Mathematical FrontierNetworkSubmit movement

Temporal frontier state

Frontier feed

Every movement, report, reproduction, dispute, correction, and invalidation remains part of the record.

All eventsFrontier deltasReportedReconciledCorrectionsInvalidations
combinatoricsFeb 1, 2016Significance 90/100

Erdős discrepancy problem solution

Open→Solved

The cited peer-reviewed publication presents a proof of unbounded discrepancy for every sequence of signs.

Published / acceptedErdős discrepancy problem →
SourceReplayReproducedFormal proofStatement auditExternal checkExpert reviewPeer review
geometry-topologyAug 30, 2026Significance 82/100

Exact maxout (3,5) certificate suite replay passed

Prior state unknown→Exact certificate replay supports a reported maximum of 42 vertices under stated reductions

The pinned public repository passed 132,869 checks with zero failures in its generation-independent upper-bound audit. Certificate transport passed 2,220 exact group-item checks, and the rational 42-vertex witness passed. The record remains scoped to artifact replay pending an independent audit of the geometric reductions.

ReplayedMaximum vertex number of a (3,5)-zonoboxtope →
SourceReplayReproducedFormal proofStatement auditExternal checkExpert reviewPeer review
combinatoricsAug 29, 2026Significance 78/100

Both claw-free Schur counterexample coefficients independently reproduced

Prior state unknown→Both published negative Schur coefficients independently reproduced exactly

An independent exact enumerator rebuilt both line graphs from the published edge lists and obtained the stated coefficients −64 and −40. VibeMathed's external label remains unreviewed; this is a separate local verification event.

Independently reproducedStanley's Claw-Free Schur-Positivity Conjecture →
SourceReplayReproducedFormal proofStatement auditExternal checkExpert reviewPeer review
combinatoricsAug 30, 2026Significance 76/100

Hales–Jewett lower bound HJ(3,3) ≥ 22 independently reproduced

Prior state unknown→HJ(3,3) ≥ 22 independently reproduced

A clean-room standard-library verifier authenticated the frozen 253-cell coloring and exhaustively checked all 1,771 corner triples. None was monochromatic. The independent implementation also rejected an incomplete coloring and a deliberately monochromatic diagonal.

Independently reproducedHales–Jewett number HJ(3,3) →
SourceReplayReproducedFormal proofStatement auditExternal checkExpert reviewPeer review
geometry-topologyAug 29, 2026Significance 76/100

Exact Kusner counterexample certificate replay passed

Prior state unknown→Exact rational-box certificate replay passed for 58 equilateral points in l_5^56

The Zenodo archive matched its published checksums. Its quick check covered all 1,653 point pairs, and the full FLINT-backed verifier passed exact claims C1-C6 in 80 seconds. The result is recorded as an artifact replay, not peer review.

ReplayedKusner's Conjecture on Equilateral Sets in ℓpn\ell_p^n →
SourceReplayReproducedFormal proofStatement auditExternal checkExpert reviewPeer review
algebraAug 29, 2026Significance 74/100

Five-variable Hessian counterexample artifact replay passed

Prior state unknown→Explicit construction artifact replay passed; symbolic determinant proof remains source-level

The pinned no-CAS checker reproduced the degree, monomial count, exact rational gradient collision, and deterministic exact Hessian evaluations. VibeMathed's external label remains unreviewed; this is a separate local replay record.

ReplayedA Five-Variable Counterexample to the Hessian Conjecture →
SourceReplayReproducedFormal proofStatement auditExternal checkExpert reviewPeer review
combinatoricsAug 30, 2026Significance 72/100

Hales–Jewett lower bound HJ(4,2) ≥ 14 independently reproduced

Prior state unknown→HJ(4,2) ≥ 14 independently reproduced

A clean-room standard-library verifier authenticated the frozen 560-cell coloring and exhaustively checked all 1,820 corner quadruples. None was monochromatic. The independent implementation also rejected an incomplete coloring and a deliberately monochromatic diagonal.

Independently reproducedHales–Jewett number HJ(4,2) →
SourceReplayReproducedFormal proofStatement auditExternal checkExpert reviewPeer review
combinatoricsAug 30, 2026Significance 68/100

WOWII Conjecture 200 counterexample independently reproduced

Prior state unknown→Explicit 14-vertex counterexample independently reproduced under the frozen statement

A clean standard-library implementation reconstructed the public 14-vertex graph and exhaustively checked all 16,383 nonempty induced vertex sets. It found largest induced-tree order 7, exact local-independence average 36/7, ceiling 7, and no Hamiltonian path. Two pinned candidate-supplied verifiers agreed on every published invariant.

Independently reproducedWritten on the Wall II Conjecture 200 →
SourceReplayReproducedFormal proofStatement auditExternal checkExpert reviewPeer review
number-theoryAug 10, 2026Significance 68/100Registry: lean checked

The Proportion of Zeta Zeros on the Critical Line

Prior state unknown→proved

An unconditional record, not a resolution: the Riemann hypothesis is untouched, and Anthropic states it does not expect these techniques to lead to a proof of it. The paper is explicit that these are lower bounds only - the remaining third of the zeros are not shown to be off the line, merely not reached by the certificate. What it does settle is a question that was posed. Goldston and Suriajaya had reduced Montgom…

Source authenticatedThe Proportion of Zeta Zeros on the Critical Line →
SourceReplayReproducedFormal proofStatement auditExternal checkExpert reviewPeer review
algebraic complexityJul 19, 2026Significance 68/100

Exact rank-48 continuation artifacts opened for independent replay

A publication-worthy computational note and its exact certificates are now indexed with open replay tasks. The record does not promote the construction beyond reported status until the full CRT certificate is independently rerun.

ReportedAlgebraic continuation around rational rank-48 schemes for 4×4 matrix multiplication →
SourceReplayReproducedFormal proofStatement auditExternal checkExpert reviewPeer review
geometry-topologyAug 24, 2026Significance 65/100Registry: unreviewed

The (3,4,∞)(3,4,\infty) Modular Family of 2-Tori as a Complex Structure on S6S^6

Prior state unknown→proved

Claims an explicit compact complex threefold XX, fibred over P1\mathbb{P}^1 by complex 2-tori via period functions on the (3,4,∞)(3,4,\infty) orbifold, degenerating to a del Pezzo-of-degree-six fibre (identified opposite sides of its hexagon) at one point and to bielliptic multiple fibres of multiplicities 3 and 4 at the other two. Argues XX is simply connected with H∗(X;Z)=H∗(S6;Z)H_*(X;\mathbb{Z})=H_*(S^6;\mathbb{Z}), hence diffeo…

Source authenticatedThe (3,4,∞)(3,4,\infty) Modular Family of 2-Tori as a Complex Structure on S6S^6 →
SourceReplayReproducedFormal proofStatement auditExternal checkExpert reviewPeer review
algebraJul 20, 2026Significance 65/100Registry: expert verified

Jacobian Conjecture

Prior state unknown→disproved

n ≥ 3; plane case open

Source authenticatedJacobian Conjecture →
SourceReplayReproducedFormal proofStatement auditExternal checkExpert reviewPeer review
number-theorySep 3, 2026Significance 62/100Registry: lean checked

Prime Gaps at Most 186

Prior state unknown→proved

Assuming three explicit input statements, the project proves DHL[40,2]\mathrm{DHL}[40,2]: every admissible 4040-tuple contains at least two primes infinitely often after translation. An explicit admissible 4040-tuple of diameter 186186 then gives lim inf⁡n→∞(pn+1−pn)≤186.\liminf_{n\to\infty}(p_{n+1}-p_n)\le186. The Lean proof of the implication from the three inputs to the final theorem is kernel-checked. Two inputs are Kloosterman-type estimat…

Source authenticatedPrime Gaps at Most 186 →
SourceReplayReproducedFormal proofStatement auditExternal checkExpert reviewPeer review
number-theorySep 3, 2026Significance 62/100Registry: lean checked

A new bound for small gaps between primes: H1≤212H_1 \le 212

Prior state unknown→proved

H1≤212H_1 \le 212, improving Stadlmann's 240240 of three days earlier and the 246246 of Polymath8b that had stood since 2014. The twin prime conjecture, H1=2H_1 = 2, is untouched. Held the record for hours at most: OpenAI's paper claiming 186186 is dated 30 August, four days before this one, though its Lean development appeared on 2 September.

Source authenticatedA new bound for small gaps between primes: H1≤212H_1 \le 212 →
SourceReplayReproducedFormal proofStatement auditExternal checkExpert reviewPeer review
algebraic complexityJul 18, 2026Significance 62/100

Fixed rank-49 matrix circuit scheduled at peak 38

Prior state unknown→Certified fused order peaks at 38 working values

Lean certificates and an exact independent replay establish a 38-value fused schedule for the public 205-operation DAG, down from its 80-value sequential order. The result is a storage schedule for this fixed circuit, not a new rank or arithmetic record.

Formally verifiedLow-memory scheduling of the public rank-49 4×4 matrix multiplication circuit →
SourceReplayReproducedFormal proofStatement auditExternal checkExpert reviewPeer review
algebraSep 3, 2026Significance 60/100Registry: lean verified

Köthe Conjecture

Prior state unknown→disproved

GPT-6 Astra constructs a unital algebra R=k⊕AR=k\oplus A over the countable field k=F2‾, k=\overline{\mathbb F_2}, with I=AI=A a nil two-sided ideal, together with a matrix W∈M2(I) W\in M_2(I) that is not nilpotent. The algebra AA is generated by three weighted backward shifts. A diagonal construction chooses the weights so that every element of AA is nilpotent. At the same time, a suitable polynomial combination of the…

Source authenticatedKöthe Conjecture →
SourceReplayReproducedFormal proofStatement auditExternal checkExpert reviewPeer review
number-theorySep 3, 2026Significance 60/100Registry: site confirmed

Improved maximal prime-gap lower bound

Prior state unknown→proved

The paper proves G(X)≫log⁡X (log⁡2X)2 log⁡4X(log⁡3X)2 G(X)\gg \frac{\log X\,(\log_2 X)^2\,\log_4 X}{(\log_3 X)^2} for all sufficiently large XX. Its main new ingredient is a short-translates theorem: for any sufficiently small set S⊆[1,H]S\subseteq[1,H] with ∣S∣≤δx|S|\le\delta x, one can find a short translate making every corresponding linear form composite. Combining this with an Erdős--Rankin covering argument produces prime-free intervals of the clai…

Source authenticatedImproved maximal prime-gap lower bound →
SourceReplayReproducedFormal proofStatement auditExternal checkExpert reviewPeer review
number-theoryAug 25, 2026Significance 60/100Registry: expert verified

A tilted residue-class construction for long prime-free intervals

Prior state unknown→proved

Claims G(T)≫log⁡Tlog⁡2T/log⁡4TG(T)\gg\log T\log_2 T/\log_4 T against FGKMT's log⁡Tlog⁡2Tlog⁡4T/log⁡3T\log T\log_2 T\log_4 T/\log_3 T, a gain of log⁡3T/(log⁡4T)2\log_3 T/(\log_4 T)^2, together with Y(X)≫Xlog⁡X/log⁡3XY(X)\gg X\log X/\log_3 X for the covering problem behind it. Ben Green resists calling it incremental: "The sieving procedure is different to the Erdos-Rankin one which underpinned all bounds on the problem since 1938, and it wins log⁡3N\log_3 N over that procedure. The [FGKMT] p…

Source authenticatedA tilted residue-class construction for long prime-free intervals →
SourceReplayReproducedFormal proofStatement auditExternal checkExpert reviewPeer review
geometry-topologyAug 19, 2026Significance 60/100Registry: unreviewed

The Yau–Tian–Donaldson Conjecture for Constant Scalar Curvature Kähler Metrics

Prior state unknown→disproved

The paper's appendix draws a distinction worth keeping: a counterexample may reduce to a finite certificate, checkable once the object is written down, or it may itself be a theorem quantified over all degenerations. This is the second kind. The method field records construction, because the resolution exhibits an explicit fivefold, but the difficulty lay elsewhere - candidate manifolds of this shape have been avail…

Source authenticatedThe Yau–Tian–Donaldson Conjecture for Constant Scalar Curvature Kähler Metrics →
SourceReplayReproducedFormal proofStatement auditExternal checkExpert reviewPeer review
combinatoricsAug 26, 2026Significance 58/100Registry: lean verified

Erdős Problem #548: the Erdős–Sós conjecture

Prior state unknown→proved

For every n,kn,k with k+1≤nk+1\le n, every simple graph GG on nn vertices satisfying ∣E(G)∣≥k−12n+1 |E(G)|\ge \frac{k-1}{2}n+1 contains every tree on k+1k+1 vertices. The proof counts pairs (π,j)(\pi,j) where π=(v1,…,vn)\pi=(v_1,\ldots,v_n) is an ordering of the host vertices and v1vjv_1v_j is an edge. There are exactly 2∣E(G)∣(n−1)! 2|E(G)|(n-1)! such pairs. An induction on the target tree bounds this quantity by a rooted-copy count plus…

Source authenticatedErdős Problem #548: the Erdős–Sós conjecture →
SourceReplayReproducedFormal proofStatement auditExternal checkExpert reviewPeer review
geometry-topologySep 6, 2026Significance 58/100

A 44-vertex triangulation of RP6\mathbb{RP}^{6} reported

Prior state unknown→44-vertex RP6\mathbb{RP}^{6} construction source-authenticated; vertex minimum remains open

VibeMathed recorded: A 44-vertex triangulation of RP6\mathbb{RP}^{6} reported. Consult the linked registry entry for the original report and scope.

Source authenticatedA 44-vertex triangulation of RP6\mathbb{RP}^{6} →
SourceReplayReproducedFormal proofStatement auditExternal checkExpert reviewPeer review
geometry-topologyAug 30, 2026Significance 55/100

Conditional complex structure on S2×S4S^{2}\times S^{4} reported

Prior state unknown→Conditional complex structure on S2×S4S^{2}\times S^{4} source-authenticated; cited S6S^{6} input and independent verification remain open

VibeMathed recorded: Conditional complex structure on S2×S4S^{2}\times S^{4} reported. Consult the linked registry entry for the original report and scope.

Source authenticatedA complex structure on S2×S4S^{2}\times S^{4} →
SourceReplayReproducedFormal proofStatement auditExternal checkExpert reviewPeer review
geometry-topologyAug 19, 2026Significance 55/100Registry: lean checked

The C∞C^\infty Carathéodory Conjecture on Umbilic Points

Prior state unknown→disproved

Only the smooth case falls. Hamburger's real-analytic theorem is untouched, and the counterexample is explicitly a C∞C^\infty object, so the conjecture's classical analytic form remains true. The gap between the two is the whole content of the result.

Source authenticatedThe C∞C^\infty Carathéodory Conjecture on Umbilic Points →
SourceReplayReproducedFormal proofStatement auditExternal checkExpert reviewPeer review
theoretical-computer-scienceAug 17, 2026Significance 55/100Registry: unreviewed

The Matrix Multiplication Exponent

Prior state unknown→proved

A record, not a resolution, and a small one by design. The bound moves from 2.3713392.371339 to 2.3711772.371177, about 1.6×10−41.6 \times 10^{-4}, and the authors describe it as a small step. Whether ω=2\omega = 2 is untouched, and nothing here suggests the laser method can reach it. The interesting claim is methodological rather than numerical. The bottleneck in this line of work is a hard optimization problem, and the paper report…

Source authenticatedThe Matrix Multiplication Exponent →
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combinatoricsJul 10, 2026Significance 55/100Registry: lean checked

Cycle Double Cover Conjecture

Prior state unknown→proved

VibeMathed recorded: Cycle Double Cover Conjecture. Consult the linked registry entry for the original report and scope.

Source authenticatedCycle Double Cover Conjecture →
SourceReplayReproducedFormal proofStatement auditExternal checkExpert reviewPeer review
analysisAug 5, 2026Significance 53/100Registry: lean verified

Schiffer's Conjecture and the Pompeiu Problem

Prior state unknown→disproved

Also refutes the 1929 Pompeiu problem: Corollary 1.2 applies Williams' classical 1976 equivalence to the same constructed domains, so this is one construction settling both, not two separate results.

Source authenticatedSchiffer's Conjecture and the Pompeiu Problem →
SourceReplayReproducedFormal proofStatement auditExternal checkExpert reviewPeer review
combinatoricsAug 26, 2026Significance 52/100Registry: lean verified

Erdős Problem #571: rational exponents for bipartite Turán numbers

Prior state unknown→proved

For every rational α\alpha satisfying 1≤α<2, 1\le\alpha<2, the Lean theorem constructs some finite qq and a bipartite graph G:SimpleGraph⁡(Fin⁡q) G:\operatorname{SimpleGraph}(\operatorname{Fin} q) such that ex⁡(n;G)=Θ(nα) \operatorname{ex}(n;G)=\Theta(n^\alpha) as n→∞n\to\infty. This resolves the single-graph rational-exponents conjecture. Earlier work of Bukh and Conlon proved the corresponding statement only for a finite family of forbid…

Source authenticatedErdős Problem #571: rational exponents for bipartite Turán numbers →
SourceReplayReproducedFormal proofStatement auditExternal checkExpert reviewPeer review
combinatoricsAug 28, 2026Significance 50/100Registry: lean verified

Erdős Problem #1: sum-distinct sets

Prior state unknown→disproved

The formal theorem proves that no universal constant C>0C>0 can satisfy N>C 2∣A∣ N>C\,2^{|A|} for every nonempty interval bound NN and every sum-distinct A⊆{1,…,N}A\subseteq\{1,\dots,N\}. Equivalently, for every ε>0\varepsilon>0 there are arbitrarily large nn and sum-distinct nn-element sets contained in {1,…,N}\{1,\dots,N\} with N≤ε2n. N\le\varepsilon 2^n. The proof is ineffective: it establishes the existence of arbitrarily la…

Source authenticatedErdős Problem #1: sum-distinct sets →
SourceReplayReproducedFormal proofStatement auditExternal checkExpert reviewPeer review
number-theoryAug 23, 2026Significance 50/100Registry: unreviewed

A Rank-3131 Record for an Elliptic Curve over Q\mathbb{Q}

Prior state unknown→proved

Two tiers, and only the first is the record. Rank ≥31\ge 31 is unconditional: 31 explicit points, independence asserted via the leaderboard's stated general practice of exact 2-descent (not reproduced here - see the verification note). Rank exactly 31 is conditional on GRH and BSD, per the submitters' commentary, in the same style as the sibling record's Bober-bound argument; no numeric derivation has been published…

Source authenticatedA Rank-3131 Record for an Elliptic Curve over Q\mathbb{Q} →
SourceReplayReproducedFormal proofStatement auditExternal checkExpert reviewPeer review
theoretical-computer-scienceAug 20, 2026Significance 50/100Registry: unreviewed

Whether Marton's Inner Bound Achieves the Broadcast Channel Capacity Region

Prior state unknown→disproved

VibeMathed recorded: Whether Marton's Inner Bound Achieves the Broadcast Channel Capacity Region. Consult the linked registry entry for the original report and scope.

Source authenticatedWhether Marton's Inner Bound Achieves the Broadcast Channel Capacity Region →
SourceReplayReproducedFormal proofStatement auditExternal checkExpert reviewPeer review
number-theoryAug 20, 2026Significance 50/100Registry: unreviewed

Record Rank for an Elliptic Curve over Q\mathbb{Q}

Prior state unknown→proved

Two tiers, and only the first is the record. Rank ≥30\ge 30 is unconditional, being thirty explicit independent points. Rank exactly 30 is conditional: applying Bober's bound (arXiv:1112.1503) with Δ=4.25\Delta = 4.25 gives an analytic rank of at most 31, and the root number is +1+1 so the rank is even, hence 30 - but that argument assumes GRH, and equating analytic rank with rank assumes BSD. The entry is a partial resu…

Source authenticatedRecord Rank for an Elliptic Curve over Q\mathbb{Q} →
SourceReplayReproducedFormal proofStatement auditExternal checkExpert reviewPeer review
combinatoricsSep 6, 2026Significance 48/100

Fixed 19-dimensional added-vector code optimum reported

Prior state unknown→Fixed added-vector code optimum α5(D)=1280\alpha_5(D)=1280 source-authenticated; unrestricted k(19)k(19) remains open

VibeMathed recorded: Fixed 19-dimensional added-vector code optimum reported. Consult the linked registry entry for the original report and scope.

Source authenticatedOptimality of the added-vector code in the 19-dimensional kissing construction →
SourceReplayReproducedFormal proofStatement auditExternal checkExpert reviewPeer review
analysisSep 6, 2026Significance 45/100Registry: lean verified

Smale’s Mean Value Conjecture (K=1K=1)

disproved→disproved

The formal proof constructs a complex polynomial pp such that p(0)=0,p′(0)=1, p(0)=0,\qquad p'(0)=1, and for every critical point cc of pp, ∣p(c)c∣>1. \left|\frac{p(c)}{c}\right|>1. Thus at z=0z=0 there is no critical point satisfying ∣p(0)−p(c)∣∣c∣≤∣p′(0)∣=1, \frac{|p(0)-p(c)|}{|c|}\le |p'(0)|=1, which disproves Smale's conjectured universal constant K=1K=1. The counterexample has very large unspecified degree and violates the bound only by a smal…

Source authenticatedSmale’s Mean Value Conjecture (K=1K=1) →
SourceReplayReproducedFormal proofStatement auditExternal checkExpert reviewPeer review
analysisSep 3, 2026Significance 45/100

Smale’s Mean Value Conjecture (K=1K=1)

Prior state unknown→disproved

The formal proof constructs a complex polynomial pp such that p(0)=0,p′(0)=1, p(0)=0,\qquad p'(0)=1, and for every critical point cc of pp, ∣p(c)c∣>1. \left|\frac{p(c)}{c}\right|>1. Thus at z=0z=0 there is no critical point satisfying ∣p(0)−p(c)∣∣c∣≤∣p′(0)∣=1, \frac{|p(0)-p(c)|}{|c|}\le |p'(0)|=1, which disproves Smale's conjectured universal constant K=1K=1. The counterexample has very large unspecified degree and violates the bound only by a smal…

Source authenticatedSmale’s Mean Value Conjecture (K=1K=1) →
SourceReplayReproducedFormal proofStatement auditExternal checkExpert reviewPeer review
analysisAug 1, 2026Significance 45/100Registry: lean verified

Connes' Rigidity Conjecture

Prior state unknown→disproved

VibeMathed recorded: Connes' Rigidity Conjecture. Consult the linked registry entry for the original report and scope.

Source authenticatedConnes' Rigidity Conjecture →
SourceReplayReproducedFormal proofStatement auditExternal checkExpert reviewPeer review
geometry-topologyJul 27, 2026Significance 45/100Registry: unreviewed

KLS Conjecture for Quadratic Forms

Prior state unknown→proved

with constant 2; also improves the global KLS bound to O(log⁡1/4n)O(\log^{1/4} n)

Source authenticatedKLS Conjecture for Quadratic Forms →
SourceReplayReproducedFormal proofStatement auditExternal checkExpert reviewPeer review
analysisAug 24, 2026Significance 45/100Registry: unreviewed

Nevanlinna’s half-plane omitted-values problem

Prior state unknown→disproved

The paper constructs a real meromorphic function FF on C\mathbb{C} satisfying F−1({0,1,∞})⊂RF^{-1}(\{0,1,\infty\})\subset\mathbb{R}, with each of the three fibers F−1(0)F^{-1}(0), F−1(1)F^{-1}(1), and F−1(∞)F^{-1}(\infty) infinite, such that for every a∈C^∖{0,1,∞}a\in\widehat{\mathbb{C}}\setminus\{0,1,\infty\}, the aa-point divisor in each of the upper and lower half-planes fails the Blaschke condition. Consequently, FF is not of bounded type in…

Source authenticatedNevanlinna’s half-plane omitted-values problem →
SourceReplayReproducedFormal proofStatement auditExternal checkExpert reviewPeer review
probability-statisticsSep 5, 2026Significance 40/100Registry: lean checked

The Ibragimov–Iosifescu conjecture for φ-mixing sequences

Prior state unknown→disproved

Astra constructs a strictly stationary real process Xt=ξt+K(ξt−1,ξt−2,…), X_t=\xi_t+K(\xi_{t-1},\xi_{t-2},\ldots), where the innovations ξt\xi_t are i.i.d. Gaussian variables convolved with a symmetric rare-spike law, and KK is bounded, continuous, and odd. The process is φ\varphi-mixing, centered, square-integrable, and satisfies Var⁡(Sn)→∞. \operatorname{Var}(S_n)\to\infty. Nevertheless there are times nj→∞n_j\to\infty such that…

Source authenticatedThe Ibragimov–Iosifescu conjecture for φ-mixing sequences →
SourceReplayReproducedFormal proofStatement auditExternal checkExpert reviewPeer review
combinatoricsJul 23, 2026Significance 40/100Registry: site confirmed

Petersen Coloring Conjecture

Prior state unknown→disproved

This preprint was not the first disproof. A 68-vertex counterexample was posted to X on 23 July 2026 by @NeuralReformist, credited to GPT-5.6 Sol Ultra, sixteen days earlier. This site decoded that sparse6 string and checked it independently: 68 vertices, 102 edges, simple, cubic, connected, bridgeless, girth five, and no Petersen coloring under the same encoder used for the 112-vertex graph. Whether the two are ind…

Source authenticatedPetersen Coloring Conjecture →
SourceReplayReproducedFormal proofStatement auditExternal checkExpert reviewPeer review
combinatoricsJul 2, 2026Significance 40/100Registry: unreviewed

White's Conjecture on Matroids

Prior state unknown→disproved

VibeMathed recorded: White's Conjecture on Matroids. Consult the linked registry entry for the original report and scope.

Source authenticatedWhite's Conjecture on Matroids →
SourceReplayReproducedFormal proofStatement auditExternal checkExpert reviewPeer review
analysisAug 5, 2026Significance 40/100Registry: lean verified

Sendov's Conjecture

Prior state unknown→proved

Sendov's conjecture is resolved for every degree n >= 2, closing a gap that had stood since 1959: degrees up to eight were settled piecemeal between 1969 and 1999, and Tao's 2020 result covered all sufficiently large degrees without ever specifying the threshold, leaving the middle range open. Tao's digestion establishes the stronger interior form of the statement, which resolves the Phelps-Rodriguez conjecture in f…

Source authenticatedSendov's Conjecture →
SourceReplayReproducedFormal proofStatement auditExternal checkExpert reviewPeer review
geometry-topologyMay 1, 2026Significance 40/100Registry: expert verified

Erdős's Planar Unit Distance Conjecture

Prior state unknown→disproved

VibeMathed recorded: Erdős's Planar Unit Distance Conjecture. Consult the linked registry entry for the original report and scope.

Source authenticatedErdős's Planar Unit Distance Conjecture →
SourceReplayReproducedFormal proofStatement auditExternal checkExpert reviewPeer review
analysisAug 13, 2026Significance 40/100Registry: unreviewed

Banach's isometric conjecture

Prior state unknown→proved

The paper proves the previously unresolved odd-dimensional real cases of Banach's isometric conjecture. Combined with Gromov's earlier theorem for even dimensions and previous results, this completes the conjecture for real Banach spaces.

Source authenticatedBanach's isometric conjecture →
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combinatoricsAug 28, 2026Significance 38/100Registry: lean verified

Erdős Problem #74: locally almost bipartite graphs of infinite chromatic number

Prior state unknown→disproved

Astra proves that there exists a divergent function f:N→N,f(n)→∞, f:\mathbb N\to\mathbb N,\qquad f(n)\to\infty, such that no graph GG of infinite chromatic number can satisfy dbip(H)≤f(n) d_{\mathrm{bip}}(H)\le f(n) for every finite nn-vertex subgraph H⊆GH\subseteq G, where dbip(H)d_{\mathrm{bip}}(H) is the minimum number of edges that must be deleted to make HH bipartite. In fact, every included resolution proves the stronger state…

Source authenticatedErdős Problem #74: locally almost bipartite graphs of infinite chromatic number →
SourceReplayReproducedFormal proofStatement auditExternal checkExpert reviewPeer review
geometry-topologyAug 13, 2026Significance 38/100Registry: unreviewed

Universal volume growth bounds from positive intermediate curvature

Prior state unknown→proved

Gromov's 1986 question drew three independent proofs within about 24 hours, two with AI in the loop. Ge's non-AI proof (heat-kernel Fisher metric, Nash entropy) came first, 13 August. Antonelli's proof here (14 August, GPT-5.6 Sol) takes a different route, Hodge obstruction and rank improvement, and its headline addition is the general family: for every 0≤m≤n−20\le m\le n-2, nonnegative Ricci plus positive (m+1)(m{+}1)-inte…

Source authenticatedUniversal volume growth bounds from positive intermediate curvature →
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combinatoricsAug 4, 2026Significance 38/100Registry: lean verified

Asymptotically attaining the Moore bound

Prior state unknown→proved

Settles two conjectures. Theorem 1.1 proves Bollobas's asymptotic degree-diameter conjecture, in the stronger liminf form rather than the conjectured limsup. Corollary 1.2 proves Conjecture 3 of Cambie, Cames van Batenburg, de Joannis de Verclos and Kang on the edge variant, again in the stronger liminf form, and is tight for bipartite graphs.

Source authenticatedAsymptotically attaining the Moore bound →
SourceReplayReproducedFormal proofStatement auditExternal checkExpert reviewPeer review
analysisAug 18, 2026Significance 38/100Registry: unreviewed

Stein’s dimension-free weak-(1,1) Riesz transform problem

Prior state unknown→proved

The theorem is the vector-valued endpoint bound ∥Rf∥L1,∞≤2∥f∥L1\|Rf\|_{L^{1,\infty}} \le 2\|f\|_{L^1} for R=(R1,…,Rn)R = (R_1,\ldots,R_n), so the constant 2 also serves each component RjR_j uniformly in the dimension; the best previously known component bound grew like clog⁡nc\log n. The mechanism is a decomposition theorem stated as Theorem 1.2: for every nonnegative f∈L1∩L2f \in L^1 \cap L^2 and every λ>0\lambda > 0, write…

Source authenticatedStein’s dimension-free weak-(1,1) Riesz transform problem →
SourceReplayReproducedFormal proofStatement auditExternal checkExpert reviewPeer review
probability-statisticsAug 16, 2026Significance 37/100Registry: unreviewed

Talagrand’s convolution conjecture

Prior state unknown→proved

The claim is the exact conjectured decay: ψμa(u)≤Ca/log⁡u\psi_{\mu_a}(u) \le C_a/\sqrt{\log u} for every u>1u > 1 and every nn, with CaC_a dimension-free - concretely ≲κa2(log⁡κaκa−1)1/2\lesssim \kappa_a^2(\log\frac{\kappa_a}{\kappa_a-1})^{1/2} where κa=(1+a)/(1−a)\kappa_a = (1+a)/(1-a). What is new is one step in a three-paper chain rather than a proof from scratch, and the paper is explicit about it. Chen's reverse-heat and Boolean-bridge framework and X…

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probability-statisticsMay 11, 2026Significance 37/100Registry: unreviewed

Talagrand's Convexity Problem

Prior state unknown→proved

VibeMathed recorded: Talagrand's Convexity Problem. Consult the linked registry entry for the original report and scope.

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quantum-information-computingSep 2, 2026Significance 36/100Registry: unreviewed

A Quantum Oracle Separation Between QMA(2)\mathsf{QMA}(2) and QMA\mathsf{QMA}

Prior state unknown→proved

The authors construct a unitary oracle UU such that QMAU≠QMA(2)U. \mathsf{QMA}^{U}\neq\mathsf{QMA}(2)^{U}. Their black-box problem is solvable by a QMA(2)\mathsf{QMA}(2) verifier with one oracle query and linear-size unentangled proofs, whereas any QMA\mathsf{QMA} verifier must use either exponentially many queries or an exponentially large witness. As a non-oracle consequence, they prove that for every fixed…

Source authenticatedA Quantum Oracle Separation Between QMA(2)\mathsf{QMA}(2) and QMA\mathsf{QMA} →
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number-theoryAug 28, 2026Significance 35/100Registry: lean verified

Erdős Problem #126: prime divisors of pairwise sums

Prior state unknown→proved

For f(n)=min⁡∣A∣=n∣{p prime:p∣a+b for some distinct a,b∈A}∣, f(n)=\min_{|A|=n}\left|\left\{p\text{ prime}:p\mid a+b\text{ for some distinct }a,b\in A\right\}\right|, Astra formally proves f(n)log⁡n→∞. \frac{f(n)}{\log n}\to\infty. The repository contains substantially stronger proofs. In particular, one verified alternate resolution establishes n≤3r2, n\le 3r^2, where rr is the number of supporting primes, yielding f(n)≫n1/2. f(n)\gg n^{1/2}. Other independent resolutions give…

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algebraJul 22, 2026Significance 35/100Registry: unreviewed

An Explicit Counterexample to the Rank-Two Poisson Conjecture

Prior state unknown→disproved

The paper constructs explicit R,T,D,S∈Q[x,q,p,z]R,T,D,S\in\mathbb Q[x,q,p,z] defining a Poisson endomorphism of the canonical rank-two Poisson algebra P2\mathcal P_2 that is not an automorphism. Its associated polynomial map preserves the canonical symplectic form, has Jacobian determinant 11, and has an explicit fiber of exactly three points. Thus PC(2)\mathrm{PC}(2) is false, and stabilization gives failure of PC(n)\mathrm{PC}(n) for…

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theoretical-computer-scienceJul 20, 2026Significance 35/100Registry: unreviewed

Optimality of Greedy for Single-Pass Semi-Streaming Matching

Prior state unknown→proved

VibeMathed recorded: Optimality of Greedy for Single-Pass Semi-Streaming Matching. Consult the linked registry entry for the original report and scope.

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probability-statisticsJul 27, 2026Significance 35/100Registry: lean verified

Feige's Conjecture

Prior state unknown→proved

VibeMathed recorded: Feige's Conjecture. Consult the linked registry entry for the original report and scope.

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number-theoryMay 20, 2026Significance 35/100Registry: unreviewed

Erdős Problem #387

Prior state unknown→disproved

false in general; the positive direction holds for k large relative to n

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logic-foundationsAug 13, 2026Significance 35/100Registry: unreviewed

SOP2=SOP3SOP_2 = SOP_3

Prior state unknown→proved

The new content is SOP2⇒SOP3SOP_2 \Rightarrow SOP_3; the converse implication was known from the start. Dzamonja and Shelah asked whether either implication in SOP3⇒SOP2⇒SOP1SOP_3 \Rightarrow SOP_2 \Rightarrow SOP_1 reverses: Mutchnik answered the second (SOP1=SOP2SOP_1 = SOP_2), and this answers the first, collapsing the bottom of the hierarchy to SOP1=SOP2=SOP3SOP_1 = SOP_2 = SOP_3. The SOPnSOP_n hierarchy for n≥3n \ge 3 remains, as does everything above…

Source authenticatedSOP2=SOP3SOP_2 = SOP_3 →
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quantum-information-computingJul 23, 2026Significance 35/100Registry: unreviewed

Two-Copy Distillability of Werner States

Prior state unknown→disproved

Two-copy distillable if and only if already one-copy distillable. Four independent papers settled this within five days of each other, and no single one of them is the account of record. Fu, Gao and Park posted first on 23 July (arXiv:2607.21367), followed by Song and Chen on 26 July (arXiv:2607.23416), then on 27 July both Fraser, Huber, Pozsgay and Vona (arXiv:2607.24309) and Bharti, Gajjala and Haug (arXiv:2607.…

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theoretical-computer-scienceAug 1, 2026Significance 35/100Registry: lean verified

Polynomial-Factor Hardness for the Closest Vector Problem

Prior state unknown→proved

VibeMathed recorded: Polynomial-Factor Hardness for the Closest Vector Problem. Consult the linked registry entry for the original report and scope.

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analysisAug 12, 2026Significance 35/100Registry: unreviewed

Gamow liquid-drop minimizer conjecture

Prior state unknown→proved

The complete fixed-volume picture, closing a gap that partial results had narrowed from both ends without meeting: balls uniquely minimize for every volume up to V_* = 3.51..., and above it no minimizer exists at all. Before this the best minimality range was V <= 1 (Chodosh-Ruohoniemi, 2025) and the best nonexistence bound V >= 7.5 (Schulz, posted two days earlier), so the open middle ran from 1 to 7.5. Frank-Nam h…

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analysisJul 31, 2026Significance 35/100Registry: unreviewed

Pólya's Conjecture for Neumann Balls in Dimensions Three and Higher

Prior state unknown→proved

The ball case. For arbitrary domains Pólya's conjecture remains open; this continues the authors' programme after the planar disk, circular sectors, and the Dirichlet case in arbitrary dimensions.

Source authenticatedPólya's Conjecture for Neumann Balls in Dimensions Three and Higher →
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analysisJul 27, 2026Significance 35/100Registry: expert verified

Crouzeix's Conjecture

Prior state unknown→proved

Two independent proofs within eight days, both with AI in the loop. Jin's (posted 27 July, preprints.org, submitted to Annals) is the first: its decisive theorem came out of an autonomous GPT-5.6 Sol run, and it is the proof Townsend, Greenbaum and Crouzeix have checked. Lorist and Schwenninger's five-page argument (arXiv, 4 August) is a genuinely different route - double-layer potentials plus a perturbation lemma f…

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mathematical-physicsJul 20, 2026Significance 35/100Registry: unreviewed

Full-RSB in the Sherrington–Kirkpatrick spin glass

Prior state unknown→proved

For the Sherrington-Kirkpatrick model with no external field, at zero temperature β=∞\beta=\infty, the paper proves that the zero-temperature Parisi minimizer is absolutely continuous with a smooth density and has support [0,1)[0,1) - full replica symmetry breaking, confirming the Parisi picture at the ground state. It also shows qβ→1q_\beta\to1 as β→∞\beta\to\infty. The positive-temperature input is Lopatto's: for every…

Source authenticatedFull-RSB in the Sherrington–Kirkpatrick spin glass →
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analysisAug 5, 2026Significance 33/100Registry: unreviewed

HRT Conjecture

Prior state unknown→disproved

VibeMathed recorded: HRT Conjecture. Consult the linked registry entry for the original report and scope.

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combinatoricsAug 29, 2026Significance 32/100Registry: site confirmed

Dean's conjecture for k=5k=5

Prior state unknown→proved

Claims the last open case of Dean's conjecture: every finite simple graph with minimum degree at least five contains a cycle of length divisible by five. Verified independently against Luo, Ma and Zhao (arXiv:2601.13552), whose abstract confirms the landscape - the conjecture was known for k∈{3,4}k\in\{3,4\} and they proved every k≥6k\ge6, leaving k=5k=5 open. If this proof stands, Dean's conjecture holds for all k≥3k\ge3.…

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probability-statisticsAug 6, 2026Significance 32/100Registry: unreviewed

Central limit theorem for the random assignment problem

Prior state unknown→proved

Claims the central limit theorem for the bipartite random assignment problem with bounded uniform costs: n (Cn−ζ(2))⇒N(0, 4ζ(2)−4ζ(3))\sqrt{n}\,(C_n-\zeta(2)) \Rightarrow \mathcal{N}(0,\,4\zeta(2)-4\zeta(3)). The limiting constant is not itself new - Wästlund had computed exactly 4ζ(2)−4ζ(3)4\zeta(2)-4\zeta(3) for the mean-one exponential model, and Malatesta, Parisi and Sicuro derived the non-bipartite analogue by replicas - but neither is a proof fo…

Source authenticatedCentral limit theorem for the random assignment problem →
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algorithms-optimizationOct 27, 2025Significance 32/100Registry: unreviewed

Point Convergence of Nesterov's Accelerated Gradient Method

Prior state unknown→proved

VibeMathed recorded: Point Convergence of Nesterov's Accelerated Gradient Method. Consult the linked registry entry for the original report and scope.

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differential-equationsSep 2, 2026Significance 30/100Registry: unreviewed

Interior Curvature Estimates for the Graphical Scalar Curvature Equation in All Dimensions

Prior state unknown→proved

For every n≥3n\geq3, let u∈C∞(B2)u\in C^\infty(B_2) be an admissible solution of κ[u]∈Γ2,σ2(κ[u])=1, \kappa[u]\in\Gamma_2,\qquad \sigma_2(\kappa[u])=1, with ∥u∥L∞(B2)+∥Du∥L∞(B2)≤K. \|u\|_{L^\infty(B_2)}+\|Du\|_{L^\infty(B_2)}\le K. Then sup⁡B1/2∣κ[u]∣≤C(n,K). \sup_{B_{1/2}}|\kappa[u]|\le C(n,K). Thus an admissible graph of constant scalar curvature with bounded height and slope cannot develop unbounded interior curvature, in any dimension n≥3n\ge3. Prior unrestricted qua…

Source authenticatedInterior Curvature Estimates for the Graphical Scalar Curvature Equation in All Dimensions →
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combinatoricsAug 12, 2026Significance 30/100Registry: site confirmed

Hadamard Matrix of Order 668

Prior state unknown→proved

Explicit construction of a Hadamard matrix of order 668, the smallest previously unresolved order, verified exactly by this site from the announcement plus its decoder reply. The same post encodes matrices for all twelve previously-open admissible orders below 2000 (668, 716, 892, 1132, 1244, 1388, 1436, 1676, 1772, 1916, 1948, 1964), and this site verified every one of them. The entry records the order-668 existenc…

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combinatoricsMay 13, 2026Significance 30/100Registry: unreviewed

Ben Green's Open Problem 90

Prior state unknown→proved

VibeMathed recorded: Ben Green's Open Problem 90. Consult the linked registry entry for the original report and scope.

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mathematical-physicsAug 3, 2026Significance 30/100Registry: unreviewed

Autonomous Lipschitz Fast Dynamo on the Three-Torus

Prior state unknown→proved

One variant case of Arnold's 1994 fast-dynamo problem, not the problem itself. Arnold asks for a single velocity field on T^3 that is smooth, divergence-free, autonomous and deterministic, fixed independently of the magnetic diffusivity, and that grows the magnetic field exponentially at every small enough diffusivity. The field constructed here is all of that except smooth: it is Lipschitz, not C^1. The sibling ent…

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probability-statisticsJun 21, 2026Significance 30/100Registry: lean verified

Kannan–Tetali–Vempala conjecture (bipartite/binary-matrix case)

Prior state unknown→proved

VibeMathed recorded: Kannan–Tetali–Vempala conjecture (bipartite/binary-matrix case). Consult the linked registry entry for the original report and scope.

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quantum-information-computingAug 9, 2026Significance 30/100Registry: unreviewed

Finite-Copy Distillability of NPT States in the DiVincenzo Family

Prior state unknown→disproved

Partial deliberately: the NPT bound entanglement problem itself is untouched. What falls is the conjecture about the canonical family, and the paper is explicit that a substantial neighbouring region remains unresolved while another is known two-copy undistillable.

Source authenticatedFinite-Copy Distillability of NPT States in the DiVincenzo Family →
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probability-statisticsAug 10, 2026Significance 30/100Registry: unreviewed

The Ellipsoid Fitting Conjecture

Prior state unknown→proved

Closes both gaps left open by Bandeira and Maillard: exact fitting, and removal of the operator-norm constraint. The threshold turns out to be governed by the statistical dimension d(d+1)/4 of the PSD cone.

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mathematical-physicsAug 9, 2026Significance 30/100Registry: unreviewed

Talagrand's critical Sherrington-Kirkpatrick overlap conjecture

Prior state unknown→proved

Talagrand's Conjecture 11.7.5 is resolved affirmatively for the critical Ising Sherrington-Kirkpatrick model: N2/3E⟨R1,22⟩N^{2/3}\mathbb{E}\langle R_{1,2}^2\rangle converges to a positive finite constant. The paper proves substantially more, showing that the entire quenched distribution of N1/3R1,2N^{1/3}R_{1,2} converges to an explicit random probability measure defined from the reflected Airy1\mathrm{Airy}_1 point process. The same…

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combinatoricsAug 6, 2026Significance 30/100Registry: unreviewed

Babai's Minimal Cayley Graph Problem

Prior state unknown→disproved

VibeMathed recorded: Babai's Minimal Cayley Graph Problem. Consult the linked registry entry for the original report and scope.

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analysisAug 20, 2026Significance 30/100Registry: unreviewed

Smooth Random Fast Dynamo on the Three-Torus

Prior state unknown→proved

Constructed an explicit class of smooth random, time-dependent incompressible velocity fields on T^3, obtained by alternating smooth shear flows with iid random phases on finite time blocks. For every fixed sufficiently small resistivity, the magnetic field has an almost-sure exponential growth rate at least 1/2, together with a time-uniform lower bound whose random prefactor has a resistivity-uniform inverse-moment…

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analysisAug 5, 2026Significance 30/100Registry: unreviewed

Rivière’s regularity question for critical nn-Laplace systems with antisymmetric potentials

Prior state unknown→disproved

For every n>2n>2, the paper constructs a bounded map U∈W1,n(Bn,Rn+2)U\in W^{1,n}(B^n,\mathbb{R}^{n+2}), smooth on Bn∖{0}B^n\setminus\{0\} but discontinuous at the origin, together with an antisymmetric potential Ω∈Ln(Bn,so(n+2)⊗Rn) \Omega\in L^n(B^n,so(n+2)\otimes\mathbb{R}^n) such that −Div(∣∇U∣n−2∇U)=Ω⋅∣∇U∣n−2∇Uin D′(Bn). -\mathrm{Div}\bigl(|\nabla U|^{n-2}\nabla U\bigr) = \Omega\cdot|\nabla U|^{n-2}\nabla U \qquad\text{in }D'(B^n). Moreover, the potential satisfies the sh…

Source authenticatedRivière’s regularity question for critical nn-Laplace systems with antisymmetric potentials →
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algebraAug 4, 2026Significance 30/100Registry: unreviewed

The Period-Index Conjecture

Prior state unknown→disproved

VibeMathed recorded: The Period-Index Conjecture. Consult the linked registry entry for the original report and scope.

Source authenticatedThe Period-Index Conjecture →
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combinatoricsAug 11, 2026Significance 30/100Registry: site confirmed

Albertson–Berman Induced Forest Conjecture

Prior state unknown→disproved

The ratio 15/31 is not claimed to be optimal, and the paper makes no claim that 31 vertices is the smallest possible counterexample. The construction produces separating triangles by design, so it says nothing about the 4-connected case.

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