Tracked problems: 1284; 517 fully resolved · 1 under review
Combined years open: 11,490; before AI closed them
Lean-verified: 159; machine-checked
Community members: 561; 7 verified · 339 votes · 96 comments

Latest activity

Edits, submissions and discussion · all members

Rasmus Lindahl approved Codimension-one holomorphic Anosov diffeomorphisms1 day ago
Rasmus Lindahl approved A lower bound of 21 for 3 × 3 matrix multiplication over every field1 day ago
Rasmus Lindahl approved Dow–Tkachuk Question 4.8: pseudocomplete countable closures in $C_p(X)$ force $X$ to be a $P$ -space1 day ago
Rasmus Lindahl approved Expected finite k-adic complexity in general integer bases1 day ago
Rasmus Lindahl approved Maximum size of a three-sunflower-free family of four-element sets1 day ago
Rasmus Lindahl approved Strict Neumann Pólya inequality on simply connected planar Lipschitz domains1 day ago
Rasmus Lindahl approved Strict Dirichlet Pólya inequality on simply connected planar domains1 day ago
Rasmus Lindahl approved Nonquadraticity of Lambert series values and improved bounds for Erdős Problem 10491 day ago
Rasmus Lindahl changed the what was actually shown on Jacobsthal's quadratic-bound question: is h(k) = O(k^2)? to “Theorem 1.1: h(k)≤ Ck^2/(loglog(3k))^2 for an absolute const…”1 day ago
FrostyBadger576 submitted Codimension-one holomorphic Anosov diffeomorphisms1 day ago

Latest

Most recent results

This week

Most significant of the last seven days

All entries

Sort by

Period

all 1284 entries

Compact-range Vietoris powers: from $\omega+1$ to all countable ordinals above $\omega$

ProvedUnder reviewAI co-developed

(First, On the Vietoris Power of a Convergent Sequence settles the highlighted case α=+1: the space is second-countable and Lindelöf, but not -compact. Compact families of surjections have finite bounds on where each isolated value occurs; a delayed diagonal defeats every countable compact cover. The stronger manuscript proves, for every <α<_1, second countability and Lindelöfness, together with failure of both -compactness and the Menger property. It combines a countable-base argument, transfer through a closed copy of [0,], and a direct fixed-compact-image diagonal. Explicit increasing open covers and the same diagonal defeat all finite selections, proving non-Mengerness. Thus the original question has a positive answer for Lindelöfness and a negative answer for -compactness. “Proved” refers to this classification theorem.)

General topology; covering and selection properties

Question 1 on page 11 of Christopher Caruvana and Jared Holshouser, An Adaptation of the Vietoris Topology for Ordered Compact Sets (arXiv:2507.17936v3), asks whether $\mathbb K(\alpha,\mathrm{ord})$ is $\sigma$ -compact or Lindelöf for $\omega<\alpha<\omega_1$ , explicitly singling out $\alpha=\omega+1$ . Here the carrier consists of arbitrary functions with actual compact image, equipped with the paper’s Vietoris-power topology; for countable ordinals the default index is $\omega$ .

Posed by Christopher Caruvana and Jared Holshouser, Question 1, p. 11, An Adaptation of the Vietoris Topology for Ordered Compact Sets, arXiv:2507.17936v3, 2025Open 1yModel ChatGPT Astra (OpenAI)Solved 2026

Lean-checked, statement unauditedSignificance 12 Submitted by StormyGander827 on 30 Sep 20263

Mean and variance of finite 2-adic complexity (Question 3.1)

ProvedUnder reviewAI-assisted

(The manuscript claims both requested results: the exact mean upper bound for every N>=1 and an explicit finite arithmetic-logarithmic variance formula obtained from the complete distribution. It also evaluates the limiting variance as 0.892302571204274625294755... . The all-length mean proof combines a written lattice argument with exact finite certificates. The variance and asymptotic arguments use finite counting, marked-lattice equidistribution and uniform integrability. The limiting geometric law is explicitly credited to Boca-Gologan. The finite answer is a sum, not a short elementary expression in N; no convergence rate or raw-height variance theorem is claimed.)

Finite 2-adic complexity, rational approximation and probability

For a uniformly random binary string of length $N\ge1$ , identified with $r\in\{0,\ldots,2^N-1\}$ , let $R_N(r)=\min\{\max(|a|,b):a,b\in\mathbb Z,\ a\equiv rb\pmod{2^N},\ b>0\text{ odd}\}$ and let $L_N=\log_2 R_N(r)$ , without rounding. Prove $\mathbb E[L_N]\le N/2$ and compute $\operatorname{Var}(L_N)$ . This is Question 3.1 of Lin, Xiao and Chen (2026). The probability space includes all length-N binary strings, including zero.

Posed by Hezheng Lin, Lingmei Xiao and Zhixiong Chen, Question 3.1, AIMS Mathematics 11(7) (2026), p.20282, DOI 10.3934/math.2026823., 2026Open 0yModel OpenAI GPT models; GPT-6.1 Sol (OpenAI)Solved 2026-10

AnnouncedSignificance 4 Submitted by Oleksiy Babanskyy on 4 Oct 2026

Expected finite k-adic complexity in general integer bases

ProvedUnder reviewAI-assisted

(The manuscript claims E_N^k-adic<N/2+0.241933621228105 for every integer k2 and N1. It gives E_N^k-adic≤ N/2 for prime-power bases and a strict inequality below N/2 for bases with exactly two distinct prime factors. The example k=30,N=2 disproves a universal zero-correction bound. Section 1 identifies the standard rational observable with the source question. Written arguments combine classical imports, finite certificates and infinite-tail estimates. The optimal additive constant and finite-length attainment remain open. Constrained FCSR connection integers and rounded register lengths are not identified with this observable. The earlier binary theorem is credited and imported.)

Rational reconstruction; k-adic complexity of finite strings

For an integer $k\ge3$ , let $E_N^{k\text{-adic}}$ be the expected $N$ th $k$ -adic complexity of a uniformly random $k$ -ary sequence of length $N\ge1$ . Lin, Xiao and Chen derive a lower bound of the form $N/2-O(1)$ and ask, as Question 3.2: estimate an upper bound on $E_N^{k\text{-adic}}$ .

The $N$ th $k$ -adic complexity of a sequence is the least $\log_k\max(|f|,|q|)$ over reduced rationals $f/q$ whose $k$ -adic expansion begins with it (unrounded; the convention fixed in Section 1 of the manuscript, following Goresky and Klapper).

Posed by Hezheng Lin, Lingmei Xiao and Zhixiong Chen, Question 3.2, AIMS Mathematics 11(7) (2026), p. 20282; DOI 10.3934/math.2026823., 2026Open 0yModel OpenAI GPT models (versions unspecified) (OpenAI)Solved 2026-10

AnnouncedSignificance 4 Submitted by Oleksiy Babanskyy on 12 Oct 2026

Maximum size of a three-sunflower-free family of four-element sets

ProvedPartialAI-assisted

(The manuscript claims a computer-assisted exclusion of 69-member three-sunflower-free four-uniform families, giving 54 <= M(4,3) <= 68. It improves the upper bound 83 in Axante et al., arXiv:2609.06175v1, Theorem 6.9. The classical lower construction is reproduced, not claimed as new. The proof combines their intersecting-family bound 27, structural point-link reductions, Jin's graph-homomorphism theorem, exact certificates and finite exhaustive checks. A local cap of 61 requires a complete R16 point-link and is not a global bound. Cardinality 68 is not excluded, the exact value is not determined, and the general sunflower conjecture (Erdős #20) remains open.)

Extremal set theory; sunflowers and finite set systems

How large can a family of distinct four-element sets be if no three of its members form a sunflower, that is, three distinct members whose pairwise intersections are all equal (an empty common intersection included)? Write $f(4,3)$ for this maximum.

Abbott, Hanson and Sauer (1972) gave $f(4,3)\le142$ and Abbott and Hanson (1974) gave $f(4,3)\ge54$ . Axante, Budala, Chitic, Dumitru and Nacu (arXiv:2609.06175v1, September 2026, Table 1 and Theorem 6.9) improved the upper bound to 83, so $54\le f(4,3)\le83$ . The exact value is open. The forcing-threshold convention used for Erdős Problem #20 is one larger than $f(4,3)$ .

Posed by Abbott, Hanson and Sauer (1972, upper bound 142) and Abbott and Hanson (1974, lower bound 54), as recorded in Axante et al., arXiv:2609.06175v1, Table 1Open —Model Sol 5.6; Astra 6; Pro 6; Sol 6.1 (OpenAI)Solved 2026-10

AnnouncedSignificance 9 Submitted by Oleksiy Babanskyy on 9 Oct 2026

Zhi-Hong Sun's Conjecture 3.7: two Apéry-like sums at discriminant -20

ProvedUnder reviewAI-assisted

(Theorem 1.1 claims the complete conjecture: both signed sums, every prime except 2 and 5, all representation classes, and modulus p squared. Split primes use two modular periods, an integral Hecke polynomial and separate finite specializations, including the degree-p endpoint and nonprincipal CM sign. For inert primes, a degree-five endomorphism with square [-5] forces Hasse vanishing, then a differential invariant lifts each original sum. The prime 3 is checked directly. The paper credits Sun's mod-p and Legendre antecedents, Straub's coefficient congruence and Beukers's framework. The stronger Sun-Ye quartic congruence concerns an auxiliary sum; the available finite transports alone have only mod-p precision. No uniform mod-p-cubed equality is claimed: at p=7 the unsigned target sums are 192 and 290 modulo 343.)

Supercongruences, modular forms and elliptic curves

Let $f_n=\sum_{j=0}^n\binom{n}{j}^3$ , $a_n=\sum_{j=0}^n\binom{n}{j}^2\binom{2j}{j}$ , $Q_n=\sum_{j=0}^n\binom{n}{j}(-8)^{n-j}f_j$ and $\varepsilon_p=(-1)^{(p-1)/2}$ . For every prime $p\ne2,5$ , is it true that $\varepsilon_p\sum_{n=0}^{p-1}\frac{\binom{2n}{n}a_n}{20^n}\equiv\varepsilon_p\sum_{n=0}^{p-1}\frac{\binom{2n}{n}Q_n}{(-16)^n}\equiv\begin{cases}4x^2-2p,&p=x^2+5y^2,\ p\equiv1,9\pmod{20},\\2x^2-2p,&2p=x^2+5y^2,\ p\equiv3,7\pmod{20},\\0,&p\equiv11,13,17,19\pmod{20}\end{cases}\pmod{p^2}?$ Here $x,y$ are integers in the representation cases. This is Conjecture 3.7 of Zhi-Hong Sun, AIMS Mathematics 7(2) (2022), p. 2742, published online on 18 November 2021.

Posed by Zhi-Hong Sun, Conjecture 3.7, AIMS Mathematics 7(2) (2022), p. 2742; DOI 10.3934/math.2022153. Published online 18 November 2021., 2021Open 5yModel OpenAI Codex; GPT-6.1 Sol (OpenAI)Solved 2026-10

AnnouncedSignificance 5 Submitted by Oleksiy Babanskyy on 4 Oct 2026

Sun's Conjecture 3.5 on an Apéry-like supercongruence

ProvedUnder reviewAI-assisted

(Theorem 1.1 claims the complete stated conjecture modulo p^2 for every prime p>3, covering both residue classes modulo 3, with p=5 treated directly. For inert primes, a finite quadratic differential invariant lifts Sun's published modulo-p vanishing. For split primes, certified modular periods, an integral Hecke polynomial with a degree bound and complex multiplication give the evaluation. The Franel sum with denominator 50 is evaluated separately for p>5 and credited to Sun's different 2018 Conjecture 3.5. No modulo-p^3 result, resolution of adjacent conjectures, or first-proof claim is made. Candidate reflects pending authoritative review of a claimed complete proof of this specific question.)

Supercongruences, Apéry-like sequences and modular forms

Let $a_n=\sum_{j=0}^{n}\binom{n}{j}^{2}\binom{2j}{j}$ . For every prime $p>3$ , is it true that
$(-1)^{(p-1)/2}\sum_{n=0}^{p-1}\binom{2n}{n}\frac{a_n}{54^n}\equiv \begin{cases} 4x^2-2p,&p=x^2+3y^2\equiv1\pmod{3},\\ 0,&p\equiv2\pmod{3} \end{cases} \pmod{p^2}?$
This is Conjecture 3.5 on p. 2742 of Zhi-Hong Sun's AIMS Mathematics paper, DOI 10.3934/math.2022153. It is distinct from Theorem 3.5 in the same article, which gives a congruence modulo $p$ .

Posed by Zhi-Hong Sun, Conjecture 3.5, AIMS Mathematics 7(2) (2022), p. 2742; published online 18 November 2021. DOI: 10.3934/math.2022153., 2021Open 5yModel OpenAI Codex; GPT-6.1 Sol (OpenAI)Solved 2026-10

AnnouncedSignificance 5 Submitted by Oleksiy Babanskyy on 3 Oct 2026

Zhi-Hong Sun's Conjecture 3.6 on two Apéry-like binomial sums

ProvedUnder reviewAI-assisted

(Theorem 1.1 claims the complete stated conjecture: both sums, every odd prime and modulus p squared. For inert primes, a finite differential invariant lifts Sun's mod-p result, with an autonomous singular treatment at p=5. For split primes, certified eta identities, a generic integral Hecke polynomial, an explicit degree bound and three CM matrix cases yield the result. An independently sourced coefficient congruence removes the degree-p boundary; p=3 is checked directly. The paper credits Sun's mod-p input, the Straub-Gorodetsky endpoint and Beukers's modular framework. It does not claim a uniform mod-p-cubed extension: at p=7 the two sums are 294 and 49 modulo 343. The full conjecture is claimed, but authoritative mathematical review is pending.)

Supercongruences, Apéry-like sequences and modular forms

Let $f_n=\sum_{j=0}^n\binom{n}{j}^3$ and $Q_n=\sum_{k=0}^n\binom{n}{k}(-8)^{n-k}f_k$ . Is it true that, for every odd prime $p$ , $\sum_{n=0}^{p-1}\frac{\binom{2n}{n}Q_n}{(-32)^n}\equiv\sum_{n=0}^{p-1}\frac{\binom{2n}{n}Q_n}{64^n}\equiv\begin{cases}4x^2-2p,&p=x^2+2y^2\equiv1,3\pmod8,\\0,&p\equiv5,7\pmod8\end{cases}\pmod{p^2}?$ Here $x,y$ are integers in the first case. This is Conjecture 3.6 of Zhi-Hong Sun, AIMS Mathematics 7(2) (2022), p. 2742, published online on 18 November 2021. The question includes both sums and the primes 3 and 5.

Posed by Zhi-Hong Sun, Conjecture 3.6, AIMS Mathematics 7(2) (2022), p. 2742; DOI 10.3934/math.2022153. Published online 18 November 2021., 2021Open 5yModel OpenAI Codex; GPT-6.1 Sol (OpenAI)Solved 2026-10

AnnouncedSignificance 5 Submitted by Oleksiy Babanskyy on 3 Oct 2026

A lower bound of 21 for 3 × 3 matrix multiplication over every field

ProvedPartialAI co-developed

(The formal proof establishes R_K⟨3,3,3⟩21 for every field K, including finite and infinite fields in every characteristic. The input linear forms and output matrices are unrestricted; there are no minimality, nonzero-term, symmetry or coefficient-rank assumptions. The argument passes to an algebraic closure, chooses a decomposition minimizing a coefficient-rank sum, and uses conductor growth and residual reconstruction to rule out at most 20 terms. This is partial progress on exact rank: it does not determine the exact rank over every field and gives no border-rank or asymptotic-exponent bound. Results establishing 22 or 23 over F_2 or for integer-coefficient algorithms do not imply the same numerical bound over arbitrary fields. The README places this result alongside the classical general lower bound 19 and recent field-specific results.)

Algebraic complexity; exact bilinear complexity and tensor rank

For a field $K$ , what is the smallest number $r$ of products of input linear forms needed to multiply two arbitrary $3\times3$ matrices exactly? Equivalently, how small can $r$ be in an identity $AB=\sum_{i=1}^{r}a_i(A)b_i(B)W_i$ holding for every pair $A,B$ , with $K$ -linear forms $a_i,b_i$ and fixed matrices $W_i$ ? The minimum is the exact bilinear complexity $R_K\langle3,3,3\rangle$ . Which lower bounds hold uniformly over all fields?

Posed by —Open —Model GPT-6 Astra (gpt-6-astra), with GPT-6 Sol (gpt-6-sol) used for part of the work (OpenAI)Solved 2026-10-12

Lean-checked, statement unauditedSignificance 30 Submitted by VelvetCobra335 on 13 Oct 2026

Dow–Tkachuk Question 4.8: pseudocomplete countable closures in $C_p(X)$ force $X$ to be a $P$ -space

ProvedUnder reviewAI-assisted

(Yes, in ZFC. The paper proves a stronger statement: if a Tychonoff space X is not a P-space, then C_p(X) contains a closed, separable, metrizable linear subspace that is not pseudocomplete. A closed separable subspace is the closure of a countable subset, so if all such closures are pseudocomplete, X must be a P-space. Dow and Tkachuk had shown this under the extra assumption that X is functionally countable (their Theorem 3.10); the result removes that assumption. The companion Question 4.9 (whether X must also be functionally countable) is not addressed.)

General topology; $C_p$-theory

Dow and Tkachuk asked (Question 4.8): assume that the closure of $A$ is pseudocomplete for every countable $A \subset C_p(X)$ . Must the Tychonoff space $X$ be a $P$ -space? Here $C_p(X)$ is the space of continuous real-valued functions on $X$ with the topology of pointwise convergence, pseudocompleteness is in Oxtoby's sense (a sequence of $\pi$ -bases along which every closure-nested choice has nonempty intersection), and a $P$ -space is one in which every $G_\delta$ set is open.

Posed by Alan Dow and Vladimir V. Tkachuk, "Reflecting topological properties in closed separable subspaces", European Journal of Mathematics 11 (2025), Question 4.8, 2025Open 1yModel ChatGPT Astra (proof); Claude Opus 5.5 (Lean formalization) (OpenAI; Anthropic)Solved 2026-10-12

Lean-checked, statement unauditedSignificance 8 Submitted by StormyGander827 on 12 Oct 2026

Strict Neumann Pólya inequality on simply connected planar Lipschitz domains

ProvedPartialAI-assisted

(Main theorem: for every bounded simply connected planar Lipschitz domain Ω, |Ω|_j(Ω)<4π j for all j1, with 0=_0<_1≤·s counted with multiplicity; equivalently N_N(E)>|Ω|E/(4π) for E>0, where N_N counts _j<E including _0. It contains the convex planar case, for which Filonov (CPAM 2025) had only a weaker constant, and the disk case of Filonov, Levitin, Polterovich and Sher (2023). Multiply connected domains, non-Lipschitz domains and higher dimensions remain open. The manuscript is not yet on arXiv; a draft is in the registered snapshot. The companion strict Dirichlet result is a separate entry.)

Spectral geometry

Does the Neumann Pólya conjecture hold for every bounded, connected planar Lipschitz domain $\Omega\subset\mathbb R^2$ ? More precisely, is it true that $|\Omega|\mu_j(\Omega)\le 4\pi j \qquad(j\ge1),$ where $0=\mu_0(\Omega)<\mu_1(\Omega)\le\mu_2(\Omega)\le\cdots$ are the Neumann eigenvalues of $\Omega$ , counted with multiplicity?

Posed by George Pólya, Mathematics and Plausible Reasoning (1954); tiling case proved by Pólya (1961) and Kellner (1966), 1954Open 72yModel ChatGPT; Aristotle; Codex (OpenAI; Harmonic)Solved 2026-10-09

Lean-checked, statement unauditedSignificance 42 Submitted by StormyCobra617 on 9 Oct 2026

Strict Dirichlet Pólya inequality on simply connected planar domains

ProvedPartialAI-assisted

(Theorem 1.1 of arXiv:2610.13139: for every bounded nonempty simply connected planar domain Ω, with no boundary regularity assumed, |Ω|λ_j(Ω)>4π j for all j1, equivalently N_D(E)<|Ω|E/(4π) for E>0 (eigenvalues counted with multiplicity, endpoint included). This is the strict form of Pólya's Dirichlet inequality on the whole simply connected planar class. Previously it was known for tiling domains (Pólya 1961), disks and sectors (Filonov, Levitin, Polterovich and Sher 2023), annuli and a few further families. Multiply connected planar domains and higher dimensions remain open. The companion strict Neumann result for simply connected Lipschitz domains is a separate entry.)

Spectral geometry

Does the Dirichlet Pólya conjecture hold for every bounded planar domain $\Omega\subset\mathbb R^2$ ? Equivalently, is it true that $4\pi j\leq |\Omega|\lambda_j(\Omega) \qquad(j\ge1),$ where $\lambda_j(\Omega)$ denotes the $j$ -th Dirichlet eigenvalue of $\Omega$ , counted with multiplicity?

Posed by George Pólya, Mathematics and Plausible Reasoning (1954); he proved it for plane-tiling domains in 1961, 1954Open 72yModel ChatGPT; Aristotle; Codex (OpenAI; Harmonic)Solved 2026-10-09

Lean-checked, statement unauditedSignificance 45 Submitted by StormyCobra617 on 9 Oct 2026

Nonquadraticity of Lambert series values and improved bounds for Erdős Problem 1049

ProvedPartialAI-discovered

(For rational bases t=a/b>1, F(t)=_n1(t^n-1)^-1 is irrational whenever log b/log a<0.5728, and a formalized refinement reaches 0.573290. The previous thresholds were 1/2-1/π^2≈0.3987 (Bundschuh and Väänänen, 1994) and 0.405683 (W. Cook, September 2026, self-published). Problem 1049 itself stays open: t=3/2, with log2/log3≈0.631, is still outside the region. Along the way, for every integer q2 the numbers 1,F(q),F(q)^2 are linearly independent over Q, so the Erdős-Borwein constant F(2) is not quadratic irrational. Both follow from one degree criterion for real algebraic bases: no relation of degree at most e over Q(q) when eλlog M(q)<log q, with λ=(42963+428220/π^2)/202140≈0.42718.)

Erdős #1049 · Irrationality and algebraic degree

Let $t>1$ be a rational number. Is $\sum_{n\ge1}\frac{1}{t^n-1}=\sum_{n\ge1}\frac{\tau(n)}{t^n}$ irrational, where $\tau(n)$ counts the divisors of $n$ ? A conjecture of Chowla; Erdős (1948) proved it for every integer $t\ge2$ .

Posed by Chowla (conjecture); Erdős, Problem 1049 (Er88c, p. 102), 1988Open 38yModel Generative AI, model not named in the source (submitter: GPT-6 Astra via OpenAI Codex) (OpenAI)Solved 2026-10-07

Lean-checked, statement unauditedSignificance 12 Submitted by CobaltMongoose239 on 8 Oct 2026

Zak's modified integer round-down conjecture for the skiving stock problem

DisprovedUnder reviewAI co-developed

(Theorem 1: for every integer c0 there is a rational instance with 5B items, sizes in (53/270,11/54) for threshold 1, LP_SSP=LP_SSP,prop=B and OPT_SSP≤ B-c-1; c=1 refutes MIRDP with no rounding ambiguity. Theorem 2 and Corollary 3: fixed additive approximation is NP-hard. The result is conditional on OpenAI's bin-packing theorem (itself a Candidate here). Not shown: any growth rate of the gap in terms of instance size.)

Skiving stock and bin covering; pattern LP integrality gap

In the one-dimensional skiving stock problem (bin covering), items of given sizes are grouped into as many disjoint groups as possible, each group reaching a threshold $L$ ; items may be left unused. The pattern linear programming relaxation gives an upper bound $LP_{\mathrm{SSP}}$ on the optimum $OPT_{\mathrm{SSP}}$ . The integer round-down property $OPT_{\mathrm{SSP}}=\lfloor LP_{\mathrm{SSP}}\rfloor$ fails, but the largest known gap $LP_{\mathrm{SSP}}-OPT_{\mathrm{SSP}}$ was about $1.18$ . The modified integer round-down property (MIRDP) conjectures that every instance satisfies $LP_{\mathrm{SSP}}-OPT_{\mathrm{SSP}}<2$ . Is that true?

Posed by E. J. Zak (2003), as cited by the manuscript; stated as an open conjecture by Martinovic and Scheithauer (Discrete Optimization 2016; Pesquisa Operacional 2019), 2003Open 23yModel Anthropic Fable 5.1 and OpenAI GPT-5.6 Sol Pro (Anthropic / OpenAI)Solved 2026-10-07

AnnouncedSignificance 14 Submitted by LucidStoat857 on 7 Oct 2026

Wall's D(2) problem

DisprovedUnder reviewAI-discovered

(Theorem 1.1: there is a finite connected three-dimensional CW complex X with H_i( X; Z)=0 for i>2 and H^3(X;M)=0 for every Z[_1X]-module M that is not homotopy equivalent to any finite complex of dimension at most two. The obstruction: on a finite 2-complex, a rank-one local system with vanishing twisted H_2 is trivial on every finite-order element of _1. _1(X) is infinite. It does not settle the D(2) problem for finite fundamental groups, the relation gap problem, or the Eilenberg-Ganea problem.)

Algebraic topology; homotopy types of 2-complexes

A finite connected CW complex $X$ with fundamental group $G$ satisfies the $D(2)$ condition if $H_i(\widetilde X;\mathbb Z)=0$ for $i>2$ and $H^3(X;M)=0$ for every finitely generated $\mathbb ZG$ -module $M$ . Wall showed such $X$ has a finite three-dimensional model and asked whether the three-cells can be removed. Stably the answer is yes (Cohen), and it is yes for many finite groups (Johnson, Hambleton, Hofmann-Nicholson); candidate counterexamples (Cohen-Dyer, Bridson-Tweedale) stayed unresolved or conditional on relation-gap assertions. Wall's finite $D(2)$ problem: is every finite $D(2)$ complex homotopy equivalent to a finite CW complex of dimension at most two?

Posed by C. T. C. Wall, 1965Open 61yModel Unreleased internal OpenAI model (OpenAI)Solved 2026-10-06

AnnouncedSignificance 46

At least five colours for the regular fourteen-gon Minkowski plane

ProvedPartialAI-discovered

(We prove ( R^2,P_14)5 using a graph with 1,540 vertices and 13,755 unit edges. Every edge difference is a rotated point on a side of the unit polygon with an exact algebraic parameter in [0,1]. A kernel-checked LRAT refutation of the graph's four-colouring CNF rules out every four-colouring, and restricting a colouring of the plane to these points gives the contradiction. The whole geometric and logical chain is in Lean. With Gehér's upper bound, 5≤( R^2,P_14)6; whether five colours suffice remains open, as does Problem 5.1 for other n.)

Discrete geometry; unit-distance graph colouring

Let $C$ be a centrally symmetric convex curve and let $\chi(\mathbb R^2,C)$ be the least number of colours needed to colour the plane so that no two points at $C$ -norm distance one share a colour. Exoo, Fisher and Ismailescu proved $\chi(\mathbb R^2,C)\ge5$ for the regular octagon, decagon and dodecagon and asked (Problem 5.1) whether $\chi(\mathbb R^2,C)\ge5$ for every regular $2n$ -gon with $n\ge4$ . This entry is the case $n=7$ : with $P_{14}=\operatorname{conv}\{z\in\mathbb C:z^{14}=1\}$ as the unit ball, is $\chi(\mathbb R^2,P_{14})\ge5$ ?

Posed by Geoffrey Exoo, David Fisher and Dan Ismailescu, The chromatic number of the Minkowski plane: the regular polygon case, Problem 5.1 (arXiv:2108.12861, 2021), 2021Open 5yModel OpenAI Codex (GPT-6 Astra) (OpenAI)Solved 2026-10-06

Lean-checked, statement unauditedSignificance 8 Submitted by ZestyRaven517 on 6 Oct 2026

Log abundance for threefolds in characteristic $p>3$ in numerical dimension one

ProvedPartialAI-discovered

(Principal Theorem 1.1: over an algebraically closed field of characteristic p>3, if (X,B) is a projective log canonical threefold pair with effective Q-boundary, K_X+B is Q-Cartier, nef and of numerical dimension one, then K_X+B is semiample; neither terminality nor Q-factoriality is needed. The companion proves the terminal Q-factorial case (K_X nef with =1 gives =1 and semiampleness). Characteristics 2 and 3, numerical dimension zero and higher dimensions are not addressed.)

Birational geometry in positive characteristic; minimal model program

Posed by The abundance conjecture of the minimal model program (classical; the manuscript names no poser); the remaining numerical-dimension-one case is isolated by Zheng Xu's 2024 and 2026 papersOpen —Model Unreleased internal OpenAI model (OpenAI)Solved 2026-10-05

AnnouncedSignificance 18

The BFSS exclusion of positive-energy normalizable states in the relative SU(2) model

DisprovedUnder reviewAI-discovered

(Theorem 1.1: the relative SU(2) BFSS Hamiltonian has infinitely many positive eigenvalues E_j→∞ with square-integrable eigenvectors, so _p(H)(0,∞) is infinite. This contradicts, for this exact operator at N=2, the BFSS paper's sentence excluding other normalizable bound states. It concerns only N=2, says nothing about N3, and does not bear on the zero-energy state (the companion entry) or on large-N matrix theory.)

Supersymmetric matrix quantum mechanics; embedded eigenvalues

For the relative $\mathrm{SU}(2)$ BFSS Hamiltonian (supersymmetric quantum mechanics of three $9$ -vectors with quartic potential $\sum_{a<b}|x^a\wedge x^b|^2$ and sixteen fermions per color), de Wit, Luscher and Nicolai (1989) proved the spectrum is $[0,\infty)$ and left open whether normalizable eigenstates lie inside it. Banks, Fischler, Shenker and Susskind (1997, Section 5, after Equation (5.4)) stated that besides the zero-energy threshold states 'No other normalizable bound states can occur'. Danielsson, Ferretti and Sundborg had found metastable excited states in a Born-Oppenheimer approximation. Does the relative $\mathrm{SU}(2)$ Hamiltonian have normalizable eigenstates of positive energy?

Posed by T. Banks, W. Fischler, S. H. Shenker and L. Susskind, M theory as a matrix model: a conjecture, Phys. Rev. D (1997), Section 5; open question in de Wit, Luscher and Nicolai (1989), 1997Open 29yModel Unreleased internal OpenAI model (OpenAI)Solved 2026-10-05

AnnouncedSignificance 25

Quasipolynomial-time solution of simple stochastic and turn-based stochastic mean-payoff games

ProvedPartialAI-discovered

(Claims a deterministic algorithm with 2^O((log(L+2))^2) bit operations that outputs the set of vertices of nonnegative value in finite turn-based stochastic mean-payoff games with signed binary rewards and arbitrary rational chance probabilities, and hence decides the threshold-1/2 problem for Condon's simple stochastic games in deterministic quasipolynomial time. The proof chooses an explicit discount of polynomial bit length and finds the discounted fixed point by a recursive labeling procedure. It does NOT give a polynomial-time algorithm, and it decides the threshold set rather than outputting values or strategies in the stated theorem.)

Algorithmic game theory; stochastic games; complexity of NP and coNP problems

Posed by Anne Condon, The complexity of stochastic games (1992), as cited by the manuscript; the question is the polynomial-time status of a problem in NP and coNP, 1992Open 34yModel Unreleased internal OpenAI model (OpenAI)Solved 2026-10-05

AnnouncedSignificance 44

Marginal irrelevance of weak bond disorder at the planar Ising critical point: quenched SLE3 interfaces for weak random bonds

ProvedVariant onlyAI-discovered

(Theorem 1.1: for any bounded mean-zero nondegenerate law on [-1,1] there is ε_0()>0 such that for fixed 0<ε<ε_0, at the spontaneous-magnetization threshold _c^(ε)=12log(1+√2)+O(ε^2), the quenched Dobrushin spin interface in any bounded Jordan domain converges to chordal SLE_3 in probability over environments, in the bounded-Lipschitz metric on oriented curves. A companion proves, conditionally on stated deterministic reference estimates, that the relative second moment of the quenched spin correlation grows as (log r)^1/4+o(1) for fair ±ε bonds. Not shown: strong disorder, the double-logarithmic specific heat, or unconditional logarithmic corrections.)

Disordered systems; two-dimensional Ising model; SLE

Posed by Vik. S. Dotsenko and Vl. S. Dotsenko (1983); B. N. Shalaev (1984); R. Shankar (1987); A. W. W. Ludwig (1990): renormalization-group predictions of marginal irrelevance, 1983Open 43yModel Unreleased internal OpenAI model (OpenAI)Solved 2026-10-05

AnnouncedSignificance 22

The Ascoli-He-Park-Talagrand graph-decomposition conjecture at the expectation threshold

ProvedUnder reviewAI-discovered

(Theorem 1.1: there are absolute k2 and L1 such that for every n2 and every graph H on at most n vertices, E(H)=E(H_1)… E(H_k) with p_c(H_i;n)≤ Lq(H;n) for each i. Pieces may be empty or share vertices, and the partition depends only on H and n. It does not give a single copy of H at threshold Lq(H;n) (the separate embeddings need not glue), and k, L are not explicit.)

Random graphs; thresholds for subgraph containment

For a graph $H$ on at most $n$ vertices let $p_c(H;n)$ be the least $p$ at which $G(n,p)$ contains a copy of $H$ with probability $1/2$ , and $q(H;n)$ its integral expectation threshold. The Park-Pham theorem gives $p_c(H;n)\le Cq(H;n)\log|E(H)|$ , and the logarithm cannot be dropped in general. Ascoli, He, Park and Talagrand (2026, Conjecture 8.1) proposed that it can be removed after cutting $H$ into boundedly many pieces, and proved bounded-degeneracy cases. Are there absolute constants $k$ and $L$ such that the edges of every graph $H$ on at most $n$ vertices split into $k$ subgraphs $H_1,\dots,H_k$ with $p_c(H_i;n)\le L\,q(H;n)$ for each $i$ ?

Posed by Ruben Ascoli, Xiaoyu He, Jinyoung Park and Michel Talagrand (Conjecture 8.1, arXiv 2608.11183, August 2026), 2026Open 0yModel Unreleased internal OpenAI model (OpenAI)Solved 2026-10-05

AnnouncedSignificance 15

Uchida's conjecture on open homomorphisms of Galois groups of solvably closed extensions of number fields

ProvedUnder reviewAI-discovered

(Claims Uchida's conjecture in full: for number fields F_1,F_2 and possibly infinite solvably closed Galois extensions E_i/F_i, every continuous open homomorphism Gal(E_1/F_1)(E_2/F_2) is induced by a unique field embedding E_2 E_1, with no hypothesis on the kernel and no separate cyclotomic-compatibility assumption. The key steps derive full cyclotomic compatibility from openness, through a uniform (in the auxiliary prime) Kummer bound and Frobenius-average character tests, and then apply Hoshi's criterion. It does NOT treat function fields, finite solvable quotients (Saidi-Tamagawa), or non-open homomorphisms.)

Anabelian geometry; Galois groups of number fields

Posed by Koji Uchida, Homomorphisms of Galois groups of solvably closed Galois extensions, J. Math. Soc. Japan 33 (1981), p. 595, 1981Open 45yModel Unreleased internal OpenAI model (OpenAI)Solved 2026-10-05

AnnouncedSignificance 22

The index conjecture for semi-log-canonical log Calabi-Yau pairs with finite rational coefficient sets

ProvedUnder reviewAI-discovered

(Theorem 1.1: for every d4 and finite [0,1] Q there is a(d,)>0 such that a(d,)(K_X+B) is Cartier with trivial line bundle for every connected projective equidimensional slc log Calabi-Yau pair of dimension d with coefficients in , independently of the number of components. With Jiang-Liu's theorem for d3 this gives the conjecture in the cited form. Not shown: slc pairs with infinite DCC coefficient sets, or explicit values of a(d,).)

Birational geometry; log Calabi-Yau pairs

Posed by Index conjecture for log Calabi-Yau pairs; finite-rational-coefficient slc form stated by C. Jiang and H. Liu (Algebra and Number Theory, 2021), Conjecture 1.5Open —Model Unreleased internal OpenAI model (OpenAI)Solved 2026-10-05

ContestedClaim issueSignificance 25

The Alon-Krivelevich-Sudakov conjecture: graphs with a fixed forbidden subgraph have chromatic number O(Delta/log Delta)

ProvedUnder reviewAI-discovered

(Theorem 1.1: for every r4 there are C_r and Δ_r such that every K_r-free graph of maximum degree Δ≥Δ_r has correspondence (DP) chromatic number at most ⌈ C_rΔ/logΔ⌉; Corollary 1.2 gives the same for any fixed forbidden subgraph F, which implies the conjecture for ordinary and list coloring. The proof is a round-based random partial coloring with weighted lists, controlled by an entropy potential and local adjustments that offset triangle correlations. Constants and thresholds are not optimized; no explicit leading constant is claimed.)

Graph coloring: sparse and F-free graphs

Every graph of maximum degree $\Delta$ has $\chi(G)\le\Delta+1$ . Johansson (1996) proved $\chi(G)=O(\Delta/\log\Delta)$ for triangle-free graphs, and Alon, Krivelevich and Sudakov (1999) extended this to graphs with sparse neighborhoods. For general fixed $F$ , and already for $F=K_4$ , the best bounds carried an extra $\log\log\Delta$ factor. Alon, Krivelevich and Sudakov (1999, Conjecture 3.1) conjectured: for every fixed graph $F$ there is $C_F$ such that every $F$ -free graph of maximum degree $\Delta$ has chromatic number at most $C_F\Delta/\log\Delta$ . Is this true?

Posed by N. Alon, M. Krivelevich and B. Sudakov, Coloring graphs with sparse neighborhoods, J. Combin. Theory Ser. B 77 (1999), Conjecture 3.1, 1999Open 27yModel Unreleased internal OpenAI model (OpenAI)Solved 2026-10-05

AnnouncedSignificance 42

Critical two-point decay $r^{-1/4}(\log r)^{1/8}$ in the planar XY model

ProvedUnder reviewAI-discovered

(Log-correction paper, Theorem 1.1: C_b_c(r)=B_XYr^-1/4(log r)^1/8(1+o(1)) as r→∞ along the axis, with B_XY∈(0,∞), for the free-box limit. Exponent paper (September 24): C_b_c(r)=r^-1/4+o(1). The constant is not computed; off-axis directions, other lattices and the Villain model at criticality are not covered here, and the next-order loglog r/log r correction is not claimed. Companions prove a Gaussian free field limit for discrete Gaussian heights through the roughening threshold with universal value 8π, and (conditional on stated inputs) the critical center magnetization n^-1/8(log n)^1/16 and spin-field scaling limit.)

Statistical mechanics; Berezinskii-Kosterlitz-Thouless transition

Posed by J. M. Kosterlitz (renormalization-group prediction), building on Berezinskii and Kosterlitz-Thouless, 1974Open 52yModel Unreleased internal OpenAI model (OpenAI)Solved 2026-10-05

AnnouncedSignificance 50

The stable coordinate conjecture in four variables

DisprovedUnder reviewAI-discovered

(Theorem 1.1: with Q=x_2^2-x_4^2+x_1x_3 and f=x_1-2Q(Q(x_2+x_4)+x_1x_4), a degree-five polynomial in C[x_1,…,x_4], some automorphism of C[x_1,…,x_4,w] sends f to x_1 but none of C[x_1,…,x_4] does. Since the conjecture holds in at most three variables, this is the smallest possible dimension, and one added variable suffices. Corollary 1.2: every fibre of f is A^3 and none is rectifiable. Not shown: anything about dimensions above four, which the release's earlier five-variable example already covered for five; the result is one explicit polynomial, not a classification.)

Affine algebraic geometry; polynomial automorphisms

Posed by V. Shpilrain and J.-T. Yu, Affine varieties with equivalent cylinders, J. Algebra 251 (2002), Conjecture 3, 2002Open 24yModel Unreleased internal OpenAI model (OpenAI)Solved 2026-10-05

Lean-checked, statement unauditedSignificance 22

…

Math problems solved with AI

Latest activity

Latest

This week

All entries