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New proofs, counterexamples, bounds and formalizations reported by people and AI systems, newest first.

Reported developments, attributed to their sources. Inclusion is not an OpenMath assessment; evidence slots show what each source reports. Updated Oct 13, 2026.

October 2026 40

New proofTheoretical computer scienceopenai/math #111

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.

Unreleased OpenAI model
LFCR
New proofTheoretical computer scienceopenai/math #110

Optimal-order randomized k-server on arbitrary metrics

Establishes a randomized competitive ratio O(log⁡2(k+1))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)\log(t+1) at request t, with a finite instance-dependent additive movement constant.

Unreleased OpenAI model
LFCR
CounterexampleTheoretical computer scienceopenai/math #109

Integer multiplication below nlog⁡nn\log n

Multiplies two n-bit integers exactly at every input length in deterministic worst-case time O(n(log⁡n)1−κ)O(n(\log n)^{1-\kappa}), with κ=2−182\kappa=2^{-182}, on one fixed finite-alphabet Turing machine with finitely many one-dimensional tapes. This disproves the Schönhage–Strassen nlog⁡nn\log n optimality conjecture in the ordinary multitape bit model.

Unreleased OpenAI model
LFCR
Improved boundTheoretical computer scienceopenai/math #108

A cubic permanent–determinant lower bound

Proves an Ω(n3)\Omega(n^3) lower bound for the border determinantal complexity of the n×nn\times n permanent over ℂ. Even coefficientwise limits of determinants of affine-linear matrices require matrix size at least cn3cn^3, for an absolute c > 0 and all sufficiently large n; the same bound therefore holds for exact representations.

Unreleased OpenAI model
LFCR
New proofTheoretical computer scienceopenai/math #106

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<δ<1/30\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 δn\delta n otherwise, where n is the number of vertices.

Unreleased OpenAI model
LFCR
New proofTheoretical computer scienceopenai/math #105

Perfect completeness for 2-to-1 games

Proves Khot's 2-to-1 Games Conjecture with perfect completeness: for every fixed rational δ∈(0,1)\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.

Unreleased OpenAI model
LFCR
New proofTheoretical computer scienceopenai/math #104

Quasipolynomial algorithms for mean-payoff, stochastic and parity games

Gives deterministic algorithms using 2O((log⁡(L+2))2)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.

Unreleased OpenAI model
LFCR
New proofGeometry and topologyopenai/math #101

The sharp simplex conjecture for isotropic constants

Proves that simplices uniquely maximize the isotropic constant among convex bodies in every dimension, resolving the strong isotropic constant conjecture. Also establishes the sharp entropy lower bound for log-concave probability densities, with equality precisely for invertible affine images of products of one-sided exponential laws.

Unreleased OpenAI model
LFCR
CounterexampleGeometry and topologyopenai/math #100

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.

Unreleased OpenAI model
LFCR
New proofGeometry and topologyopenai/math #099

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⁡(Θ(log⁡d log⁡log⁡d))\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.

Unreleased OpenAI model
LFCR
CounterexampleGeometry and topologyopenai/math #098

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.

Unreleased OpenAI model
LFCR
Improved boundGeometry and topologyopenai/math #097

The Euclidean Steinitz–Bergström bound

Proves that any finite sequence in the Euclidean unit ball of ℝd admits signs keeping every partial sum within CdC\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 S2(d)=Θ(d)S_2(d)=\Theta(\sqrt d).

Unreleased OpenAI model
LFCR
New proofGeometry and topologyopenai/math #096

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/(8π)9/(8\pi). In dimension at least two, three planar sectors of angle 2π/32\pi/3, extended orthogonally, attain the bound. Combined with the separate Unique Games theorem, this proves NP-hardness of improving the loss factor (8π/9)(1−1/k)(8\pi/9)(1-1/k) for identity-target kernel clustering with fixed k ≥ 3 on rational centered positive semidefinite inputs.

Unreleased OpenAI model
LFCR
CounterexampleGeometry and topologyopenai/math #095

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.

Unreleased OpenAI model
LFCR
New proofGeometry and topologyopenai/math #094

Subpolynomial dimension reduction in Lp

For every fixed 1<p<∞1\lt p\lt \infty and distortion D > 1, every n-point subset of real Lp embeds into ℓpd\ell_p^d with distortion at most D and dimension d=no(1)d=n^{o(1)}, answering Naor's sublinear-dimension question for p ≠ 2. In contrast, exact embeddings require worst-case dimension Θ(n2)\Theta(n^2) when p ≠ 2.

Unreleased OpenAI model
LFCR
New proofGeometry and topologyopenai/math #092

The optimal order of convex-body covering density

Determines the optimal worst-case covering density as Θ(nlog⁡n)\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 Cnlog⁡nCn\log n; centrally symmetric examples in every sufficiently large dimension require at least cnlog⁡ncn\log n even without the lattice restriction, for absolute c,C>0c,C\gt 0.

Unreleased OpenAI model
LFCR
New proofGeometry and topologyopenai/math #091

Logarithmic and Lp 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 Lp Brunn–Minkowski inequality for full-dimensional origin-symmetric convex bodies throughout 0<p<10\lt p\lt 1.

Unreleased OpenAI model
LFCR
New proofGeometry and topologyopenai/math #090

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<s<20\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.

Unreleased OpenAI model
LFCR
New proofGeometry and topologyopenai/math #089

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.

Unreleased OpenAI model
LFCR
CounterexampleGeometry and topologyopenai/math #088

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.

Unreleased OpenAI model
LFCR
New proofGeometry and topologyopenai/math #087

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×K∘K\times K^\circ in dimension 2n2n has Gromov width 4.

Unreleased OpenAI model
LFCR
Improved boundAnalysisopenai/math #086

An L3 bound for the trilinear Hilbert transform

Proves that the principal-value trilinear Hilbert transform with shifts x−tx-t, x−2tx-2t, x−3tx-3t is bounded from L3(R)3L^3(\mathbb R)^3 to L1(R)L^1(\mathbb R). This resolves the L3 exponent case of the standard conjecture for slopes 1, 2, 3.

Unreleased OpenAI model
LFCR
New proofAnalysisopenai/math #085

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∈W1,1(R2)f\in W^{1,1}(\mathbb R^2), the centered disk maximal function satisfies ∥∇Mf∥1≤C∥∇f∥1\|\nabla Mf\|_1\le C\|\nabla f\|_1 with an absolute constant. It belongs locally to W1,1W^{1,1} and has a globally integrable weak gradient.

Unreleased OpenAI model
LFCR
New proofAnalysisopenai/math #084

The geometric case of the Erdős similarity conjecture

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

Unreleased OpenAI model
LFCR
New proofAnalysisopenai/math #083

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.

Unreleased OpenAI model
LFCR
Improved boundAnalysisopenai/math #082

Annular variation and dyadic absolute bounds for the triangular Hilbert transform

Proves maximal and annular r-variation bounds, for every r > 2, from complex L3(R2)×L3(R2)L^3(\mathbb R^2)\times L^3(\mathbb R^2) to L3/2(R2)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 L3×L3×L3L^3\times L^3\times L^3 point.

Unreleased OpenAI model
LFCR
New proofAnalysisopenai/math #081

Riesz transforms and rectifiability in higher codimension

Resolves the remaining higher-codimension Riesz-transform rectifiability problem: for d ≥ 4 and 2≤n≤d−22\le n\le d-2, an n-Ahlfors–David regular Radon measure on ℝd 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.

Unreleased OpenAI model
LFCR
New proofAnalysisopenai/math #080

The exact Sobolev endpoint for Schrödinger convergence

Proves almost-everywhere convergence eitΔf→fe^{it\Delta}f\to f as t↓0t\downarrow0 for every f∈Hn/(2(n+1))(Rn)f\in H^{n/(2(n+1))}(\mathbb R^n) and every dimension n ≥ 2. This attains the sharp Sobolev equality case of Carleson's Schrödinger convergence problem, including the planar endpoint H1/3.

Unreleased OpenAI model
LFCR
New proofAnalysisopenai/math #079

Local smoothing in three dimensions

Resolves Sogge's local smoothing conjecture for the Euclidean wave equation in three spatial dimensions. The estimate holds throughout the full strict range 2<p<∞2\lt p\lt \infty, with Sobolev regularity above max⁡{0,1−3/p}\max\{0,1-3/p\}. In particular, the critical L3 estimate holds with every positive Sobolev loss.

Unreleased OpenAI model
LFCR
New proofAnalysisopenai/math #078

The three-dimensional Bochner–Riesz conjecture

Resolves the three-dimensional Bochner–Riesz conjecture in its strict range: the Bochner–Riesz multipliers of order δ are bounded on Lp(R3)L^p(\mathbb R^3) for every 1≤p≤∞1\le p\le\infty whenever δ>max⁡{3∣1/p−1/2∣−1/2,0}\delta\gt \max\{3|1/p-1/2|-1/2,0\}.

Unreleased OpenAI model
LFCR
New proofAnalysisopenai/math #077

Fourier restriction for positively curved surfaces

Proves the diagonal Fourier extension conjecture for positively curved surfaces in three dimensions. For every compact smooth positively curved surface Σ⊂R3\Sigma\subset\mathbb R^3, including surfaces with boundary, the extension operator is bounded from Lp(Σ)L^p(\Sigma) to Lp(R3)L^p(\mathbb R^3) for every p > 3.

Unreleased OpenAI model
LFCR
CounterexampleAnalysisopenai/math #076

Real ultraflat Littlewood polynomials and unbounded binary merit factors

Constructs polynomials with N consecutive coefficients in {−1,1}\{-1,1\} whose modulus is (1+o(1))N(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.

Unreleased OpenAI model
LFCR
New proofAnalysisopenai/math #075

The Llog⁡LL\log L Fourier-convergence conjecture

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

Unreleased OpenAI model
LFCR
New proofAnalysisopenai/math #073

The Falconer distance conjecture

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

Unreleased OpenAI model
LFCR
CounterexampleAnalysisopenai/math #072

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, ∣ϕ′∣s|\phi'|^s is area-integrable for 4/3<s<44/3\lt s\lt 4. The sharp universal integral-means identity is BS(t)=∣t∣−1B_{\mathcal S}(t)=|t|-1 for t ≤ −2. A strict bound Bb(−1)<1/4B_b(-1)\lt 1/4 for bounded univalent functions disproves Kraetzer's prediction at that parameter.

Unreleased OpenAI model
LFCR
New proofAnalysisopenai/math #071

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.

Unreleased OpenAI model
LFCR
New proofGeometry and topologyopenai/math #069

Global quantum geometric Langlands at irrational level

Proves the unramified de Rham quantum geometric Langlands equivalence for every connected simple complex algebraic group on every smooth projective connected complex curve, at every shifted level c∈C∖Qc\in\mathbb C\setminus\mathbb Q. It identifies the full derived categories of twisted D-modules for the group and its Langlands dual, retaining all global forms and connected components.

Unreleased OpenAI model
LFCR
New proofGeometry and topologyopenai/math #068

Anticanonical nonvanishing in every dimension

If X is a smooth connected complex projective variety and −KX-K_X admits a smooth Hermitian metric with nonnegative curvature, then H0(X,−mKX)≠0H^0(X,-mK_X)\ne0 for some m > 0. Thus smooth semipositivity forces a nonzero section of a positive tensor power of the anticanonical bundle in every dimension.

Unreleased OpenAI model
LFCR
New proofGeometry and topologyopenai/math #067

The Campana–Peternell conjecture in dimension six

Proves the Campana–Peternell conjecture in complex dimension six: every smooth connected complex projective Fano sixfold with nef tangent bundle is rational homogeneous.

Unreleased OpenAI model
LFCR