Lean-checked results

OpenAI lists a formal proof for these results: a proof written in the Lean language and checked by a computer, step by step. What exactly Lean checked is stated on each entry.

242 AI-made results with this trust label, from 475 papers in the OpenAI math collection, each explained in plain English. All of them have a formal Lean proof listed. They span 17 fields, led by theoretical computer science (34), combinatorics (33) and probability and statistical mechanics (19). The oldest question here dates back to 1896: The quasi-Riemann hypothesis. Results rated Landmark include Nagata’s conjecture and maximal Seshadri constants and Koebe’s circle-domain conjecture.

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Field

Number theory17

003Sep 30, 2026 Explained

The quasi-Riemann hypothesis

Proves that every Dirichlet L-function, including $\zeta(s)$ , is zero-free in $\Re s\gt 7/8$ , resolving the quasi-Riemann hypothesis. The same half-plane is zero-free for every finite-order Hecke L-function over $\mathbb Q(\sqrt{-3})$ . A companion gives a different proof of the zero-free half-plane $\Re s\gt 11/12$ .

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005Sep 24, 2026 Explained

Irrationality of Catalan’s constant

Proves that Catalan's constant $G=\sum_{j\ge0}(-1)^j/(2j+1)^2$ is irrational.

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007Sep 24, 2026 Explained

Ordinary two-point correlations and the corrected Elliott conjecture

Proves the ordinary two-point Chowla conjecture, with a bound $O(X/(\log X)^c)$ for Liouville correlation sums along fixed nonproportional affine forms, where c > 0 is absolute. More generally, proves the binary corrected Elliott conjecture for complex multiplicative functions bounded by one when one factor is uniformly nonpretentious against each fixed Dirichlet character times $n^{it}$ for $|t|\le X$ .

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008Sep 23, 2026 Explained

The Deligne–Drinfeld conjecture

Proves that the rational Grothendieck–Teichmüller Lie algebra, with the Ihara bracket, is freely generated by one element in each odd weight 3, 5, 7, …, resolving the Deligne–Drinfeld conjecture.

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009Sep 23, 2026 Explained

Function-field reconstruction from Milnor K-theory and Galois data

Reconstructs function fields of transcendence degree at least two over algebraically closed constants from $K^{\mathrm M}_1/\ell$ , $K^{\mathrm M}_2/\ell$ , and their product. These data recover the perfect closure and constants when ℓ differs from the characteristic, and the original field and its named base in equal characteristic. Also proves Bogomolov–Pop reconstruction from abelian-by-central pro-ℓ Galois data away from the characteristic.

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012Sep 24, 2026 Explained

Independent largest prime factors of consecutive integers

Resolves the Erdős–Pomerance joint Dickman conjecture: the logarithmic sizes of the largest prime factors of n and $n+1$ are asymptotically independent in ordinary natural density. In particular, the integers satisfying $P^+(n)\lt P^+(n+1)$ have density 1/2.

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013Sep 24, 2026 Explained

Ostmann’s inverse Goldbach conjecture

Proves that no finite modification of the primes can be written as $A+B$ with $A,B\subseteq\mathbb Z_{\ge0}$ each containing at least two elements. This resolves Ostmann's inverse Goldbach conjecture on additive indecomposability.

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015Sep 24, 2026 Explained

Torus-packet equidistribution in prime, quartic, and sextic degrees

Proves Haar equidistribution without escape of mass for complete volume-weighted torus packets from totally real fields: arbitrary lattices and prescribed local types in fixed prime degree at least five, arbitrary-order Picard packets in primitive quartic fields, and maximal-order ideal-class packets in primitive sextic fields. Here primitive means having no proper intermediate field; the relevant order or field discriminant tends to infinity.

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017Sep 24, 2026 Explained

The irrationality exponent of π is 2

Proves that the irrationality exponent of π is exactly 2: for every ε > 0 and all sufficiently large denominators q, every rational $p/q$ satisfies $|\pi-p/q|\ge q^{-2-\varepsilon}$ . This also proves convergence of the Flint–Hills series $\sum_{n\ge1}1/(n^3\sin^2 n)$ , with angles in radians.

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020Sep 24, 2026 Explained

Squarefree quartics and power-free polynomial values

Proves the squarefree-values conjecture for irreducible integer quartics with no fixed prime-square divisor: squarefree values on positive integers have the predicted positive Euler-product density. More generally, establishes the $(d-2)$ -power-free density for irreducible integer polynomials of degrees four through eight under the necessary local condition; together with Browning's higher-degree theorem, this covers every d ≥ 4.

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021Sep 25, 2026 Explained

A quadratic bound for Jacobsthal’s function

Answers Jacobsthal's quadratic-bound question: every interval of $Ck^2$ consecutive integers contains an integer coprime to any prescribed positive integer with at most k distinct prime divisors, for an absolute constant C. The bound is uniform over prime sets and interval positions and removes the classical logarithmic loss.

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023Sep 25, 2026 Explained

Patterson's first moment for cubic Gauss sums

Proves unconditionally the all-primary-prime form of Patterson’s first-moment asymptotic: normalized cubic Gauss sums over primary Eisenstein primes of norm at most X, including both conjugates, have an explicit positive main term of order $X^{5/6}/\log X$ . Every fixed nonzero prime-angle Fourier mode has smaller order.

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024Sep 25, 2026 Explained

An asymptotic formula for the number of totients

Gives an asymptotic equivalent for the number $V(x)$ of distinct totient values up to x, with a positive bounded phase-dependent factor determined by convergent arithmetic approximations. In particular, $V(cx)/V(x)\to c$ for every fixed c > 0, answering Erdős and Hall’s scaling question.

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025Sep 25, 2026 Explained

Short Egyptian fractions

Every rational $a/b$ with $1\le a\lt b$ is a sum of $O(\log\log b)$ distinct positive unit fractions. The worst-case minimum number of terms has the same order, resolving Erdős’s conjecture on short Egyptian fractions.

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026Sep 25, 2026 Explained

Positive lower density of large prime gaps

For every fixed C > 0, a positive proportion of consecutive prime gaps exceed $C\log p_n$ , throughout every sufficiently large initial segment of the primes. The proportion may depend on C. Consequently, the indices where $p_n/n$ increases have positive lower density, answering Erdős and Prachar.

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027Sep 25, 2026 Explained

Potential integral density on curve character varieties

Resolves the determinant-one curve case of Litt's integral-density question. For every smooth connected complex algebraic curve and every rank, integral points become Zariski dense in every component of its SL_r character variety over the full ring of integers of one number field. Prescribed quasi-unipotent boundary conjugacy classes are allowed, including nonsemisimple classes.

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028Sep 26, 2026 Explained

Uniformly bounded components of Gaussian-prime graphs

Proves the Gaussian moat conjecture: no infinite walk through distinct Gaussian primes can have uniformly bounded steps. More strongly, for every distance bound D, the graph joining Gaussian primes at distance at most D has uniformly bounded finite component sizes, depending only on D, including primes on the coordinate axes.

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Algebraic and complex geometry8

033Sep 10, 2026 Explained

Iitaka subadditivity, variation, and logarithmic additivity

Proves Campana's orbifold Iitaka subadditivity conjecture for smooth Fujiki-class- $\mathcal C$ manifolds with rational simple-normal-crossing boundaries. For projective fibrations $f:U\to V$ of smooth complex quasi-projective varieties with connected fibers, general fiber F, and $\bar\kappa(V)\ge0$ , proves Popa's inequality $\bar\kappa(U)\ge\kappa(F)+\max\{\bar\kappa(V),\mathop{\mathrm{Var}}\nolimits (f)\}$ , where variation measures the whole geometric generic fiber.

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039Sep 23, 2026 Explained

Nagata’s conjecture and maximal Seshadri constants

Proves Nagata's strict inequality $\sum_i m_i\lt d\sqrt r$ for every nonzero effective plane curve of degree d through r ≥ 10 very general complex points, with arbitrary multiplicities m_i. It also proves maximal multipoint Seshadri constants $(L^n/r)^{1/n}$ for every smooth polarized projective variety of dimension n ≥ 2 and all sufficiently large r: at very general points over ℂ, and at the geometric generic tuple over any algebraically closed field of positive characteristic.

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047Sep 23, 2026 Explained

Zariski cancellation and affine fibrations over the complex numbers

Constructs an integral complex affine fourfold $X\not\cong\mathbb A^4$ with $X\times\mathbb A^1\cong\mathbb A^5$ , disproving affine-space cancellation over ℂ in dimension four. It also disproves the Dolgachev–Weisfeiler affine-fibration conjecture: smooth surjections $X\to\mathbb A^1$ and $\mathbb A^5\to\mathbb A^2$ have every residue-field fiber isomorphic to affine three-space but are not Zariski-locally trivial.

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049Sep 24, 2026 Explained

A stable-coordinate counterexample in four variables

Constructs a polynomial in four complex variables that is not a coordinate but becomes one after adjoining a single variable, disproving the Stable Coordinate conjecture in four variables. Every fiber is affine three-space, yet none of its embeddings is rectifiable, also disproving the Abhyankar–Sathaye conjecture even when all fibers are affine spaces.

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050Sep 24, 2026 Explained

A counterexample to Griffiths’ positivity conjecture

Constructs ample rank-two bundles on $\mathbb P^1\times\mathbb P^1$ with no smooth Hermitian metric of strictly Griffiths-positive curvature, disproving Griffiths' positivity conjecture already on the quadric surface.

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052Sep 23, 2026 Explained

Tangent splittings and product decompositions

A splitting of the tangent bundle of a compact Kähler manifold into two integrable holomorphic subbundles induces a compatible product decomposition of its universal cover, proving the two-summand form of Beauville's splitting conjecture. On smooth rationally connected projective manifolds, both summands are automatically integrable, establishing Höring's conjecture and the corresponding product decomposition.

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058Sep 24, 2026 Explained

Semialgebraic universal covers and bounded domains

Proves the Kollár–Pardon conjecture: the semialgebraic universal covers of connected normal projective complex varieties are exactly products $D\times\mathbb C^m\times F$ , with D bounded symmetric and F simply connected, normal, and projective. A universal cover is quasi-projective exactly when the bounded symmetric factor is absent. In particular, a smooth projective variety covered by ℂⁿ has a finite étale cover by an abelian variety.

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066Sep 25, 2026 Explained

Bounded klt complements for Fano contractions

Proves the finite-rational-coefficient form of Shokurov’s bounded-klt-complement conjecture for ϵ-lc complex Fano-type pairs with nef anti-log-canonical divisor. For ϵ-lc Fano contractions over any algebraically closed characteristic-zero field, it gives klt complements near every base point, with index bounded only by dimension and positive rational ϵ.

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Real and complex analysis10

071Sep 23, 2026 Explained

Koebe’s circle-domain conjecture

Resolves the existence part of Koebe's circle-domain conjecture: every domain in the Riemann sphere is conformally equivalent to a domain whose complementary components are round disks or points. It also proves that circle domains with conformally removable boundary are rigid, meaning every conformal equivalence to another circle domain is Möbius, establishing this direction of the He–Schramm conjecture.

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072Sep 24, 2026 Explained

Brennan's conjecture and the integral-means spectrum

Proves Brennan's conjecture: for every conformal bijection ϕ from a simply connected plane domain onto the disk, $|\phi'|^s$ is area-integrable for $4/3\lt s\lt 4$ . The sharp universal integral-means identity is $B_{\mathcal S}(t)=|t|-1$ for t ≤ −2. A strict bound $B_b(-1)\lt 1/4$ for bounded univalent functions disproves Kraetzer's prediction at that parameter.

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073Sep 23, 2026 Explained

The Falconer distance conjecture

Resolves the Falconer distance conjecture in every dimension d ≥ 2: every compact set $E\subset\mathbb R^d$ with Hausdorff dimension greater than $d/2$ determines a set of Euclidean distances of positive Lebesgue measure.

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075Sep 23, 2026 Explained

The $L\log L$ Fourier-convergence conjecture

Proves that the ordinary symmetric Fourier partial sums of every complex-valued function in $L\log L(\mathbb T)$ converge almost everywhere along the full sequence. This resolves the classical sufficiency conjecture at the $L\log L$ scale.

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076Sep 23, 2026 Explained

Real ultraflat Littlewood polynomials and unbounded binary merit factors

Constructs polynomials with N consecutive coefficients in $\{-1,1\}$ whose modulus is $(1+o(1))\sqrt N$ uniformly on the entire unit circle, for every sufficiently large integer length N. Thus real Littlewood polynomials are ultraflat, including at the real endpoints. Their binary merit factors tend to infinity, disproving Turyn's bounded-merit-factor conjecture.

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081Sep 24, 2026 Explained

Riesz transforms and rectifiability in higher codimension

Resolves the remaining higher-codimension Riesz-transform rectifiability problem: for d ≥ 4 and $2\le n\le d-2$ , an n-Ahlfors–David regular Radon measure on ℝ^d is uniformly n-rectifiable whenever its n-dimensional Riesz transform is uniformly L²-bounded over all positive hard truncations. The rectifiability bounds depend only on dimension, regularity and operator bounds.

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082Sep 24, 2026 Explained

Annular variation and dyadic absolute bounds for the triangular Hilbert transform

Proves maximal and annular r-variation bounds, for every r > 2, from complex $L^3(\mathbb R^2)\times L^3(\mathbb R^2)$ to $L^{3/2}(\mathbb R^2)$ . The maximal estimate controls both hard truncation endpoints and gives almost-everywhere and norm convergence. Pairing with a third input settles the triangular Hilbert transform estimate at the symmetric $L^3\times L^3\times L^3$ point.

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083Sep 25, 2026 Explained

Hilbert transforms along Lipschitz directions

Proves a uniform strong L² bound for the planar Hilbert transform along any Lipschitz unit vector field, at integration lengths bounded by an absolute multiple of its reciprocal Lipschitz constant. The estimate is uniform over inner truncations and yields an L²-bounded principal-value operator, establishing Stein's weak-type conjecture at this short scale.

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084Sep 25, 2026 Explained

The geometric case of the Erdős similarity conjecture

For every fixed $q\in(0,1)$ , constructs compact subsets of $[0,1]$ with measure arbitrarily close to one containing no translated and nontrivially dilated copy of $\{q^n:n\ge1\}$ , with dilations of either sign. This resolves the geometric-progression case of the Erdős similarity conjecture for every ratio.

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085Sep 26, 2026 Explained

Endpoint Sobolev regularity of centered disk averages

Resolves the planar centered-disk case of the Hajłasz–Onninen maximal-function regularity problem. For every real $f\in W^{1,1}(\mathbb R^2)$ , the centered disk maximal function satisfies $\|\nabla Mf\|_1\le C\|\nabla f\|_1$ with an absolute constant. It belongs locally to $W^{1,1}$ and has a globally integrable weak gradient.

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Convex and metric geometry13

087Sep 22, 2026 Explained

The Mahler conjectures, functional inequalities and polar-product symplectic width

Resolves the symmetric and nonsymmetric geometric Mahler conjectures in every dimension, with Hanner polytopes and simplices as the respective volume-product minimizers and all equality cases classified. The corresponding sharp functional Mahler inequalities also hold. For n ≥ 2, every symmetric polar product $K\times K^\circ$ in dimension $2n$ has Gromov width 4.

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088Sep 24, 2026 Explained

Sharp projection-body inequalities and a counterexample to simplex maximization

Proves Petty's projection-volume conjecture in the remaining dimensions n ≥ 4: ellipsoids uniquely minimize projection-body volume at fixed body volume. Also establishes the full Lutwak–Petty projection inequalities. In contrast, products of simplices exceed Brannen's proposed simplex maximum for normalized projection-body volume by an exponential factor in every sufficiently large dimension.

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089Sep 23, 2026 Explained

Bounded-distortion L₁ embeddings of planar and bounded-treewidth graphs

Resolves the planar and bounded-treewidth cases of the Gupta–Newman–Rabinovich–Sinclair conjecture. Shortest-path metrics of finite connected graphs with arbitrary positive edge lengths embed into real L₁ with universal distortion for planar graphs, and distortion depending only on treewidth for bounded-treewidth graphs. The corresponding multicommodity flow–cut gaps are uniformly bounded.

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090Sep 23, 2026 Explained

Triangular-lattice optimality, long-range Riesz and Coulomb energies, and spherical logarithmic energy

Proves that the triangular lattice minimizes the lower limit of energy per particle for every nonnegative completely monotone potential of squared distance among locally finite planar configurations of centered-disk density one. It also minimizes unit-background renormalized Riesz energies for $0\lt s\lt 2$ and Coulomb energy, resolving Sandier–Serfaty and the two-dimensional Brauchart–Hardin–Saff conjecture on the linear term of optimal spherical logarithmic energy.

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091Sep 23, 2026 Explained

Logarithmic and L_p Brunn–Minkowski inequalities and the B-conjecture

Proves the logarithmic Brunn–Minkowski inequality for origin-symmetric convex bodies in every dimension, and the scalar-dilation B-conjecture for all even log-concave Radon measures. For Lebesgue volume it also proves the additive L_p Brunn–Minkowski inequality for full-dimensional origin-symmetric convex bodies throughout $0\lt p\lt 1$ .

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092Sep 23, 2026 Explained

The optimal order of convex-body covering density

Determines the optimal worst-case covering density as $\Theta(n\log n)$ , for both lattice and unrestricted translative coverings. Every convex body in ℝⁿ, n ≥ 2, admits a lattice covering of density at most $Cn\log n$ ; centrally symmetric examples in every sufficiently large dimension require at least $cn\log n$ even without the lattice restriction, for absolute $c,C\gt 0$ .

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094Sep 23, 2026 Explained

Subpolynomial dimension reduction in L_p

For every fixed $1\lt p\lt \infty$ and distortion D > 1, every n-point subset of real L_p embeds into $\ell_p^d$ with distortion at most D and dimension $d=n^{o(1)}$ , answering Naor's sublinear-dimension question for p ≠ 2. In contrast, exact embeddings require worst-case dimension $\Theta(n^2)$ when p ≠ 2.

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095Sep 24, 2026 Explained

Hyperbolicity cones without semidefinite lifts

Disproves the Projected Lax conjecture: some hyperbolicity cones are not spectrahedral shadows. The examples admit no exact finite affine semidefinite lift, regardless of the number of auxiliary variables or the real coefficients used. This also disproves the generalized Lax conjecture that every hyperbolicity cone is spectrahedral.

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096Sep 24, 2026 Explained

The Gaussian propeller conjecture in every dimension

Proves that the sum of squared Gaussian first moments of any finite measurable partition is at most $9/(8\pi)$ . In dimension at least two, three planar sectors of angle $2\pi/3$ , extended orthogonally, attain the bound. Combined with the separate Unique Games theorem, this proves NP-hardness of improving the loss factor $(8\pi/9)(1-1/k)$ for identity-target kernel clustering with fixed k ≥ 3 on rational centered positive semidefinite inputs.

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097Sep 24, 2026 Explained

The Euclidean Steinitz–Bergström bound

Proves that any finite sequence in the Euclidean unit ball of ℝ^d admits signs keeping every partial sum within $C\sqrt d$ , independently of length. Consequently every zero-sum family can be reordered with the same bound on unsigned partial sums. A matching lower bound gives the optimal order $S_2(d)=\Theta(\sqrt d)$ .

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098Sep 25, 2026 Explained

Compact counterexamples to bi-Lipschitz dimension reduction

Every infinite-dimensional real Banach space contains a compact doubling set that admits no bi-Lipschitz embedding into any finite-dimensional normed space. The doubling constant is universal. This answers the Lang–Plaut problem negatively, even for compact subsets of Hilbert space.

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099Sep 27, 2026 Explained

The sharp exponential scale of edit-distance distortion

Determines the least distortion of embedding edit distance on words of length at most d into real ℓ₁: it is $\exp(\Theta(\sqrt{\log d\,\log\log d}))$ . Insertions, deletions and substitutions have unit cost. The constants are uniform over all finite alphabets with at least two symbols, even when the alphabet grows with d; binary words already force the lower bound.

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100Sep 27, 2026 Explained

Cylinder coverings below the half-area bound

Covers the entire closed regular tetrahedron by finitely many cylinders with compact triangular perpendicular bases whose total area is less than half its smallest orthogonal projection area. This disproves Bang's half-area cylinder-covering bound and the stronger directionwise normalized conjecture in dimension three.

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Theoretical computer science34

102Sep 23, 2026 Explained

The Unique Games Conjecture and optimal approximation thresholds

Proves Khot's Unique Games Conjecture. Independent direct reductions also establish NP-hardness, on unweighted graphs, of approximation beyond the Goemans–Williamson ratio for Max-Cut, below factor two for Vertex Cover, and within any fixed constant factor for Min-UnCut and directed feedback vertex set. These direct proofs use established PCP and Label Cover hardness results.

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103Sep 23, 2026 Explained

Exact derandomization of logarithmic space: $\mathsf L=\mathsf{RL}=\mathsf{BPL}$

Proves $\mathsf L=\mathsf{RL}=\mathsf{BPL}$ , resolving derandomization for bounded-error logarithmic-space computation. An effective compiler converts each randomized polynomial-time logarithmic-space machine deciding a language with one-sided or two-sided error into a deterministic logarithmic-space decider with explicit polynomial running-time bounds.

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104Sep 25, 2026 Explained

Quasipolynomial algorithms for mean-payoff, stochastic and parity games

Gives deterministic algorithms using $2^{O((\log(L+2))^2)}$ bit operations, for complete binary input length L, for ordinary mean-payoff games and two separate extensions. They compute exact values and optimal positional strategies in ordinary games, the nonnegative expectation-of-liminf value set in turn-based stochastic games, and the winning set for nonnegative liminf mean payoff conjoined with parity. Signed rewards, rational chance probabilities, and parity priorities are unrestricted and binary-encoded.

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105Sep 23, 2026 Explained

Perfect completeness for 2-to-1 games

Proves Khot's 2-to-1 Games Conjecture with perfect completeness: for every fixed rational $\delta\in(0,1)$ , it is NP-hard to distinguish satisfiable games from games whose optimum is at most δ, on explicit unweighted instances. The alphabet depends only on δ, and every right-hand label has exactly two preimages under each constraint map.

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106Sep 24, 2026 Explained

Hardness of coloring three-colorable graphs

It is NP-hard to color a three-colorable graph using any fixed number c ≥ 3 of colors. More strongly, for every fixed $0\lt \delta\lt 1/3$ , a deterministic polynomial-time reduction from 3SAT produces simple unweighted graphs that are three-colorable in the satisfiable case and have no independent set of size $\delta n$ otherwise, where n is the number of vertices.

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107Sep 24, 2026 Explained

Matrix multiplication with exponent at most 9/4

Proves $\omega\le9/4$ over ℂ, giving $O_\varepsilon(n^{9/4+\varepsilon})$ arithmetic operations for square matrix multiplication. In characteristic zero, some inner dimension n^a with a > 0.465 permits $n^{2+o(1)}$ rectangular multiplication. Further square bounds give ω < 2.258 outside finitely many positive characteristics and ω < 2.371054886006746 over every fixed field.

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108Sep 24, 2026 Explained

A cubic permanent–determinant lower bound

Proves an $\Omega(n^3)$ lower bound for the border determinantal complexity of the $n\times n$ permanent over ℂ. Even coefficientwise limits of determinants of affine-linear matrices require matrix size at least $cn^3$ , for an absolute c > 0 and all sufficiently large n; the same bound therefore holds for exact representations.

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110Sep 24, 2026 Explained

Optimal-order randomized k-server on arbitrary metrics

Establishes a randomized competitive ratio $O(\log^2(k+1))$ for k-server on every metric space, matching the worst-case lower-bound order. One policy serves every finite oblivious request sequence, including on infinite unbounded metrics. On finite rational metrics, a uniform implementation has polynomial preprocessing and per-request bit cost in the input length and $\log(t+1)$ at request t, with a finite instance-dependent additive movement constant.

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111Sep 23, 2026 Explained

One-sample matroid prophet inequalities against an almighty adversary

For every finite matroid known in advance, gives a distribution-independent online rule using one independent sample per element and earning a universal constant fraction of the expected offline optimum. Values are independent and nonnegative, with finite expected optimum. The guarantee holds even when the arrival-order adversary sees all samples, values, and the rule's entire random seed; no polynomial-time implementation is asserted.

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112Sep 23, 2026 Explained

Beyond the square-root exponent for depth-three circuits

Constructs a single language in deterministic polynomial time whose n-bit membership function requires $2^{\omega(\sqrt n)}$ total gates in unbounded-fan-in OR–AND–OR circuits, at every sufficiently large input length. This crosses the square-root-exponent threshold for explicit depth-three Boolean circuit lower bounds.

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113Sep 23, 2026 Explained

Approximate counting and entropy of perfect matchings

Gives a fully polynomial randomized approximation scheme for counting perfect matchings in arbitrary finite simple graphs, with exact detection of zero counts. Also proves the perfect-matching entropy conjecture of Anari, Oveis Gharan, and Vinzant, bounding the maximum entropy of a matching law at every feasible edge-marginal vector in a loopless labelled multigraph, including boundary points.

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114Sep 23, 2026 Explained

Approximate counting of common integer polymatroid bases

Gives a fully polynomial randomized approximation scheme for counting common integer bases of two integral polymatroids of equal total rank, supplied by exact rank-value oracles. Capacities are binary-encoded, each integer vector counts once, and oracle calls and bit operations outside the oracles are polynomial on every execution. For matroids presented by independence oracles, the results also cover common independent sets of prescribed, unrestricted, or maximum cardinality, even when the ranks differ.

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115Sep 24, 2026 Explained

Sampling and counting contingency tables with arbitrary margins

For nonnegative integer matrices with prescribed row and column sums, gives exact uniform sampling in expected polynomial bit time and almost-uniform sampling in worst-case polynomial bit time. The dimensions and binary-encoded margins are unrestricted. Also gives a fully polynomial randomized approximation scheme for counting such tables with arbitrary individual cell bounds, including structural zeros, with polynomial cost on every execution.

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116Sep 24, 2026 Explained

Uniform black-box noncommutative identity testing across characteristics

For each characteristic, constructs in deterministic polynomial bit time a polynomial-dimensional matrix tuple detecting every nonzero division-free noncommutative formula of bounded size over any field of that characteristic. Rational formulas over ℚ also admit polynomial-size hitting lists whenever they have a defined rational-matrix evaluation.

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117Sep 24, 2026 Explained

Uniform sparsest cut: hardness and semidefinite gaps

Proves that approximating Uniform Sparsest Cut within any fixed constant factor is NP-hard, even with nonnegative rational capacities and unit demands. The Goemans–Linial semidefinite relaxation also has integrality gaps of order at least $\sqrt{\log n}/(\log\log n)^3$ , approaching the square-root-logarithmic upper bound.

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118Sep 24, 2026 Explained

Bin packing and unbounded configuration-LP gaps

Disproves the modified integer round-up conjecture of Scheithauer and Terno: the integral bin-packing optimum can exceed its configuration linear-programming value by an arbitrarily large additive constant. Approximating the optimum within any fixed additive constant is also NP-hard, even when every item exceeds 1/6 and each bin holds at most five items.

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119Sep 24, 2026 Explained

The Courtade–Kumar and Hellinger conjectures

Proves the Courtade–Kumar conjecture: among Boolean functions of independent uniform bits, a single coordinate retains the most mutual information after independent bit-flip noise. A stronger theorem treats randomized binary summaries at fixed initial information. The Hellinger conjecture is also proved for every Boolean output bias and noise correlation.

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121Sep 24, 2026 Explained

Almost-linear approximation of edit distance

For every fixed rational $\varepsilon\in(0,1)$ , gives a randomized $(1+\varepsilon)$ approximation to unit-cost edit distance in worst-case expected time $N^{1+o(1)}$ , with success probability at least 2/3. The strings have total length N and polynomially bounded integer symbols. This is an asymptotic guarantee at fixed accuracy.

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122Sep 24, 2026 Explained

Quantitative trace-reconstruction bounds with a uniform decoder

At every fixed deletion probability in $(0,1)$ , reconstructing an arbitrary length-n binary string requires $n^{\Omega(\log\log n)}$ independent traces, ruling out polynomial-sample reconstruction. A uniform decoder achieves quasipolynomial sample and running-time bounds for known fixed rational retention probabilities. When the deletion probability is at most $n^{-\varepsilon}$ for fixed ε > 0, both bounds become polynomial in the input and parameter encoding.

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124Sep 24, 2026 Explained

Polynomial-time scheduling on three identical machines

Resolves the three-processor unit-job scheduling problem of Garey and Johnson: a deterministic polynomial-time algorithm minimizes makespan for nonpreemptive unit-length jobs with arbitrary precedence constraints on three identical parallel machines. For an explicitly given precedence graph, it decides deadline feasibility exactly and constructs a feasible schedule.

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125Sep 24, 2026 Explained

The metric k-median approximation threshold and recovery

Gives a deterministic polynomial-time $(1+2/e+\varepsilon)$ -approximation for finite rational metric k-median with specified candidate facilities, for every fixed ε > 0. Assuming $P\ne NP$ , the optimal infimum approximation factor is $1+2/e$ .

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126Oct 5, 2026 Explained

Exponential semidefinite complexity of perfect matching

Proves that every exact semidefinite lift of the perfect matching polytope has exponential size, answering Rothvoss's polynomial-size lift question negatively. The bound holds even for the positive semidefinite rank of its odd-cut slack matrix after any fixed shift $0\lt \rho\lt 1$ , allowing arbitrary real positive semidefinite factors.

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127Sep 25, 2026 Explained

Average sensitivity of polynomial threshold functions

Proves that a degree-at-most-d polynomial threshold function on the uniform n-dimensional Boolean cube has average sensitivity at most $8d\sqrt n$ , uniformly for $1\le d\le n$ . Average sensitivity counts expected output changes under single-bit flips. This establishes the asymptotic Gotsman–Linial conjecture, allowing polynomial zeros with $\mathop{\mathrm{sign}}\nolimits (0)=1$ .

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128Sep 24, 2026 Explained

A factor-two approximation for shortest common superstring

Gives a deterministic polynomial-time algorithm constructing a common superstring of length at most twice the optimum for every finite family of explicitly represented strings. The running time is polynomial in the full encoded input length, including symbol labels.

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129Sep 25, 2026 Explained

Exponential state costs for two-way automata

Proves exponential lower bounds both for complementing two-way nondeterministic finite automata and for simulating one-way nondeterministic automata by two-way deterministic ones. The latter resolves the Sakoda–Sipser state-succinctness conjecture over growing finite alphabets; both results rule out polynomial state bounds independent of alphabet size.

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130Sep 25, 2026 Explained

Exact Fourier transforms below $n\log n$

Gives a deterministic length-n discrete Fourier transform algorithm using $O(n(\log n)^{1-\delta})$ operations for every n, with explicit $\delta=10^{-13}$ . The model uses exact complex arithmetic, unrestricted coefficients and a supplied root of unity, and counts scalar preparation and logarithmic-word indexing.

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131Sep 25, 2026 Explained

Rapid mixing of graph switches for every degree sequence

Resolves the simple-undirected Kannan–Tetali–Vempala conjecture: the lazy edge-switch chain mixes in $O(n^8)$ time for every graphical labeled degree sequence. The same degree-constrained graphs can also be sampled exactly uniformly by an almost-surely terminating algorithm with expected polynomial bit running time.

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132Sep 25, 2026 Explained

A superquadratic separation of sensitivity and block sensitivity

Constructs total Boolean functions with block sensitivity $\mathop{\mathrm{bs}}\nolimits (f)\ge s(f)^\alpha$ for a fixed α > 2, disproving the quadratic strengthening of the Sensitivity Conjecture. Here $s(f)$ counts influential individual-bit flips, while block sensitivity allows disjoint groups of bits to change together.

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133Sep 25, 2026 Explained

The computational complexity of Weisfeiler–Leman refinement

Proves unconditional $n^{\Omega(k)}$ deterministic time lower bounds for joint and separate k-dimensional Weisfeiler–Leman equivalence, for sufficiently large fixed k in the specified sequential adjacency-matrix models. With dimension as input, joint equivalence is EXPTIME-complete even on subcubic graphs; deciding whether refinement identifies a graph is also EXPTIME-complete.

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134Sep 25, 2026 Explained

Generalized star height at most three

Every regular language over a finite alphabet has a generalized regular expression with at most three nested Kleene stars, allowing union, concatenation and complement over the same alphabet. This establishes an absolute bound independent of automaton size, resolving the uniform-boundedness version of the generalized star-height problem.

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135Sep 25, 2026 Explained

Homogeneous depth-five lower bounds for iterated matrix multiplication

Over every characteristic-zero field, the $(1,1)$ entry of a product of n independent $n\times n$ variable matrices requires $n^{\Theta(\sqrt n)}$ gates in homogeneous depth-five sum–product circuits. This sharp bound allows shared gates and bottom linear forms involving all variables.

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137Sep 25, 2026 Explained

One-tape time simulation in two-fifths-power space

Determines the halting and finite-control outcome of a fixed deterministic one-writable-tape machine up to time T using $O(T^{2/5}\log^C(T+2))$ space, improving the square-root exponent. Heads move at most one cell per step; finitely many read-only input heads are allowed. Initial contents are independent of T, and contents and input symbols have polylogarithmic-space access. Simulation time is unrestricted.

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139Sep 26, 2026 Explained

Subpolynomial query complexity for log-concave sampling

For C² potentials with a supplied minimizer and $I\preceq\nabla^2V\preceq2I$ , proves that sampling within total variation 1/10 requires only $C_\varepsilon d^\varepsilon$ exact value-and-gradient queries for every fixed ε > 0. The bound holds on every run, with unrestricted computation between queries. A logarithmic lower bound also holds, so the optimal power-law exponent in this oracle model is zero.

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140Sep 27, 2026 Explained

Memory–sample lower bounds for noiseless Gaussian regression

For fixed A > 0, a one-pass learner with $Ad^2$ persistent bits needs $\Omega_A(d\log(1/\epsilon))$ noiseless Gaussian samples to recover a unit vector to angular error $0\lt \epsilon\le1/10$ with probability 2/3, uniformly in accuracy for large d. Computation and randomized updates are unrestricted, but output uses only the terminal state, stopping index and fresh randomness.

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Dynamical systems and ergodic theory9

143Sep 24, 2026 Explained

Hilbert's sixteenth problem: uniform bounds for limit cycles

Resolves the uniform boundedness assertion in Hilbert's sixteenth problem: the number of isolated periodic orbits of a real planar polynomial vector field is bounded by a finite constant depending only on its degree. For classical quintic Liénard systems, the exact maximum is two limit cycles.

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144Sep 23, 2026 Explained

Banach’s simple Lebesgue-spectrum problem

Resolves the probability-preserving form of Banach's simple Lebesgue-spectrum problem within smooth dynamics. A smooth volume-preserving diffeomorphism of the standard-volume three-torus has simple Lebesgue spectrum on its entire complex mean-zero L² space: the bilateral iterates of one real observable form an orthonormal basis of that space.

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145Sep 23, 2026 Explained

Rokhlin’s multiple-mixing problem

Proves that every invertible mixing probability-preserving transformation is mixing of all finite orders, resolving Rokhlin's multiple-mixing problem for a single transformation. Correlations among any finite collection of measurable sets converge to the product of their measures whenever all pairwise time separations diverge.

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146Sep 23, 2026 Explained

Positive metric entropy for the standard map

Proves that the standard sine map on the two-dimensional torus has positive metric entropy with respect to area for every sufficiently large positive parameter. This establishes Sinai's positive-parameter-measure conjecture for the original family, with the stronger conclusion of a full parameter tail.

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148Sep 24, 2026 Explained

The entropy-rate dimension formula for self-similar measures

For every self-similar measure on the line generated by finitely many contracting similarities, proves $\dim_{\mathrm H}\mu=\min\{1,h_{\mathrm{RW}}/\chi\}$ , where $h_{\mathrm{RW}}$ is the entropy rate of random composed maps and χ the average logarithmic contraction. This resolves the entropy-rate dimension conjecture without a separation assumption, allowing exact overlaps and unequal contraction ratios.

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149Sep 25, 2026 Explained

Classwise permanence for weakly reversible mass-action systems

Proves the permanence conjecture for every finite weakly reversible mass-action system with fixed positive rate constants. Every positive stoichiometric compatibility class, even an unbounded one, has a common compact convex forward-invariant absorbing set. All positive trajectories in that class therefore eventually share positive lower and finite upper concentration bounds.

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150Sep 25, 2026 Explained

Weak mixing of triangular billiards with an irrational angle

Proves that the billiard flow in every nondegenerate Euclidean triangle with at least one angle irrational relative to π is weakly mixing for normalized area times uniform direction. This strengthens ergodicity on the entire irrational-angle class, with no genericity or Diophantine restrictions.

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151Sep 25, 2026 Explained

A C¹ counterexample to the entropy conjecture

Constructs a noninvertible C¹ self-map of a compact smooth manifold with zero topological entropy but eigenvalue 2 on second homology. This disproves the homological entropy lower bound for general C¹ self-maps: homological growth need not force positive orbit complexity.

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152Sep 25, 2026 Explained

Zero entropy does not guarantee a smooth positive-volume model

Constructs a zero-entropy ergodic invertible transformation of a standard nonatomic probability space that is not measurably conjugate to any C^∞ diffeomorphism preserving a strictly positive smooth probability density on a compact finite-dimensional manifold. One example rules out every finite dimension.

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Combinatorics33

155Sep 23, 2026 Explained

A counterexample to periodic tiling in dimension three

Constructs a finite translational tile in ℤ³ that tiles space but admits no fully periodic tiling, disproving the periodic tiling conjecture in the smallest possible lattice dimension. Its unit-cube thickening gives the same counterexample in ℝ³, even with arbitrary real translation vectors.

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156Sep 23, 2026 Explained

Borsuk's conjecture fails in dimension nine

Constructs a compact subset of ℝ⁹ that cannot be covered by ten sets of strictly smaller diameter, disproving Borsuk's covering assertion already in dimension nine. The example consists of rank-one orthogonal projectors onto lines in ℝ⁴, with the Frobenius metric.

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157Sep 23, 2026 Explained

Graph coloring, clique minors, and Colin de Verdière invariants

Disproves Hadwiger's conjecture even for fractional coloring: arbitrarily large finite simple graphs with independence number at most two satisfy $\chi_f(G)\gt h(G)$ , where $h(G)$ is the largest clique-minor order. Also disproves the fractional Colin de Verdière chromatic bound $\chi_f(G)\le\mu(G)+1$ . In the positive direction, every finite nonempty graph satisfies $\chi_{\mathrm{list}}(G)\le C h(G)$ for a universal constant C.

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158Sep 23, 2026 Explained

The Euclidean plane cannot be colored with five colors

Proves that every five-coloring of the Euclidean plane has a monochromatic pair at distance one, with no restriction on the color classes. This advances the Hadwiger–Nelson problem: together with the classical seven-coloring, only six and seven remain possible chromatic numbers of the plane.

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159Sep 23, 2026 Explained

Erdős’s reciprocal-sum conjecture and quasipolynomial Szemerédi bounds

Proves Erdős's conjecture that every set of positive integers with divergent reciprocal sum contains arithmetic progressions of every finite length. Quantitatively, for each fixed k ≥ 3, every subset of $\{1,\ldots,N\}$ with no nonconstant k-term progression has size at most $C_kN\exp[-c_k(\log N)^{\varepsilon_k}]$ , with positive constants depending only on k.

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160Sep 23, 2026 Explained

Superexponential van der Waerden numbers

Resolves Erdős's superexponential-growth question for van der Waerden numbers. If $W_r(k)$ is the least interval length forcing a monochromatic k-term progression in every r-coloring, then $W_r(k)\gt k^{ck\lfloor\log_2 r\rfloor}$ for an absolute c > 0, all r ≥ 2 and sufficiently large k, uniformly in r. In particular, $W_r(k)^{1/k}\to\infty$ for each fixed r.

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161Sep 23, 2026 Explained

Counterexamples to Sidorenko’s conjecture and the forcing conjecture

Disproves Sidorenko's conjecture with a connected bipartite pattern on 35 vertices and 66 edges that occurs less frequently than in a random graph of the same edge density. The same pattern disproves the forcing conjecture of Skokan and Thoma: matching its density and the edge density of a constant graphon need not force quasirandomness.

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162Sep 23, 2026 Explained

Counterexamples to Ryser’s covering conjecture

Disproves Ryser's covering conjecture by constructing intersecting $(q+1)$ -partite, $(q+1)$ -uniform hypergraphs with covering number $q+1$ , rather than the predicted bound q, for every sufficiently large prime q. A separate construction over extension fields also disproves Gyárfás's monochromatic tree-cover conjecture.

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165Sep 23, 2026 Explained

The Harary–Hill and Zarankiewicz crossing-number formulas

Resolves the Harary–Hill conjecture and Turán's brickyard problem in the Zarankiewicz formulation, determining the crossing numbers of every complete and complete bipartite graph. The result proves the optimality of the classical drawings among all plane drawings with continuous edge arcs.

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167Sep 23, 2026 Explained

Planar distinct distances and unit-distance bounds

Proves the weak pinned Erdős distance conjecture: for every fixed ε > 0, all but $o(n)$ points of any n-point planar set determine at least $n^{1-\varepsilon}$ distinct nonzero distances. A complementary theorem bounds the number of unit-distance pairs by $O(n^{4/3-\delta})$ for an absolute δ > 0.

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168Sep 24, 2026 Explained

Combinatorial invariance of Kazhdan–Lusztig polynomials

Resolves the full combinatorial invariance conjecture: isomorphic Bruhat intervals in arbitrary Coxeter systems have identical equal-parameter Kazhdan–Lusztig polynomials. Thus the abstract order of the interval determines the polynomial, even across different Coxeter systems.

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169Sep 24, 2026 Explained

Shareshian–Wachs elementary positivity

Resolves the elementary-positivity part of the Shareshian–Wachs conjecture: the chromatic quasisymmetric function of every natural unit interval graph has elementary-basis coefficients in $\mathbb N[q]$ . The coefficients count explicitly described permutations, giving a combinatorial explanation of positivity.

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170Sep 24, 2026 Explained

Sharp logarithmic exponents for off-diagonal Ramsey numbers

For every fixed integer s ≥ 5, proves $r(s,t)=t^{s-1}/(\log t)^{s-2+o(1)}$ as $t\to\infty$ , determining the logarithmic exponent and matching the classical upper bound at that scale. Here $r(s,t)$ is the least number of vertices forcing an s-clique or a t-vertex independent set.

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172Sep 23, 2026 Explained

Classification of finite Euclidean Ramsey configurations

Classifies finite point configurations that occur monochromatically, at their original scale, in every finite coloring of sufficiently high-dimensional Euclidean space. The characterization is an algebraic condition over the coordinate field. It also disproves the Leader–Russell–Walters conjecture that every such configuration is a subset of a finite transitive set.

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173Sep 23, 2026 Explained

Seymour’s second-neighborhood conjecture

Proves Seymour's second-neighborhood conjecture: every nonempty finite oriented graph has a vertex with at least as many vertices at directed distance exactly two as at directed distance one. Oriented graphs may be arbitrary apart from the exclusion of loops and oppositely directed edge pairs.

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174Sep 23, 2026 Explained

Deterministic construction of strong thin spanning trees

Resolves the strong thin-tree conjecture constructively. Every finite loopless k-edge-connected multigraph on at least two vertices has a spanning tree containing at most a universal $C/k$ fraction of the edges of every cut. Such a tree can be found deterministically in polynomial time, even with binary-encoded parallel-edge multiplicities.

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175Sep 23, 2026 Explained

Talagrand’s expectation thresholds, discrete convexity, and graph decompositions

Proves that integral and fractional expectation thresholds differ by at most a universal factor, and resolves Talagrand's discrete-convexity conjecture. An application proves the Ascoli–He–Park–Talagrand graph-decomposition conjecture: every graph's edges split into a universally bounded number of fixed pieces, each with containment threshold at most a universal constant times the original graph's integral expectation threshold. The pieces' embeddings need not agree on shared vertices.

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176Sep 24, 2026 Explained

The second Kahn–Kalai conjecture with an edge-count bound

Proves the second Kahn–Kalai conjecture: for every finite simple graph H with h ≥ 1 edges and at most n vertices, its appearance threshold in $G(n,p)$ is at most $C p_{\mathrm E}(n,H)(1+\log_2 h)$ , with universal C. Here $p_{\mathrm E}$ is the least density at which every subgraph of H has expected copy count at least 1/2.

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177Sep 24, 2026 Explained

Bounded-degree coboundary expanders

Constructs arbitrarily large finite d-dimensional simplicial complexes, for every d ≥ 3, with uniformly bounded vertex degrees and uniform 𝔽₂ coboundary expansion in every degree below d. Together with the known graph and two-dimensional cases, this establishes the existence of such expanders in every positive dimension.

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179Sep 23, 2026 Explained

The circulant Hadamard and Barker-sequence conjectures

Proves that real circulant Hadamard matrices exist exactly in orders 1 and 4, resolving the circulant Hadamard conjecture. Together with classical Barker-sequence results, this shows that binary sequences whose nontrivial aperiodic autocorrelations have magnitude at most 1 exist at lengths n > 1 exactly when $n\in\{2,3,4,5,7,11,13\}$ .

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180Sep 24, 2026 Explained

Barnette’s Hamiltonian-cycle conjecture

Proves that every finite simple cubic bipartite planar 3-vertex-connected graph has a Hamiltonian cycle, resolving Barnette's conjecture. Equivalently, every three-edge path in a finite simple cubic 3-vertex-connected bipartite Pfaffian graph lies in a Hamiltonian cycle.

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181Sep 24, 2026 Explained

The Erdős–Gallai cycle-decomposition conjecture

Proves that the edges of every finite simple undirected graph on n vertices can be partitioned into at most $Cn$ simple cycles and single edges, for an absolute constant C. This resolves the Erdős–Gallai cycle-decomposition conjecture, bounding the number of pieces linearly even for dense graphs.

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182Sep 24, 2026 Explained

Power savings for intersective polynomial differences and prime arguments

For every fixed intersective integer polynomial h of degree k ≥ 2 with positive leading coefficient, proves that a subset of $\{1,\ldots,N\}$ avoiding nonzero values $h(1),h(2),\ldots$ as differences has size $O_h(N^{1-c_k})$ , with $c_k\gt 0$ depending only on degree. Here intersective means having a root modulo every modulus. For prime arguments, a power saving also holds when h has a unit root modulo every modulus, with exponent allowed to depend on h.

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183Sep 25, 2026 Explained

Power savings for planar halving lines and k-sets

Improves the planar halving-line bound to $O(n^{4/3-\varepsilon})$ for sets with no three collinear and an absolute ε > 0. More generally, an n-point set with no three collinear has $O(n(k+1)^{1/3-\varepsilon_0})$ strictly separable k-subsets for $1\le k\le n/2$ , with an absolute $\varepsilon_0\gt 0$ . The constants and positive exponents are nonquantitative.

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184Sep 25, 2026 Explained

Correspondence coloring with a fixed forbidden subgraph

Proves the Alon–Krivelevich–Sudakov coloring conjecture in correspondence-coloring form: graphs avoiding any fixed subgraph F need $O_F(\Delta/\log\Delta)$ colors when their maximum degree Δ is sufficiently large. Also proves the Ajtai–Erdős–Komlós–Szemerédi independence conjecture: for fixed r ≥ 4, every n-vertex K_r-free graph of average degree d ≥ 2 has an independent set of size $\Omega_r(n\log d/d)$ .

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185Sep 24, 2026 Explained

Counterexamples to infinite matroid intersection and packing/covering

Disproves the unrestricted infinite matroid intersection and packing/covering conjectures in ZFC, using two self-dual partitional matroids on a countably infinite ground set. The same examples answer Joó’s partitional-matroid question negatively. They are neither finitary nor cofinitary, so Nash-Williams’ original finitary conjecture remains outside the result.

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186Sep 25, 2026 Explained

Uniform influence and sharp thresholds for graph and hypergraph properties

Proves the Friedgut–Kalai threshold-width conjectures for graphs and fixed-uniformity hypergraphs. For fixed $0\lt \varepsilon\lt 1/2$ , every nontrivial increasing relabeling-invariant property crosses from probability ε to $1-\varepsilon$ within width $O((\log n)^{-2})$ for graphs and $O_r((\log n)^{-r/(r-1)})$ for r-uniform hypergraphs, r ≥ 3. The hypergraph influence bound also applies to nonmonotone properties.

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187Sep 25, 2026 Explained

Snaky in 21 Maker moves

Settles the Snaky achievement problem: Maker can force the six-cell Snaky shape within 21 of its own moves on the initially empty infinite square board. Maker moves first, each player claims one free cell per turn, and translations, rotations and reflections count as wins.

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188Sep 25, 2026 Explained

The sharp terminal leave in random triangle removal

Starting from the complete graph on n vertices, repeatedly delete a uniformly chosen remaining triangle. The terminal edge count is asymptotic to $n^{3/2}/(2\sqrt2)$ , with mean-square convergence after normalization by n^3/2. This proves the triangle case of the Joos–Kühn sharp-constant conjecture.

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189Sep 25, 2026 Explained

Cycle–clique Ramsey numbers

Proves the Erdős–Faudree–Rousseau–Schelp conjecture: $R(C_m,K_n)=(m-1)(n-1)+1$ for every $m\ge n\ge3$ , except $R(C_3,K_3)=6$ . This is the exact threshold forcing a red m-cycle or a blue n-clique in every red–blue coloring of a complete graph.

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190Sep 25, 2026 Explained

Polynomial removal fails for ordered binary matrices

Disproves polynomial ordered binary matrix removal with one fixed $66\times66$ zero–one pattern. Matrices can require many binary-entry changes to become pattern-free while their copy density is smaller than every proposed polynomial bound in that distance. Copies preserve row and column orders and match both zeros and ones.

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191Sep 25, 2026 Explained

A power improvement in the Heilbronn triangle lower bound

For every sufficiently large n, constructs n points in the unit square such that every triangle has area at least $n^{-2+c}$ for one absolute c > 0. This disproves the conjectured almost-n⁻² upper bound in Heilbronn's triangle problem, which asks how large the smallest determined triangle can be.

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192Sep 26, 2026 Explained

Boolean functions violate the square-root degree bound by arbitrary factors

Disproves the proposed square-root bound relating a Boolean function's linear Fourier coefficients to its polynomial degree. For every C > 0, there is a sign-valued Boolean function f with $\sum_i\widehat f(\{i\})\gt C\sqrt{\deg(f)}$ . Thus its total signed correlation with individual input bits can exceed the proposed bound by an arbitrary factor.

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Algebra10

194Sep 23, 2026 Explained

Lech’s multiplicity conjecture

Proves $e(R)\le e(S)$ for every flat local homomorphism of nonzero Noetherian local rings, where e is Hilbert–Samuel multiplicity. This resolves Lech's conjecture in every dimension and characteristic.

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195Sep 23, 2026 Explained

A counterexample to the small Cohen–Macaulay module conjecture

Constructs a three-dimensional complete Noetherian normal local domain over ℂ with no nonzero finitely generated maximal Cohen–Macaulay module. A three-dimensional local domain essentially of finite type over ℂ has the same property, disproving the domain form of the small Cohen–Macaulay module conjecture.

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196Sep 23, 2026 Explained

A counterexample to Kaplansky’s zero-divisor conjecture

Constructs a finitely presented torsion-free group G whose group algebra $\mathbb F_2[G]$ has nonzero zero divisors, disproving Kaplansky's zero-divisor conjecture. The group has a finite two-dimensional classifying space.

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197Sep 23, 2026 Explained

A torsion-free group algebra that is not directly finite

Constructs a finitely presented torsion-free nonsofic group whose group algebra over 𝔽₂ is not directly finite, disproving Kaplansky's conjecture even without torsion. Companion examples give injective nonsurjective cellular automata on all configurations, refuting Gottschalk's surjunctivity conjecture. Another counterexample is an integral group-ring matrix, invertible over the rational group ring, with Fuglede–Kadison determinant strictly between zero and one, disproving the unrestricted Determinant Conjecture.

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198Sep 23, 2026 Explained

A counterexample to finitistic-dimension finiteness

Constructs a finite-dimensional complex algebra whose finite-dimensional modules have unbounded finite projective dimensions. This disproves the little finitistic-dimension conjecture.

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199Sep 23, 2026 Explained

Counterexamples to Auslander–Reiten, Tachikawa and related homological conjectures

Constructs finite-dimensional algebras over a characteristic-two rational-function field that disprove the Auslander–Reiten and Gorenstein-projective conjectures, and Tachikawa's second conjecture. An associated endomorphism algebra also disproves the classical, generalized and strong Nakayama conjectures, the Auslander–Gorenstein conjecture, and the Wakamatsu tilting conjecture. The counterexamples persist under every extension of the base field.

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205Sep 24, 2026 Explained

Saxl’s conjecture and universal tensor squares

Proves Saxl's conjecture: the tensor square of every staircase representation contains every irreducible complex representation of the corresponding symmetric group. More generally, every S_n with $n\notin\{2,4,9\}$ has an irreducible representation whose tensor square contains all irreducibles.

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206Sep 24, 2026 Explained

Finite lattice representation and undecidability

Some finite lattices are not congruence lattices of any finite algebra, answering the finite lattice representation problem negatively. Moreover, no algorithm decides whether a finite lattice has such a representation, or whether it is a full subgroup interval of a finite group.

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207Sep 24, 2026 Explained

The ℓ¹-Bass conjecture for all discrete groups

Proves the ℓ¹-Bass conjecture for every discrete group: Hattori–Stallings traces of idempotent matrices over $\ell^1(G)$ are supported on finitely many finite-order conjugacy classes. The algebraic companion proves the integral Bass trace conjecture and Kaplansky's idempotent conjecture for torsion-free groups over every commutative unital characteristic-zero domain.

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210Sep 25, 2026 Explained

Foulkes' conjecture for sixth powers and quadratic stabilization

Proves the sixth case of Foulkes’ conjecture: $\mathop{\mathrm{Sym}}\nolimits ^6(\mathop{\mathrm{Sym}}\nolimits ^bV)$ embeds equivariantly in $\mathop{\mathrm{Sym}}\nolimits ^b(\mathop{\mathrm{Sym}}\nolimits ^6V)$ for every b ≥ 6 and finite-dimensional complex V. More generally, the canonical multiplication map $\mathop{\mathrm{Sym}}\nolimits ^b(\mathop{\mathrm{Sym}}\nolimits ^aV)\to\mathop{\mathrm{Sym}}\nolimits ^a(\mathop{\mathrm{Sym}}\nolimits ^bV)$ is surjective for a ≥ 2 and $b\ge a(a-1)$ , giving dimension-independent quadratic stabilization.

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Probability and statistical mechanics19

211Sep 24, 2026 Explained

The geometric phase diagram, diffusion, and spectra of random planar maps

Critical Fortuin–Kasteleyn planar maps converge to Liouville quantum gravity spheres for $0\lt q\le4$ and to the Brownian continuum random tree for q > 4, establishing the surface-to-tree geometric transition. For FK–Ising and spanning-tree-weighted maps, stationary random walks converge to Liouville Brownian motion on the limiting sphere. The FK–Ising spectral result also gives convergence of eigenvalues and heat traces, using the stated Brownian/LQG inputs.

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212Sep 24, 2026 Explained

Planar first-passage geometry and the absence of bigeodesics

Proves that planar first-passage percolation has no doubly infinite geodesic for iid nonnegative nonatomic edge weights when the minimum of four weights has finite second moment. For exponential weights, the limit shape is strictly convex with C¹ boundary. Differentiability also holds for every Gamma law with positive shape and rate.

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213Sep 24, 2026 Explained

Critical percolation on every quasi-transitive graph

Resolves the Benjamini–Schramm criticality conjecture for bond percolation on every infinite connected locally finite quasi-transitive graph with $p_c\lt 1$ : at the critical probability, there is almost surely no infinite cluster. The family also establishes this conclusion for both nearest-neighbor bond and site percolation on ℤ³.

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214Sep 24, 2026 Explained

The Benjamini–Schramm nonuniqueness conjecture

Proves $p_c\lt p_u$ for Bernoulli bond percolation on every infinite connected locally finite nonamenable quasi-transitive graph, resolving the Benjamini–Schramm nonuniqueness conjecture. Thus there is a nonempty range of probabilities with infinitely many infinite clusters. A stronger operator bound also establishes the critical triangle condition.

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215Sep 23, 2026 Explained

Canonical $O(3)$ continuum limit and exact $O(4)$ mass asymptotics

Constructs the canonical continuum limit of the two-dimensional nearest-neighbor $O(3)$ model: a non-Gaussian local relativistic theory with a unique vacuum and a positive mass gap. For the square-lattice $O(4)$ model, determines the exact leading asymptotic of the full transfer gap, $m_{\mathrm{lat}}(\beta)\sim32e^{\pi/4-1/2}\sqrt\beta\,e^{-\pi\beta}$ . The family also proves exponential spin-correlation decay for two-dimensional nearest-neighbor $O(n)$ models with n ≥ 3 at every positive temperature.

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218Sep 23, 2026 Explained

Conformal universality for weakly interacting and random-bond Ising models

Weak finite-range square-symmetric even multispin perturbations of the square-lattice Ising model preserve critical bulk spin and energy limits; weak square-symmetric contour interactions also yield chordal SLE₃ interface limits. With sufficiently weak iid bond disorder of any fixed bounded nondegenerate mean-zero law, critical spin interfaces converge to the same law in probability over environments.

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220Sep 23, 2026 Explained

Directional zero–one laws beyond iid environments and iid ballisticity

On ℤ^d, d ≥ 3, directional escape has probability zero or one for iid strictly elliptic nearest-neighbor environments, and for stationary ergodic finite-range-dependent environments under uniform ellipticity. In iid uniformly elliptic environments with d ≥ 2, almost-sure directional transience implies a deterministic limiting velocity with positive projection in that direction, resolving the ballisticity conjecture.

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221Sep 23, 2026 Explained

The Mézard–Parisi formula for diluted spin glasses

Proves the Mézard–Parisi hierarchical cavity formula for Poisson-diluted even-arity Ising models satisfying the Panchenko–Talagrand factorization and positivity assumptions, with only first-moment integrability. The limiting free energy equals the infimum over finite-depth hierarchical trial laws. This includes the Viana–Bray model, symmetric diluted even-spin models, and weighted soft even-K satisfiability.

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222Sep 24, 2026 Explained

Perceptron free energies and microscopic jamming exponents

Determines finite-temperature variational free energies for Gaussian Ising perceptrons with bounded Borel log-potentials and Gaussian spherical perceptrons with bounded continuous potentials, at every positive pattern density. A spherical extension treats bi-orthogonally invariant disorder with compact limiting singular-value distributions and no outliers. At margin −1, the quadratic-penalty spherical model has a sharp feasibility threshold and limiting gap and force laws, with system size, zero temperature, and critical density taken in that order.

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227Sep 24, 2026 Explained

Critical SK autocorrelation processes and dynamics across the temperature transition

For zero-field Gaussian SK heat-bath dynamics with rate-one updates per spin, proves worst-start cutoff on the $\log n$ scale for fixed $0\le\beta\lt 1$ , mixing time $n^{2/3+o(1)}$ at β = 1, and stretched-exponential mixing from a Gibbs-sampled fixed starting configuration for β > 1, in probability over disorder. At criticality, rescaled stationary and quench autocorrelation processes have universal random limits for Gaussian and Rademacher disorder; the quench limit relaxes to the stationary limit.

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228Sep 24, 2026 Explained

Continuum phase transitions for radial pair potentials

Constructs stable distance-dependent pair interactions for three-dimensional classical particles with a first-order phase transition: the canonical free energy has a derivative jump at one inverse temperature throughout an open density interval. One potential has a divergent repulsive core; another is bounded and continuous with an integrable power-law tail, realizing the type of transition sought in Simon's continuum problem.

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229Sep 25, 2026 Explained

Exact three- and four-state reconstruction thresholds and four-state tree capacity

Proves the exact reconstruction threshold $d\lambda^2\gt 1$ , with nonreconstruction at equality, for three-state symmetric and four-state ferromagnetic broadcasting on regular trees (d ≥ 2) and observed Poisson trees (mean d > 1 and d > 0, respectively), with Poisson advantage averaged without conditioning on survival. The three-state theorem allows both signs of λ and gives the exact weak-recovery threshold for the symmetric three-community stochastic block model.

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230Sep 25, 2026 Explained

Exact Hausdorff gauges for SLE

Resolves Schramm’s Hausdorff-measure question for chordal SLE_κ, $0\lt \kappa\lt 8$ . The explicit gauge $r^d(\log\log(1/r))^{(2-d)/2}$ , $d=1+\kappa/8$ , gives almost surely positive finite measure to every trace segment $\gamma([s,t])$ with $0\lt s\lt t\lt \infty$ , and finite expected measure to the trace in every bounded disk.

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231Sep 25, 2026 Explained

The free uniform spanning forest is a factor of IID

On every infinite connected locally finite simple unweighted graph, the free uniform spanning forest is a factor of independent vertex labels, by one isomorphism-equivariant rule using no root. Translation-invariant strongly Rayleigh binary processes on every countable group, including invariant determinantal processes with Hermitian positive-contraction kernels, are also factors of IID.

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234Sep 25, 2026 Explained

All-temperature pressure of orthogonally invariant Ising spin glasses

Gives an exact variational formula for the limiting pressure of orthogonally invariant Ising spin glasses at every fixed temperature, both almost surely and in expectation. The coupling matrix is a Haar-random rotation of a deterministic spectrum converging to a compactly supported law, with extreme eigenvalues converging to its support edges. The zero-field ground-state energy follows as temperature tends to zero.

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235Sep 25, 2026 Explained

Limiting random SAT thresholds, sharp variance and computability

For random k-SAT with independent uniformly signed proper clauses sampled with replacement, proves finite positive limiting thresholds and hitting-time variance $\Theta_k(n)$ for every fixed k ≥ 3, and computability of the 3-SAT threshold. We credit Gaia Carenini with priority for resolving the threshold-existence conjecture in her concurrent [ECCC TR26-229](https://eccc.weizmann.ac.il/report/2026/229/), made public October 5, 2026; this family supplies another proof and the sharper variance and computability results.

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236Sep 26, 2026 Explained

The exact factor-of-IID threshold for free Ising spins on trees

Determines when the free zero-field ferromagnetic Ising state on the infinite d-regular tree is a factor of independent vertex labels: exactly when $\tanh\beta\le(d-1)^{-1/2}$ , including equality, for d ≥ 3 and β ≥ 0. The construction uses no root and is almost surely equivariant for each fixed tree automorphism, resolving the ferromagnetic case of Lyons's question.

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237Sep 26, 2026 Explained

The three-quarter exponent for honeycomb self-avoiding walk

Proves the diameter form of Nienhuis's predicted three-quarter exponent: a uniformly chosen n-step self-avoiding walk on the honeycomb lattice has diameter $n^{3/4+o(1)}$ . Its local mass and covering numbers have exponent 4/3. These estimates hold at every sufficiently large fixed length, simultaneously across scales, with arbitrarily high polynomial probability.

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238Sep 26, 2026 Explained

Optimal logarithmic mixing of the Thorp shuffle

Proves that the Thorp shuffle randomizes $N=2^d$ labeled cards in $\Theta(\log N)$ physical shuffles, settling its optimal mixing order for power-of-two deck sizes. Convergence is in total variation from the worst initial ordering and concerns the entire permutation, not just individual card positions.

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Mathematical logic6

240Sep 24, 2026 Explained

Shelah's eventual categoricity and the prescribed-threshold obstruction

Proves Shelah's eventual categoricity conjecture in ZFC: for each bound on the Löwenheim–Skolem number, a uniform threshold makes categoricity of an abstract elementary class in one cardinal above that threshold imply categoricity throughout the same tail. Categoricity means uniqueness up to isomorphism at a given cardinality. Under the continuum hypothesis, a proposed specific Hanf threshold need not suffice.

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241Sep 24, 2026 Explained

Rigidity of the Turing degrees

Every order automorphism of the Turing degrees is the identity, resolving their rigidity problem. Thus no nontrivial relabeling of degrees preserves the ordering by relative computability.

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242Sep 24, 2026 Explained

Single-fold Diophantine representations and undecidability under an at-most-one-solution promise

Every recursively enumerable set of tuples of natural numbers has a Diophantine representation with exactly one auxiliary solution for each member and none for nonmembers. This proves the single-fold conjecture and hence the finite-fold conjecture. Diophantine solvability over the nonnegative integers remains undecidable even with an at-most-one-solution promise.

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243Sep 23, 2026 Explained

Separating choiceless counting from polynomial time and witnessed choice

Confirms the Blass–Gurevich–Shelah noncapture conjecture: consistency of a linear system over 𝔽₃ defines a polynomial-time query on unordered finite structures that choiceless polynomial time with counting cannot express. A separate result shows that adding witnessed symmetric choice strictly increases expressive power. Both separations hold for the full counting formalism, allowing hereditarily finite sets of arbitrary finite rank.

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244Sep 24, 2026 Explained

The Partition Principle does not imply Choice

Assuming ZF is consistent, constructs a model in which every surjective image of a set injects into that set, yet the axiom of choice fails. Choice for ordinal-indexed families still holds. From any countable transitive model of ZFC, a separate construction gives a transitive symmetric extension with these properties and no new countable sequences of ground-model elements.

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245Sep 25, 2026 Explained

Weak normalization implies strong normalization in pure type systems

Proves that weak normalization implies strong normalization for every pure type system: if every legal expression in every valid context has a β-normal form, every β-reduction sequence terminates. This resolves the β-Barendregt–Geuvers–Klop conjecture, including nonfunctional rules and open contexts.

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Group theory12

246Sep 23, 2026 Explained

Cannon's conjecture

Every word-hyperbolic group with boundary homeomorphic to S² admits a proper cocompact isometric action on hyperbolic three-space with finite kernel, proving Cannon's conjecture. Every torsion-free such group is therefore the fundamental group of a closed hyperbolic three-manifold.

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247Sep 23, 2026 Explained

An infinite finitely presented residually finite 2-group and a finitely presented nil algebra

Constructs an infinite finitely presented residually finite group whose elements all have finite 2-power order, answering the finitely presented Burnside problem negatively even in this class. The construction also yields an infinite-dimensional finitely presented nil associative 𝔽₂-algebra and a finitely presented infinite-dimensional algebraic unitization, giving negative answers to the corresponding nilpotence and Kurosh finiteness questions.

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248Sep 23, 2026 Explained

Thompson's group F is nonamenable

Proves that Thompson's group F, the group of dyadic piecewise linear homeomorphisms of the interval, is nonamenable, resolving its longstanding amenability problem.

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249Sep 23, 2026 Explained

A finitely generated Eilenberg–Ganea counterexample

Constructs a finitely generated residually finite group with integral cohomological dimension two and geometric dimension three, disproving the Eilenberg–Ganea conjecture. It has no two-dimensional classifying space, even with infinitely many cells.

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250Sep 23, 2026 Explained

Boone–Higman embeddings with higher finiteness

A finitely generated group has decidable word problem exactly when it embeds in a finitely presented simple group, proving the Boone–Higman conjecture. The target can have type F_∞: a classifying space with finitely many cells in each dimension. A single group of type F_∞ can also contain every finitely presented group.

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251Sep 23, 2026 Explained

Amenability, unitarizability, and strong Ulam stability

Resolves Dixmier's problem for all discrete groups: amenability is equivalent to every uniformly bounded Hilbert-space representation being similar to a unitary representation. For countable discrete groups, amenability is also equivalent to strong Ulam stability: sufficiently accurate unitary approximate representations are uniformly close in operator norm to genuine representations on the same, possibly infinite-dimensional, Hilbert space.

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252Sep 23, 2026 Explained

A torsion-free hyperbolic group that is neither residually finite nor linear over any field

Constructs a torsion-free word-hyperbolic group that is not residually finite, answering the residual-finiteness question negatively. One fixed nonidentity element is killed by every finite-dimensional linear representation over every commutative field, so the group is not linear over any such field.

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253Sep 23, 2026 Explained

An infinite finitely presented simple amenable group

Constructs an infinite finitely presented simple amenable group, answering the longstanding question of whether these properties can occur simultaneously.

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254Sep 23, 2026 Explained

Classifying spaces and geometric obstructions for Artin groups

The Salvetti complex of every finite-rank Artin group is aspherical, proving the Artin $K(\pi,1)$ conjecture. Arbitrary intersections of its parabolic subgroups are parabolic, proving the Parabolic Intersection Conjecture. An explicit Artin group admits no proper cocompact isometric action on any nonempty proper CAT $(0)$ space.

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255Sep 24, 2026 Explained

Quasi-isometric recognition of virtually polycyclic groups

Proves that every finitely generated group quasi-isometric to a finitely generated virtually polycyclic group is virtually polycyclic, resolving the Eskin–Fisher–Whyte lattice-recognition conjecture. Equivalently, a group quasi-isometric to a lattice in a connected simply connected solvable Lie group is virtually a uniform lattice in some such Lie group, possibly a different one.

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256Sep 24, 2026 Explained

Nonsingular systems of equations over arbitrary groups

Proves that every finite system of equations over an arbitrary group whose exponent-sum matrix has full row rank over ℚ has a simultaneous solution in an overgroup, resolving Howie's conjecture. The coefficient group embeds in the presented quotient. A companion proves Kervaire's conjecture: adjoining one generator and one relation cannot trivialize a nontrivial group.

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257Sep 25, 2026 Explained

A hyperbolic group without a geometric CAT(0) action

Answers negatively whether every word-hyperbolic group is a CAT(0) group. Constructs one with a finite classifying space but no proper cocompact isometric action on any proper complete $\mathop{\mathrm{CAT}}\nolimits (0)$ space, in any dimension.

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Mathematical physics17

260Sep 27, 2026 Explained

Spacetime Penrose inequalities: enclosing area, charge, rotation, and anti-de Sitter extensions

Proves the sharp enclosing-area spacetime Penrose inequality for smooth one-ended asymptotically flat initial data in every spatial dimension n ≥ 3, under dominant energy, weak future trapping, positive enclosing area, and the stated decay assumptions. It bounds invariant ADM mass below using minimum enclosing area, with equality rigidity under additional horizon hypotheses. Charged upper-area bounds treat dyonic three-dimensional data; the higher-dimensional purely electric extension uses the matched neutral theorem.

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261Sep 23, 2026 Explained

Localization and delocalization in the Anderson model

Resolves the predicted spectral contrast for the lattice Anderson model with independent uniform site potentials. In dimension two, every positive disorder strength gives almost surely pure-point spectrum. In every fixed dimension d ≥ 3, sufficiently weak positive disorder gives purely absolutely continuous spectrum on a fixed open interval with nonzero spectral weight.

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262Sep 23, 2026 Explained

Sharp finite-matrix Lieb–Thirring inequalities and all equality cases

Proves the sharp one-dimensional Lieb–Thirring inequality for $1/2\lt \gamma\lt 3/2$ and arbitrary finite-matrix potentials W ≥ 0 with $\int\mathop{\mathrm{tr}}\nolimits (W^{\gamma+1/2})\lt \infty$ : the optimal constant is the scalar one-bound-state value, independent of matrix size. All equality cases are direct sums, in one constant unitary basis, of scalar sech² solitons with independent scales and centers, and zero channels.

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263Sep 24, 2026 Explained

The ionization and generalized ionization conjectures

For the full nonrelativistic Coulomb model with two electron spin states, proves that a molecule with M fixed nuclei of charges at least one and total charge Z strictly binds at most $Z+CM$ electrons. Neutral-atom first ionization energies and radii containing all but an expected half-electron have universal positive upper and lower bounds. The energy cost of removing m electrons has Thomas–Fermi asymptotics as $m\to\infty$ and $Z/m\to\infty$ ; neutral-atom outer radii have the corresponding iterated-limit asymptotics, taking $Z\to\infty$ first.

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266Sep 24, 2026 Explained

Exactly three mutually unbiased bases in dimension six

Proves $N(6)=3$ , resolving Zauner's dimension-six mutually unbiased bases conjecture: three such bases exist in ℂ⁶, but four cannot. The exclusion is a complete certified computation under the stated binary64 arithmetic and compiler conditions. An independent companion proves the Matolcsi–Ruzsa–Weiner Fourier-vanishing conjecture for order-six complex Hadamard matrices outside Tao's cubic equivalence class.

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267Sep 24, 2026 Explained

Positive-temperature Bose–Einstein condensation and exact quantum depletion

Proves Bose–Einstein condensation for the exact canonical Gibbs state of the three-dimensional hard-sphere gas: each fixed exclusion distance and sufficiently small fixed density admit a strictly positive temperature, independent of volume, with positive condensate fraction in the thermodynamic limit. At zero temperature, proves the Bogoliubov leading quantum-depletion law for hard spheres and fixed bounded nonnegative radial finite-range potentials of positive scattering length, taking the thermodynamic limit before the dilute limit.

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269Sep 24, 2026 Explained

Uniform Laughlin gap and stability under bounded scalar disorder

Proves the fermionic Laughlin spectral-gap conjecture for the full V₁ interaction at filling 1/3 on the round sphere. The unique ground state remains uniformly gapped under sufficiently weak bounded real scalar one-body potentials projected to the lowest Landau level. Both the gap and disorder threshold are uniform over all sufficiently large particle numbers and all normalized potential profiles.

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271Sep 24, 2026 Explained

Bloch's law, its lattice correction, and the spherical magnetization law

Proves Bloch's T^3/2 law with its exact coefficient for three-dimensional quantum Heisenberg ferromagnets at every positive quantum spin, allowing nonnegative symmetric finite-range couplings whose support generates ℤ³. The thermodynamic limit precedes the zero-field derivative and low-temperature limit. The family also proves spontaneous magnetization for nearest-neighbor models in every dimension d ≥ 3 and determines the first lattice correction for three-dimensional nearest-neighbor couplings.

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272Sep 27, 2026 Explained

Entanglement without distillable secret key

Constructs an entangled state on $\mathbb C^{10}\otimes\mathbb C^{10}$ with zero distillable secret key for the specified local-instrument protocols that complete almost surely. These allow joint local processing and authenticated two-way public communication, with no other shared private resource and an eavesdropper holding the input purification and public record. A trace-preserving PPT channel on $M_{21}(\mathbb C)$ whose square is not entanglement breaking disproves Christandl's PPT-square conjecture.

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273Sep 24, 2026 Explained

The entropy photon-number inequality

Proves the entropy photon-number inequality for beam-splitter mixing of two independent finite-energy bosonic inputs in any finite number of modes: the output's entropy photon number is at least the transmissivity-weighted average of the inputs'. Arbitrary entanglement within each input is allowed, and product thermal inputs attain equality even when their entropies differ.

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274Sep 24, 2026 Explained

Parity is not in QAC⁰

Resolves Moore's parity conjecture in the measured-output model: constant-depth quantum circuits with arbitrary one-qubit gates, unbounded-arity Toffoli gates and polynomially many total qubits cannot compute parity with any fixed positive worst-case advantage. Ancillas start in zero, one output qubit is measured, and all other registers may be discarded. Xu–Li's reductions give the same bounded-error obstruction for strict majority.

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275Sep 24, 2026 Explained

QMA-hardness of continuum Coulomb energy

Proves QMA-hardness of approximating the electronic Coulomb energy infimum in three dimensions, minimizing over the full spinful fermionic continuum space. Deterministic polynomial-time reductions work even with only unit-charge nuclei at distinct rational positions, polynomially many electrons and an energy-threshold separation of at least one.

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276Sep 24, 2026 Explained

Classical capacity of generalized amplitude damping

Determines the unassisted classical capacity of every qubit generalized amplitude-damping channel, including all damping and thermal parameters. An explicit one-variable optimization gives the capacity, attained by independent two-state signal ensembles with collective decoding. Holevo capacity, minimum output entropy and regularized classical capacity are additive when tensoring with any finite-dimensional quantum channel.

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277Sep 25, 2026 Explained

Threshold repetition for entangled games

Proves exponential threshold repetition for every finite two-player one-round game: if its entangled value is v < 1, the probability of winning at least a fraction $v+\delta$ of k independent repetitions decays exponentially in k, for $0\lt \delta\lt 1-v$ . Arbitrary joint finite-dimensional entangled strategies and correlated question distributions are allowed.

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279Sep 25, 2026 Explained

Exact quantum factoring over a fixed finite gate set

Gives a polynomial-time uniform quantum circuit family that outputs the complete prime factorization of every integer with probability one. Both gate count and qubit count are polynomial in the input length, and one fixed finite gate set suffices.

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280Sep 25, 2026 Explained

Unitary vertex operator algebras and conformal nets

Proves the strongly rational case of the strong-locality conjecture: every simple unitary strongly rational complex vertex operator algebra generates a completely rational conformal net. Its simple modules are unitarizable, and its representation category agrees with the net’s finite-index sectors as a braided unitary tensor category.

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281Sep 25, 2026 Explained

QAOA attains the SK optimum in the thermodynamic-first limit

Proves that QAOA approaches the ground-state energy of the Gaussian zero-field Sherrington–Kirkpatrick model when system size tends to infinity before circuit depth. For every accuracy, finite depth and deterministic angles independent of size and disorder achieve the required limiting expected energy per spin. This also yields leading-order optimal expected MaxCut values on large-degree random regular graphs, with size tending to infinity before degree.

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Operator algebras15

287Sep 23, 2026 Explained

Isomorphism of the free group factors

Resolves the free group factor isomorphism problem: $L(\mathbb F_2)\cong L(\mathbb F_3)$ , and hence all interpolated free group factors, including $L(\mathbb F_\infty)$ , are isomorphic. Their common factor has fundamental group $\mathbb R_{\gt 0}$ .

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288Sep 23, 2026 Explained

Kadison's similarity conjecture

Proves Kadison's similarity conjecture: every bounded complex-linear unital algebra homomorphism from a unital complex C^∗-algebra to operators on a Hilbert space becomes a $*$ -homomorphism after conjugation by a bounded invertible operator.

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289Sep 23, 2026 Explained

Strong Kadison–Kastler stability and its spatial boundaries

Proves that sufficiently close unital von Neumann algebras on the same Hilbert space are conjugate by a unitary arbitrarily close to the identity, with a universal tolerance in the operator-norm distance between unit balls. Counterexamples show that near-identity conjugacy fails for one-sided near inclusions, and that arbitrarily close norm-separable C^∗-algebras need not be ambiently unitarily conjugate.

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290Sep 23, 2026 Explained

Relative bicentralizers and modular spectral recovery

Proves Connes' bicentralizer conjecture for every type III₁ factor with separable predual and every faithful normal state. More generally, for every inclusion $N\subset M$ of von Neumann algebras with separable preduals admitting a faithful normal conditional expectation, constructs an amenable expected subalgebra $P\subset N$ with $P'\cap c(M)=N'\cap c(M)$ , resolving the relative bicentralizer conjecture.

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291Sep 23, 2026 Explained

Cuntz comparison, nuclear dimension, and equivariant Jiang–Su stability

Proves equivariant Jiang–Su stability for every countable discrete amenable group action on a simple separable unital infinite-dimensional nuclear stably finite Jiang–Su-stable C^∗-algebra, resolving this case of Szabó's conjecture without restrictions on trace dynamics. The family also proves the unital Toms–Winter conjecture, equating strict comparison, finite nuclear dimension and Jiang–Su stability in the simple separable unital infinite-dimensional nuclear setting.

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292Sep 23, 2026 Explained

Kirchberg's $\mathcal O_2$ norm-ultrapower embedding problem

Constructs an explicit separable unital full group C^∗-algebra that cannot embed unitally into the norm ultrapower of any fixed nonzero unital nuclear C^∗-algebra, for any free ultrafilter on the natural numbers. Taking the target to be $\mathcal O_2$ answers Kirchberg's norm-ultrapower embedding problem negatively.

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293Sep 27, 2026 Explained

Invariant projections, hyperinvariant subspaces, and transitive algebras

Constructs a nonzero norm-quasinilpotent operator on every infinite-dimensional separable complex Hilbert space with no nonzero proper closed subspace invariant under every commuting operator. The construction also gives operators with no nontrivial invariant projection in the hyperfinite type II₁ factor.

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294Sep 23, 2026 Explained

Kaplansky's quasitrace conjecture and failure of tensor-product stable finiteness

Disproves Kaplansky's quasitrace conjecture by constructing a separable unital complex C^∗-algebra admitting normalized 2-quasitraces, all of which are nonadditive. As a consequence, two unital simple stably finite C^∗-algebras can have a properly infinite minimal tensor product, with one factor $C_r^*(\mathbb F_2)$ .

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295Sep 23, 2026 Explained

The Kadison–Ringrose cohomology conjecture

Proves that every bounded Hochschild cocycle of degree at least two on a complex von Neumann algebra, with coefficients in the algebra itself, has a bounded primitive. Equivalently, all higher bounded Hochschild cohomology groups vanish, resolving the Kadison–Ringrose conjecture.

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296Sep 23, 2026 Explained

The generator problem for finite factors

Proves that every type II₁ factor with separable predual is generated by a single operator, equivalently by two self-adjoint operators, resolving the generator problem. More strongly, for every irreducible inclusion $P\subset M$ of such factors, the unitaries u with $M=W^*(P,u)$ form a dense G_δ subset in the trace 2-norm topology.

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297Sep 24, 2026 Explained

A ZFC counterexample to Naimark's problem

Gives an alternative to [Tanaka's ZFC construction](https://arxiv.org/abs/2609.26930v1) of a unital infinite-dimensional simple complex C^∗-algebra with a faithful tracial state and exactly one nonzero irreducible representation up to unitary equivalence. Thus the unrestricted compact-operator characterization fails without additional set-theoretic assumptions; the counterexample is nonseparable.

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298Sep 25, 2026 Explained

Two notions of free entropy differ even when both are finite

Constructs a bounded self-adjoint tuple in a tracial von Neumann algebra whose microstates and nonmicrostates free entropies satisfy $-\infty\lt \chi\lt \chi^*\lt \infty$ . This answers Voiculescu’s finite-entropy equality question negatively: the matrix-approximation and free-Fisher-information definitions differ even when both are finite.

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299Sep 25, 2026 Explained

The Kirchberg–Rørdam character criterion and infinite tensor-power Jiang–Su stability

A nonzero unital separable complex C^∗-algebra is Jiang–Su stable exactly when its norm central-sequence algebra has no characters, for every free ultrafilter. This answers the Kirchberg–Rørdam character question. Also, the infinite minimal tensor power of every such algebra without characters is Jiang–Su stable, answering the Dadarlat–Toms question.

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301Sep 25, 2026 Explained

Trace cones and Razak–Jacelon stabilization

Classifies separable nuclear complex C^∗-algebras after tensoring with the Razak–Jacelon algebra and the compact operators, using their full topological cones of extended lower-semicontinuous tracial weights. This answers Robert’s trace-cone question, including algebras with arbitrary ideal structure and both finite and infinite subquotients.

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303Sep 25, 2026 Explained

Weak pure infiniteness and Cuntz-algebra absorption

Resolves the ordinary-to-strong pure-infiniteness question of Kirchberg and Rørdam for complex C^∗-algebras. For exact algebras, proper infiniteness of one fixed finite amplification of every positive element also suffices. Consequently, every separable nuclear algebra with this property absorbs $\mathcal O_\infty$ , without unitality or simplicity assumptions.

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Topology3

307Sep 23, 2026 Explained

Failure of rational injectivity for maximal coarse assembly

Constructs a uniformly discrete bounded-geometry space whose maximal coarse assembly map is not rationally injective. The example is a coarse disjoint union of finite connected graphs of uniformly bounded degree, with an infinite-order kernel class. A companion gives the analogous failure for reduced coarse assembly, disproving the rational coarse Novikov conjecture.

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312Sep 24, 2026 Explained

The Grothendieck homotopy hypothesis

Proves the Grothendieck homotopy hypothesis for ∞-groupoids associated with every Grothendieck coherator in the Ara–Henry convention: these algebraic objects recover the homotopy theory of spaces.

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317Sep 25, 2026 Explained

Thomason model structures in all strict higher dimensions

Resolves the Ara–Maltsiniotis conjecture: for every n ≥ 1 and n = ω, small strict globular n-categories admit proper combinatorial Thomason model structures Quillen equivalent to simplicial sets. Thus strict higher categories model the homotopy theory of spaces in every stated dimension.

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Functional analysis10

322Sep 23, 2026 Explained

Tingley’s sphere-isometry problem

Resolves Tingley's problem: every surjective isometry between the unit spheres of nonzero real Banach spaces extends uniquely to a surjective real-linear isometry of the whole spaces. No dimension restriction is imposed, so the metric geometry of the unit sphere determines the Banach space up to linear isometry.

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323Sep 23, 2026 Explained

Independence of the separable quotient problem

Establishes, relative to the consistency of a measurable cardinal, that the separable quotient problem is independent of ZFC. The assertion that every infinite-dimensional Banach space has a separable infinite-dimensional quotient can hold for all real and complex Banach spaces, whereas the continuum hypothesis yields counterexamples over both fields.

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324Sep 24, 2026 Explained

Lipschitz equivalent Banach spaces need not be linearly isomorphic

Constructs separable real Banach spaces that are globally bi-Lipschitz equivalent but not linearly isomorphic, resolving the separable Lipschitz-isomorphism problem negatively. Thus even the complete metric structure up to bi-Lipschitz equivalence does not determine a separable Banach space's linear isomorphism class.

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325Sep 23, 2026 Explained

The complete Crouzeix conjecture

Resolves the complete Crouzeix conjecture: for every bounded operator A on a complex Hilbert space and every finite matrix-valued polynomial P, one has $\lVert P[A]\rVert\le2\sup_{z\in W(A)}\lVert P(z)\rVert$ , where $W(A)$ is the numerical range. The constant 2 is sharp, independent of the matrix size, and valid in infinite dimensions.

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326Sep 23, 2026 Explained

The cotype–cotype conjecture under the approximation property

Resolves the cotype–cotype conjecture for real Banach spaces with the approximation property. Such a nonzero space is K-convex if and only if both it and its dual have finite Rademacher cotype, with possibly different exponents. Equivalently, these cotype assumptions force nontrivial Rademacher type.

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327Sep 23, 2026 Explained

Markov type characterizes superreflexivity

Proves that every real Banach space with Markov type p for some p > 1 admits an equivalent uniformly convex norm, answering Naor's renorming question. Together with the known converse, this characterizes superreflexivity by nontrivial Markov type.

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328Sep 24, 2026 Explained

Nonexpansive fixed points in reflexive Banach spaces

Resolves Kirk's reflexive-space fixed-point problem: every nonexpansive selfmap of a nonempty closed bounded convex subset of a real reflexive Banach space has a fixed point. The result uses the original norm, without assuming uniform convexity.

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329Sep 24, 2026 Explained

A counterexample to metric-entropy duality

Disproves Pietsch's dimension-free duality conjecture for metric entropy. Origin-symmetric convex bodies violate every proposed choice of universal constants in the conjectured comparison between covering numbers and those of the polar bodies, even when the covering body is a cube.

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330Sep 26, 2026 Explained

A uniformly discrete counterexample to bounded approximation in Lipschitz-free spaces

Constructs a countable uniformly discrete metric space whose real Lipschitz-free Banach space has the approximation property but not the bounded approximation property, answering Kalton's question negatively. Finite-rank operators approximate the identity on every compact set, but their norms cannot share a finite bound.

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331Sep 27, 2026 Explained

Reflexive midpoint convexity and diamond distortion

Constructs a real reflexive Banach space with an asymptotically midpoint uniformly convex norm but no equivalent asymptotically uniformly convex norm, extending Baudier's separation to reflexive spaces. In the same space, depth-k countably branching diamonds require distortion at least $\sqrt{1+k/12}$ , so midpoint uniform convexity does not force uniformly bounded diamond distortion even under reflexivity.

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Differential geometry15

333Sep 23, 2026 Explained

Smooth isometric immersions of surfaces into ℝ⁴

Every closed smooth Riemannian surface admits a smooth isometric immersion into ℝ⁴, resolving the closed-surface form of the four-dimensional isometric-immersion problem. This includes nonorientable surfaces and metrics of arbitrary Gaussian curvature.

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334Sep 24, 2026 Explained

A smooth surface metric with no local isometric immersion in ℝ³

Constructs a smooth positive-definite metric on $(-1,1)^2$ for which no neighborhood of the origin admits a smooth isometric immersion into ℝ³. This answers the unrestricted smooth local isometric realization problem for surfaces negatively, even after shrinking the neighborhood.

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337Sep 23, 2026 Explained

Sharp Cartan–Hadamard isoperimetry and rigidity

Proves generalized Cartan–Hadamard isoperimetry in every dimension: in a complete simply connected manifold with sectional curvature at most κ ≤ 0, every finite-volume finite-perimeter set satisfies the sharp comparison with the equal-volume model ball. Bounded positive-volume equality regions for κ = 0 are Euclidean balls. Also proves sharp Euclidean filling bounds for compactly supported integral n-cycles, n ≥ 2, in arbitrary proper CAT $(0)$ spaces.

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342Oct 6, 2026 Explained

Donaldson's tamed-to-compatible conjecture

Proves Donaldson's tamed-to-compatible conjecture: every smooth almost complex structure on a closed four-manifold that is tamed by a symplectic form admits a compatible symplectic form. The almost complex structure stays fixed; the form's cohomology class may change.

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343Sep 23, 2026 Explained

Symplectic ball packing in higher dimensions

Resolves the Siegel–Yao conjecture for arbitrary capacities in every dimension $2n\ge6$ . Finitely many closed symplectic balls of capacities $R_1,\ldots,R_k$ embed disjointly into an open ball of capacity R exactly when $\sum_iR_i^n\lt R^n$ and $R_i+R_j\lt R$ for every distinct pair i, j.

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347Sep 23, 2026 Explained

Counterexamples to stable-Morse and strong Arnold fixed-point bounds

Disproves stable-Morse lower bounds for nondegenerate Hamiltonian fixed points: on simply connected closed Kähler manifolds of real dimension 22, the deficit below the stable Morse number is unbounded. A separate Hamiltonian diffeomorphism of the complex quadric threefold has exactly three fixed points, fewer than the four critical points required of every smooth function.

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348Sep 23, 2026 Explained

Nonnegative-curvature Einstein classification and an L² topological gap

Classifies closed connected Einstein four-manifolds with positive Einstein constant and nonnegative sectional curvature: up to scaling, their universal Riemannian covers are the round S⁴, Fubini–Study $\mathbb{CP}^2$ , or a product of equal round two-spheres. A closed simply connected nonnegatively curved four-manifold is diffeomorphic to one of these whenever the scale-invariant L² norm of its trace-free Ricci curvature lies below a universal positive constant.

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350Sep 23, 2026 Explained

Yau’s nodal bounds: surfaces and higher dimensions

Proves the sharp $C\sqrt\lambda$ upper bound for nodal length on every fixed smooth closed surface, completing Yau's conjecture there. The upper bound fails for fixed smooth metrics in dimensions three and four, including metrics on S³ arbitrarily close to round. In dimension five, nodal measure can grow faster than $\lambda^{1/2+\varepsilon_0}$ for some fixed $\varepsilon_0\gt 0$ , ruling out even arbitrarily small power losses.

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353Sep 24, 2026 Explained

Affine Bernstein rigidity through dimension nine and a smooth dimension-ten counterexample

Proves that every smooth locally uniformly convex affine-maximal graph of dimension three through nine, complete for its induced Euclidean metric, is an elliptic paraboloid. A smooth entire nonquadratic example in dimension ten makes this range sharp. In dimensions three through nine, the paraboloid classification also holds for connected open locally uniformly convex affine-maximal hypersurfaces complete for the affine Berwald–Blaschke metric.

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354Sep 24, 2026 Explained

The isoperimetric profile of the cubic three-torus

Determines the isoperimetric profile of the unit cubic flat three-torus and classifies every finite-perimeter minimizer: balls, circular tubes around shortest closed geodesics, coordinate slabs, and their complements. The transition volumes are $4\pi/81$ and $1/\pi$ , with exactly the adjacent two types minimizing at each transition.

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356Sep 24, 2026 Explained

Gigli’s characterization of Alexandrov curvature

Proves Gigli's conjecture: in every integer dimension n ≥ 2, Alexandrov curvature at least κ is characterized by the full-support $\mathop{\mathrm{RCD}}\nolimits ((n-1)\kappa,n)$ condition with reference measure $\mathcal H^n$ and distributional sectional curvature at least κ in the original global test classes. The RCD condition is unreduced.

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358Sep 24, 2026 Explained

A three-manifold without conjugate points or nonpositive curvature

Constructs a closed connected orientable smooth three-manifold that admits a metric without conjugate points but no metric of nonpositive sectional curvature. This answers negatively, already in dimension three, whether the first metric-existence property implies the second.

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359Sep 25, 2026 Explained

Negative Kähler curvature without bounded holomorphic coordinates

Constructs a contractible domain in ℂ³ with a complete negatively pinched Kähler metric but no bounded holomorphic coordinates, disproving bounded-domain uniformization in this setting. A higher-dimensional example has sectional curvature at most −1 and only constant bounded holomorphic functions; its curvature is not bounded below.

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360Sep 25, 2026 Explained

Weak MTW curvature gives convexity and regular optimal transport

On every closed connected Riemannian manifold of dimension at least two satisfying weak Ma–Trudinger–Wang curvature, all tangent injectivity domains are convex, resolving Villani’s conjecture in this setting. For squared-distance transport between measurable probability densities bounded above and away from zero, the optimal map and its inverse are Hölder continuous.

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361Sep 25, 2026 Explained

Failure of integer-degree harmonic dimension comparison

Disproves Yau's proposed Euclidean dimension bound for harmonic functions of integer growth on manifolds with nonnegative Ricci curvature. For every sufficiently large integer k, a complete smooth metric on ℝ³ has at least $(k+2)^2$ independent harmonic functions of growth at most k, exceeding the Euclidean count $(k+1)^2$ . The metric may depend on k.

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Partial differential equations11

362Sep 23, 2026 Explained

Global smoothness for relativistic Vlasov–Maxwell

Proves large-data global existence and uniqueness for the three-dimensional, one-species relativistic Vlasov–Maxwell system. Smooth admissible initial data may be arbitrary provided the particle density is compactly supported and the electromagnetic fields have finite energy and bounded derivatives of every order; the solution remains smooth on every finite time interval.

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363Sep 23, 2026 Explained

Nonuniqueness with local conservation for the hard-sphere Boltzmann equation

Constructs two distinct global entropy solutions of the three-dimensional periodic hard-sphere Boltzmann equation from the same nonnegative initial density, with bounded velocity support and finite mass, energy and absolute entropy. Both are strongly continuous in L¹ and satisfy exact local conservation of mass, momentum and kinetic energy.

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365Sep 23, 2026 Explained

Joint metric and connection recovery from one boundary patch

Zero-frequency measurements on any nonempty open boundary patch determine a smooth metric and smooth unitary connection on a trivial Hermitian rank-two bundle over a compact connected manifold of dimension at least three, up to diffeomorphism and gauge fixed on that patch. Inputs and observations use the same patch. In contrast, distinct uniformly positive bounded measurable scalar conductivities on a three-dimensional ball can have identical full-boundary data.

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367Sep 24, 2026 Explained

The critical dimension for the one-phase Bernoulli problem

Establishes seven as the first dimension admitting a nonflat, one-homogeneous global minimizer of the one-phase Bernoulli energy. Consequently, minimizing free boundaries are smooth through dimension six, and their singular sets have dimension at most $n-7$ in higher dimensions.

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369Sep 24, 2026 Explained

The hot spots conjecture for simply connected planar domains

Proves a strict form of Burdzy's simply connected hot spots conjecture. On every smooth bounded simply connected planar domain, each nonzero eigenfunction for the first positive Neumann eigenvalue has no interior critical point, so all global extrema lie on the boundary. Eigenvalue multiplicity is allowed.

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370Sep 24, 2026 Explained

The Lane–Emden and Hénon–Lane–Emden conjectures

Resolves the subcritical Lane–Emden conjecture and its weighted Hénon extension. For n ≥ 2, $p,q\gt 0$ and real A, B, the system $-\Delta u=|x|^A v^p$ , $-\Delta v=|x|^B u^q$ has no positive entire solution when $(n+A)/(p+1)+(n+B)/(q+1)\gt n-2$ , with solutions continuous at the origin and classical elsewhere. No symmetry or growth assumption is needed. Known radial existence gives the exact existence criterion for n ≥ 3 and $A,B\gt -2$ .

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371Sep 24, 2026 Explained

Stable blowup for the defocusing Schrödinger equation

For a sufficiently large odd nonlinearity power, constructs a nonempty open set of initial data in $H^k(\mathbb T^{12})$ , with k > 8, whose solutions of the scalar defocusing nonlinear Schrödinger equation blow up in finite time. Thus finite-time blowup is stable under Sobolev perturbations in this supercritical regime.

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372Sep 24, 2026 Explained

Global uniqueness in smooth isotropic elasticity

Proves that full static boundary displacement-to-traction data determine both smooth real Lamé moduli on every bounded connected smooth domain in ℝ³, provided μ > 0 and $3\lambda+2\mu\gt 0$ on the closure. Neither analyticity, proximity to constant coefficients nor prior knowledge near the boundary is required.

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373Sep 25, 2026 Explained

Nonattainment of the three-marginal Coulomb Monge problem

An optimal three-particle Coulomb configuration need not be a deterministic function of the first particle, even for smooth identical spatial densities. The Monge and Kantorovich infima agree, but the Monge infimum is not attained. The same phenomenon occurs for every inverse-power Riesz exponent in each dimension at least two, with a suitable density in each case.

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374Sep 25, 2026 Explained

Sharp one-third stability of Brenier maps

For uniform source measure ρ on a compact convex body with interior in dimension at least two, quadratic optimal transport maps satisfy $\|T_\mu-T_\nu\|_{L^2(\rho)}\le C W_2(\mu,\nu)^{1/3}$ uniformly over targets in a fixed compact set. The exponent is sharp, even for three-atom targets, disproving Letrouit's conjectured square-root bound.

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376Sep 27, 2026 Explained

Universal computation in forced Navier–Stokes flows

Constructs viscous incompressible flows starting from rest on a fixed flat three-dimensional domain that perform universal computation under smooth external forcing. A terminating compiler turns a Turing machine and input into a finite program for the force, so a designated particle reaches a fixed region exactly when the machine halts. The viscosity is fixed, positive and computable.

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