-68 decomposition of <3,5,6> over Z" is a decomposition of that tensor of rank 68 over that ring | AlphaEvolve (DeepMind, 2025; arXiv:2506.13131) | CERTIFIED | every one of the 8,100 tensor-identity equations, exactly, over Z (an integer identity, so the scheme holds over every commutative ring) — C-layout CA; rank 68 < 90 naive | — | the record |
| the notebook's "Rank-80 decomposition of <3,5,7> over Z" is a decomposition of that tensor of rank 80 over that ring | AlphaEvolve (DeepMind, 2025; arXiv:2506.13131) | CERTIFIED | every one of the 11,025 tensor-identity equations, exactly, over Z (an integer identity, so the scheme holds over every commutative ring) — C-layout CA; rank 80 < 105 naive | — | the record |
| the notebook's "Rank-96 decomposition of <4,4,8> over 0.5*C" is a decomposition of that tensor of rank 96 over that ring | AlphaEvolve (DeepMind, 2025; arXiv:2506.13131) | CERTIFIED | every one of the 16,384 tensor-identity equations, exactly, over (1/2)Z[i], decided as the Gaussian-integer identity 8·T after the factors were scaled by 2·2·2, imaginary part exactly 0 — C-layout CA; rank 96 < 128 naive; the object is not printed; rebuilt here from the printed recipe and the printed <4,4,4> | — | the record |
| the notebook's "Rank-61 decomposition of <4,4,5> over Z" is a decomposition of that tensor of rank 61 over that ring | AlphaEvolve (DeepMind, 2025; arXiv:2506.13131) | CERTIFIED | every one of the 6,400 tensor-identity equations, exactly, over Z (an integer identity, so the scheme holds over every commutative ring) — C-layout CA; rank 61 < 80 naive | — | the record |
| the notebook's "Rank-85 decomposition of <4,4,7> over Z" is a decomposition of that tensor of rank 85 over that ring | AlphaEvolve (DeepMind, 2025; arXiv:2506.13131) | CERTIFIED | every one of the 12,544 tensor-identity equations, exactly, over Z (an integer identity, so the scheme holds over every commutative ring) — C-layout CA; rank 85 < 112 naive | — | the record |
| the notebook's "Rank-90 decomposition of <4,5,6> over Z" is a decomposition of that tensor of rank 90 over that ring | AlphaEvolve (DeepMind, 2025; arXiv:2506.13131) | CERTIFIED | every one of the 14,400 tensor-identity equations, exactly, over Z (an integer identity, so the scheme holds over every commutative ring) — C-layout CA; rank 90 < 120 naive | — | the record |
| the notebook's "Rank-93 decomposition of <5,5,5> over Z" is a decomposition of that tensor of rank 93 over that ring | AlphaEvolve (DeepMind, 2025; arXiv:2506.13131) | CERTIFIED | every one of the 15,625 tensor-identity equations, exactly, over Z (an integer identity, so the scheme holds over every commutative ring) — C-layout CA; rank 93 < 125 naive | — | the record |
| the factors stored under key 2,2,3 of alphatensor_f2.npz are a decomposition of the ⟨2,2,3⟩ tensor over F2 of the rank the array's shape states | AlphaTensor (DeepMind, Nature 610, 2022) | CERTIFIED | every one of the 144 tensor-identity equations, exactly, over F2 — C-layout CA; rank 11 < 12 naive; the same factors are refuted over Q | — | the record |
| the factors stored under key 2,2,4 of alphatensor_f2.npz are a decomposition of the ⟨2,2,4⟩ tensor over F2 of the rank the array's shape states | AlphaTensor (DeepMind, Nature 610, 2022) | CERTIFIED | every one of the 256 tensor-identity equations, exactly, over F2 — C-layout CA; rank 14 < 16 naive; the same factors are refuted over Q | — | the record |
| the factors stored under key 2,3,3 of alphatensor_f2.npz are a decomposition of the ⟨2,3,3⟩ tensor over F2 of the rank the array's shape states | AlphaTensor (DeepMind, Nature 610, 2022) | CERTIFIED | every one of the 324 tensor-identity equations, exactly, over F2 — C-layout CA; rank 15 < 18 naive; the same factors are refuted over Q | — | the record |
| the factors stored under key 2,2,5 of alphatensor_f2.npz are a decomposition of the ⟨2,2,5⟩ tensor over F2 of the rank the array's shape states | AlphaTensor (DeepMind, Nature 610, 2022) | CERTIFIED | every one of the 400 tensor-identity equations, exactly, over F2 — C-layout CA; rank 18 < 20 naive; the same factors are refuted over Q | — | the record |
| the factors stored under key 2,3,4 of alphatensor_f2.npz are a decomposition of the ⟨2,3,4⟩ tensor over F2 of the rank the array's shape states | AlphaTensor (DeepMind, Nature 610, 2022) | CERTIFIED | every one of the 576 tensor-identity equations, exactly, over F2 — C-layout CA; rank 20 < 24 naive; the same factors are refuted over Q | — | the record |
| the factors stored under key 2,3,5 of alphatensor_f2.npz are a decomposition of the ⟨2,3,5⟩ tensor over F2 of the rank the array's shape states | AlphaTensor (DeepMind, Nature 610, 2022) | CERTIFIED | every one of the 900 tensor-identity equations, exactly, over F2 — C-layout CA; rank 25 < 30 naive; the same factors are refuted over Q | — | the record |
| the factors stored under key 2,4,4 of alphatensor_f2.npz are a decomposition of the ⟨2,4,4⟩ tensor over F2 of the rank the array's shape states | AlphaTensor (DeepMind, Nature 610, 2022) | CERTIFIED | every one of the 1,024 tensor-identity equations, exactly, over F2 — C-layout CA; rank 26 < 32 naive; the same factors are refuted over Q | — | the record |
| the factors stored under key 3,3,4 of alphatensor_f2.npz are a decomposition of the ⟨3,3,4⟩ tensor over F2 of the rank the array's shape states | AlphaTensor (DeepMind, Nature 610, 2022) | CERTIFIED | every one of the 1,296 tensor-identity equations, exactly, over F2 — C-layout CA; rank 29 < 36 naive; the same factors are refuted over Q | — | the record |
| the factors stored under key 2,2,3 of alphatensor_r.npz are a decomposition of the ⟨2,2,3⟩ matrix-multiplication tensor of the rank the array's shape states, over the ring the paper names | AlphaTensor (DeepMind, Nature 610, 2022) | CERTIFIED | every one of the 144 tensor-identity equations, exactly, over Z (an integer identity, so the scheme holds over every commutative ring) — C-layout CA; rank 11 < 12 naive | — | the record |
| the factors stored under key 2,2,4 of alphatensor_r.npz are a decomposition of the ⟨2,2,4⟩ matrix-multiplication tensor of the rank the array's shape states, over the ring the paper names | AlphaTensor (DeepMind, Nature 610, 2022) | CERTIFIED | every one of the 256 tensor-identity equations, exactly, over Z (an integer identity, so the scheme holds over every commutative ring) — C-layout CA; rank 14 < 16 naive | — | the record |
| the factors stored under key 2,3,3 of alphatensor_r.npz are a decomposition of the ⟨2,3,3⟩ matrix-multiplication tensor of the rank the array's shape states, over the ring the paper names | AlphaTensor (DeepMind, Nature 610, 2022) | CERTIFIED | every one of the 324 tensor-identity equations, exactly, over Z (an integer identity, so the scheme holds over every commutative ring) — C-layout CA; rank 15 < 18 naive | — | the record |
| the factors stored under key 2,2,5 of alphatensor_r.npz are a decomposition of the ⟨2,2,5⟩ matrix-multiplication tensor of the rank the array's shape states, over the ring the paper names | AlphaTensor (DeepMind, Nature 610, 2022) | CERTIFIED | every one of the 400 tensor-identity equations, exactly, over Z (an integer identity, so the scheme holds over every commutative ring) — C-layout CA; rank 18 < 20 naive | — | the record |
| the factors stored under key 2,2,6 of alphatensor_r.npz are a decomposition of the ⟨2,2,6⟩ matrix-multiplication tensor of the rank the array's shape states, over the ring the paper names | AlphaTensor (DeepMind, Nature 610, 2022) | CERTIFIED | every one of the 576 tensor-identity equations, exactly, over Z (an integer identity, so the scheme holds over every commutative ring) — C-layout CA; rank 21 < 24 naive | — | the record |
| the factors stored under key 2,3,4 of alphatensor_r.npz are a decomposition of the ⟨2,3,4⟩ matrix-multiplication tensor of the rank the array's shape states, over the ring the paper names | AlphaTensor (DeepMind, Nature 610, 2022) | CERTIFIED | every one of the 576 tensor-identity equations, exactly, over Z (an integer identity, so the scheme holds over every commutative ring) — C-layout CA; rank 20 < 24 naive | — | the record |
| the factors stored under key 2,2,7 of alphatensor_r.npz are a decomposition of the ⟨2,2,7⟩ matrix-multiplication tensor of the rank the array's shape states, over the ring the paper names | AlphaTensor (DeepMind, Nature 610, 2022) | CERTIFIED | every one of the 784 tensor-identity equations, exactly, over Z (an integer identity, so the scheme holds over every commutative ring) — C-layout CA; rank 25 < 28 naive | — | the record |
| the factors stored under key 2,3,5 of alphatensor_r.npz are a decomposition of the ⟨2,3,5⟩ matrix-multiplication tensor of the rank the array's shape states, over the ring the paper names | AlphaTensor (DeepMind, Nature 610, 2022) | CERTIFIED | every one of the 900 tensor-identity equations, exactly, over Z (an integer identity, so the scheme holds over every commutative ring) — C-layout CA; rank 25 < 30 naive | — | the record |
| the factors stored under key 2,2,8 of alphatensor_r.npz are a decomposition of the ⟨2,2,8⟩ matrix-multiplication tensor of the rank the array's shape states, over the ring the paper names | AlphaTensor (DeepMind, Nature 610, 2022) | CERTIFIED | every one of the 1,024 tensor-identity equations, exactly, over Z (an integer identity, so the scheme holds over every commutative ring) — C-layout CA; rank 28 < 32 naive | — | the record |
| the factors stored under key 2,4,4 of alphatensor_r.npz are a decomposition of the ⟨2,4,4⟩ matrix-multiplication tensor of the rank the array's shape states, over the ring the paper names | AlphaTensor (DeepMind, Nature 610, 2022) | CERTIFIED | every one of the 1,024 tensor-identity equations, exactly, over Z (an integer identity, so the scheme holds over every commutative ring) — C-layout CA; rank 26 < 32 naive | — | the record |
| the factors stored under key 3,3,4 of alphatensor_r.npz are a decomposition of the ⟨3,3,4⟩ matrix-multiplication tensor of the rank the array's shape states, over the ring the paper names | AlphaTensor (DeepMind, Nature 610, 2022) | CERTIFIED | every one of the 1,296 tensor-identity equations, exactly, over Z (an integer identity, so the scheme holds over every commutative ring) — C-layout CA; rank 29 < 36 naive | — | the record |
| the factors stored under key 2,4,5 of alphatensor_r.npz are a decomposition of the ⟨2,4,5⟩ matrix-multiplication tensor of the rank the array's shape states, over the ring the paper names | AlphaTensor (DeepMind, Nature 610, 2022) | CERTIFIED | every one of the 1,600 tensor-identity equations, exactly, over Z (an integer identity, so the scheme holds over every commutative ring) — C-layout CA; rank 33 < 40 naive | — | the record |
| the object is a feasible solution of "Erdős Minimum Overlap (Upper Bound)" and the platform's score of it is 0.3808585748578584 (minimize) | CodexProLong (EinsteinArena, solution 2507, 2026-08-15) | REPAIRED | as published, Σh − n/2 = 0.000000000000192824623888, not 0; the platform's verifier has no tolerance on the sum — in every public version (corpus/sources/easota-platform/erdos-min-overlap_9cd6fbfb.ts) it rescales h to Σh = n/2 in float64 before scoring, the repair below in floating point — and the 1e-6 is the Together repository notebook's np.isclose (corpus/easota), not the platform's. The printed 0.3808585748578584 agrees with the as-published bound 0.38085857485785828368… (above); the exact witness built from the same bytes (h scaled down to Σh = n/2) certifies C5 ≤ 0.38085857485785830932…, 0.000000000000000025639955 from the as-published value — at or below the printed 0.3808585748578584, so the printed upper bound itself is proved by the repaired witness | tolerance witness | the record |
| the object is a feasible solution of "First Autocorrelation Inequality (Upper Bound)" and the platform's score of it is 1.5027436492326165 (minimize) | CodexProLong (EinsteinArena, solution 2504, 2026-08-14) | CERTIFIED | f ≥ 0 at all 65536 values, exactly: C1 ≤ 1.50274364923261637372… exactly (the maximum of the autoconvolution decided at 16684 candidate indexes the float screen could not separate; the printed 1.5027436492326165 agrees, above) | — | the record |
| the object is a feasible solution of "Flat Polynomials (degree 69)" and the platform's score of it is 1.2807274949642549 (minimize) | white (EinsteinArena, solution 2475, 2026-08-10) | CERTIFIED | 70 coefficients ±1: C⁺ = max_{|z|=1}|g(z)|/√71 ∈ [1.28072749505304782858, 1.28072749505304790673], the supremum certified (the platform's 10⁶-point grid value 1.2807274949642549 is below it, within the platform's resolution) | — | the record |
| the object is a feasible solution of "Edges vs Triangles (Minimal Triangle Density)" and the platform's score of it is -0.7117091757692579 (maximize) | CHRONOS (EinsteinArena, solution 2367, 2026-06-27) | CERTIFIED | 500 rows of 20 weights: the platform's envelope score −(area + 10·max gap) = -0.71170917576925610657… exactly (the printed -0.7117091757692579 agrees, below) | — | the record |
| the object is a feasible solution of "Circles in a Rectangle (n = 21)" and the platform's score of it is 2.3658323759098576 (maximize) | ExoMind-TTS (EinsteinArena, solution 2544, 2026-08-30) | CERTIFIED | 21 circles, no two overlapping and the bounding box at w + h ≤ 2, exactly: Σr = 2.36583237590985849000… (the printed 2.3658323759098576 agrees, below) | — | the record |
| the object is a feasible solution of "Kissing Number in Dimension 12 (n=841)" and the platform's score of it is 2 (minimize) | CHRONOS (EinsteinArena, solution 2081, 2026-04-23) | CERTIFIED | NOT a kissing configuration of 841: 1 pair (0 & 840) point in the same direction (cos = 1 exactly) and every other pair is at 60° or more, exactly — 840 distinct directions form a kissing configuration (40,992 exact contacts); the platform's score 2 is exactly the penalty of the repeat (2 per repeated pair) | — | the record |
| the object is a feasible solution of "Kissing Number in Dimension 11 (n=605)" and the platform's score of it is 1.7102381876301676 (minimize) | ExoMind-TTS (EinsteinArena, solution 2560, 2026-09-05) | CERTIFIED | NOT a kissing configuration of 605 (as the platform's score itself says): 9,510 of 182,710 pairs are closer than 60°, decided exactly; the worst, vectors 601 and 604, has cos²θ = 1/4 + δ with δ = an exact rational (in the ledger), cos θ − 1/2 ∈ [0.26329908059535543918, 0.26329908059535543919] (θ ≈ 40.2441°, 19.7559° short of 60°); the platform's score 1.7102381876301676 is the overlap penalty Σ max(0, 2 − |c_i − c_j|), enclosed here in [1.71023818763016759486, 1.71023818763016759487] | — | the record |
| the object is a feasible solution of "Kissing Number in Dimension 12 (n=842)" and the platform's score of it is 0.5469062458557645 (minimize) | ExoMind-TTS (EinsteinArena, solution 2561, 2026-09-05) | CERTIFIED | NOT a kissing configuration of 842 (as the platform's score itself says): 237 of 354,061 pairs are closer than 60°, decided exactly; the worst, vectors 510 and 547, has cos²θ = 1/4 + δ with δ = an exact rational (in the ledger), cos θ − 1/2 ∈ [0.04327239768587444034, 0.04327239768587444035] (θ ≈ 57.0933°, 2.9067° short of 60°); the platform's score 0.5469062458557645 is the overlap penalty Σ max(0, 2 − |c_i − c_j|), enclosed here in [0.54690624585576453779, 0.54690624585576453780] | — | the record |
| the polynomial map with the sparse rational coefficients given has the Jacobian property claimed and the collision witnesses listed collide | The Station agents (dualverse-ai; station_data_v2) | CERTIFIED | every fact the claim needs, as an exact identity or evaluation: det JF = -6 identically (the 3×3 Jacobian determinant expanded over Q), components of degrees 4, 6, 7 with 3, 6, 7 terms, and three distinct rational points with the one image (1, 7/3, 0) — a non-injective Keller map of C^3, a counterexample to the Jacobian conjecture in dimension 3 (the claim the register already certifies for Gao's G, here with 16 terms and degrees 4, 6, 7); the file's family checks for a = 1, 2, -3, 3 (a3_transfer) name points of maps it does not print and are not decided here | — | the record |
The scope column is the load-bearing one. "CERTIFIED" without it says something the record does not: several of these rows certify a computational fragment of a claim whose analytic core was never touched, and the row says which. 2 rows are NEEDS DATA — a headline result whose coordinates have never been published, so it cannot be decided by anyone but its authors. That verdict measures the claimant, not the claim.