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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

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

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

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

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

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

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

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

The Friedgut-Kalai threshold-width conjecture for r-uniform hypergraph properties

ProvedUnder reviewAI-discovered

(Theorem 1.1: for each r3 there is C_r with Var_p(f)≤ C_rI_p(f)/(log n)^r/(r-1) for every relabelling-invariant Boolean property of r-uniform hypergraphs, every n≥ r and every 0<p<1, without monotonicity. Corollary 4.1: for nontrivial increasing properties the width from ε to 1-ε is O_r(log(1/ε)/(log n)^r/(r-1)). Not shown: explicit constants C_r or sharpness examples for each r.)

Random hypergraphs, analysis of Boolean functions

For a nontrivial increasing property of $r$ -uniform hypergraphs on $n$ vertices invariant under relabelling, the transitive symmetry of the $\binom nr$ edge coordinates gives threshold width $O(1/\log n)$ by the Friedgut-Kalai method. Friedgut and Kalai (1996, Section 5) conjectured that the vertex symmetry does better: for each fixed $r$ the width from probability $\varepsilon$ to $1-\varepsilon$ should be $O_{r,\varepsilon}((\log n)^{-r/(r-1)})$ , the graph case being $r=2$ . Bourgain and Kalai reached exponent $r/(r-1)-\delta$ for the influence bound at the uniform measure. For every fixed $r\ge3$ , is the threshold width of every such property $O_{r,\varepsilon}((\log n)^{-r/(r-1)})$ ?

Posed by Ehud Friedgut and Gil Kalai, 1996Open 30yModel Unreleased internal OpenAI model (OpenAI)Solved 2026-10-05

AnnouncedSignificance 25

The Burger-Ozawa-Thom question: does strong Ulam stability characterize amenability?

ProvedUnder reviewAI-discovered

(Theorem 1.1: a countable discrete group is strongly Ulam stable if and only if it is amenable. For every nonamenable countable G and δ>0 there is a separable H and a normalized :G→ U(H) with defect below δ that stays at uniform distance more than 1/10 from every unitary representation on H. Not shown: anything about finite-dimensional (ordinary) Ulam stability, or uncountable groups.)

Group theory; stability of approximate representations

Posed by Marc Burger, Narutaka Ozawa and Andreas Thom, On Ulam stability, Israel J. Math. 193 (2013), p. 111, 2013Open 13yModel Unreleased internal OpenAI model (OpenAI)Solved 2026-10-05

AnnouncedSignificance 25

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

The Hasenfratz-Maggiore-Niedermayer exact mass gap for the two-dimensional O(4) lattice model

ProvedUnder reviewAI-discovered

(Theorem 1.1: for the periodic infinite-volume state, the gap of the full Osterwalder-Schrader transfer operator, including rotation-invariant sectors and measured in original lattice time steps, satisfies _β→∞e^πββ^-1/2m_lat(β)=32(π/4-1/2). The input companion gives c√β e^-πβ≤ m_lat≤ C√β e^-πβ and a positive full gap at every β>0. Not shown: the corresponding exact constant for O(3) or general O(n), a continuum O(4) theory, or any statement about nonperiodic Gibbs states.)

Statistical mechanics; asymptotically free sigma models, mass gap

Posed by Peter Hasenfratz, Michele Maggiore and Ferenc Niedermayer (Phys. Lett. B, 1990); extended to all O(n), n >= 3, by Hasenfratz and Niedermayer (1990), 1990Open 36yModel Unreleased internal OpenAI model (OpenAI)Solved 2026-10-05

AnnouncedSignificance 28

The codimension-one bound for singular sets of stationary integral varifolds

ProvedUnder reviewAI-discovered

(Theorem 1.1: for every stationary integral m-varifold V in an open U R^m+n, m,n1, _ HSing V≤ m-1, with no stability, minimizing, orientability, density or codimension hypothesis; crossing planes show it is sharp. This implies H^m(Sing V)=0. The proof combines the companion paper's signed-excess estimates with a fitting and frequency analysis around a smooth center and a measure-valued blow-up at higher-multiplicity planar tangents. It does not prove rectifiability or finer structure of the singular set, and does not cover Riemannian ambient metrics.)

Geometric measure theory: regularity of stationary varifolds

Posed by Camillo Brena, Stefano Decio and Camillo De Lellis, Remarks and conjectures on stationary varifolds dedicated to Enrico Bombieri, Rend. Lincei Mat. Appl. 36 (2025), Conjecture 1.1, 2025Open 1yModel Unreleased internal OpenAI model (OpenAI)Solved 2026-10-05

AnnouncedSignificance 50

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

Small spatial embeddings for one-sided near inclusions of von Neumann algebras

DisprovedUnder reviewAI-discovered

(Theorem 1.1: there are ε_0>0 and pairs of von Neumann algebras on separable Hilbert spaces with common identity and γ(M_n,N_n)0 for which spatial embeddings exist but every implementing unitary satisfies ‖u-1‖≥ε_0. So the unrestricted one-sided question has a negative answer, in contrast with Christensen's injective case and with the two-sided strong Kadison-Kastler theorem in the same family. The obstruction is to small unitaries, not to the existence of embeddings.)

Operator algebras; near inclusions, perturbation theory

Posed by J. Cameron, E. Christensen, A. M. Sinclair, R. R. Smith, S. White and A. Wiggins, Type II_1 factors satisfying the spatial isomorphism conjecture, PNAS 109 (2012), 2012Open 14yModel Unreleased internal OpenAI model (OpenAI)Solved 2026-10-05

AnnouncedSignificance 20

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

Szabo's Conjecture A on equivariant Jiang-Su stability, in the unital stably finite case

ProvedPartialAI-discovered

(Theorem 1.1: for A simple, separable, unital, infinite-dimensional, nuclear, stably finite and Z-stable, every action α of a countable discrete amenable group is cocycle conjugate to α_ Z, with no restriction on how α acts on the trace simplex, and including finite groups and trivial actions. It does not treat nonunital algebras, and stably infinite (purely infinite) algebras are not part of this theorem.)

Operator algebras; C*-dynamics of amenable group actions

Posed by Gabor Szabo, 2021Open 5yModel Unreleased internal OpenAI model (OpenAI)Solved 2026-10-05

AnnouncedSignificance 22

Randomized versus quantum query complexity of total Boolean functions: is R(f) = O(Q(f)^3)?

DisprovedUnder reviewAI-discovered

(Theorem 1.1: for each fixed k2 there are total Boolean functions F_k,m with Q(F_k,m)≤ C_k√ m(log m)^b_k and R(F_k,m)≥ c_km^2-1/k/(log m)^2, a power 4-2/k separation up to logarithms, so no bound R=O((1+Q)^α) holds with α<4. With the known R=O(Q^4) the optimal exponent is exactly 4, and the cubic conjecture is false. The functions are cheat-sheet-style columns whose certificate cells are addressed by answers to partial-function instances. Not shown: a separation with exponent exactly 4 for a single family, or new results for partial functions.)

Query complexity

Posed by Scott Aaronson, Shalev Ben-David, Robin Kothari, Shravas Rao and Avishay Tal, 2021Open 5yModel Unreleased internal OpenAI model (OpenAI)Solved 2026-10-05

AnnouncedSignificance 38

Does the perfect matching polytope have a polynomial-size semidefinite lift?

DisprovedUnder reviewAI-discovered

(Theorem 1.1: there is c>0 such that for every fixed 0<<1 the odd-cut matrix with entries |Mδ(U)|-1+ has real PSD rank at least 2^cn for large even n, with no symmetry or precision restriction on the factors. Corollary 1.2: every exact semidefinite lift of the perfect matching polytope has size 2^Ω(n). The method follows Lee-Raghavendra-Steurer with a Grigoriev-type parity functional and new averaging lemmas. The endpoint =1 has a polynomial factorization, and approximate lifts are not excluded (Kaniewski-Lee-de Wolf give subexponential approximations).)

Extended formulations; positive semidefinite rank, lower bounds

Posed by G. Braun, J. Brown-Cohen, A. Huq, S. Pokutta, P. Raghavendra, A. Roy, B. Weitz, D. Zink, The matching problem has no small symmetric SDP, SODA 2016 / Math. Program. 165 (2017), 2015Open 11yModel Unreleased internal OpenAI model (OpenAI)Solved 2026-10-05

Lean-checked, statement unauditedSignificance 35

The trilinear Hilbert transform conjecture at the L3 x L3 x L3 to L1 point (slopes 1, 2, 3)

ProvedPartialAI-discovered

(Theorem 1.1: there is a finite C with ‖T(f_1,f_2,f_3)‖_L^1≤ C_j‖f_j‖_L^3 for Schwartz functions, so T extends uniquely to a bounded trilinear map L^3× L^3× L^3→ L^1. The proof combines the Leng-Sah-Sawhney inverse theorem with new compression, curvature-localization and sparse counting lemmas that sum the scales without loss. It concerns only the fixed slopes 1, 2, 3 and the single exponent point (3,3,3;1): the rest of the conjectured range, other slopes, maximal or truncation-uniform versions and higher-degree multilinear Hilbert transforms are not treated, as the manuscript itself says.)

Harmonic analysis: multilinear singular integrals

Posed by Standard conjecture after Lacey and Thiele's bilinear theorem; the manuscript cites it as Conjecture 1.7 of B. Hu and V. Lie, On the curved trilinear Hilbert transform (arXiv:2308.10706, 2023)Open —Model Unreleased internal OpenAI model (OpenAI)Solved 2026-10-05

AnnouncedSignificance 50

The generalized Ramanujan conjecture for globally generic cuspidal representations of exceptional groups over function fields

ProvedPartialAI-discovered

(Theorem 1.1: for G split connected adjoint absolutely simple of type G_2,F_4,E_6,E_7 or E_8 over a global function field, every complex globally generic cuspidal automorphic representation is tempered at every place. The companion (September 24) gives temperedness at unramified places for all split adjoint absolutely simple groups from genericity at one unramified place. Nothing is claimed over number fields, for non-generic representations, or for ramified places of classical groups beyond what Lomeli already proved.)

Automorphic representations over function fields; temperedness

Posed by Generic form of the generalized Ramanujan conjecture (Arthur's framework; Shahidi)Open —Model Unreleased internal OpenAI model (OpenAI)Solved 2026-10-05

AnnouncedSignificance 34

The Bogoliubov quantum-depletion law for the dilute Bose gas in the thermodynamic limit

ProvedUnder reviewAI-discovered

(For hard spheres (October 5): for every bounded continuous f and ε>0, at all sufficiently small fixed , every thermodynamic accumulation value of 1/N√( a^3)_k0f(k/√(8π a)) n(k) is within ε of ∫ f d_Bog, uniformly over ground states; taking f=1 gives depletion 8/3√π√( a^3)+o(√( a^3)). A companion proves the same depletion asymptotic for fixed bounded, nonnegative, radial, finite-range potentials with positive scattering length. Not covered: positive temperature, the next-order correction, or unbounded non-hard-core potentials.)

Quantum many-body theory; dilute Bose gases

Bogoliubov's theory (1947), carried to the hard-sphere gas by Lee, Huang and Yang (1957), predicts that at zero temperature a dilute three-dimensional Bose gas with scattering length $a$ and density $\rho$ has a fraction $\frac{8}{3\sqrt\pi}\sqrt{\rho a^3}$ of its particles outside the condensate, distributed in momentum by an explicit Bogoliubov profile on the scale $\sqrt{8\pi\rho a}$ . The energy to Lee-Huang-Yang order is now proved, but energy bounds do not control occupation of modes whose energy vanishes with the volume, and the depletion had been derived only in Gross-Pitaevskii-type scalings or for approximate equations. In the thermodynamic limit at fixed density, followed by $\rho a^3\to0$ , is the ground-state depletion $\frac{8}{3\sqrt\pi}\sqrt{\rho a^3}+o(\sqrt{\rho a^3})$ ?

Posed by Prediction of N. N. Bogoliubov (1947) and T. D. Lee, K. Huang and C. N. Yang (1957), 1957Open 69yModel Unreleased internal OpenAI model (OpenAI)Solved 2026-10-05

AnnouncedSignificance 35

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

Howie's conjecture: every nonsingular system of equations over a group is solvable in an overgroup

ProvedUnder reviewAI-discovered

(Claims Theorem 1: for every group G and every finite system w_1,…,w_m∈ G*F_n whose exponent-sum matrix has rank m over Q, the canonical map G→(G*F_n)/⟨⟨ w_1,…,w_m⟩⟩ is injective, so the system has a simultaneous solution in an overgroup. This contains the Kervaire-Laudenbach conjecture (one equation, one unknown, nonzero exponent sum) and the Kervaire conjecture. It does NOT control the overgroup (no finiteness, solvability or other structure of the solution group), and says nothing about singular systems.)

Combinatorial group theory; equations over groups

Posed by James Howie (1981), On pairs of 2-complexes and systems of equations over groups; name and formulation as in Klyachko, Mikheenko and Roman'kov (2024), 1981Open 45yModel Unreleased internal OpenAI model (OpenAI)Solved 2026-10-05

AnnouncedSignificance 45

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

ProvedUnder reviewAI-assisted

Supercongruences, modular forms and elliptic curves

AnnouncedSignificance 5 Submitted by Oleksiy Babanskyy on 4 Oct 2026

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