Problems

number-theory

Riemann Hypothesis

Every nontrivial zero of the Riemann zeta function has real part 1/2.

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theoretical-computer-science

P versus NP

Determine whether every problem whose solution can be verified in polynomial time can also be solved in polynomial time.

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combinatorics

Erdős discrepancy problem

For every sequence of signs, some homogeneous arithmetic progression has unbounded discrepancy.

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combinatorics / Ramsey theory

Hales–Jewett number HJ(3,3)

Determine the least dimension n such that every three-coloring of [3]^n contains a monochromatic combinatorial line.

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geometry-topology / Polyhedral geometry

Maximum vertex number of a (3,5)-zonoboxtope

Determine the largest possible number of vertices of a (3,5)-zonoboxtope, equivalently a maxout polytope of type (3,5,1). A published preprint gave the upper bound 44, asserted tightness, and conjectured the value 44 as part of its odd-n formula.

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combinatorics / Ramsey theory

Hales–Jewett number HJ(4,2)

Determine the least dimension n such that every two-coloring of [4]^n contains a monochromatic combinatorial line.

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number-theory / Analytic number theory

The Proportion of Zeta Zeros on the Critical Line

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 Montgomery's conditional $\tfrac23$ to a single obstruction, the termwise positivity that fails for zeros off the line, and asked what would follow without it. Theorem A replaces that positivity with a rank-trace inequality on a finite compression of Weil's Hermitian form, with Sylvester's law of inertia handling off-line pairs; reading the negative index of truncations as a count of off-line pairs is Bombieri's device. The paper also proves the bound sharp for this route: improving on $\tfrac23$ this way would need pair-correlation information beyond Fourier support 1.

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algebraic complexity / matrix multiplication

Algebraic continuation around rational rank-48 schemes for 4×4 matrix multiplication

Characterize exact algebraic families, boundary degenerations, and lower-bound obstructions around known rational rank-48 decompositions of 4×4 matrix multiplication.

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geometry-topology / Complex geometry; differential topology

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

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 diffeomorphic to $S^6$, with algebraic dimension exactly 1. This directly contradicts [CDP20, Cor. 2.3], a published (and once-corrected) theorem; the paper states this and argues where the two accounts diverge, rather than overlooking it. Posted hours before this entry, with no independent check, no formalisation, and no refutation yet in any venue found. Filed as a candidate specifically because none of that has happened, not because a problem with the argument has been found.

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algebra / Algebraic Geometry

Jacobian Conjecture

n ≥ 3; plane case open

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number-theory / Analytic number theory

Bounded prime gaps: $H_1 \le 236$

$H_1 \le 236$, between Stadlmann's $240$ of 31 August and Axiom Math's $212$ of 3 September. Superseded within two days, which Kintali noted himself.

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number-theory / Analytic number theory

Prime Gaps at Most 186

Assuming three explicit input statements, the project proves $\mathrm{DHL}[40,2]$: every admissible $40$-tuple contains at least two primes infinitely often after translation. An explicit admissible $40$-tuple of diameter $186$ then gives $$\liminf_{n\to\infty}(p_{n+1}-p_n)\le186.$$ The Lean proof of the implication from the three inputs to the final theorem is kernel-checked. Two inputs are Kloosterman-type estimates cited to Katz/Deligne and Fouvry--Kowalski--Michel; the third consists of finitely many numerical integral and cap inequalities backed by a Python/FLINT certificate.

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number-theory / Analytic number theory

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

$H_1 \le 212$, improving Stadlmann's $240$ of three days earlier and the $246$ of Polymath8b that had stood since 2014. The twin prime conjecture, $H_1 = 2$, is untouched. Held the record for hours at most: OpenAI's paper claiming $186$ is dated 30 August, four days before this one, though its Lean development appeared on 2 September.

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algebraic complexity / matrix multiplication

Low-memory scheduling of the public rank-49 4×4 matrix multiplication circuit

Determine low-peak-live-storage topological schedules for the fixed public 49-product, 156-addition arithmetic DAG under the stated scalar overwrite-on-final-use model.

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algebra / Noncommutative ring theory

Köthe Conjecture

GPT-6 Astra constructs a unital algebra $R=k\oplus A$ over the countable field $$k=\overline{\mathbb F_2},$$ with $I=A$ a nil two-sided ideal, together with a matrix $$W\in M_2(I)$$ that is not nilpotent. The algebra $A$ is generated by three weighted backward shifts. A diagonal construction chooses the weights so that every element of $A$ is nilpotent. At the same time, a suitable polynomial combination of the shifts fixes a nonzero vector; this yields a companion-type matrix with a nonzero eigenvalue, and hence a nonnilpotent matrix whose entries lie in $I$. This formally disproves Krempa's matrix formulation of Köthe's conjecture.

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number-theory / Analytic number theory; large gaps between primes

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

Claims $G(T)\gg\log T\log_2 T/\log_4 T$ against FGKMT's $\log T\log_2 T\log_4 T/\log_3 T$, a gain of $\log_3 T/(\log_4 T)^2$, together with $Y(X)\gg X\log X/\log_3 X$ 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 $\log_3 N$ over that procedure. The [FGKMT] paper also wins a $\log_3 N$. These wins are essentially independent of one another so one now wins basically $(\log_3 N)^2$ over Rankin's 1938 bound." He adds that the new sieve "by itself could have claimed the Erdos 10000 dollars for this question". What is new is narrow: only the intermediate sieve is replaced, its hard cutoff smoothed into a probabilistic tilt; the hypergraph covering theorem and Maynard weight come from FGKMT. Readers on the thread note that several later sections reproduce FGKMT with no new content, and Green calls the exposition "horrific".

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geometry-topology / Kähler geometry

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

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 available since 2008, and what was missing was the proof that the mechanism works.

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number-theory / Analytic number theory

Improved maximal prime-gap lower bound

The paper proves $$G(X)\gg \frac{\log X\,(\log_2 X)^2\,\log_4 X}{(\log_3 X)^2}$$ for all sufficiently large $X$. Its main new ingredient is a short-translates theorem: for any sufficiently small set $S\subseteq[1,H]$ with $|S|\le\delta x$, 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 claimed length. This directly and asymptotically improves the August 2026 GPT-5.6 Sol bound $$G(X)\gg\frac{\log X\log_2 X}{\log_4 X}$$ by the unbounded factor $$\frac{\log_2 X(\log_4 X)^2}{(\log_3 X)^2}.$$

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algebra / Geometric group theory

Gromov and Weiss's Question on Sofic Groups

answered no: non-sofic groups exist

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combinatorics / Graph theory

Erdős Problem #548: the Erdős–Sós conjecture

For every $n,k$ with $k+1\le n$, every simple graph $G$ on $n$ vertices satisfying $$|E(G)|\ge \frac{k-1}{2}n+1$$ contains every tree on $k+1$ vertices. The proof counts pairs $(\pi,j)$ where $\pi=(v_1,\ldots,v_n)$ is an ordering of the host vertices and $v_1v_j$ is an edge. There are exactly $$2|E(G)|(n-1)!$$ such pairs. An induction on the target tree bounds this quantity by a rooted-copy count plus $$(k-1)n!.$$ If the target tree is absent, the rooted-copy term vanishes and one obtains $$2|E(G)|\le (k-1)n,$$ contradicting the density hypothesis.

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geometry-topology / Complex geometry; six-manifolds; birational geometry

A complex structure on $S^{2}\times S^{4}$

Determine whether the standard smooth manifold $S^{2}\times S^{4}$ admits a complex structure; the source constructs one conditionally on the cited compact complex threefold on $S^{6}$.

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combinatorics / Graph Theory

Cycle Double Cover Conjecture

Registry resolution: resolved. Reported result type: proved. The registry record and named primary source provide the statement and scope.

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combinatorics / Additive combinatorics

The Sum-Product Conjecture over the Reals

the model's contribution is one simplifying lemma; the authors state the main ideas are human

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geometry-topology / Differential geometry

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

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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theoretical-computer-science / Algebraic complexity

The Matrix Multiplication Exponent

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 reports progress by reformulating that problem and then improving the optimizer, with AlphaEvolve doing the final refinement. That is a different kind of contribution from a new mathematical identity, and it is why the entry is filed as computation.

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combinatorics / Graph theory

Written on the Wall II Conjecture 200

For every finite connected simple graph G with more than one vertex, does equality between the largest induced-tree order and the ceiling of one plus the average independence number of the open neighborhoods imply that G has a Hamiltonian path?

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analysis / Spectral geometry

Schiffer's Conjecture and the Pompeiu Problem

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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combinatorics / Graph theory

Erdős Problem #571: rational exponents for bipartite Turán numbers

For every rational $\alpha$ satisfying $$1\le\alpha<2,$$ the Lean theorem constructs some finite $q$ and a bipartite graph $$G:\operatorname{SimpleGraph}(\operatorname{Fin} q)$$ such that $$\operatorname{ex}(n;G)=\Theta(n^\alpha)$$ as $n\to\infty$. This resolves the single-graph rational-exponents conjecture. Earlier work of Bukh and Conlon proved the corresponding statement only for a finite family of forbidden bipartite graphs, and subsequent work realized many large classes of individual rational exponents. Astra's theorem covers every rational $\alpha\in[1,2)$ with one forbidden graph for each exponent.

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combinatorics / Additive combinatorics

Erdős Problem #1: sum-distinct sets

The formal theorem proves that no universal constant $C>0$ can satisfy $$N>C\,2^{|A|}$$ for every nonempty interval bound $N$ and every sum-distinct $A\subseteq\{1,\dots,N\}$. Equivalently, for every $\varepsilon>0$ there are arbitrarily large $n$ and sum-distinct $n$-element sets contained in $\{1,\dots,N\}$ with $$N\le\varepsilon 2^n.$$ The proof is ineffective: it establishes the existence of arbitrarily large such $n$ but gives no explicit bound for how large $n$ must be in terms of $\varepsilon$.

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theoretical-computer-science / Network information theory

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

Registry resolution: candidate. Reported result type: disproved. The registry record and named primary source provide the statement and scope.

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number-theory / Elliptic curves

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

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 result because the open question is whether ranks are unbounded at all, which no single record answers. Superseded three days later by this project's own rank $\ge 31$ record (see the related entry); left unedited otherwise as a record of what was known at the time.

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geometry-topology / Discrete geometry

Upper Bounds for High-Dimensional Sphere Packing

upper bounds reach the Cohn-Elkies threshold; the true asymptotic density remains open

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number-theory / Elliptic curves

A Rank-$31$ Record for an Elliptic Curve over $\mathbb{Q}$

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 published for this curve specifically. The entry is a partial result because the open question - whether ranks are unbounded at all - remains unanswered by any single record.

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analysis / Complex analysis

Nevanlinna’s half-plane omitted-values problem

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 in either half-plane. This gives a negative answer to Nevanlinna’s century-old question asking whether an entire-plane meromorphic function that omits three distinct values in a half-plane must be of bounded type there. By postcomposition with Möbius transformations, the three exceptional values $\{0,1,\infty\}$ may be replaced by any prescribed triple of distinct values in $\widehat{\mathbb{C}}$. By affine change of variables, the construction applies to any Euclidean half-plane.

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analysis / Operator algebras

Connes' Rigidity Conjecture

Registry resolution: candidate. Reported result type: disproved. The registry record and named primary source provide the statement and scope.

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analysis / Complex polynomials

Smale’s Mean Value Conjecture ($K=1$)

The formal proof constructs a complex polynomial $p$ such that $$p(0)=0,\qquad p'(0)=1,$$ and for every critical point $c$ of $p$, $$\left|\frac{p(c)}{c}\right|>1.$$ Thus at $z=0$ there is no critical point satisfying $$\frac{|p(0)-p(c)|}{|c|}\le |p'(0)|=1,$$ which disproves Smale's conjectured universal constant $K=1$. The counterexample has very large unspecified degree and violates the bound only by a small margin, so it is consistent with Smale's original $K=4$ theorem, the known low-degree positive cases, and previous asymptotic improvements toward $1$.

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geometry-topology / Asymptotic convex geometry

KLS Conjecture for Quadratic Forms

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

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probability-statistics / Stationary stochastic processes

The Ibragimov–Iosifescu conjecture for φ-mixing sequences

Astra constructs a strictly stationary real process $$X_t=\xi_t+K(\xi_{t-1},\xi_{t-2},\ldots),$$ where the innovations $\xi_t$ are i.i.d. Gaussian variables convolved with a symmetric rare-spike law, and $K$ is bounded, continuous, and odd. The process is $\varphi$-mixing, centered, square-integrable, and satisfies $$\operatorname{Var}(S_n)\to\infty.$$ Nevertheless there are times $n_j\to\infty$ such that $$\frac{S_{n_j}}{\sqrt{\operatorname{Var}(S_{n_j})}}\to0$$ in probability. Therefore the normalized sums cannot converge in distribution to $N(0,1)$. The same example also rules out Iosifescu's stronger weak invariance-principle conjecture, since Brownian convergence would imply the CLT at time $1$.

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combinatorics / Graph theory

Petersen Coloring Conjecture

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 independent is unknown - the preprint does not cite the post. The headline axes still record the preprint, the only complete writeup with certificates. The implication runs one way: the Petersen coloring conjecture implies Berge-Fulkerson and the 5-cycle-double-cover conjecture, so refuting it leaves both of those open. The paper does not claim 112 is minimum, and it supplies a second, nonisomorphic $D_3$-symmetric 112-vertex counterexample. Combined with a theorem of Ma, Mattiolo, Steffen and Wolf, one counterexample yields infinitely many.

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geometry-topology / Combinatorial Geometry

Erdős's Planar Unit Distance Conjecture

Registry resolution: resolved. Reported result type: disproved. The registry record and named primary source provide the statement and scope.

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analysis / Functional analysis

Banach's isometric conjecture

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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combinatorics / Matroid theory

White's Conjecture on Matroids

Registry resolution: resolved. Reported result type: disproved. The registry record and named primary source provide the statement and scope.

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analysis / Complex analysis

Sendov's Conjecture

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 full generality as a consequence - a second conjecture falling out of the same argument, and one that likely merits its own entry. Two independent Lean developments now exist: Mazur's original at roughly 90,000 lines and Tao's streamlined version at about 15,000.

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theoretical-computer-science / Coding theory

Upper Bounds for Binary and Spherical Codes

exponential improvement over the 1977 MRRW bounds; the exact rate-distance trade-off remains open

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analysis / Harmonic analysis

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

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 $f = \mu + (-\Delta)^{\alpha/2}u$ with $\mu \le \lambda$ and $u$ in the fractional Sobolev space $H^\alpha$, obtained from an obstacle problem for the fractional Laplacian together with a Lewy-Stampacchia type estimate on an unbounded domain. That replaces the Calderon-Zygmund decomposition, whose cube geometry is where the dimensional loss enters.

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geometry-topology / Differential geometry

Universal volume growth bounds from positive intermediate curvature

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 $(m{+}1)$-intermediate curvature forces at most $m$-dimensional growth - linear growth under biRicci curvature at $m=1$, plus a noncollapsed Urysohn-width bound; the author states this extension is his own, not the model's. Kong and Zhu's proof (also 14 August; GPT-5.6 Sol Ultra and Codex, "essential ideas were generated by AI") proves the same case plus a related codimension-one conjecture via a heat-transport deficit, close to Ge's method by its own account, produced before Ge's went public and not derived from it. Headline axes stay Antonelli's, for the broader scope.

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combinatorics / Graph theory

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

Astra proves that there exists a divergent function $$f:\mathbb N\to\mathbb N,\qquad f(n)\to\infty,$$ such that no graph $G$ of infinite chromatic number can satisfy $$d_{\mathrm{bip}}(H)\le f(n)$$ for every finite $n$-vertex subgraph $H\subseteq G$, where $d_{\mathrm{bip}}(H)$ is the minimum number of edges that must be deleted to make $H$ bipartite. In fact, every included resolution proves the stronger statement that graphs satisfying the constructed local bound are 3-colorable. The formal challenge advertises only the weaker conclusion that their chromatic number must be finite.

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theoretical-computer-science / Algebraic complexity

Lower Bounds for the Permanent in Arithmetic Circuits

an n^4/log n formula lower bound; VP vs VNP remains wide open

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combinatorics

Asymptotically attaining the Moore bound

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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probability-statistics / Analysis of Boolean functions

Talagrand’s convolution conjecture

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 Xiang-Zhang's localized terminal-discrepancy method are taken as given; the addition is a power coupling that splits each reverse edge ratio into two geometric powers, producing a switched exponential weight that restores the exact reverse jump rate of the perturbed coordinate. Because the frozen exponent then has a fixed numerator, no growing stopping buffer is needed and the $\log\log u$ factor disappears. That last $\log\log$ is what stood between the previous work and Talagrand's statement.

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probability-statistics / High-dimensional probability

Talagrand's Convexity Problem

Registry resolution: resolved. Reported result type: proved. The registry record and named primary source provide the statement and scope.

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quantum-information-computing / Quantum complexity theory

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

The authors construct a unitary oracle $U$ such that $$\mathsf{QMA}^{U}\neq\mathsf{QMA}(2)^{U}.$$ Their black-box problem is solvable by a $\mathsf{QMA}(2)$ verifier with one oracle query and linear-size unentangled proofs, whereas any $\mathsf{QMA}$ verifier must use either exponentially many queries or an exponentially large witness. As a non-oracle consequence, they prove that for every fixed $\varepsilon,\delta\ge 0$ with $\varepsilon+\delta<1$, any $(\varepsilon,\delta)$-disentangler requires exponentially many input qubits in the number of output qubits, resolving Watrous's no-disentanglers conjecture.

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combinatorics / Matroid theory

Rota's Unimodality Conjecture for Matroid Flats

Registry resolution: resolved. Reported result type: disproved. The registry record and named primary source provide the statement and scope.

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probability-statistics / Probability

Feige's Conjecture

Registry resolution: resolved. Reported result type: proved. The registry record and named primary source provide the statement and scope.

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logic-foundations / Model theory

$SOP_2 = SOP_3$

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 above it.

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combinatorics / Zero-error information theory

Record Lower Bounds for the Shannon Capacity of Odd Cycles

record lower bounds for seven odd cycles; the exact capacities remain open for every odd cycle beyond C5

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analysis / Matrix analysis

Crouzeix's Conjecture

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 for 2-dilations - produced with ChatGPT 5.6 Pro exploring proof strategies. The entry's headline axes record Jin's proof; the earlier version of this entry recorded Lorist-Schwenninger's as primary while Jin's AI provenance was still unknown.

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probability-statistics / Probability

Krauth-Mezard Storage Capacity of the Ising Perceptron

the last conditional step in the Ding-Sun and Huang bounds is discharged

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mathematical-physics

Full-RSB in the Sherrington–Kirkpatrick spin glass

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 every $\beta>1$ the Parisi measure is supported on the closed interval $[0,q_\beta]$, with a smooth density on $[0,q_\beta)$ and a single atom at the right endpoint $q_\beta$. That endpoint atom is what the paper's own quantitative estimates target, so it is not incidental. This entry is cited as Theorem 1.1 rather than reproved. Remark 1.4 is worth reading beside the support claim: the half-open interval is essential, because the zero-temperature functional cannot see an endpoint atom at all, so there is no canonical mass there to converge to.

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analysis / Spectral Geometry, Laplace Eigenvalues

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

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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number-theory / Number theory

Erdős Problem #387

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

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quantum-information-computing / Entanglement theory

Two-Copy Distillability of Werner States

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:2607.24479). The headline axes here follow the first posting, which is a filing convention and not a claim about who solved it. Pozsgay has stated publicly that his group had their AI-assisted proof before the first paper appeared. This site cannot verify a private completion date, so that is recorded as his account rather than as a finding. All four are linked below.

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geometry-topology / 4-manifold topology

The Kinoshita Conjecture and Kirby Problem 4.37

Registry resolution: resolved. Reported result type: disproved. The registry record and named primary source provide the statement and scope.

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analysis / Calculus of variations and geometric measure theory

Gamow liquid-drop minimizer conjecture

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 had already proved existence up to V_*, and the new proof uses it; the fresh content is uniqueness of the ball across the whole range and nonexistence immediately above the threshold. A corollary settles the minimal binding energy question of Frank-Lieb: the infimum of E(Omega)/|Omega| is 3(9pi/5)^(1/3), attained exactly at balls of volume 5/2. The mechanism is a capacitary estimate that sharpens an Agostiniani-Mazzieri monotonicity formula using Gauss-Bonnet, an improvement the authors note applies only to this particular weight and only in three dimensions.

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number-theory / Elementary number theory

Erdős Problem #126: prime divisors of pairwise sums

For $$f(n)=\min_{|A|=n}\left|\left\{p\text{ prime}:p\mid a+b\text{ for some distinct }a,b\in A\right\}\right|,$$ Astra formally proves $$\frac{f(n)}{\log n}\to\infty.$$ The repository contains substantially stronger proofs. In particular, one verified alternate resolution establishes $$n\le 3r^2,$$ where $r$ is the number of supporting primes, yielding $$f(n)\gg n^{1/2}.$$ Other independent resolutions give exponents $1/3$, $1/5$, and $1/8$. Thus the formal work goes well beyond the qualitative conjecture, although only the limit statement is the registered benchmark theorem.

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theoretical-computer-science / Streaming algorithms

Optimality of Greedy for Single-Pass Semi-Streaming Matching

Registry resolution: resolved. Reported result type: proved. The registry record and named primary source provide the statement and scope.

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algebra / Operator algebras

Connes' Rigidity Conjecture for ICC Property (T) Groups

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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algebra / Poisson algebra

An Explicit Counterexample to the Rank-Two Poisson Conjecture

The paper constructs explicit $R,T,D,S\in\mathbb Q[x,q,p,z]$ defining a Poisson endomorphism of the canonical rank-two Poisson algebra $\mathcal P_2$ that is not an automorphism. Its associated polynomial map preserves the canonical symplectic form, has Jacobian determinant $1$, and has an explicit fiber of exactly three points. Thus $\mathrm{PC}(2)$ is false, and stabilization gives failure of $\mathrm{PC}(n)$ for every $n\ge2$. An appendix further constructs an explicit nonautomorphic endomorphism of the fourth Weyl algebra, proving $\mathrm{DC}(4)$ false.

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theoretical-computer-science / Lattices & cryptography

Polynomial-Factor Hardness for the Closest Vector Problem

Registry resolution: candidate. Reported result type: proved. The registry record and named primary source provide the statement and scope.

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analysis / Harmonic Analysis

HRT Conjecture

Registry resolution: resolved. Reported result type: disproved. The registry record and named primary source provide the statement and scope.

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probability-statistics / Random combinatorial optimization

Central limit theorem for the random assignment problem

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 for the bounded bipartite model, and Wästlund's zero-free-disk conjecture, which would imply a Gaussian limit, remains open. So the value was expected; the proof of convergence to it is what is claimed. The route is an exact change of variables on an optimal dual potential, after which the residual dependence is a single directed-tree factor whose matrix-tree determinant becomes triangular once the potentials are ordered.

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combinatorics / Graph Theory

Dean's conjecture for $k=5$

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 $k\in\{3,4\}$ and they proved every $k\ge6$, leaving $k=5$ open. If this proof stands, Dean's conjecture holds for all $k\ge3$. Nine finite proposition families in the bipartite and triangle-free branches are computer-assisted, with verifiers and certificates in a separately archived supplement. The reductions from arbitrary graphs to those finite state spaces are prose arguments in the paper and are not machine-checked. The claim has not been refereed, and no independent mathematician has audited the graph-theoretic core. The author describes extensive model-assisted hostile auditing of his own argument, which is worth something but is not external review, and says so plainly.

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algorithms-optimization / Convex optimization

Point Convergence of Nesterov's Accelerated Gradient Method

Registry resolution: resolved. Reported result type: proved. The registry record and named primary source provide the statement and scope.

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combinatorics / Graph theory

Seymour's Second Neighborhood Conjecture

minimum out-degree 7; the conjecture is open in general

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quantum-information-computing / Entanglement theory

Finite-Copy Distillability of NPT States in the DiVincenzo Family

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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combinatorics / Extremal graph theory

Tuza's Conjecture for Maximum Degree at Most Seven

Settles a class, not the conjecture: Tuza's conjecture remains open in general.

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combinatorics / Design theory

Kotzig's Perfect 1-Factorisation Conjecture, Asymptotically

asymptotic form only; Kotzig's conjecture itself remains far from solved

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combinatorics / Glauber dynamics

Rapid mixing for spin systems on graphs of girth at least five

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 $q\ge(1+\delta)(1-\beta)\Delta$. What it does and does not improve. On girth it is a large gain: previous results near the $(1+\delta)\Delta$ threshold needed girth at least eleven (Hayes-Vigoda, extended to constant degrees by Jain-Mizgerd-Vigoda). On the mixing rate it is weaker - those give optimal $O(n\log n)$, this gives $O(n^2\log q)$. It does not touch the folklore conjecture that mixing is rapid on every graph for $q\ge\Delta+2$; Remark 7 names spanning 4-cycles as the obstruction.

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mathematical-physics / Dynamo theory; spectral PDE

Autonomous Lipschitz Fast Dynamo on the Three-Torus

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 entry on this site relaxes the opposite hypothesis, keeping a smooth field but making it random and time-dependent. Neither settles Arnold's problem as posed, which remains open.

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geometry-topology / Convex geometry

Bellman's Lost-in-a-Forest Problem for the Golden Gnomon

Bellman's problem for general regions remains open

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probability-statistics / Markov chain mixing time

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

Registry resolution: resolved. Reported result type: proved. The registry record and named primary source provide the statement and scope.

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probability-statistics / Statistics

Universal Multiplicative FDR Bound for Benjamini-Hochberg

Registry resolution: resolved. Reported result type: disproved. The registry record and named primary source provide the statement and scope.

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combinatorics / Additive combinatorics

Ben Green's Open Problem 90

Registry resolution: resolved. Reported result type: proved. The registry record and named primary source provide the statement and scope.

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combinatorics / Algebraic graph theory

Babai's Minimal Cayley Graph Problem

Registry resolution: resolved. Reported result type: disproved. The registry record and named primary source provide the statement and scope.

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algebra / Algebraic Geometry

Grothendieck's Finite Flat Group Scheme Order Question

Registry resolution: resolved. Reported result type: disproved. The registry record and named primary source provide the statement and scope.

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combinatorics / Additive combinatorics

The Largest Sum-Free Subset of the Lattice Cube

Registry resolution: resolved. Reported result type: proved. The registry record and named primary source provide the statement and scope.

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mathematical-physics / Spin glass theory

Talagrand's critical Sherrington-Kirkpatrick overlap conjecture

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 same limiting distribution and second-moment constant are obtained for the spherical SK model. This does not resolve the broader low-temperature overlap structure of the SK model.

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number-theory / Diophantine approximation

The Lonely Runner Conjecture for Nine and Ten Runners

Settles 9 and 10 runners only; the general conjecture remains open.

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theoretical-computer-science / Average-case complexity

The Polynomial-Time Low-Degree Conjecture

the construction is probabilistic; an explicit uniformly samplable example remains open

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probability-statistics / Random matrix theory

The Ellipsoid Fitting Conjecture

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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combinatorics / Polyhedral combinatorics

The Mihail-Vazirani Conjecture

Registry resolution: resolved. Reported result type: disproved. The registry record and named primary source provide the statement and scope.

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combinatorics / Matroid theory

The Divisible Rank-Three Case of the Kajitani–Ueno–Miyano Conjecture

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 Heuvel and Thomasse, this covers every finite uniformly dense rank-three matroid. The conjecture remains open in higher rank.

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combinatorics / Combinatorial design theory

Hadamard Matrix of Order 668

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 existence question, which this fully resolves; the general Hadamard conjecture - existence for ALL admissible orders - remains open, with the smallest unknown order now 2004 or beyond.

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combinatorics / Discrete geometry

Borsuk Conjecture lowest-ever counterexample (N=63)

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 submitter of this entry surfaced the earlier work themselves after publication. Dimension 63 is the current record; whether Borsuk's conjecture fails for any dimension in 4..62 remains open.

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analysis / Dynamo theory

Smooth Random Fast Dynamo on the Three-Torus

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 estimate. This is one variant case of Arnold's 1994 fast-dynamo problem, not the problem itself: Arnold asks for a field that is smooth, autonomous and deterministic all at once, and this one keeps the smoothness while giving up the other two. The sibling entry on this site relaxes the opposite hypothesis, keeping an autonomous deterministic field at Lipschitz regularity. Neither settles Arnold's problem as posed, which remains open.

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combinatorics / Graph theory

Seymour's Second Neighborhood Conjecture

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 the missing edges.

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geometry-topology / Discrete geometry

Kusner's Conjecture on Equilateral Sets in $\ell_p^n$

Registry resolution: resolved. Reported result type: disproved. The registry record and named primary source provide the statement and scope.

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analysis / Complex analysis

Phelps–Rodriguez Conjecture

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 conjectures fell out of one argument, and the strict form was not the announced target - Tao's digestion of Mazur's proof turned out to establish it, which is how a 1972 conjecture was resolved as a by-product of resolving a 1959 one.

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combinatorics / Matroid theory

Log-Concavity of Flats of Matroids

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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combinatorics / Graph theory

Albertson–Berman Induced Forest Conjecture

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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