Erdős discrepancy problem solution
Open→Solved
The cited peer-reviewed publication presents a proof of unbounded discrepancy for every sequence of signs.
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Temporal frontier state
Frontier feed
Every movement, report, reproduction, dispute, correction, and invalidation remains part of the record.
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.
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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.
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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 Montg…
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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.
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The $(3,4,\infty)$ Modular Family of 2-Tori as a Complex Structure on $S^6$
Prior state unknown→proved
Claims an explicit compact complex threefold $X$, fibred over $\mathbb{P}^1$ by complex 2-tori via period functions on the $(3,4,\infty)$ 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 $X$ is simply connected with $H_*(X;\mathbb{Z})=H_*(S^6;\mathbb{Z})$, hence diff…
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Jacobian Conjecture
Prior state unknown→disproved
n ≥ 3; plane case open
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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.
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Gromov and Weiss's Question on Sofic Groups
Prior state unknown→disproved
answered no: non-sofic groups exist
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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 ava…
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The $C^\infty$ 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 $C^\infty$ object, so the conjecture's classical analytic form remains true. The gap between the two is the whole content of the result.
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The Matrix Multiplication Exponent
Prior state unknown→proved
A record, not a resolution, and a small one by design. The bound moves from $2.371339$ to $2.371177$, about $1.6 \times 10^{-4}$, and the authors describe it as a small step. Whether $\omega = 2$ 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 repo…
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The Sum-Product Conjecture over the Reals
Prior state unknown→disproved
the model's contribution is one simplifying lemma; the authors state the main ideas are human
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Cycle Double Cover Conjecture
Prior state unknown→proved
Conjectures that every bridgeless graph has a collection of cycles covering each edge exactly twice.
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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.
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A Rank-$31$ Record for an Elliptic Curve over $\mathbb{Q}$
Prior state unknown→proved
Two tiers, and only the first is the record. Rank $\ge 31$ 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 publishe…
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Upper Bounds for High-Dimensional Sphere Packing
Prior state unknown→proved
upper bounds reach the Cohn-Elkies threshold; the true asymptotic density remains open
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Record Rank for an Elliptic Curve over $\mathbb{Q}$
Prior state unknown→proved
Two tiers, and only the first is the record. Rank $\ge 30$ is unconditional, being thirty explicit independent points. Rank exactly 30 is conditional: applying Bober's bound (arXiv:1112.1503) with $\Delta = 4.25$ gives an analytic rank of at most 31, and the root number is $+1$ 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 re…
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Whether Marton's Inner Bound Achieves the Broadcast Channel Capacity Region
Prior state unknown→disproved
Marton's inner bound, proposed in 1979, is the best known achievable region for a general discrete memoryless broadcast channel, and whether it always achieves the capacity region had been open ever since. It does not: there is a finite two-receiver discrete memoryless broadcast channel whose two-letter Marton value strictly exceeds twice its one-letter value, so the complete one-letter Marton region is strictly c…
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KLS Conjecture for Quadratic Forms
Prior state unknown→proved
with constant 2; also improves the global KLS bound to $O(\log^{1/4} n)$
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Connes' Rigidity Conjecture
Prior state unknown→disproved
Are ICC property (T) groups remembered by their von Neumann algebras - if $L(\Gamma) \cong L(\Lambda)$ for such groups, must $\Gamma \cong \Lambda$? A counterexample refutes Connes' conjecture that these groups are uniquely determined by their group von Neumann algebras.
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Nevanlinna’s half-plane omitted-values problem
Prior state unknown→disproved
The paper constructs a real meromorphic function $F$ on $\mathbb{C}$ satisfying $F^{-1}(\{0,1,\infty\})\subset\mathbb{R}$, with each of the three fibers $F^{-1}(0)$, $F^{-1}(1)$, and $F^{-1}(\infty)$ infinite, such that for every $a\in\widehat{\mathbb{C}}\setminus\{0,1,\infty\}$, the $a$-point divisor in each of the upper and lower half-planes fails the Blaschke condition. Consequently, $F$ is not of bounded type…
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Erdős's Planar Unit Distance Conjecture
Prior state unknown→disproved
Conjectured upper bound on how many pairs among $n$ points in the plane can be exactly one unit apart.
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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.
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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…
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White's Conjecture on Matroids
Prior state unknown→disproved
White conjectured that the symmetric exchange binomials generate the toric ideal of a matroid. This is now known to be false; a rank $9$ binary matroid constitutes a counterexample.
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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 i…
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Upper Bounds for Binary and Spherical Codes
Prior state unknown→proved
exponential improvement over the 1977 MRRW bounds; the exact rate-distance trade-off remains open
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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.
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Lower Bounds for the Permanent in Arithmetic Circuits
Prior state unknown→proved
an n^4/log n formula lower bound; VP vs VNP remains wide open
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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 $0\le m\le n-2$, nonnegative Ricci plus positive $(m{+}1)$-in…
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Stein’s dimension-free weak-(1,1) Riesz transform problem
Prior state unknown→proved
The theorem is the vector-valued endpoint bound $\|Rf\|_{L^{1,\infty}} \le 2\|f\|_{L^1}$ for $R = (R_1,\ldots,R_n)$, so the constant 2 also serves each component $R_j$ uniformly in the dimension; the best previously known component bound grew like $c\log n$. The mechanism is a decomposition theorem stated as Theorem 1.2: for every nonnegative $f \in L^1 \cap L^2$ and every $\lambda > 0$, write $f = \mu + (-\Delta…
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Talagrand's Convexity Problem
Prior state unknown→proved
Talagrand's convexity problem asks whether a universal number of Minkowski sum operations turns any set of large Gaussian measure into one containing a convex body of comparable measure. It is equivalent to a question about subgaussian vectors: is every centered $1$-subgaussian random vector in $\mathbb{R}^n$ the sum of a universal number of standard Gaussian vectors? Both are answered affirmatively, via the sharp…
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Talagrand’s convolution conjecture
Prior state unknown→proved
The claim is the exact conjectured decay: $\psi_{\mu_a}(u) \le C_a/\sqrt{\log u}$ for every $u > 1$ and every $n$, with $C_a$ dimension-free - concretely $\lesssim \kappa_a^2(\log\frac{\kappa_a}{\kappa_a-1})^{1/2}$ where $\kappa_a = (1+a)/(1-a)$. 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…
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Rota's Unimodality Conjecture for Matroid Flats
Prior state unknown→disproved
Is the sequence $W_0, W_1, \dots, W_n$ counting the flats of each rank of a matroid always unimodal? Rota conjectured yes in 1970.
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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…
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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:260…
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$SOP_2 = SOP_3$
Prior state unknown→proved
The new content is $SOP_2 \Rightarrow SOP_3$; the converse implication was known from the start. Dzamonja and Shelah asked whether either implication in $SOP_3 \Rightarrow SOP_2 \Rightarrow SOP_1$ reverses: Mutchnik answered the second ($SOP_1 = SOP_2$), and this answers the first, collapsing the bottom of the hierarchy to $SOP_1 = SOP_2 = SOP_3$. The $SOP_n$ hierarchy for $n \ge 3$ remains, as does everything abo…
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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…
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Krauth-Mezard Storage Capacity of the Ising Perceptron
Prior state unknown→proved
the last conditional step in the Ding-Sun and Huang bounds is discharged
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Erdős Problem #387
Prior state unknown→disproved
false in general; the positive direction holds for k large relative to n
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Connes' Rigidity Conjecture for ICC Property (T) Groups
Prior state unknown→disproved
Settles the ICC property (T) case. The paper records that the result was obtained independently of and concurrently with work by OpenAI.
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Full-RSB in the Sherrington–Kirkpatrick spin glass
Prior state unknown→proved
For the Sherrington-Kirkpatrick model with no external field, at zero temperature $\beta=\infty$, the paper proves that the zero-temperature Parisi minimizer is absolutely continuous with a smooth density and has support $[0,1)$ - full replica symmetry breaking, confirming the Parisi picture at the ground state. It also shows $q_\beta\to1$ as $\beta\to\infty$. The positive-temperature input is Lopatto's: for ever…
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Feige's Conjecture
Prior state unknown→proved
Let $X_1,\ldots,X_n$ be independent nonnegative random variables with $\mathbb{E}X_i \le 1$, and let $S$ be their sum. Is $\mathbb{P}(S < \mathbb{E}S + 1) \ge 1/e$? Feige proved the constant $1/13$ and conjectured the sharp $1/e$. Three independent July 2026 proofs settle it, both building on the Vlassis-Thomas calibration theorem; the sharper one determines the optimal small-deviation bound for every deviation $\…
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The Kinoshita Conjecture and Kirby Problem 4.37
Prior state unknown→disproved
Kinoshita conjectured that every embedded projective plane in $S^4$ is reducible. False: an irreducible embedded projective plane exists in $S^4$. The construction also answers both parts of Problem 4.37 of the Kirby problem list.
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Polynomial-Factor Hardness for the Closest Vector Problem
Prior state unknown→proved
Is the closest vector problem NP-hard to approximate within polynomial factors $n^c$? Yes for some $c > 0$: hardness of approximation reaches polynomial factors, with consequences for decoding and related lattice problems - a foundational question underpinning post-quantum cryptography where hardness had stalled at almost-polynomial factors since the late 1990s.
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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.
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Record Lower Bounds for the Shannon Capacity of Odd Cycles
Prior state unknown→proved
record lower bounds for seven odd cycles; the exact capacities remain open for every odd cycle beyond C5
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Optimality of Greedy for Single-Pass Semi-Streaming Matching
Prior state unknown→proved
Can any single-pass semi-streaming algorithm beat the naive greedy $1/2$-approximation for maximum matching? No. No single-pass semi-streaming algorithm, deterministic or randomized, achieves a better-than-half approximation, so greedy is optimal. The same construction settles the optimal competitive ratio of online matching with preemption at $1/2$.
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HRT Conjecture
Prior state unknown→disproved
Heil, Ramanathan and Topiwala conjectured in 1996 that any finite set of time-frequency shifts of a nonzero square-integrable function is linearly independent. This refutes it: there is a Schwartz function admitting 12 linearly dependent time-frequency shifts.
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Point Convergence of Nesterov's Accelerated Gradient Method
Prior state unknown→proved
Nesterov's accelerated gradient method (1983) is a cornerstone of optimization, yet whether its iterates themselves converge to a minimizer, rather than just the function values, stayed open for over forty years. Jang and Ryu resolve it in the affirmative. Ryu first announced the continuous-time result on X; Bot, Fadili and Nguyen's concurrent human proof of the critical-regime case (answering a decade-old conject…
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Seymour's Second Neighborhood Conjecture
Prior state unknown→proved
minimum out-degree 7; the conjecture is open in general
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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: $\sqrt{n}\,(C_n-\zeta(2)) \Rightarrow \mathcal{N}(0,\,4\zeta(2)-4\zeta(3))$. The limiting constant is not itself new - Wästlund had computed exactly $4\zeta(2)-4\zeta(3)$ for the mean-one exponential model, and Malatesta, Parisi and Sicuro derived the non-bipartite analogue by replicas - but neither is a proof…
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The Divisible Rank-Three Case of the Kajitani–Ueno–Miyano Conjecture
Prior state unknown→proved
Proves the divisible rank-three case: rank exactly three and ground-set size a multiple of three, with no restriction to simple, paving, representable or graphic matroids. The part not previously in the literature is the non-simple sub-case, since McGuinness had settled all paving matroids and a rank-three matroid is paving exactly when it has no parallel pairs. Combined with the coprime-case theorem of van den He…
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Bellman's Lost-in-a-Forest Problem for the Golden Gnomon
Prior state unknown→proved
Bellman's problem for general regions remains open
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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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Phelps–Rodriguez Conjecture
Prior state unknown→proved
Phelps-Rodriguez implies Sendov, so this entry records the stronger of the pair; the companion Sendov entry records the weaker statement and Mazur's original formalization, which proved Sendov but never stated the equality classification. The exceptional family is genuinely attained rather than an artefact of the proof: for p = z^n - 1 and a = 1 the only critical point is the origin, at distance exactly 1. Both co…
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Tuza's Conjecture for Maximum Degree at Most Seven
Prior state unknown→proved
Settles a class, not the conjecture: Tuza's conjecture remains open in general.
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Kusner's Conjecture on Equilateral Sets in $\ell_p^n$
Prior state unknown→disproved
Kusner conjectured in 1983 that the maximum number of points in $\mathbb{R}^n$ that are pairwise at $\ell_p$-distance one is exactly $n+1$ for every $2 < p < \infty$, as in the Euclidean case. False: an explicit configuration of $n+2$ equilateral points exists for some exponent, placing the infimum of exponents at which the conjecture fails in $[4,5)$. The configuration is the unique solution of an explicit polyno…
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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: $N^{2/3}\mathbb{E}\langle R_{1,2}^2\rangle$ converges to a positive finite constant. The paper proves substantially more, showing that the entire quenched distribution of $N^{1/3}R_{1,2}$ converges to an explicit random probability measure defined from the reflected $\mathrm{Airy}_1$ point process. The sam…
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Grothendieck's Finite Flat Group Scheme Order Question
Prior state unknown→disproved
Grothendieck asked whether every finite locally free group scheme of order $n$ is killed by $n$ (its $n$-th convolution power map equals the unit). The counterexample is an order-4 group scheme not killed by 4 (killed only by 8); since Deligne settled the commutative case, it is necessarily non-commutative over a non-reduced base.
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The Matrix Spencer Conjecture for Finite Groups
Prior state unknown→proved
the group case; the full Matrix Spencer conjecture remains open
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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.
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Borsuk Conjecture lowest-ever counterexample (N=63)
Prior state unknown→disproved
Priority: the result was first obtained by Max Grinsztajn with GPT-5.5 Pro assistance, published 26 May 2026 and recorded as the current best bound on Tao's optimization-problems ledger. The same construction was found again independently in August 2026 by Nicholas Konz working with Claude, with a different derivation and a fuller AI disclosure; the two efforts were evidently unaware of each other, and the submitt…
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The Period-Index Conjecture
Prior state unknown→disproved
For a Brauer class on a variety, the period-index conjecture bounds the index in terms of the period and the dimension. Disproved: for any uncountable algebraically closed field $k$ of characteristic $0$ and any $d \geq 3$ there is a $d$-dimensional variety over $k$ carrying a Brauer class that violates it, for Hodge-theoretic reasons. For $d = 3$ the construction needs no uncountability, so the conjecture fails a…
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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 e…
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Kotzig's Perfect 1-Factorisation Conjecture, Asymptotically
Prior state unknown→proved
asymptotic form only; Kotzig's conjecture itself remains far from solved
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Rivière’s regularity question for critical $n$-Laplace systems with antisymmetric potentials
Prior state unknown→disproved
For every $n>2$, the paper constructs a bounded map $U\in W^{1,n}(B^n,\mathbb{R}^{n+2})$, smooth on $B^n\setminus\{0\}$ but discontinuous at the origin, together with an antisymmetric potential $$ \Omega\in L^n(B^n,so(n+2)\otimes\mathbb{R}^n) $$ such that $$ -\mathrm{Div}\bigl(|\nabla U|^{n-2}\nabla U\bigr) = \Omega\cdot|\nabla U|^{n-2}\nabla U \qquad\text{in }D'(B^n). $$ Moreover, the potential satisfies the…
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The Generalized Busemann-Petty Problem in Dimensions 2 and 3
Prior state unknown→proved
Settles subspace dimensions 2 and 3; the generalized problem stays open for larger m.
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The Mihail-Vazirani Conjecture
Prior state unknown→disproved
Mihail and Vazirani conjectured that the graph of every $0/1$-polytope has edge expansion at least one. Disproved by a family of $0/1$-polytopes whose edge expansion decreases exponentially in the dimension.
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The Polynomial-Time Low-Degree Conjecture
Prior state unknown→disproved
the construction is probabilistic; an explicit uniformly samplable example remains open
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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 existe…
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Babai's Minimal Cayley Graph Problem
Prior state unknown→disproved
A Cayley graph is minimal when no proper subset of its connection set generates the group. Babai asked whether minimal Cayley graphs have bounded chromatic number. Resolved negatively: finite minimal Cayley graphs exist with arbitrarily large chromatic number.
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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-mome…
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A Five-Variable Counterexample to the Hessian Conjecture
Prior state unknown→disproved
The paper exhibits an explicit integer polynomial in five variables, of total degree 14 with constant Hessian determinant 128, whose gradient is not injective. Its formal Legendre transform is therefore not a polynomial, so the Hessian conjecture $\mathrm{HC}_5$ is false. The counterexample comes from a one-variable Schur descent applied to the six-variable doubling of Alpöge's 2026 Jacobian counterexample.
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The Largest Sum-Free Subset of the Lattice Cube
Prior state unknown→proved
How dense can a sum-free subset of the lattice cube $\{1,\dots,n\}^d$ be? Aydinian and Cameron asked for the limiting density, which is also Problem 6 in Ben Green's list of 100 open problems. The natural conjecture is that the optimum is a slice $\{x : 1 \le L(x) < 2\}$ for a linear map $L$, previously known only for $d \le 4$. Proved for all $d$. The paper also shows the same phenomenon fails if the cube is repl…
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Maxwell's Conjecture on Point-Charge Equilibria
Prior state unknown→disproved
Do $n$ point charges whose electrostatic potential has only non-degenerate critical points always have at most $(n-1)^2$ of them? A configuration of five charges - three at the vertices of an equilateral triangle plus two small central charges pulled apart into a shallow bipyramid - has at least $24 > 16$ non-degenerate critical points, so the conjecture is false.
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Shokurov's Global Index Conjecture for Foliations
Prior state unknown→proved
proved in dimension at most three
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Kannan–Tetali–Vempala conjecture (bipartite/binary-matrix case)
Prior state unknown→proved
The swap chain flips checkerboard 2×2 blocks to sample 0/1 matrices with fixed row and column sums. Kannan, Tetali and Vempala conjectured in 1997 that it mixes in polynomial time for all feasible margins; the lazy chain is shown to have spectral gap at least $\binom{m}{2}^{-1}\binom{n}{2}^{-1}$ on $m \times n$ matrices, which is worst-case tight and settles the bipartite case.
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Log-Concavity of Flats of Matroids
Prior state unknown→disproved
Refuting log-concavity of the flat counts is weaker than refuting their unimodality, since log-concavity is the stronger property. A counterexample to unimodality followed three weeks later and is tracked separately as Rota's Unimodality Conjecture for Matroid Flats; this paper came first.
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Seymour's Second Neighborhood Conjecture
Prior state unknown→proved
A dense case, not the conjecture: it remains open in general. The concrete gain is on the size of any counterexample - combined with the known minimum-outdegree results, this raises the best known lower bound on the order of a counterexample from 16 to 17, and to 19 conditional on the 2026 preprint of Sadhukhan, Sandeep and Sen. The novelty against the earlier dense-case work is that no structure is prescribed on…
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The Lonely Runner Conjecture for Nine and Ten Runners
Prior state unknown→proved
Settles 9 and 10 runners only; the general conjecture remains open.
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Universal Multiplicative FDR Bound for Benjamini-Hochberg
Prior state unknown→disproved
The Benjamini-Hochberg procedure is known not to control the false discovery rate at its nominal level under arbitrary dependence. A folklore conjecture in the FDR literature held that it must at least control the FDR up to a universal multiplicative constant. It does not: there are finite Gaussian models whose FDR divided by $q$ diverges as $q \downarrow 0$, with an explicit two-sided lower bound $q\sqrt{\log(1/q…
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Ben Green's Open Problem 90
Prior state unknown→proved
For $A \subset \mathbb{F}_p$ of density $1/2$, call $A$ almost affine invariant under $\varphi(x) = ax+b$ if $|A \triangle \varphi(A)| = o(p)$. Problem 90 asks for the threshold $K$ below which $A$ can be almost affine invariant simultaneously under all such $\varphi$ with $|a|, |b| \le K$ and $a \ne 0$. The threshold is $K = o(\log p)$.
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Rapid mixing for spin systems on graphs of girth at least five
Prior state unknown→proved
For fixed $\delta\in(0,1)$ and all sufficiently large $\Delta$ depending only on $\delta$, Glauber dynamics for proper $q$-colorings mixes rapidly on every graph of girth at least $5$ whenever $q\ge(1+\delta)\Delta$: spectral gap $\Omega_\delta(1/n)$ and $t_{\mathrm{mix}}(\varepsilon)=O_\delta(n^2\log q+n\log(1/\varepsilon))$. An analogous theorem holds for the anti-ferromagnetic Potts model at $q\ge(1+\delta)(1-\…
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The Matrix Spencer Conjecture for C*-Algebra Contractions
Prior state unknown→proved
a structured special case, proved the same week as the independent group version
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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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Counting Linear Extensions Below the $2^n$ Barrier
Prior state unknown→proved
Koivisto asked at Dagstuhl in 2013 whether the linear extensions of an arbitrary $n$-element poset can be counted exactly in time $O^*(c^n)$ for some $c < 2$. Yes: a deterministic exact algorithm runs in $O^*(1.89^n)$, breaking the $2^n$ barrier for the general problem.
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The Kára–Pór–Wood Big-Line-Big-Clique Conjecture: Four Collinear Points or a Six-Clique
Prior state unknown→proved
The big-line-big-clique conjecture of Kára, Pór and Wood asserts that for all $k, \ell$ there is an $n$ such that every finite point set of at least $n$ points contains $\ell$ collinear points or $k$ points that pairwise see each other. True for $\ell = 4$, $k = 6$, the first case left open: every finite point set of size at least $10^{11055931}$ has four collinear points or six pairwise visible points.
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The Gardner Transition in the Ising pure $p$-spin glass
Prior state unknown→proved
For the Ising pure $p$-spin glass where $p \geq 3$, it was predicted by Gardner that there exists critical inverse temperatures $0<\beta_1^p<\beta_2^p <\infty$ such that: (1) When $0<\beta\leq \beta_1^p$, the Parisi measure is replica symmetric (RS); (2) When $\beta_1^p<\beta \leq \beta_2^p$, the Parisi measure is one-step replica symmetry breaking (1-RSB); (3) When $\beta>\beta_p^2$, the Parisi measure is full r…
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The Optimal Approximation Ratio for Permanents of PSD Matrices
Prior state unknown→proved
What is the best deterministic polynomial-time approximation ratio for the permanent of a Hermitian positive semidefinite matrix? Resolved up to lower-order terms in the exponent: an explicit concave maximisation $\widehat P(A)$ satisfies $e^{-\gamma n}\widehat P(A) \le \mathrm{per}(A) \le \widehat P(A)$, giving a deterministic $e^{(\gamma+\varepsilon)n}$-approximation for every $\varepsilon > 0$ and matching the…
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Sparse domination implies convex body domination
Prior state unknown→proved
If a bilinear form admits an $(r,s)$-sparse bound, its coordinate-wise extension to $\mathbb C^n$-valued functions admits an $(r,s)$-convex body sparse bound, for $1\le r,s<\infty$ with $\tfrac1r+\tfrac1s>1$. It holds both in a fixed dyadic lattice (Theorem 2.6, constants independent of the ambient dimension) and for arbitrary cubes (Theorem 3.4), via a randomization of the good part of the form. On scope, two th…
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Feige's Hypergraph Moore-Bound Conjecture
Prior state unknown→proved
At the conjectured density, must every $k$-uniform hypergraph contain a short nontrivial even cover - a set of hyperedges covering each vertex an even number of times - with no superfluous polylogarithmic factors? Known up to polylog factors since 2022; now proved exactly for every $k \ge 3$.
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Gaussian Mass Maximality of the Integer Lattice
Prior state unknown→proved
integral unimodular lattices of rank at most 32; the general conjecture is open
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Batyrev's Stringy Hodge Number Conjecture
Prior state unknown→disproved
For a projective variety $X$ with at worst Gorenstein canonical singularities whose stringy $E$-function $E_{\mathrm{st}}(X; u, v)$ is a polynomial, all stringy Hodge numbers $h^{p,q}_{\mathrm{st}}(X)$ are non-negative. (Batyrev 1998, Conjecture 3.10.)
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The Banks-Martin Conjecture on Primitive Sets
Prior state unknown→proved
Banks and Martin conjectured in 2013 that for a primitive set $A$ and any set $Q$ of primes, the Erdos sum of the members of $A$ composed only of primes in $Q$ is at most the corresponding sum over $Q$ itself. The unrestricted form turned out to be false once $Q$ is allowed to contain $2$; Lichtman proposed a revised form restricted to odd primes. That revised conjecture, long viewed as a unifying master theorem f…
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Brezis-Mironescu Open Problems 23 and 24 on Minimizing Maps to the Circle
Prior state unknown→proved
in the range s in (1/4,1); Brezis's Problem 5.4 outside that range is untouched
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Bosonic Quantum Communication Beyond the Thermal Threshold
Prior state unknown→disproved
Holevo and Werner's 1999 lower bound on the quantum capacity of the bosonic thermal attenuator comes from thermal inputs. Is it optimal? The paper proves it is exactly the supremum over single-mode Gaussian states, then exhibits a non-Gaussian state that beats it, giving positive quantum capacity in a region where every single-mode Gaussian input yields none.
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The Erdos-Hajnal High-Girth Subgraph Conjecture
Prior state unknown→proved
in polynomial edge-density regimes; the general question remains open
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Bipartite Exact Matching in P
Prior state unknown→proved
The Exact Matching problem asks whether a bipartite graph with edges colored red and blue admits a perfect matching with exactly $t$ red edges. Introduced by Papadimitriou and Yannakakis in 1982, it has been in randomized polynomial time since Mulmuley-Vazirani-Vazirani (1987) while membership in P stayed open for four decades. The paper claims a deterministic polynomial-time algorithm, replacing probabilistic amp…
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