{ "schema": "mobcorp-research/1.0", "checked_at": "2026-10-16T13:32:32.368985+00:00", "revision": "1581d4cbdd69a0faf34b2089614fe2a814316a6a", "source": "https://github.com/openai/math", "release_url": "https://openai.com/index/sharing-ai-progress-in-mathematics/", "license": "Apache-2.0", "counts": { "families": 372, "active_manuscripts": 719, "withdrawals": 3, "formalization_catalogue_sources": 173, "matched_active_formalization_sources": 172 }, "publisher_formalization_claim": "300 / 719 top-line results (~42%); this is a publisher metric, not a count of families or locally verified proofs.", "formalization_review": "unchecked", "families": [ { "id": "001", "title": "Milne’s rationality conjecture and algebraic specialization", "subject": "Number theory", "description": "Proves Milne's rationality conjecture for abelian varieties over overline{mathbb Q} with good reduction: specialized Hodge classes pair rationally with complementary divisor products, independently of cohomology theory. Together with result 032, every specialized Hodge class is represented by a single rational algebraic cycle simultaneously in all prime-to-p and crystalline realizations, for every residue characteristic p.", "scope_url": null }, { "id": "002", "title": "The full BSD formula from low Selmer corank", "subject": "Number theory", "description": "Proves the full Birch–Swinnerton-Dyer leading-term formula for every elliptic curve over ℚ whose full q-power Selmer group has corank zero or one for some prime q, including finiteness of the Tate–Shafarevich group. With result 006, this gives full BSD for a density-one set of quadratic twists of every elliptic curve over ℚ.", "scope_url": null }, { "id": "003", "title": "The quasi-Riemann hypothesis", "subject": "Number theory", "description": "Proves that every Dirichlet L-function, including zeta(s), is zero-free in Re sgt 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 sgt 11/12.", "scope_url": "https://github.com/openai/math/blob/1581d4cbdd69a0faf34b2089614fe2a814316a6a/lean/docs/003.md" }, { "id": "004", "title": "Hilbert’s tenth problem over ℚ", "subject": "Number theory", "description": "Proves that no algorithm decides whether an integer-coefficient polynomial in an arbitrary number of variables has a rational zero, resolving Hilbert's tenth problem over ℚ negatively.", "scope_url": null }, { "id": "005", "title": "Irrationality of Catalan’s constant", "subject": "Number theory", "description": "Proves that Catalan's constant G=sum_{jge0}(-1)^j/(2j+1)^2 is irrational.", "scope_url": "https://github.com/openai/math/blob/1581d4cbdd69a0faf34b2089614fe2a814316a6a/lean/docs/005.md" }, { "id": "006", "title": "Goldfeld’s conjecture: densities and mean analytic rank", "subject": "Number theory", "description": "Proves Goldfeld's conjecture for quadratic twists of every elliptic curve over ℚ: analytic ranks zero and one each have density 1/2, and the mean analytic rank tends to 1/2. Both statements order signed squarefree twist parameters by absolute value.", "scope_url": null }, { "id": "007", "title": "Ordinary two-point correlations and the corrected Elliott conjecture", "subject": "Number theory", "description": "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.", "scope_url": "https://github.com/openai/math/blob/1581d4cbdd69a0faf34b2089614fe2a814316a6a/lean/docs/007.md" }, { "id": "008", "title": "The Deligne–Drinfeld conjecture", "subject": "Number theory", "description": "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.", "scope_url": "https://github.com/openai/math/blob/1581d4cbdd69a0faf34b2089614fe2a814316a6a/lean/docs/008.md" }, { "id": "009", "title": "Function-field reconstruction from Milnor K-theory and Galois data", "subject": "Number theory", "description": "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.", "scope_url": "https://github.com/openai/math/blob/1581d4cbdd69a0faf34b2089614fe2a814316a6a/lean/docs/009.md" }, { "id": "010", "title": "Unrestricted pro-modularity at the prime two", "subject": "Number theory", "description": "Every continuous odd absolutely irreducible two-dimensional 2-adic representation of G_{mathbb Q} unramified outside finitely many primes occurs in a completed Hecke algebra at some odd tame level. Also proves classical modularity up to Tate twist for irreducible odd representations with these finiteness conditions that are de Rham at 2 with distinct Hodge–Tate weights, resolving the dyadic Fontaine–Mazur case without residual restrictions.", "scope_url": null }, { "id": "011", "title": "Prime-factor statistics of p-1", "subject": "Number theory", "description": "Proves that the normalized ordered logarithms of the prime factors of p-1, counted with multiplicity, converge jointly to the Poisson–Dirichlet law mathrm{PD}(1) as p ranges uniformly over primes up to x and xtoinfty. This resolves the Ford–Konyagin–Luca conjecture. It also proves that infinitely many integers n have more than n^{1-varepsilon} totient preimages, for every ε > 0.", "scope_url": null }, { "id": "012", "title": "Independent largest prime factors of consecutive integers", "subject": "Number theory", "description": "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.", "scope_url": "https://github.com/openai/math/blob/1581d4cbdd69a0faf34b2089614fe2a814316a6a/lean/docs/012.md" }, { "id": "013", "title": "Ostmann’s inverse Goldbach conjecture", "subject": "Number theory", "description": "Proves that no finite modification of the primes can be written as A+B with A,Bsubseteqmathbb Z_{ge0} each containing at least two elements. This resolves Ostmann's inverse Goldbach conjecture on additive indecomposability.", "scope_url": "https://github.com/openai/math/blob/1581d4cbdd69a0faf34b2089614fe2a814316a6a/lean/docs/013.md" }, { "id": "014", "title": "Restricted geometric Langlands, global Arthur enhancements, and generic Ramanujan", "subject": "Number theory", "description": "Proves the restricted geometric Langlands equivalence for connected reductive groups on smooth projective connected curves over overline{mathbb F}_q under the four stated Lie-theoretic characteristic hypotheses. Over arbitrary algebraically closed fields of characteristic p > 0, the same conclusion holds assuming additionally that p is very good for the group and pnmid |W_G|. Over global function fields, proves Ramanujan at every place for globally generic cuspidal representations of split adjoint absolutely simple exceptional groups, without characteristic or ramification-depth restrictions, and at every unramified place for cuspidal representations of split adjoint absolutely simple groups with a generic unramified component. Assuming the finite-level Ramanujan–Arthur decomposition, constructs global Arthur enhancements of occurring cuspidal excursion parameters for split connected semisimple groups at full finite level, recovering the given parameters by diagonal specialization on the entire Weil group, including inertia.", "scope_url": null }, { "id": "015", "title": "Torus-packet equidistribution in prime, quartic, and sextic degrees", "subject": "Number theory", "description": "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.", "scope_url": "https://github.com/openai/math/blob/1581d4cbdd69a0faf34b2089614fe2a814316a6a/lean/docs/015.md" }, { "id": "016", "title": "Zilber–Pink in abelian varieties and the Siegel threefold", "subject": "Number theory", "description": "Proves the abelian Zilber–Pink conjecture over overline{mathbb Q}: every irreducible subvariety has finitely many maximal atypical subvarieties relative to its smallest containing torsion coset. It also proves the full curve case in the Siegel threefold mathcal A_2 for Hodge-generic curves defined over overline{mathbb Q}, without boundary or reduction assumptions.", "scope_url": null }, { "id": "017", "title": "The irrationality exponent of π is 2", "subject": "Number theory", "description": "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_{nge1}1/(n^3sin^2 n), with angles in radians.", "scope_url": "https://github.com/openai/math/blob/1581d4cbdd69a0faf34b2089614fe2a814316a6a/lean/docs/017.md" }, { "id": "018", "title": "The Margulis–Platonov conjecture over global fields", "subject": "Number theory", "description": "Proves the Margulis–Platonov conjecture over every global field, including function fields of characteristic two. For an absolutely almost simple simply connected algebraic group G over k, every noncentral abstract normal subgroup of G(k) is the inverse image of an open normal subgroup in the finite product of its anisotropic nonarchimedean local groups.", "scope_url": null }, { "id": "019", "title": "The local p-adic section conjecture and global consequences", "subject": "Number theory", "description": "Proves that rational points on every smooth proper geometrically connected curve of genus at least two over a finite extension of ℚp correspond bijectively to conjugacy classes of sections of its full arithmetic étale fundamental group. It also proves Grothendieck's section conjecture over ℚ for the modular curves X_0(N) and X_1(N) of genus at least two.", "scope_url": null }, { "id": "020", "title": "Squarefree quartics and power-free polynomial values", "subject": "Number theory", "description": "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.", "scope_url": "https://github.com/openai/math/blob/1581d4cbdd69a0faf34b2089614fe2a814316a6a/lean/docs/020.md" }, { "id": "021", "title": "A quadratic bound for Jacobsthal’s function", "subject": "Number theory", "description": "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.", "scope_url": "https://github.com/openai/math/blob/1581d4cbdd69a0faf34b2089614fe2a814316a6a/lean/docs/021.md" }, { "id": "022", "title": "The weak inhomogeneous Duffin–Schaeffer conjecture", "subject": "Number theory", "description": "Proves that for every real shift γ and finite-valued psi:mathbb Nto[0,infty), divergence of sum_qphi(q)psi(q)/q implies |qx-gamma|lt psi(q) for infinitely many q, for almost every x. Here ϕ is Euler's totient and the norm is distance to the nearest integer. Numerators are unrestricted; no monotonicity or Diophantine condition on γ is needed.", "scope_url": null }, { "id": "023", "title": "Patterson's first moment for cubic Gauss sums", "subject": "Number theory", "description": "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.", "scope_url": "https://github.com/openai/math/blob/1581d4cbdd69a0faf34b2089614fe2a814316a6a/lean/docs/023.md" }, { "id": "024", "title": "An asymptotic formula for the number of totients", "subject": "Number theory", "description": "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.", "scope_url": "https://github.com/openai/math/blob/1581d4cbdd69a0faf34b2089614fe2a814316a6a/lean/docs/024.md" }, { "id": "025", "title": "Short Egyptian fractions", "subject": "Number theory", "description": "Every rational a/b with 1le alt b is a sum of O(loglog b) distinct positive unit fractions. The worst-case minimum number of terms has the same order, resolving Erdős’s conjecture on short Egyptian fractions.", "scope_url": "https://github.com/openai/math/blob/1581d4cbdd69a0faf34b2089614fe2a814316a6a/lean/docs/025.md" }, { "id": "026", "title": "Positive lower density of large prime gaps", "subject": "Number theory", "description": "For every fixed C > 0, a positive proportion of consecutive prime gaps exceed Clog 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.", "scope_url": "https://github.com/openai/math/blob/1581d4cbdd69a0faf34b2089614fe2a814316a6a/lean/docs/026.md" }, { "id": "027", "title": "Potential integral density on curve character varieties", "subject": "Number theory", "description": "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 SLr character variety over the full ring of integers of one number field. Prescribed quasi-unipotent boundary conjugacy classes are allowed, including nonsemisimple classes.", "scope_url": "https://github.com/openai/math/blob/1581d4cbdd69a0faf34b2089614fe2a814316a6a/lean/docs/027.md" }, { "id": "028", "title": "Uniformly bounded components of Gaussian-prime graphs", "subject": "Number theory", "description": "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.", "scope_url": "https://github.com/openai/math/blob/1581d4cbdd69a0faf34b2089614fe2a814316a6a/lean/docs/028.md" }, { "id": "029", "title": "Primitive roots for every admissible integer base", "subject": "Number theory", "description": "Proves the infinitude assertion in Artin's primitive root conjecture for every integer a that is neither −1 nor a square. For each such base, at least c_a x/(log x)^2 primes in every sufficiently large interval (x,2x) have primitive root a, with c_agt 0.", "scope_url": null }, { "id": "030", "title": "Modularity of elliptic curves over imaginary quadratic fields", "subject": "Number theory", "description": "Proves the modularity conjecture for elliptic curves over imaginary quadratic fields: every elliptic curve over every imaginary quadratic field is modular, with matching local parameters at every place.", "scope_url": null }, { "id": "031", "title": "Uchida’s conjecture for open homomorphisms of Galois groups", "subject": "Number theory", "description": "Proves Uchida's conjecture: every continuous open homomorphism between Galois groups of possibly infinite solvably closed Galois extensions of number fields comes from a unique equivariant field embedding in the opposite direction. No restriction on the kernel or separate cyclotomic-compatibility assumption is needed.", "scope_url": null }, { "id": "032", "title": "The rational Hodge conjecture for CM abelian varieties", "subject": "Algebraic and complex geometry", "description": "Proves the rational Hodge conjecture for every complex CM abelian variety, in every dimension and codimension. Through Milne's theorems, this also gives the Tate conjecture for all abelian varieties over finite fields and the Hodge standard conjecture for abelian varieties in every characteristic.", "scope_url": null }, { "id": "033", "title": "Iitaka subadditivity, variation, and logarithmic additivity", "subject": "Algebraic and complex geometry", "description": "Proves Campana's orbifold Iitaka subadditivity conjecture for smooth Fujiki-class-mathcal C manifolds with rational simple-normal-crossing boundaries. For projective fibrations f:Uto V of smooth complex quasi-projective varieties with connected fibers, general fiber F, and barkappa(V)ge0, proves Popa's inequality barkappa(U)gekappa(F)+max{barkappa(V),mathop{mathrm{Var}}nolimits (f)}, where variation measures the whole geometric generic fiber.", "scope_url": "https://github.com/openai/math/blob/1581d4cbdd69a0faf34b2089614fe2a814316a6a/lean/docs/033.md" }, { "id": "034", "title": "Log abundance for compact Kähler spaces under logarithmic Iitaka subadditivity", "subject": "Algebraic and complex geometry", "description": "Using logarithmic Iitaka subadditivity, proves log abundance in every dimension for normal compact Kähler log canonical pairs with effective rational boundary: an analytically nef ℚ-Cartier adjoint is semiample. It also proves projective log abundance over every algebraically closed field of characteristic zero and the effective Iitaka fibration conjecture for smooth projective varieties of nonnegative Kodaira dimension in that setting. A further result resolves the finite-rational-coefficient index conjecture for connected projective semi-log-canonical log Calabi–Yau pairs over such fields in each fixed dimension and for each fixed finite set of rational boundary coefficients, with a uniform index independent of the number of components.", "scope_url": null }, { "id": "035", "title": "Log-canonical threefold abundance in numerical dimension one", "subject": "Algebraic and complex geometry", "description": "Proves log abundance for projective log canonical threefold pairs over algebraically closed fields of characteristic p > 3 when the effective boundary is rational and the ℚ-Cartier adjoint is nef of numerical dimension one. The adjoint is semiample, without requiring the original variety to be terminal or ℚ-factorial.", "scope_url": null }, { "id": "036", "title": "Numerical semiampleness and generalized minimal models", "subject": "Algebraic and complex geometry", "description": "Proves numerical semiampleness for nef adjoints K_X+B+M with K_X+B pseudo-effective and M nef rational, for projective klt rational pairs over algebraically closed characteristic-zero fields and smooth compact Kähler rational klt simple-normal-crossing pairs, using Bott–Chern cohomology in the latter case. Separately, projective generalized log canonical rational pairs over such fields admit minimal models for pseudo-effective adjoints and Mori fiber spaces otherwise, with nef b-data fixed.", "scope_url": null }, { "id": "037", "title": "The ordinary-double-point volume gap", "subject": "Algebraic and complex geometry", "description": "Proves the ordinary-double-point volume-gap conjecture: every singular complex algebraic klt germ of dimension n ≥ 2, with zero boundary, has normalized volume at most 2(n-1)^n. Equality holds precisely for an analytic ordinary double point.", "scope_url": null }, { "id": "038", "title": "Fujita’s freeness conjecture", "subject": "Algebraic and complex geometry", "description": "Proves Fujita's freeness conjecture at its sharp bound in every dimension: for a smooth projective complex variety X of dimension n and an ample line bundle L, the adjoint K_X+mL is globally generated for every integer mge n+1.", "scope_url": null }, { "id": "039", "title": "Nagata’s conjecture and maximal Seshadri constants", "subject": "Algebraic and complex geometry", "description": "Proves Nagata's strict inequality sum_i m_ilt dsqrt r for every nonzero effective plane curve of degree d through r ≥ 10 very general complex points, with arbitrary multiplicities mi. 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.", "scope_url": "https://github.com/openai/math/blob/1581d4cbdd69a0faf34b2089614fe2a814316a6a/lean/docs/039.md" }, { "id": "040", "title": "Bloch’s conjecture for complex surfaces", "subject": "Algebraic and complex geometry", "description": "Proves Bloch's conjecture: for every smooth connected projective complex surface S with p_g(S)=0, the Albanese map mathrm{CH}_0(S)^0tomathrm{Alb}(S)(mathbb C) on integral degree-zero zero-cycles is an isomorphism. This combines the new p_g=q=0 theorem with the classical theorem of Bloch, Kas, and Lieberman.", "scope_url": null }, { "id": "041", "title": "Hyperkähler SYZ and projective-space bases", "subject": "Algebraic and complex geometry", "description": "Proves the strong hyperkähler SYZ conjecture: every holomorphic line bundle with nonzero nef isotropic first Chern class on a compact irreducible holomorphic symplectic Kähler manifold is semiample. It also proves that every projective Lagrangian fibration with normal projective base has projective space as its base, in every dimension and deformation type.", "scope_url": null }, { "id": "042", "title": "Oka classification for minimal compact complex surfaces: Kodaira dimension zero and class VII", "subject": "Algebraic and complex geometry", "description": "Proves that every complex K3 surface is Oka, including nonprojective surfaces. More generally, every connected minimal compact complex surface of Kodaira dimension zero is Oka; a connected minimal compact complex surface of class VII is Oka exactly when it is a Hopf or Enoki surface.", "scope_url": null }, { "id": "043", "title": "P = W for fixed-determinant SLn moduli spaces", "subject": "Algebraic and complex geometry", "description": "Proves P_k=W_{2k}=W_{2k+1} on the full rational cohomology of smooth coprime fixed-determinant, trace-free Higgs moduli spaces and their character varieties for composite ranks over smooth projective complex curves of genus at least two. This includes variant cohomology and, together with the known prime-rank theorems, establishes the fixed-determinant P = W conjecture in every coprime rank.", "scope_url": null }, { "id": "044", "title": "The equivariant cohomological Hikita conjecture", "subject": "Algebraic and complex geometry", "description": "Proves the equivariant cohomological Hikita correspondence for every finite quiver, including loops and multiple arrows, with arbitrary dimension and framing vectors and commuting flavor torus. When every semistable point is stable and the gauge action is free, the equivariant cohomology of the Nakajima variety is canonically the coordinate ring of the scheme-theoretic cocharacter-fixed locus of its flavor-deformed Coulomb branch.", "scope_url": null }, { "id": "046", "title": "Shafarevich counterexamples in dimension two and with large fundamental group", "subject": "Algebraic and complex geometry", "description": "Constructs a smooth projective complex fourfold with large fundamental group whose universal cover contains no positive-dimensional compact analytic subvariety but is neither Stein nor holomorphically convex. A separate smooth projective complex surface already disproves unrestricted Shafarevich holomorphic convexity in dimension two.", "scope_url": null }, { "id": "047", "title": "Zariski cancellation and affine fibrations over the complex numbers", "subject": "Algebraic and complex geometry", "description": "Constructs an integral complex affine fourfold Xnotcongmathbb A^4 with Xtimesmathbb A^1congmathbb A^5, disproving affine-space cancellation over ℂ in dimension four. It also disproves the Dolgachev–Weisfeiler affine-fibration conjecture: smooth surjections Xtomathbb A^1 and mathbb A^5tomathbb A^2 have every residue-field fiber isomorphic to affine three-space but are not Zariski-locally trivial.", "scope_url": "https://github.com/openai/math/blob/1581d4cbdd69a0faf34b2089614fe2a814316a6a/lean/docs/047.md" }, { "id": "048", "title": "A characteristic-zero counterexample to Lipman–Zariski", "subject": "Algebraic and complex geometry", "description": "Constructs a singular normal affine complex surface with free rank-two tangent sheaf, disproving the characteristic-zero Lipman–Zariski conjecture.", "scope_url": null }, { "id": "049", "title": "A stable-coordinate counterexample in four variables", "subject": "Algebraic and complex geometry", "description": "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.", "scope_url": "https://github.com/openai/math/blob/1581d4cbdd69a0faf34b2089614fe2a814316a6a/lean/docs/049.md" }, { "id": "050", "title": "A counterexample to Griffiths’ positivity conjecture", "subject": "Algebraic and complex geometry", "description": "Constructs ample rank-two bundles on mathbb P^1timesmathbb P^1 with no smooth Hermitian metric of strictly Griffiths-positive curvature, disproving Griffiths' positivity conjecture already on the quadric surface.", "scope_url": "https://github.com/openai/math/blob/1581d4cbdd69a0faf34b2089614fe2a814316a6a/lean/docs/050.md" }, { "id": "051", "title": "Kobayashi’s canonical-ampleness conjecture", "subject": "Algebraic and complex geometry", "description": "Every compact connected Kähler manifold of positive complex dimension with no nonconstant entire curve has ample canonical bundle and is therefore projective. This proves Kobayashi's canonical-ampleness conjecture in the smooth compact Kähler setting.", "scope_url": null }, { "id": "052", "title": "Tangent splittings and product decompositions", "subject": "Algebraic and complex geometry", "description": "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.", "scope_url": "https://github.com/openai/math/blob/1581d4cbdd69a0faf34b2089614fe2a814316a6a/lean/docs/052.md" }, { "id": "053", "title": "A counterexample to Pixton completeness in Chow", "subject": "Algebraic and complex geometry", "description": "Constructs a tautological relation on a moduli space of stable pointed curves that vanishes in rational Chow, hence in rational cohomology, but lies outside Pixton's original relation span. This disproves the Chow and rational-cohomological forms of his original completeness conjecture.", "scope_url": null }, { "id": "054", "title": "Irrational cubic fourfolds with Hodge-theoretic and categorical K3 associations", "subject": "Algebraic and complex geometry", "description": "For every sufficiently large admissible Hassett discriminant, a very general smooth complex cubic fourfold is irrational despite having both an untwisted geometric K3 category and an integral Hodge-theoretic K3 association. This disproves Kuznetsov's rationality conjecture and the sufficiency of the associated-K3 criterion for rationality; the discriminant threshold is ineffective.", "scope_url": null }, { "id": "055", "title": "Gepner symmetry and large-volume stability on threefolds", "subject": "Algebraic and complex geometry", "description": "Proves Toda's Gepner conjecture for every smooth complex quintic threefold, constructing a numerical Bridgeland stability condition with the prescribed phase shift 2/5. Also constructs numerical Bridgeland stability conditions at every sufficiently large volume on all smooth projective complex threefolds with trivial canonical bundle, with the exact ordinary and square-root-Todd central charges.", "scope_url": null }, { "id": "056", "title": "Termination of projective and Kähler fourfold minimal model programs", "subject": "Algebraic and complex geometry", "description": "Proves termination of every existing generalized log canonical flip sequence on globally Weil ℚ-factorial compact Kähler fourfolds, with rational boundary, fixed rational analytically nef b-data, and projective small flip diagrams with the prescribed ample signs. Also proves termination of arbitrary permitted minimal model programs for projective log canonical fourfolds with rational boundary in characteristic zero.", "scope_url": null }, { "id": "057", "title": "Fundamental groups of special complex varieties and root orbifolds", "subject": "Algebraic and complex geometry", "description": "Proves Campana's abelianity conjecture: special compact Kähler manifolds have virtually abelian fundamental groups. Using this theorem, establishes the same conclusion for order-two root orbifolds of smooth projective complex fourfolds along one nonempty smooth connected divisor, when special in the stated differential-line sense. For smooth special complex quasi-projective varieties, proves that every finite-dimensional complex linear representation of the fundamental group has virtually nilpotent image of class at most two.", "scope_url": null }, { "id": "058", "title": "Semialgebraic universal covers and bounded domains", "subject": "Algebraic and complex geometry", "description": "Proves the Kollár–Pardon conjecture: the semialgebraic universal covers of connected normal projective complex varieties are exactly products Dtimesmathbb C^mtimes 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.", "scope_url": "https://github.com/openai/math/blob/1581d4cbdd69a0faf34b2089614fe2a814316a6a/lean/docs/058.md" }, { "id": "059", "title": "Counterexamples to Zariski’s multiplicity conjecture", "subject": "Algebraic and complex geometry", "description": "Disproves Zariski's multiplicity conjecture by constructing reduced holomorphic hypersurface germs that are ambiently homeomorphic but have different multiplicities. The examples include hypersurfaces in ℂ4 with isolated critical points and multiplicities four and five.", "scope_url": null }, { "id": "060", "title": "The Global Spherical Shell conjecture", "subject": "Algebraic and complex geometry", "description": "Every connected minimal compact complex surface of class VII with b_2gt 0 contains a global spherical shell, proving the positive-b2 Global Spherical Shell conjecture. Such a shell is a holomorphically embedded neighborhood of the standard three-sphere in mathbb C^2setminus{0} whose complement is connected.", "scope_url": null }, { "id": "062", "title": "Projective contact classification and the LeBrun–Salamon conjecture", "subject": "Algebraic and complex geometry", "description": "Proves the LeBrun–Salamon conjecture: every closed connected positive quaternionic-Kähler manifold of real dimension at least eight is homothetic to a compact symmetric Wolf space. It also proves contact-Fano homogeneity and classifies smooth connected complex projective contact manifolds of complex dimension at least three: those with b_2=1 are adjoint varieties with their canonical contact structures, while those with b_2ge2 have underlying manifold mathbb P(T^*Z) for a smooth projective variety Z.", "scope_url": null }, { "id": "063", "title": "The generalized Mukai conjecture", "subject": "Algebraic and complex geometry", "description": "Proves the generalized Mukai conjecture: every positive-dimensional smooth complex projective Fano manifold of dimension n, Picard number ρ and pseudoindex ι satisfies rho(iota-1)le n, with equality exactly for (mathbb P^{iota-1})^rho. Here the pseudoindex is the least anticanonical degree of a rational curve.", "scope_url": null }, { "id": "064", "title": "Topological triviality of μ-constant surface singularities", "subject": "Algebraic and complex geometry", "description": "Proves topological right-triviality for every holomorphic one-parameter family of isolated hypersurface singularities in ℂ3 with constant Milnor number, resolving the surface case of the μ-constant problem. After shrinking the parameter disk and representatives, ambient homeomorphisms vary jointly continuously, fix the origin section and preserve the defining functions.", "scope_url": null }, { "id": "065", "title": "Virasoro constraints for complete intersections and projective-bundle towers", "subject": "Algebraic and complex geometry", "description": "Proves the full ordinary unreduced descendant Virasoro conjecture for smooth complete intersections in complex projective space, in every genus and curve class with arbitrary cohomology insertions. The constraints also pass from any smooth projective complex base satisfying them to the projectivization of every algebraic vector bundle of rank at least two, and hence to projective-bundle towers.", "scope_url": null }, { "id": "066", "title": "Bounded klt complements for Fano contractions", "subject": "Algebraic and complex geometry", "description": "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.", "scope_url": "https://github.com/openai/math/blob/1581d4cbdd69a0faf34b2089614fe2a814316a6a/lean/docs/066.md" }, { "id": "067", "title": "The Campana–Peternell conjecture in dimension six", "subject": "Algebraic and complex geometry", "description": "Proves the Campana–Peternell conjecture in complex dimension six: every smooth connected complex projective Fano sixfold with nef tangent bundle is rational homogeneous.", "scope_url": null }, { "id": "068", "title": "Anticanonical nonvanishing in every dimension", "subject": "Algebraic and complex geometry", "description": "If X is a smooth connected complex projective variety and -K_X admits a smooth Hermitian metric with nonnegative curvature, then H^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.", "scope_url": null }, { "id": "069", "title": "Global quantum geometric Langlands at irrational level", "subject": "Algebraic and complex geometry", "description": "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 cinmathbb Csetminusmathbb 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.", "scope_url": null }, { "id": "071", "title": "Koebe’s circle-domain conjecture", "subject": "Real and complex analysis", "description": "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.", "scope_url": "https://github.com/openai/math/blob/1581d4cbdd69a0faf34b2089614fe2a814316a6a/lean/docs/071.md" }, { "id": "072", "title": "Brennan's conjecture and the integral-means spectrum", "subject": "Real and complex analysis", "description": "Proves Brennan's conjecture: for every conformal bijection ϕ from a simply connected plane domain onto the disk, |phi'|^s is area-integrable for 4/3lt slt 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.", "scope_url": "https://github.com/openai/math/blob/1581d4cbdd69a0faf34b2089614fe2a814316a6a/lean/docs/072.md" }, { "id": "073", "title": "The Falconer distance conjecture", "subject": "Real and complex analysis", "description": "Resolves the Falconer distance conjecture in every dimension d ≥ 2: every compact set Esubsetmathbb R^d with Hausdorff dimension greater than d/2 determines a set of Euclidean distances of positive Lebesgue measure.", "scope_url": "https://github.com/openai/math/blob/1581d4cbdd69a0faf34b2089614fe2a814316a6a/lean/docs/073.md" }, { "id": "074", "title": "Kakeya in three and four dimensions", "subject": "Real and complex analysis", "description": "Resolves the Kakeya maximal conjecture in three dimensions and the Hausdorff-dimension conjecture in four. In three dimensions, the radius-δ tube maximal operator maps L^3(mathbb R^3) to L^3(S^2) with norm O_varepsilon(delta^{-varepsilon}) for every ε > 0. In four dimensions, every set containing a unit segment in every direction has Hausdorff dimension four.", "scope_url": null }, { "id": "075", "title": "The Llog L Fourier-convergence conjecture", "subject": "Real and complex analysis", "description": "Proves that the ordinary symmetric Fourier partial sums of every complex-valued function in Llog L(mathbb T) converge almost everywhere along the full sequence. This resolves the classical sufficiency conjecture at the Llog L scale.", "scope_url": "https://github.com/openai/math/blob/1581d4cbdd69a0faf34b2089614fe2a814316a6a/lean/docs/075.md" }, { "id": "076", "title": "Real ultraflat Littlewood polynomials and unbounded binary merit factors", "subject": "Real and complex analysis", "description": "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.", "scope_url": "https://github.com/openai/math/blob/1581d4cbdd69a0faf34b2089614fe2a814316a6a/lean/docs/076.md" }, { "id": "077", "title": "Fourier restriction for positively curved surfaces", "subject": "Real and complex analysis", "description": "Proves the diagonal Fourier extension conjecture for positively curved surfaces in three dimensions. For every compact smooth positively curved surface Sigmasubsetmathbb R^3, including surfaces with boundary, the extension operator is bounded from L^p(Sigma) to L^p(mathbb R^3) for every p > 3.", "scope_url": null }, { "id": "078", "title": "The three-dimensional Bochner–Riesz conjecture", "subject": "Real and complex analysis", "description": "Resolves the three-dimensional Bochner–Riesz conjecture in its strict range: the Bochner–Riesz multipliers of order δ are bounded on L^p(mathbb R^3) for every 1le pleinfty whenever deltagt max{3|1/p-1/2|-1/2,0}.", "scope_url": null }, { "id": "079", "title": "Local smoothing in three dimensions", "subject": "Real and complex analysis", "description": "Resolves Sogge's local smoothing conjecture for the Euclidean wave equation in three spatial dimensions. The estimate holds throughout the full strict range 2lt plt infty, with Sobolev regularity above max{0,1-3/p}. In particular, the critical L3 estimate holds with every positive Sobolev loss.", "scope_url": null }, { "id": "080", "title": "The exact Sobolev endpoint for Schrödinger convergence", "subject": "Real and complex analysis", "description": "Proves almost-everywhere convergence e^{itDelta}fto f as tdownarrow0 for every fin 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.", "scope_url": null }, { "id": "081", "title": "Riesz transforms and rectifiability in higher codimension", "subject": "Real and complex analysis", "description": "Resolves the remaining higher-codimension Riesz-transform rectifiability problem: for d ≥ 4 and 2le nle 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.", "scope_url": "https://github.com/openai/math/blob/1581d4cbdd69a0faf34b2089614fe2a814316a6a/lean/docs/081.md" }, { "id": "082", "title": "Annular variation and dyadic absolute bounds for the triangular Hilbert transform", "subject": "Real and complex analysis", "description": "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^3times L^3times L^3 point.", "scope_url": "https://github.com/openai/math/blob/1581d4cbdd69a0faf34b2089614fe2a814316a6a/lean/docs/082.md" }, { "id": "083", "title": "Hilbert transforms along Lipschitz directions", "subject": "Real and complex analysis", "description": "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.", "scope_url": "https://github.com/openai/math/blob/1581d4cbdd69a0faf34b2089614fe2a814316a6a/lean/docs/083.md" }, { "id": "084", "title": "The geometric case of the Erdős similarity conjecture", "subject": "Real and complex analysis", "description": "For every fixed qin(0,1), constructs compact subsets of [0,1] with measure arbitrarily close to one containing no translated and nontrivially dilated copy of {q^n:nge1}, with dilations of either sign. This resolves the geometric-progression case of the Erdős similarity conjecture for every ratio.", "scope_url": "https://github.com/openai/math/blob/1581d4cbdd69a0faf34b2089614fe2a814316a6a/lean/docs/084.md" }, { "id": "085", "title": "Endpoint Sobolev regularity of centered disk averages", "subject": "Real and complex analysis", "description": "Resolves the planar centered-disk case of the Hajłasz–Onninen maximal-function regularity problem. For every real fin W^{1,1}(mathbb R^2), the centered disk maximal function satisfies |nabla Mf|_1le C|nabla f|_1 with an absolute constant. It belongs locally to W^{1,1} and has a globally integrable weak gradient.", "scope_url": "https://github.com/openai/math/blob/1581d4cbdd69a0faf34b2089614fe2a814316a6a/lean/docs/085.md" }, { "id": "086", "title": "An L3 bound for the trilinear Hilbert transform", "subject": "Real and complex analysis", "description": "Proves that the principal-value trilinear Hilbert transform with shifts x-t, x-2t, x-3t is bounded from L^3(mathbb R)^3 to L^1(mathbb R). This resolves the L3 exponent case of the standard conjecture for slopes 1, 2, 3.", "scope_url": null }, { "id": "087", "title": "The Mahler conjectures, functional inequalities and polar-product symplectic width", "subject": "Convex and metric geometry", "description": "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 Ktimes K^circ in dimension 2n has Gromov width 4.", "scope_url": "https://github.com/openai/math/blob/1581d4cbdd69a0faf34b2089614fe2a814316a6a/lean/docs/087.md" }, { "id": "088", "title": "Sharp projection-body inequalities and a counterexample to simplex maximization", "subject": "Convex and metric geometry", "description": "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.", "scope_url": "https://github.com/openai/math/blob/1581d4cbdd69a0faf34b2089614fe2a814316a6a/lean/docs/088.md" }, { "id": "089", "title": "Bounded-distortion L1 embeddings of planar and bounded-treewidth graphs", "subject": "Convex and metric geometry", "description": "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.", "scope_url": "https://github.com/openai/math/blob/1581d4cbdd69a0faf34b2089614fe2a814316a6a/lean/docs/089.md" }, { "id": "090", "title": "Triangular-lattice optimality, long-range Riesz and Coulomb energies, and spherical logarithmic energy", "subject": "Convex and metric geometry", "description": "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 0lt slt 2 and Coulomb energy, resolving Sandier–Serfaty and the two-dimensional Brauchart–Hardin–Saff conjecture on the linear term of optimal spherical logarithmic energy.", "scope_url": "https://github.com/openai/math/blob/1581d4cbdd69a0faf34b2089614fe2a814316a6a/lean/docs/090.md" }, { "id": "091", "title": "Logarithmic and Lp Brunn–Minkowski inequalities and the B-conjecture", "subject": "Convex and metric geometry", "description": "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 0lt plt 1.", "scope_url": "https://github.com/openai/math/blob/1581d4cbdd69a0faf34b2089614fe2a814316a6a/lean/docs/091.md" }, { "id": "092", "title": "The optimal order of convex-body covering density", "subject": "Convex and metric geometry", "description": "Determines the optimal worst-case covering density as Theta(nlog n), for both lattice and unrestricted translative coverings. Every convex body in ℝn, n ≥ 2, admits a lattice covering of density at most Cnlog n; centrally symmetric examples in every sufficiently large dimension require at least cnlog n even without the lattice restriction, for absolute c,Cgt 0.", "scope_url": "https://github.com/openai/math/blob/1581d4cbdd69a0faf34b2089614fe2a814316a6a/lean/docs/092.md" }, { "id": "093", "title": "Dimension-free logarithmic Sobolev inequality for subgaussian log-concave measures", "subject": "Convex and metric geometry", "description": "Proves a dimension-free logarithmic Sobolev inequality for centered log-concave densities with uniformly subgaussian linear marginals, with constant bounded by a universal multiple of the squared linear subgaussian parameter.", "scope_url": null }, { "id": "094", "title": "Subpolynomial dimension reduction in Lp", "subject": "Convex and metric geometry", "description": "For every fixed 1lt plt infty and distortion D > 1, every n-point subset of real Lp 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.", "scope_url": "https://github.com/openai/math/blob/1581d4cbdd69a0faf34b2089614fe2a814316a6a/lean/docs/094.md" }, { "id": "095", "title": "Hyperbolicity cones without semidefinite lifts", "subject": "Convex and metric geometry", "description": "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.", "scope_url": "https://github.com/openai/math/blob/1581d4cbdd69a0faf34b2089614fe2a814316a6a/lean/docs/095.md" }, { "id": "096", "title": "The Gaussian propeller conjecture in every dimension", "subject": "Convex and metric geometry", "description": "Proves that the sum of squared Gaussian first moments of any finite measurable partition is at most 9/(8pi). In dimension at least two, three planar sectors of angle 2pi/3, extended orthogonally, attain the bound. Combined with the separate Unique Games theorem, this proves NP-hardness of improving the loss factor (8pi/9)(1-1/k) for identity-target kernel clustering with fixed k ≥ 3 on rational centered positive semidefinite inputs.", "scope_url": "https://github.com/openai/math/blob/1581d4cbdd69a0faf34b2089614fe2a814316a6a/lean/docs/096.md" }, { "id": "097", "title": "The Euclidean Steinitz–Bergström bound", "subject": "Convex and metric geometry", "description": "Proves that any finite sequence in the Euclidean unit ball of ℝd admits signs keeping every partial sum within Csqrt 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).", "scope_url": "https://github.com/openai/math/blob/1581d4cbdd69a0faf34b2089614fe2a814316a6a/lean/docs/097.md" }, { "id": "098", "title": "Compact counterexamples to bi-Lipschitz dimension reduction", "subject": "Convex and metric geometry", "description": "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.", "scope_url": "https://github.com/openai/math/blob/1581d4cbdd69a0faf34b2089614fe2a814316a6a/lean/docs/098.md" }, { "id": "099", "title": "The sharp exponential scale of edit-distance distortion", "subject": "Convex and metric geometry", "description": "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,loglog 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.", "scope_url": "https://github.com/openai/math/blob/1581d4cbdd69a0faf34b2089614fe2a814316a6a/lean/docs/099.md" }, { "id": "100", "title": "Cylinder coverings below the half-area bound", "subject": "Convex and metric geometry", "description": "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.", "scope_url": "https://github.com/openai/math/blob/1581d4cbdd69a0faf34b2089614fe2a814316a6a/lean/docs/100.md" }, { "id": "101", "title": "The sharp simplex conjecture for isotropic constants", "subject": "Convex and metric geometry", "description": "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.", "scope_url": null }, { "id": "102", "title": "The Unique Games Conjecture and optimal approximation thresholds", "subject": "Theoretical computer science", "description": "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.", "scope_url": "https://github.com/openai/math/blob/1581d4cbdd69a0faf34b2089614fe2a814316a6a/lean/docs/102.md" }, { "id": "103", "title": "Exact derandomization of logarithmic space: mathsf L=mathsf{RL}=mathsf{BPL}", "subject": "Theoretical computer science", "description": "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.", "scope_url": "https://github.com/openai/math/blob/1581d4cbdd69a0faf34b2089614fe2a814316a6a/lean/docs/103.md" }, { "id": "104", "title": "Quasipolynomial algorithms for mean-payoff, stochastic and parity games", "subject": "Theoretical computer science", "description": "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.", "scope_url": "https://github.com/openai/math/blob/1581d4cbdd69a0faf34b2089614fe2a814316a6a/lean/docs/104.md" }, { "id": "105", "title": "Perfect completeness for 2-to-1 games", "subject": "Theoretical computer science", "description": "Proves Khot's 2-to-1 Games Conjecture with perfect completeness: for every fixed rational deltain(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.", "scope_url": "https://github.com/openai/math/blob/1581d4cbdd69a0faf34b2089614fe2a814316a6a/lean/docs/105.md" }, { "id": "106", "title": "Hardness of coloring three-colorable graphs", "subject": "Theoretical computer science", "description": "It is NP-hard to color a three-colorable graph using any fixed number c ≥ 3 of colors. More strongly, for every fixed 0lt deltalt 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.", "scope_url": "https://github.com/openai/math/blob/1581d4cbdd69a0faf34b2089614fe2a814316a6a/lean/docs/106.md" }, { "id": "107", "title": "Matrix multiplication with exponent at most 9/4", "subject": "Theoretical computer science", "description": "Proves omegale9/4 over ℂ, giving O_varepsilon(n^{9/4+varepsilon}) arithmetic operations for square matrix multiplication. In characteristic zero, some inner dimension na 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.", "scope_url": "https://github.com/openai/math/blob/1581d4cbdd69a0faf34b2089614fe2a814316a6a/lean/docs/107.md" }, { "id": "108", "title": "A cubic permanent–determinant lower bound", "subject": "Theoretical computer science", "description": "Proves an Omega(n^3) lower bound for the border determinantal complexity of the ntimes 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.", "scope_url": "https://github.com/openai/math/blob/1581d4cbdd69a0faf34b2089614fe2a814316a6a/lean/docs/108.md" }, { "id": "109", "title": "Integer multiplication below nlog n", "subject": "Theoretical computer science", "description": "Multiplies two n-bit integers exactly at every input length in deterministic worst-case time O(n(log n)^{1-kappa}), with kappa=2^{-182}, on one fixed finite-alphabet Turing machine with finitely many one-dimensional tapes. This disproves the Schönhage–Strassen nlog n optimality conjecture in the ordinary multitape bit model.", "scope_url": null }, { "id": "110", "title": "Optimal-order randomized k-server on arbitrary metrics", "subject": "Theoretical computer science", "description": "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.", "scope_url": "https://github.com/openai/math/blob/1581d4cbdd69a0faf34b2089614fe2a814316a6a/lean/docs/110.md" }, { "id": "111", "title": "One-sample matroid prophet inequalities against an almighty adversary", "subject": "Theoretical computer science", "description": "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.", "scope_url": "https://github.com/openai/math/blob/1581d4cbdd69a0faf34b2089614fe2a814316a6a/lean/docs/111.md" }, { "id": "112", "title": "Beyond the square-root exponent for depth-three circuits", "subject": "Theoretical computer science", "description": "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.", "scope_url": "https://github.com/openai/math/blob/1581d4cbdd69a0faf34b2089614fe2a814316a6a/lean/docs/112.md" }, { "id": "113", "title": "Approximate counting and entropy of perfect matchings", "subject": "Theoretical computer science", "description": "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.", "scope_url": "https://github.com/openai/math/blob/1581d4cbdd69a0faf34b2089614fe2a814316a6a/lean/docs/113.md" }, { "id": "114", "title": "Approximate counting of common integer polymatroid bases", "subject": "Theoretical computer science", "description": "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.", "scope_url": "https://github.com/openai/math/blob/1581d4cbdd69a0faf34b2089614fe2a814316a6a/lean/docs/114.md" }, { "id": "115", "title": "Sampling and counting contingency tables with arbitrary margins", "subject": "Theoretical computer science", "description": "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.", "scope_url": "https://github.com/openai/math/blob/1581d4cbdd69a0faf34b2089614fe2a814316a6a/lean/docs/115.md" }, { "id": "116", "title": "Uniform black-box noncommutative identity testing across characteristics", "subject": "Theoretical computer science", "description": "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.", "scope_url": "https://github.com/openai/math/blob/1581d4cbdd69a0faf34b2089614fe2a814316a6a/lean/docs/116.md" }, { "id": "117", "title": "Uniform sparsest cut: hardness and semidefinite gaps", "subject": "Theoretical computer science", "description": "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}/(loglog n)^3, approaching the square-root-logarithmic upper bound.", "scope_url": "https://github.com/openai/math/blob/1581d4cbdd69a0faf34b2089614fe2a814316a6a/lean/docs/117.md" }, { "id": "118", "title": "Bin packing and unbounded configuration-LP gaps", "subject": "Theoretical computer science", "description": "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.", "scope_url": "https://github.com/openai/math/blob/1581d4cbdd69a0faf34b2089614fe2a814316a6a/lean/docs/118.md" }, { "id": "119", "title": "The Courtade–Kumar and Hellinger conjectures", "subject": "Theoretical computer science", "description": "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.", "scope_url": "https://github.com/openai/math/blob/1581d4cbdd69a0faf34b2089614fe2a814316a6a/lean/docs/119.md" }, { "id": "120", "title": "Almost-linear-time exact matching and prescribed-degree factors in general graphs", "subject": "Theoretical computer science", "description": "Gives a randomized algorithm finding an exact maximum-cardinality matching in any simple undirected graph in (n+m)^{1+o(1)} word time, with success probability at least 2/3. The time bound holds on every computation path. The same guarantees apply to finding a spanning subgraph with prescribed admissible vertex degrees, or deciding that none exists.", "scope_url": null }, { "id": "121", "title": "Almost-linear approximation of edit distance", "subject": "Theoretical computer science", "description": "For every fixed rational varepsilonin(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.", "scope_url": "https://github.com/openai/math/blob/1581d4cbdd69a0faf34b2089614fe2a814316a6a/lean/docs/121.md" }, { "id": "122", "title": "Quantitative trace-reconstruction bounds with a uniform decoder", "subject": "Theoretical computer science", "description": "At every fixed deletion probability in (0,1), reconstructing an arbitrary length-n binary string requires n^{Omega(loglog 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.", "scope_url": "https://github.com/openai/math/blob/1581d4cbdd69a0faf34b2089614fe2a814316a6a/lean/docs/122.md" }, { "id": "124", "title": "Polynomial-time scheduling on three identical machines", "subject": "Theoretical computer science", "description": "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.", "scope_url": "https://github.com/openai/math/blob/1581d4cbdd69a0faf34b2089614fe2a814316a6a/lean/docs/124.md" }, { "id": "125", "title": "The metric k-median approximation threshold and recovery", "subject": "Theoretical computer science", "description": "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 Pne NP, the optimal infimum approximation factor is 1+2/e.", "scope_url": "https://github.com/openai/math/blob/1581d4cbdd69a0faf34b2089614fe2a814316a6a/lean/docs/125.md" }, { "id": "126", "title": "Exponential semidefinite complexity of perfect matching", "subject": "Theoretical computer science", "description": "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 0lt rholt 1, allowing arbitrary real positive semidefinite factors.", "scope_url": "https://github.com/openai/math/blob/1581d4cbdd69a0faf34b2089614fe2a814316a6a/lean/docs/126.md" }, { "id": "127", "title": "Average sensitivity of polynomial threshold functions", "subject": "Theoretical computer science", "description": "Proves that a degree-at-most-d polynomial threshold function on the uniform n-dimensional Boolean cube has average sensitivity at most 8dsqrt n, uniformly for 1le dle 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.", "scope_url": "https://github.com/openai/math/blob/1581d4cbdd69a0faf34b2089614fe2a814316a6a/lean/docs/127.md" }, { "id": "128", "title": "A factor-two approximation for shortest common superstring", "subject": "Theoretical computer science", "description": "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.", "scope_url": "https://github.com/openai/math/blob/1581d4cbdd69a0faf34b2089614fe2a814316a6a/lean/docs/128.md" }, { "id": "129", "title": "Exponential state costs for two-way automata", "subject": "Theoretical computer science", "description": "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.", "scope_url": "https://github.com/openai/math/blob/1581d4cbdd69a0faf34b2089614fe2a814316a6a/lean/docs/129.md" }, { "id": "130", "title": "Exact Fourier transforms below nlog n", "subject": "Theoretical computer science", "description": "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.", "scope_url": "https://github.com/openai/math/blob/1581d4cbdd69a0faf34b2089614fe2a814316a6a/lean/docs/130.md" }, { "id": "131", "title": "Rapid mixing of graph switches for every degree sequence", "subject": "Theoretical computer science", "description": "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.", "scope_url": "https://github.com/openai/math/blob/1581d4cbdd69a0faf34b2089614fe2a814316a6a/lean/docs/131.md" }, { "id": "132", "title": "A superquadratic separation of sensitivity and block sensitivity", "subject": "Theoretical computer science", "description": "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.", "scope_url": "https://github.com/openai/math/blob/1581d4cbdd69a0faf34b2089614fe2a814316a6a/lean/docs/132.md" }, { "id": "133", "title": "The computational complexity of Weisfeiler–Leman refinement", "subject": "Theoretical computer science", "description": "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.", "scope_url": "https://github.com/openai/math/blob/1581d4cbdd69a0faf34b2089614fe2a814316a6a/lean/docs/133.md" }, { "id": "134", "title": "Generalized star height at most three", "subject": "Theoretical computer science", "description": "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.", "scope_url": "https://github.com/openai/math/blob/1581d4cbdd69a0faf34b2089614fe2a814316a6a/lean/docs/134.md" }, { "id": "135", "title": "Homogeneous depth-five lower bounds for iterated matrix multiplication", "subject": "Theoretical computer science", "description": "Over every characteristic-zero field, the (1,1) entry of a product of n independent ntimes 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.", "scope_url": "https://github.com/openai/math/blob/1581d4cbdd69a0faf34b2089614fe2a814316a6a/lean/docs/135.md" }, { "id": "136", "title": "A quasilinear PCP theorem for PPAD", "subject": "Theoretical computer science", "description": "Resolves the quasilinear PCP-for-PPAD conjecture. An End-of-Line instance of length N reduces to numerical circuit constraints of total length N(log N)^{O(1)} such that any polynomially encoded rational assignment satisfying all but a fixed fraction to fixed accuracy yields an endpoint solution. Such assignments always exist, giving robust local verification with only quasilinear size overhead.", "scope_url": null }, { "id": "137", "title": "One-tape time simulation in two-fifths-power space", "subject": "Theoretical computer science", "description": "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.", "scope_url": "https://github.com/openai/math/blob/1581d4cbdd69a0faf34b2089614fe2a814316a6a/lean/docs/137.md" }, { "id": "138", "title": "Subset Sum in O(2^{0.49n}) time", "subject": "Theoretical computer science", "description": "Gives a uniform randomized classical algorithm for worst-case Subset Sum in ordinary O(2^{0.49n}) word-RAM time on polynomial-bit inputs, where n counts the integers. The time bound holds on every execution and success probability is at least 2/3 on every input. Inputs may repeat positive integers; words have O(n+b) bits for maximum input bit length b.", "scope_url": null }, { "id": "139", "title": "Subpolynomial query complexity for log-concave sampling", "subject": "Theoretical computer science", "description": "For C2 potentials with a supplied minimizer and Ipreceqnabla^2Vpreceq2I, 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.", "scope_url": "https://github.com/openai/math/blob/1581d4cbdd69a0faf34b2089614fe2a814316a6a/lean/docs/139.md" }, { "id": "140", "title": "Memory–sample lower bounds for noiseless Gaussian regression", "subject": "Theoretical computer science", "description": "For fixed A > 0, a one-pass learner with Ad^2 persistent bits needs Omega_A(dlog(1/epsilon)) noiseless Gaussian samples to recover a unit vector to angular error 0lt epsilonle1/10 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.", "scope_url": "https://github.com/openai/math/blob/1581d4cbdd69a0faf34b2089614fe2a814316a6a/lean/docs/140.md" }, { "id": "141", "title": "Existential–universal real sentences in the counting hierarchy", "subject": "Theoretical computer science", "description": "Proves that the existential theory of the reals lies in the counting hierarchy. More generally, truth of existential–universal real sentences can be decided at one fixed level of that hierarchy, even when their integer polynomials are specified by arithmetic circuits.", "scope_url": null }, { "id": "142", "title": "Deterministic polynomial factorization over prime fields", "subject": "Theoretical computer science", "description": "Gives a uniform deterministic algorithm that completely factors every nonzero dense degree-n polynomial over a prime field 𝔽p, including multiplicities, in bit complexity polynomial in (n+1)log p. The prime is supplied in binary. No randomness, integer-factorization or primitive-root oracle, or GRH assumption is required.", "scope_url": null }, { "id": "143", "title": "Hilbert's sixteenth problem: uniform bounds for limit cycles", "subject": "Dynamical systems and ergodic theory", "description": "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.", "scope_url": "https://github.com/openai/math/blob/1581d4cbdd69a0faf34b2089614fe2a814316a6a/lean/docs/143.md" }, { "id": "144", "title": "Banach’s simple Lebesgue-spectrum problem", "subject": "Dynamical systems and ergodic theory", "description": "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.", "scope_url": "https://github.com/openai/math/blob/1581d4cbdd69a0faf34b2089614fe2a814316a6a/lean/docs/144.md" }, { "id": "145", "title": "Rokhlin’s multiple-mixing problem", "subject": "Dynamical systems and ergodic theory", "description": "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.", "scope_url": "https://github.com/openai/math/blob/1581d4cbdd69a0faf34b2089614fe2a814316a6a/lean/docs/145.md" }, { "id": "146", "title": "Positive metric entropy for the standard map", "subject": "Dynamical systems and ergodic theory", "description": "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.", "scope_url": "https://github.com/openai/math/blob/1581d4cbdd69a0faf34b2089614fe2a814316a6a/lean/docs/146.md" }, { "id": "147", "title": "The near-boundary Birkhoff conjecture", "subject": "Dynamical systems and ergodic theory", "description": "Resolves the near-boundary Birkhoff conjecture for smooth strictly convex planar billiards of positive curvature. Such a billiard is an ellipse whenever a full grazing annulus is continuously foliated by individually invariant essential curves. A continuous physical collar of smooth closed convex caustics also suffices.", "scope_url": null }, { "id": "148", "title": "The entropy-rate dimension formula for self-similar measures", "subject": "Dynamical systems and ergodic theory", "description": "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.", "scope_url": "https://github.com/openai/math/blob/1581d4cbdd69a0faf34b2089614fe2a814316a6a/lean/docs/148.md" }, { "id": "149", "title": "Classwise permanence for weakly reversible mass-action systems", "subject": "Dynamical systems and ergodic theory", "description": "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.", "scope_url": "https://github.com/openai/math/blob/1581d4cbdd69a0faf34b2089614fe2a814316a6a/lean/docs/149.md" }, { "id": "150", "title": "Weak mixing of triangular billiards with an irrational angle", "subject": "Dynamical systems and ergodic theory", "description": "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.", "scope_url": "https://github.com/openai/math/blob/1581d4cbdd69a0faf34b2089614fe2a814316a6a/lean/docs/150.md" }, { "id": "151", "title": "A C1 counterexample to the entropy conjecture", "subject": "Dynamical systems and ergodic theory", "description": "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.", "scope_url": "https://github.com/openai/math/blob/1581d4cbdd69a0faf34b2089614fe2a814316a6a/lean/docs/151.md" }, { "id": "152", "title": "Zero entropy does not guarantee a smooth positive-volume model", "subject": "Dynamical systems and ergodic theory", "description": "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.", "scope_url": "https://github.com/openai/math/blob/1581d4cbdd69a0faf34b2089614fe2a814316a6a/lean/docs/152.md" }, { "id": "153", "title": "Arithmetic classification and non-Pisot singularity for Bernoulli convolutions", "subject": "Dynamical systems and ergodic theory", "description": "Classifies singular and absolutely continuous unbiased Bernoulli convolutions for every lambdain(0,1) by an infinite, one-sided approximation condition using explicit finite sets of algebraic units. It also proves singularity at reciprocals of every quartic Salem number in (1,2), giving examples beyond reciprocal Pisot parameters.", "scope_url": null }, { "id": "154", "title": "Pointwise multiple ergodic averages for mixing transformations", "subject": "Dynamical systems and ergodic theory", "description": "Proves almost-everywhere convergence of consecutive multiple ergodic averages of every finite length for invertible mixing probability-preserving transformations. For each fixed tuple of bounded functions, the limit is the product of their integrals, along all positive averaging lengths. No mixing rate or standardness assumption on the probability space is required.", "scope_url": null }, { "id": "155", "title": "A counterexample to periodic tiling in dimension three", "subject": "Combinatorics", "description": "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.", "scope_url": "https://github.com/openai/math/blob/1581d4cbdd69a0faf34b2089614fe2a814316a6a/lean/docs/155.md" }, { "id": "156", "title": "Borsuk's conjecture fails in dimension nine", "subject": "Combinatorics", "description": "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.", "scope_url": "https://github.com/openai/math/blob/1581d4cbdd69a0faf34b2089614fe2a814316a6a/lean/docs/156.md" }, { "id": "157", "title": "Graph coloring, clique minors, and Colin de Verdière invariants", "subject": "Combinatorics", "description": "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)lemu(G)+1. In the positive direction, every finite nonempty graph satisfies chi_{mathrm{list}}(G)le C h(G) for a universal constant C.", "scope_url": "https://github.com/openai/math/blob/1581d4cbdd69a0faf34b2089614fe2a814316a6a/lean/docs/157.md" }, { "id": "158", "title": "The Euclidean plane cannot be colored with five colors", "subject": "Combinatorics", "description": "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.", "scope_url": "https://github.com/openai/math/blob/1581d4cbdd69a0faf34b2089614fe2a814316a6a/lean/docs/158.md" }, { "id": "159", "title": "Erdős’s reciprocal-sum conjecture and quasipolynomial Szemerédi bounds", "subject": "Combinatorics", "description": "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_kNexp[-c_k(log N)^{varepsilon_k}], with positive constants depending only on k.", "scope_url": "https://github.com/openai/math/blob/1581d4cbdd69a0faf34b2089614fe2a814316a6a/lean/docs/159.md" }, { "id": "160", "title": "Superexponential van der Waerden numbers", "subject": "Combinatorics", "description": "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^{cklfloorlog_2 rrfloor} for an absolute c > 0, all r ≥ 2 and sufficiently large k, uniformly in r. In particular, W_r(k)^{1/k}toinfty for each fixed r.", "scope_url": "https://github.com/openai/math/blob/1581d4cbdd69a0faf34b2089614fe2a814316a6a/lean/docs/160.md" }, { "id": "161", "title": "Counterexamples to Sidorenko’s conjecture and the forcing conjecture", "subject": "Combinatorics", "description": "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.", "scope_url": "https://github.com/openai/math/blob/1581d4cbdd69a0faf34b2089614fe2a814316a6a/lean/docs/161.md" }, { "id": "162", "title": "Counterexamples to Ryser’s covering conjecture", "subject": "Combinatorics", "description": "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.", "scope_url": "https://github.com/openai/math/blob/1581d4cbdd69a0faf34b2089614fe2a814316a6a/lean/docs/162.md" }, { "id": "164", "title": "Hindman’s finite sums and products conjecture", "subject": "Combinatorics", "description": "Proves Hindman's finite sums and products conjecture: every finite coloring of the positive integers contains sets of any prescribed finite size whose nonempty subset sums and nonempty subset products all have one common color.", "scope_url": null }, { "id": "165", "title": "The Harary–Hill and Zarankiewicz crossing-number formulas", "subject": "Combinatorics", "description": "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.", "scope_url": "https://github.com/openai/math/blob/1581d4cbdd69a0faf34b2089614fe2a814316a6a/lean/docs/165.md" }, { "id": "166", "title": "The higher-dimensional Erdős distinct-distances conjecture", "subject": "Combinatorics", "description": "For every fixed d ≥ 3, any n ≥ 2 distinct points in ℝd determine at least c_dn^{2/d} distinct distances, with c_dgt 0 depending only on dimension. This matches the integer-grid order and resolves the higher-dimensional Erdős distinct-distances conjecture with a constant-factor bound.", "scope_url": null }, { "id": "167", "title": "Planar distinct distances and unit-distance bounds", "subject": "Combinatorics", "description": "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.", "scope_url": "https://github.com/openai/math/blob/1581d4cbdd69a0faf34b2089614fe2a814316a6a/lean/docs/167.md" }, { "id": "168", "title": "Combinatorial invariance of Kazhdan–Lusztig polynomials", "subject": "Combinatorics", "description": "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.", "scope_url": "https://github.com/openai/math/blob/1581d4cbdd69a0faf34b2089614fe2a814316a6a/lean/docs/168.md" }, { "id": "169", "title": "Shareshian–Wachs elementary positivity", "subject": "Combinatorics", "description": "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.", "scope_url": "https://github.com/openai/math/blob/1581d4cbdd69a0faf34b2089614fe2a814316a6a/lean/docs/169.md" }, { "id": "170", "title": "Sharp logarithmic exponents for off-diagonal Ramsey numbers", "subject": "Combinatorics", "description": "For every fixed integer s ≥ 5, proves r(s,t)=t^{s-1}/(log t)^{s-2+o(1)} as ttoinfty, 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.", "scope_url": "https://github.com/openai/math/blob/1581d4cbdd69a0faf34b2089614fe2a814316a6a/lean/docs/170.md" }, { "id": "171", "title": "The hypercube Ramsey conjecture", "subject": "Combinatorics", "description": "Resolves the Burr–Erdős hypercube Ramsey conjecture: the two-color Ramsey number of the n-dimensional cube is Theta(2^n). Thus every red-blue coloring of a complete graph on a universal constant times the cube's number of vertices contains a monochromatic copy of the cube.", "scope_url": null }, { "id": "172", "title": "Classification of finite Euclidean Ramsey configurations", "subject": "Combinatorics", "description": "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.", "scope_url": "https://github.com/openai/math/blob/1581d4cbdd69a0faf34b2089614fe2a814316a6a/lean/docs/172.md" }, { "id": "173", "title": "Seymour’s second-neighborhood conjecture", "subject": "Combinatorics", "description": "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.", "scope_url": "https://github.com/openai/math/blob/1581d4cbdd69a0faf34b2089614fe2a814316a6a/lean/docs/173.md" }, { "id": "174", "title": "Deterministic construction of strong thin spanning trees", "subject": "Combinatorics", "description": "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.", "scope_url": "https://github.com/openai/math/blob/1581d4cbdd69a0faf34b2089614fe2a814316a6a/lean/docs/174.md" }, { "id": "175", "title": "Talagrand’s expectation thresholds, discrete convexity, and graph decompositions", "subject": "Combinatorics", "description": "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.", "scope_url": "https://github.com/openai/math/blob/1581d4cbdd69a0faf34b2089614fe2a814316a6a/lean/docs/175.md" }, { "id": "176", "title": "The second Kahn–Kalai conjecture with an edge-count bound", "subject": "Combinatorics", "description": "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.", "scope_url": "https://github.com/openai/math/blob/1581d4cbdd69a0faf34b2089614fe2a814316a6a/lean/docs/176.md" }, { "id": "177", "title": "Bounded-degree coboundary expanders", "subject": "Combinatorics", "description": "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.", "scope_url": "https://github.com/openai/math/blob/1581d4cbdd69a0faf34b2089614fe2a814316a6a/lean/docs/177.md" }, { "id": "178", "title": "Deterministic nonbipartite Ramanujan graphs in every fixed degree", "subject": "Combinatorics", "description": "For every fixed d ≥ 3, constructs a simple d-regular nonbipartite Ramanujan graph on every sufficiently large even number n of vertices, with every nonconstant adjacency eigenvalue strictly between -2sqrt{d-1} and 2sqrt{d-1}. A deterministic algorithm outputs the full adjacency list in polynomial bit time, with exponent depending on d.", "scope_url": null }, { "id": "179", "title": "The circulant Hadamard and Barker-sequence conjectures", "subject": "Combinatorics", "description": "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 nin{2,3,4,5,7,11,13}.", "scope_url": "https://github.com/openai/math/blob/1581d4cbdd69a0faf34b2089614fe2a814316a6a/lean/docs/179.md" }, { "id": "180", "title": "Barnette’s Hamiltonian-cycle conjecture", "subject": "Combinatorics", "description": "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.", "scope_url": "https://github.com/openai/math/blob/1581d4cbdd69a0faf34b2089614fe2a814316a6a/lean/docs/180.md" }, { "id": "181", "title": "The Erdős–Gallai cycle-decomposition conjecture", "subject": "Combinatorics", "description": "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.", "scope_url": "https://github.com/openai/math/blob/1581d4cbdd69a0faf34b2089614fe2a814316a6a/lean/docs/181.md" }, { "id": "182", "title": "Power savings for intersective polynomial differences and prime arguments", "subject": "Combinatorics", "description": "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_kgt 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.", "scope_url": "https://github.com/openai/math/blob/1581d4cbdd69a0faf34b2089614fe2a814316a6a/lean/docs/182.md" }, { "id": "183", "title": "Power savings for planar halving lines and k-sets", "subject": "Combinatorics", "description": "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 1le kle n/2, with an absolute varepsilon_0gt 0. The constants and positive exponents are nonquantitative.", "scope_url": "https://github.com/openai/math/blob/1581d4cbdd69a0faf34b2089614fe2a814316a6a/lean/docs/183.md" }, { "id": "184", "title": "Correspondence coloring with a fixed forbidden subgraph", "subject": "Combinatorics", "description": "Proves the Alon–Krivelevich–Sudakov coloring conjecture in correspondence-coloring form: graphs avoiding any fixed subgraph F need O_F(Delta/logDelta) 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 Kr-free graph of average degree d ≥ 2 has an independent set of size Omega_r(nlog d/d).", "scope_url": "https://github.com/openai/math/blob/1581d4cbdd69a0faf34b2089614fe2a814316a6a/lean/docs/184.md" }, { "id": "185", "title": "Counterexamples to infinite matroid intersection and packing/covering", "subject": "Combinatorics", "description": "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.", "scope_url": "https://github.com/openai/math/blob/1581d4cbdd69a0faf34b2089614fe2a814316a6a/lean/docs/185.md" }, { "id": "186", "title": "Uniform influence and sharp thresholds for graph and hypergraph properties", "subject": "Combinatorics", "description": "Proves the Friedgut–Kalai threshold-width conjectures for graphs and fixed-uniformity hypergraphs. For fixed 0lt varepsilonlt 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.", "scope_url": "https://github.com/openai/math/blob/1581d4cbdd69a0faf34b2089614fe2a814316a6a/lean/docs/186.md" }, { "id": "187", "title": "Snaky in 21 Maker moves", "subject": "Combinatorics", "description": "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.", "scope_url": "https://github.com/openai/math/blob/1581d4cbdd69a0faf34b2089614fe2a814316a6a/lean/docs/187.md" }, { "id": "188", "title": "The sharp terminal leave in random triangle removal", "subject": "Combinatorics", "description": "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}/(2sqrt2), with mean-square convergence after normalization by n3/2. This proves the triangle case of the Joos–Kühn sharp-constant conjecture.", "scope_url": "https://github.com/openai/math/blob/1581d4cbdd69a0faf34b2089614fe2a814316a6a/lean/docs/188.md" }, { "id": "189", "title": "Cycle–clique Ramsey numbers", "subject": "Combinatorics", "description": "Proves the Erdős–Faudree–Rousseau–Schelp conjecture: R(C_m,K_n)=(m-1)(n-1)+1 for every mge nge3, 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.", "scope_url": "https://github.com/openai/math/blob/1581d4cbdd69a0faf34b2089614fe2a814316a6a/lean/docs/189.md" }, { "id": "190", "title": "Polynomial removal fails for ordered binary matrices", "subject": "Combinatorics", "description": "Disproves polynomial ordered binary matrix removal with one fixed 66times66 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.", "scope_url": "https://github.com/openai/math/blob/1581d4cbdd69a0faf34b2089614fe2a814316a6a/lean/docs/190.md" }, { "id": "191", "title": "A power improvement in the Heilbronn triangle lower bound", "subject": "Combinatorics", "description": "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.", "scope_url": "https://github.com/openai/math/blob/1581d4cbdd69a0faf34b2089614fe2a814316a6a/lean/docs/191.md" }, { "id": "192", "title": "Boolean functions violate the square-root degree bound by arbitrary factors", "subject": "Combinatorics", "description": "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_iwidehat f({i})gt Csqrt{deg(f)}. Thus its total signed correlation with individual input bits can exceed the proposed bound by an arbitrary factor.", "scope_url": "https://github.com/openai/math/blob/1581d4cbdd69a0faf34b2089614fe2a814316a6a/lean/docs/192.md" }, { "id": "193", "title": "Serre’s intersection-multiplicity conjecture", "subject": "Algebra", "description": "Proves strict positivity of Serre's intersection multiplicity chi^R(M,N) for nonzero finitely generated modules over any regular local ring, provided Motimes_R N has finite length and dim M+dim N=dim R. This resolves the positivity conjecture, including ramified mixed characteristic.", "scope_url": null }, { "id": "194", "title": "Lech’s multiplicity conjecture", "subject": "Algebra", "description": "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.", "scope_url": "https://github.com/openai/math/blob/1581d4cbdd69a0faf34b2089614fe2a814316a6a/lean/docs/194.md" }, { "id": "195", "title": "A counterexample to the small Cohen–Macaulay module conjecture", "subject": "Algebra", "description": "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.", "scope_url": "https://github.com/openai/math/blob/1581d4cbdd69a0faf34b2089614fe2a814316a6a/lean/docs/195.md" }, { "id": "196", "title": "A counterexample to Kaplansky’s zero-divisor conjecture", "subject": "Algebra", "description": "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.", "scope_url": "https://github.com/openai/math/blob/1581d4cbdd69a0faf34b2089614fe2a814316a6a/lean/docs/196.md" }, { "id": "197", "title": "A torsion-free group algebra that is not directly finite", "subject": "Algebra", "description": "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.", "scope_url": "https://github.com/openai/math/blob/1581d4cbdd69a0faf34b2089614fe2a814316a6a/lean/docs/197.md" }, { "id": "198", "title": "A counterexample to finitistic-dimension finiteness", "subject": "Algebra", "description": "Constructs a finite-dimensional complex algebra whose finite-dimensional modules have unbounded finite projective dimensions. This disproves the little finitistic-dimension conjecture.", "scope_url": "https://github.com/openai/math/blob/1581d4cbdd69a0faf34b2089614fe2a814316a6a/lean/docs/198.md" }, { "id": "199", "title": "Counterexamples to Auslander–Reiten, Tachikawa and related homological conjectures", "subject": "Algebra", "description": "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.", "scope_url": "https://github.com/openai/math/blob/1581d4cbdd69a0faf34b2089614fe2a814316a6a/lean/docs/199.md" }, { "id": "200", "title": "Eisenbud–Green–Harris and lex-plus-powers", "subject": "Algebra", "description": "Proves the Eisenbud–Green–Harris and lex-plus-powers conjectures over every characteristic-zero field. Any homogeneous ideal containing a regular sequence, of arbitrary length and degrees at least two, admits a lex-plus-powers ideal with the same Hilbert function and no smaller graded Betti numbers.", "scope_url": null }, { "id": "201", "title": "A counterexample to Kurosh’s division-ring problem", "subject": "Algebra", "description": "Constructs a countable characteristic-zero division ring that is algebraic over its center and generated by two elements over that center, but has infinite dimension over it. This answers Kurosh's division-ring problem on local finiteness negatively.", "scope_url": null }, { "id": "202", "title": "The blockwise Alperin weight conjecture", "subject": "Algebra", "description": "Proves the numerical blockwise Alperin weight conjecture for every prime and every finite group: the number of irreducible Brauer characters in a block equals the number of conjugacy classes of its weights.", "scope_url": null }, { "id": "203", "title": "Donovan's conjecture over fields and complete mixed-characteristic DVRs", "subject": "Algebra", "description": "Proves Donovan's conjecture: over each fixed algebraically closed field of characteristic p, blocks of finite groups with bounded defect-group order have only finitely many Morita-equivalence classes, for every prime p. Also proves the integral form over each fixed complete mixed-characteristic discrete valuation ring with algebraically closed residue field.", "scope_url": null }, { "id": "204", "title": "Tensor saturation for even spin groups", "subject": "Algebra", "description": "Proves saturation factor one for mathop{mathrm{Spin}}nolimits (2n), n ≥ 2: for three dominant integral weights whose sum lies in the root lattice, an invariant at any common positive integral dilation already gives an invariant at the original weights. This resolves the type-D part of the simply-laced saturation conjecture.", "scope_url": null }, { "id": "205", "title": "Saxl’s conjecture and universal tensor squares", "subject": "Algebra", "description": "Proves Saxl's conjecture: the tensor square of every staircase representation contains every irreducible complex representation of the corresponding symmetric group. More generally, every Sn with nnotin{2,4,9} has an irreducible representation whose tensor square contains all irreducibles.", "scope_url": "https://github.com/openai/math/blob/1581d4cbdd69a0faf34b2089614fe2a814316a6a/lean/docs/205.md" }, { "id": "206", "title": "Finite lattice representation and undecidability", "subject": "Algebra", "description": "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.", "scope_url": "https://github.com/openai/math/blob/1581d4cbdd69a0faf34b2089614fe2a814316a6a/lean/docs/206.md" }, { "id": "207", "title": "The ℓ¹-Bass conjecture for all discrete groups", "subject": "Algebra", "description": "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.", "scope_url": "https://github.com/openai/math/blob/1581d4cbdd69a0faf34b2089614fe2a814316a6a/lean/docs/207.md" }, { "id": "208", "title": "Finite symmetric tensor categories and the Verlinde tower", "subject": "Algebra", "description": "Proves that every finite symmetric tensor category over an algebraically closed field k of characteristic p > 0 admits a k-linear exact faithful strong symmetric monoidal fiber functor to a higher Verlinde category mathrm{Ver}_{p^n}. The level may depend on the category, and the theorem includes characteristic two, resolving the finite case of the Benson–Etingof–Ostrik conjecture.", "scope_url": null }, { "id": "209", "title": "Integral counterexamples to Gersten’s conjecture", "subject": "Algebra", "description": "Disproves unrestricted integral Gersten injectivity in degrees 3 and 5. Two explicit two-dimensional ramified regular local rings of mixed characteristic (0,5) have nonzero integral K-theory classes that vanish over their fraction fields.", "scope_url": null }, { "id": "210", "title": "Foulkes' conjecture for sixth powers and quadratic stabilization", "subject": "Algebra", "description": "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)tomathop{mathrm{Sym}}nolimits ^a(mathop{mathrm{Sym}}nolimits ^bV) is surjective for a ≥ 2 and bge a(a-1), giving dimension-independent quadratic stabilization.", "scope_url": "https://github.com/openai/math/blob/1581d4cbdd69a0faf34b2089614fe2a814316a6a/lean/docs/210.md" }, { "id": "211", "title": "The geometric phase diagram, diffusion, and spectra of random planar maps", "subject": "Probability and statistical mechanics", "description": "Critical Fortuin–Kasteleyn planar maps converge to Liouville quantum gravity spheres for 0lt qle4 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.", "scope_url": "https://github.com/openai/math/blob/1581d4cbdd69a0faf34b2089614fe2a814316a6a/lean/docs/211.md" }, { "id": "212", "title": "Planar first-passage geometry and the absence of bigeodesics", "subject": "Probability and statistical mechanics", "description": "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.", "scope_url": "https://github.com/openai/math/blob/1581d4cbdd69a0faf34b2089614fe2a814316a6a/lean/docs/212.md" }, { "id": "213", "title": "Critical percolation on every quasi-transitive graph", "subject": "Probability and statistical mechanics", "description": "Resolves the Benjamini–Schramm criticality conjecture for bond percolation on every infinite connected locally finite quasi-transitive graph with p_clt 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.", "scope_url": "https://github.com/openai/math/blob/1581d4cbdd69a0faf34b2089614fe2a814316a6a/lean/docs/213.md" }, { "id": "214", "title": "The Benjamini–Schramm nonuniqueness conjecture", "subject": "Probability and statistical mechanics", "description": "Proves p_clt 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.", "scope_url": "https://github.com/openai/math/blob/1581d4cbdd69a0faf34b2089614fe2a814316a6a/lean/docs/214.md" }, { "id": "215", "title": "Canonical O(3) continuum limit and exact O(4) mass asymptotics", "subject": "Probability and statistical mechanics", "description": "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}sqrtbeta,e^{-pibeta}. The family also proves exponential spin-correlation decay for two-dimensional nearest-neighbor O(n) models with n ≥ 3 at every positive temperature.", "scope_url": "https://github.com/openai/math/blob/1581d4cbdd69a0faf34b2089614fe2a814316a6a/lean/docs/215.md" }, { "id": "216", "title": "Critical and near-critical XY scaling and BKT universality", "subject": "Probability and statistical mechanics", "description": "For the square-lattice nearest-neighbor cosine XY model, proves critical axis correlations C_{beta_c}(r)sim Ar^{-1/4}(log r)^{1/8} and the Berezinskii–Kosterlitz–Thouless essential singularity sqrt{beta_c-beta}logxi(beta)to B, with A,Bgt 0 after the free-box thermodynamic limit. For finite square-symmetric interactions containing nearest neighbors, discrete Gaussian heights converge to Gaussian fields throughout the rough phase, including its threshold, along geometric torus sizes. Critical center-magnetization and spin-field conclusions retain their stated height, renormalization, and field-input assumptions.", "scope_url": null }, { "id": "217", "title": "The low-temperature Sherrington–Kirkpatrick fluctuation law", "subject": "Probability and statistical mechanics", "description": "For every fixed inverse temperature β > 1, determines the fluctuation scale and limiting law of the zero-field Gaussian Sherrington–Kirkpatrick log partition function. Its variance is asymptotic to c_beta n^{1/3}, with c_betagt 0, confirming the predicted n1/6 standard-deviation scale. Exact centering and standardization give full-sequence convergence to a uniquely characterized nondegenerate law.", "scope_url": null }, { "id": "218", "title": "Conformal universality for weakly interacting and random-bond Ising models", "subject": "Probability and statistical mechanics", "description": "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.", "scope_url": "https://github.com/openai/math/blob/1581d4cbdd69a0faf34b2089614fe2a814316a6a/lean/docs/218.md" }, { "id": "219", "title": "GOE bulk universality for regular graphs with weak Anderson disorder", "subject": "Probability and statistical mechanics", "description": "For every fixed degree d ≥ 3, the bulk adjacency-eigenvalue point process of a uniform simple random d-regular graph converges to the Gaussian orthogonal ensemble law, including for cubic graphs. The same fixed-energy universality persists under sufficiently weak fixed iid uniform diagonal disorder, throughout compact bands strictly inside the clean spectral edges.", "scope_url": null }, { "id": "220", "title": "Directional zero–one laws beyond iid environments and iid ballisticity", "subject": "Probability and statistical mechanics", "description": "On ℤd, 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.", "scope_url": "https://github.com/openai/math/blob/1581d4cbdd69a0faf34b2089614fe2a814316a6a/lean/docs/220.md" }, { "id": "221", "title": "The Mézard–Parisi formula for diluted spin glasses", "subject": "Probability and statistical mechanics", "description": "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.", "scope_url": "https://github.com/openai/math/blob/1581d4cbdd69a0faf34b2089614fe2a814316a6a/lean/docs/221.md" }, { "id": "222", "title": "Perceptron free energies and microscopic jamming exponents", "subject": "Probability and statistical mechanics", "description": "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.", "scope_url": "https://github.com/openai/math/blob/1581d4cbdd69a0faf34b2089614fe2a814316a6a/lean/docs/222.md" }, { "id": "223", "title": "Random-cluster interfaces: critical, disordered, thermal, and natural-time scaling", "subject": "Probability and statistical mechanics", "description": "Proves chordal SLEκ limits for critical square-lattice random-cluster interfaces for 0lt qle4, with kappa=4pi/arccos(-sqrt q/2): bounded Jordan domains are allowed for q ≥ 1, and smooth Jordan domains for q < 1, under the stated marked-boundary approximations. For 1le qle4, complete nested plane loops converge to CLEκ.", "scope_url": null }, { "id": "224", "title": "Critical and quenched near-critical universality for Poisson–Voronoi percolation", "subject": "Probability and statistical mechanics", "description": "Proves Cardy's formula for annealed critical Poisson–Voronoi crossing probabilities in every bounded Jordan quadrilateral. With each model normalized by its own expected unit-square pivotal count, the conditional joint near-critical crossing-threshold laws for rational polygonal quads converge in environment probability to the triangular-lattice reference law. This establishes quenched near-critical universality for crossing thresholds.", "scope_url": null }, { "id": "225", "title": "Gaussian free field limits throughout the balanced six-vertex regime", "subject": "Probability and statistical mechanics", "description": "The balanced square-lattice six-vertex height field with a=b=1 and 0lt cle2 converges to a Gaussian free field, including at the endpoint c = 2. The plane state is defined by balanced-torus limits. For unit height increments and Green kernel -(2pi)^{-1}log|x-y|, the exact variance multiplier is 1/arcsin(c/2).", "scope_url": null }, { "id": "226", "title": "The double-dimer loop ensemble converges to CLE4", "subject": "Probability and statistical mechanics", "description": "Resolves the half-plane Temperleyan form of the double-dimer scaling-limit conjecture: the complete loop ensemble formed by two independent dimer coverings of the Temperleyan square lattice converges to nested CLE4. Convergence matches every macroscopic loop as an unparametrized curve, upgrading convergence of loop observables to convergence of the loops themselves.", "scope_url": null }, { "id": "227", "title": "Critical SK autocorrelation processes and dynamics across the temperature transition", "subject": "Probability and statistical mechanics", "description": "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 0lebetalt 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.", "scope_url": "https://github.com/openai/math/blob/1581d4cbdd69a0faf34b2089614fe2a814316a6a/lean/docs/227.md" }, { "id": "228", "title": "Continuum phase transitions for radial pair potentials", "subject": "Probability and statistical mechanics", "description": "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.", "scope_url": "https://github.com/openai/math/blob/1581d4cbdd69a0faf34b2089614fe2a814316a6a/lean/docs/228.md" }, { "id": "229", "title": "Exact three- and four-state reconstruction thresholds and four-state tree capacity", "subject": "Probability and statistical mechanics", "description": "Proves the exact reconstruction threshold dlambda^2gt 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.", "scope_url": "https://github.com/openai/math/blob/1581d4cbdd69a0faf34b2089614fe2a814316a6a/lean/docs/229.md" }, { "id": "230", "title": "Exact Hausdorff gauges for SLE", "subject": "Probability and statistical mechanics", "description": "Resolves Schramm’s Hausdorff-measure question for chordal SLEκ, 0lt kappalt 8. The explicit gauge r^d(loglog(1/r))^{(2-d)/2}, d=1+kappa/8, gives almost surely positive finite measure to every trace segment gamma([s,t]) with 0lt slt tlt infty, and finite expected measure to the trace in every bounded disk.", "scope_url": "https://github.com/openai/math/blob/1581d4cbdd69a0faf34b2089614fe2a814316a6a/lean/docs/230.md" }, { "id": "231", "title": "The free uniform spanning forest is a factor of IID", "subject": "Probability and statistical mechanics", "description": "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.", "scope_url": "https://github.com/openai/math/blob/1581d4cbdd69a0faf34b2089614fe2a814316a6a/lean/docs/231.md" }, { "id": "232", "title": "Gaussian fields and interfaces for triangular-lattice Lipschitz heights", "subject": "Probability and statistical mechanics", "description": "Proves Gaussian free field limits on bounded smooth simply connected domains for triangular-lattice height models: uniform odd heights with increments 0,pm2 and two-arc boundary values pm1, and zero-boundary integer Lipschitz heights weighted by fixed xin[1/sqrt2,1]. Uniform real Lipschitz heights also converge to a Gaussian field; at a tuned opposite-boundary amplitude, their interface converges to chordal SLE4, establishing Schramm’s real-field/interface predictions.", "scope_url": null }, { "id": "233", "title": "The joint critical Ashkin–Teller current limit", "subject": "Probability and statistical mechanics", "description": "Identifies the joint scaling limit of Ashkin–Teller heights and both complete current-cluster collections throughout the critical line, including the four-state Potts endpoint. In bounded Jordan domains with admissible lattice approximations and wired primal/free dual boundaries, the height converges to the predicted Gaussian free field, and the clusters to canonical recursive sets of that same field, retaining every nesting depth.", "scope_url": null }, { "id": "234", "title": "All-temperature pressure of orthogonally invariant Ising spin glasses", "subject": "Probability and statistical mechanics", "description": "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.", "scope_url": "https://github.com/openai/math/blob/1581d4cbdd69a0faf34b2089614fe2a814316a6a/lean/docs/234.md" }, { "id": "235", "title": "Limiting random SAT thresholds, sharp variance and computability", "subject": "Probability and statistical mechanics", "description": "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.", "scope_url": "https://github.com/openai/math/blob/1581d4cbdd69a0faf34b2089614fe2a814316a6a/lean/docs/235.md" }, { "id": "236", "title": "The exact factor-of-IID threshold for free Ising spins on trees", "subject": "Probability and statistical mechanics", "description": "Determines when the free zero-field ferromagnetic Ising state on the infinite d-regular tree is a factor of independent vertex labels: exactly when tanhbetale(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.", "scope_url": "https://github.com/openai/math/blob/1581d4cbdd69a0faf34b2089614fe2a814316a6a/lean/docs/236.md" }, { "id": "237", "title": "The three-quarter exponent for honeycomb self-avoiding walk", "subject": "Probability and statistical mechanics", "description": "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.", "scope_url": "https://github.com/openai/math/blob/1581d4cbdd69a0faf34b2089614fe2a814316a6a/lean/docs/237.md" }, { "id": "238", "title": "Optimal logarithmic mixing of the Thorp shuffle", "subject": "Probability and statistical mechanics", "description": "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.", "scope_url": "https://github.com/openai/math/blob/1581d4cbdd69a0faf34b2089614fe2a814316a6a/lean/docs/238.md" }, { "id": "239", "title": "Sharp singularity rates for symmetric random sign matrices", "subject": "Probability and statistical mechanics", "description": "Determines the sharp exponential singularity rate of symmetric random sign matrices with independent entries on and above the diagonal. Uniform signs give Pr(det A_n=0)=(1/2+o(1))^n; for fixed bias pin(0,1)setminus{1/2}, the rate is (p^2+(1-p)^2+o(1))^n. In the biased case, agreeing rows attain this rate.", "scope_url": null }, { "id": "240", "title": "Shelah's eventual categoricity and the prescribed-threshold obstruction", "subject": "Mathematical logic", "description": "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.", "scope_url": "https://github.com/openai/math/blob/1581d4cbdd69a0faf34b2089614fe2a814316a6a/lean/docs/240.md" }, { "id": "241", "title": "Rigidity of the Turing degrees", "subject": "Mathematical logic", "description": "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.", "scope_url": "https://github.com/openai/math/blob/1581d4cbdd69a0faf34b2089614fe2a814316a6a/lean/docs/241.md" }, { "id": "242", "title": "Single-fold Diophantine representations and undecidability under an at-most-one-solution promise", "subject": "Mathematical logic", "description": "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.", "scope_url": "https://github.com/openai/math/blob/1581d4cbdd69a0faf34b2089614fe2a814316a6a/lean/docs/242.md" }, { "id": "243", "title": "Separating choiceless counting from polynomial time and witnessed choice", "subject": "Mathematical logic", "description": "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.", "scope_url": "https://github.com/openai/math/blob/1581d4cbdd69a0faf34b2089614fe2a814316a6a/lean/docs/243.md" }, { "id": "244", "title": "The Partition Principle does not imply Choice", "subject": "Mathematical logic", "description": "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.", "scope_url": "https://github.com/openai/math/blob/1581d4cbdd69a0faf34b2089614fe2a814316a6a/lean/docs/244.md" }, { "id": "245", "title": "Weak normalization implies strong normalization in pure type systems", "subject": "Mathematical logic", "description": "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.", "scope_url": "https://github.com/openai/math/blob/1581d4cbdd69a0faf34b2089614fe2a814316a6a/lean/docs/245.md" }, { "id": "246", "title": "Cannon's conjecture", "subject": "Group theory", "description": "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.", "scope_url": "https://github.com/openai/math/blob/1581d4cbdd69a0faf34b2089614fe2a814316a6a/lean/docs/246.md" }, { "id": "247", "title": "An infinite finitely presented residually finite 2-group and a finitely presented nil algebra", "subject": "Group theory", "description": "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.", "scope_url": "https://github.com/openai/math/blob/1581d4cbdd69a0faf34b2089614fe2a814316a6a/lean/docs/247.md" }, { "id": "248", "title": "Thompson's group F is nonamenable", "subject": "Group theory", "description": "Proves that Thompson's group F, the group of dyadic piecewise linear homeomorphisms of the interval, is nonamenable, resolving its longstanding amenability problem.", "scope_url": "https://github.com/openai/math/blob/1581d4cbdd69a0faf34b2089614fe2a814316a6a/lean/docs/248.md" }, { "id": "249", "title": "A finitely generated Eilenberg–Ganea counterexample", "subject": "Group theory", "description": "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.", "scope_url": "https://github.com/openai/math/blob/1581d4cbdd69a0faf34b2089614fe2a814316a6a/lean/docs/249.md" }, { "id": "250", "title": "Boone–Higman embeddings with higher finiteness", "subject": "Group theory", "description": "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.", "scope_url": "https://github.com/openai/math/blob/1581d4cbdd69a0faf34b2089614fe2a814316a6a/lean/docs/250.md" }, { "id": "251", "title": "Amenability, unitarizability, and strong Ulam stability", "subject": "Group theory", "description": "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.", "scope_url": "https://github.com/openai/math/blob/1581d4cbdd69a0faf34b2089614fe2a814316a6a/lean/docs/251.md" }, { "id": "252", "title": "A torsion-free hyperbolic group that is neither residually finite nor linear over any field", "subject": "Group theory", "description": "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.", "scope_url": "https://github.com/openai/math/blob/1581d4cbdd69a0faf34b2089614fe2a814316a6a/lean/docs/252.md" }, { "id": "253", "title": "An infinite finitely presented simple amenable group", "subject": "Group theory", "description": "Constructs an infinite finitely presented simple amenable group, answering the longstanding question of whether these properties can occur simultaneously.", "scope_url": "https://github.com/openai/math/blob/1581d4cbdd69a0faf34b2089614fe2a814316a6a/lean/docs/253.md" }, { "id": "254", "title": "Classifying spaces and geometric obstructions for Artin groups", "subject": "Group theory", "description": "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.", "scope_url": "https://github.com/openai/math/blob/1581d4cbdd69a0faf34b2089614fe2a814316a6a/lean/docs/254.md" }, { "id": "255", "title": "Quasi-isometric recognition of virtually polycyclic groups", "subject": "Group theory", "description": "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.", "scope_url": "https://github.com/openai/math/blob/1581d4cbdd69a0faf34b2089614fe2a814316a6a/lean/docs/255.md" }, { "id": "256", "title": "Nonsingular systems of equations over arbitrary groups", "subject": "Group theory", "description": "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.", "scope_url": "https://github.com/openai/math/blob/1581d4cbdd69a0faf34b2089614fe2a814316a6a/lean/docs/256.md" }, { "id": "257", "title": "A hyperbolic group without a geometric CAT(0) action", "subject": "Group theory", "description": "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.", "scope_url": "https://github.com/openai/math/blob/1581d4cbdd69a0faf34b2089614fe2a814316a6a/lean/docs/257.md" }, { "id": "258", "title": "Gersten’s conjecture and virtual compact specialness of one-relator groups", "subject": "Group theory", "description": "Proves Gersten's conjecture: every finitely generated one-relator group containing no Baumslag–Solitar subgroup mathrm{BS}(m,n), with m,nne0, is word-hyperbolic. It also proves that every word-hyperbolic one-relator group is virtually compact special.", "scope_url": null }, { "id": "259", "title": "A group without fixed price", "subject": "Group theory", "description": "Constructs a finitely generated group with two essentially free probability-measure-preserving actions of different costs, answering the general fixed-price problem negatively. Its Bernoulli action has cost bounded away from one, while a sequence of finite height extensions has costs tending to one.", "scope_url": null }, { "id": "260", "title": "Spacetime Penrose inequalities: enclosing area, charge, rotation, and anti-de Sitter extensions", "subject": "Mathematical physics", "description": "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.", "scope_url": "https://github.com/openai/math/blob/1581d4cbdd69a0faf34b2089614fe2a814316a6a/lean/docs/260.md" }, { "id": "261", "title": "Localization and delocalization in the Anderson model", "subject": "Mathematical physics", "description": "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.", "scope_url": "https://github.com/openai/math/blob/1581d4cbdd69a0faf34b2089614fe2a814316a6a/lean/docs/261.md" }, { "id": "262", "title": "Sharp finite-matrix Lieb–Thirring inequalities and all equality cases", "subject": "Mathematical physics", "description": "Proves the sharp one-dimensional Lieb–Thirring inequality for 1/2lt gammalt 3/2 and arbitrary finite-matrix potentials W ≥ 0 with intmathop{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.", "scope_url": "https://github.com/openai/math/blob/1581d4cbdd69a0faf34b2089614fe2a814316a6a/lean/docs/262.md" }, { "id": "263", "title": "The ionization and generalized ionization conjectures", "subject": "Mathematical physics", "description": "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 mtoinfty and Z/mtoinfty; neutral-atom outer radii have the corresponding iterated-limit asymptotics, taking Ztoinfty first.", "scope_url": "https://github.com/openai/math/blob/1581d4cbdd69a0faf34b2089614fe2a814316a6a/lean/docs/263.md" }, { "id": "264", "title": "Strong cosmic censorship near two-ended Kerr data", "subject": "Mathematical physics", "description": "Proves local strong cosmic censorship near each fixed rotating subextremal Kerr bridge. A dense Gδ subset of a weighted smooth neighborhood of smooth complete two-ended asymptotically flat vacuum data has full maximal globally hyperbolic developments with no future continuous nondegenerate extension whose weak connection is locally square-integrable. No symmetry is imposed; extensions need not satisfy the vacuum equations.", "scope_url": null }, { "id": "265", "title": "Area laws and tensor networks for two-dimensional gapped systems", "subject": "Mathematical physics", "description": "Proves an entropy area law for unique ground states of finite-range Hamiltonians on arbitrary finite induced square-lattice domains, using only a uniform full-system spectral gap and bounds on the local interactions. On open Ltimes L squares, uniformly gapped nearest-neighbor ground states also admit projected entangled-pair state approximations with polynomial bond dimension and global vector error at most L−1.", "scope_url": null }, { "id": "266", "title": "Exactly three mutually unbiased bases in dimension six", "subject": "Mathematical physics", "description": "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.", "scope_url": "https://github.com/openai/math/blob/1581d4cbdd69a0faf34b2089614fe2a814316a6a/lean/docs/266.md" }, { "id": "267", "title": "Positive-temperature Bose–Einstein condensation and exact quantum depletion", "subject": "Mathematical physics", "description": "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.", "scope_url": "https://github.com/openai/math/blob/1581d4cbdd69a0faf34b2089614fe2a814316a6a/lean/docs/267.md" }, { "id": "268", "title": "The spin-one Haldane gap", "subject": "Mathematical physics", "description": "Proves the spin-one Haldane gap conjecture for the pure antiferromagnetic Heisenberg chain on even periodic rings: the spectral gap stays uniformly positive as the chain grows. A companion establishes a gap for odd open chains with endpoint field h=3/5 and gives boundary-selected infinite-volume states with topological index −1.", "scope_url": null }, { "id": "269", "title": "Uniform Laughlin gap and stability under bounded scalar disorder", "subject": "Mathematical physics", "description": "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.", "scope_url": "https://github.com/openai/math/blob/1581d4cbdd69a0faf34b2089614fe2a814316a6a/lean/docs/269.md" }, { "id": "270", "title": "Threshold and positive-energy bound states of the BFSS matrix model", "subject": "Mathematical physics", "description": "Proves that the undeformed relative mathrm{SU}(N) BFSS model has exactly one normalizable zero-energy state for every finite N ≥ 2, resolving the threshold-bound-state conjecture. For mathrm{SU}(2), a companion proves infinitely many normalizable positive-energy eigenstates with unbounded energies, contradicting the original BFSS paper's exclusion of additional bound states at N = 2.", "scope_url": null }, { "id": "271", "title": "Bloch's law, its lattice correction, and the spherical magnetization law", "subject": "Mathematical physics", "description": "Proves Bloch's T3/2 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.", "scope_url": "https://github.com/openai/math/blob/1581d4cbdd69a0faf34b2089614fe2a814316a6a/lean/docs/271.md" }, { "id": "272", "title": "Entanglement without distillable secret key", "subject": "Mathematical physics", "description": "Constructs an entangled state on mathbb C^{10}otimesmathbb 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.", "scope_url": "https://github.com/openai/math/blob/1581d4cbdd69a0faf34b2089614fe2a814316a6a/lean/docs/272.md" }, { "id": "273", "title": "The entropy photon-number inequality", "subject": "Mathematical physics", "description": "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.", "scope_url": "https://github.com/openai/math/blob/1581d4cbdd69a0faf34b2089614fe2a814316a6a/lean/docs/273.md" }, { "id": "274", "title": "Parity is not in QAC0", "subject": "Mathematical physics", "description": "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.", "scope_url": "https://github.com/openai/math/blob/1581d4cbdd69a0faf34b2089614fe2a814316a6a/lean/docs/274.md" }, { "id": "275", "title": "QMA-hardness of continuum Coulomb energy", "subject": "Mathematical physics", "description": "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.", "scope_url": "https://github.com/openai/math/blob/1581d4cbdd69a0faf34b2089614fe2a814316a6a/lean/docs/275.md" }, { "id": "276", "title": "Classical capacity of generalized amplitude damping", "subject": "Mathematical physics", "description": "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.", "scope_url": "https://github.com/openai/math/blob/1581d4cbdd69a0faf34b2089614fe2a814316a6a/lean/docs/276.md" }, { "id": "277", "title": "Threshold repetition for entangled games", "subject": "Mathematical physics", "description": "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 0lt deltalt 1-v. Arbitrary joint finite-dimensional entangled strategies and correlated question distributions are allowed.", "scope_url": "https://github.com/openai/math/blob/1581d4cbdd69a0faf34b2089614fe2a814316a6a/lean/docs/277.md" }, { "id": "278", "title": "Failure of Kohn–Sham ensemble representation", "subject": "Mathematical physics", "description": "Constructs a three-electron Coulomb molecule with two equal positive-integer-charge nuclei whose absolute ground-state density has no noninteracting ground-state ensemble representation by a single real spin-independent local potential in L^{3/2}(mathbb R^3)+L^infty(mathbb R^3). This disproves Kohn–Sham ensemble representability for that potential class; the required nuclear charge is specified nonnumerically.", "scope_url": null }, { "id": "279", "title": "Exact quantum factoring over a fixed finite gate set", "subject": "Mathematical physics", "description": "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.", "scope_url": "https://github.com/openai/math/blob/1581d4cbdd69a0faf34b2089614fe2a814316a6a/lean/docs/279.md" }, { "id": "280", "title": "Unitary vertex operator algebras and conformal nets", "subject": "Mathematical physics", "description": "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.", "scope_url": "https://github.com/openai/math/blob/1581d4cbdd69a0faf34b2089614fe2a814316a6a/lean/docs/280.md" }, { "id": "281", "title": "QAOA attains the SK optimum in the thermodynamic-first limit", "subject": "Mathematical physics", "description": "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.", "scope_url": "https://github.com/openai/math/blob/1581d4cbdd69a0faf34b2089614fe2a814316a6a/lean/docs/281.md" }, { "id": "282", "title": "From scale symmetry to local conformal symmetry in four-dimensional QFT", "subject": "Mathematical physics", "description": "Under the stated bounded-local-net and field-reconstruction hypotheses, proves that scale symmetry implies local conformal symmetry for four-dimensional unitary positive-energy theories with a discrete bounded-below scaling spectrum of finite multiplicity, finite scaling support and a physical local scale current. The stress tensor has a traceless improvement with unchanged spacetime charges. The conclusion concerns local Ward identities, not a global conformal action on the whole net.", "scope_url": null }, { "id": "283", "title": "Polynomial-time unitary synthesis from a Boolean oracle", "subject": "Mathematical physics", "description": "Solves the constant-error Aaronson–Kuperberg unitary synthesis problem: a uniform polynomial-size quantum oracle circuit approximates every n-qubit unitary channel within diamond-norm error 1/2, after a suitable Boolean oracle is chosen. Gates, qubits, oracle calls and query length are polynomially bounded. The target-dependent oracle may have an unrestricted truth table; its efficient classical construction is not asserted.", "scope_url": null }, { "id": "284", "title": "The optimal quartic separation between randomized and quantum queries", "subject": "Mathematical physics", "description": "Shows that the universal bound R(f)=O((1+Q(f))^4) for total Boolean functions is sharp in its exponent, ruling out every smaller power and disproving the conjectured cubic relation. Here R and Q are randomized and quantum worst-case bit-query complexities with error at most 1/3; computation between queries is unrestricted.", "scope_url": null }, { "id": "285", "title": "Counterexamples to Baum–Connes and Kadison–Kaplansky", "subject": "Operator algebras", "description": "Disproves the coefficient-free reduced Baum–Connes conjecture through a failure of rational injectivity and a separate failure of surjectivity of assembly. A finitely generated torsion-free example witnesses the injectivity failure. Separately, a torsion-free group has a nontrivial projection in its reduced group C∗-algebra, disproving the Kadison–Kaplansky conjecture.", "scope_url": null }, { "id": "286", "title": "Rigidity and arithmetic of lattice von Neumann algebras", "subject": "Operator algebras", "description": "Classifies finite-index bimodules between scalar-twisted group factors of ICC groups commensurable with property-(T) lattices over characteristic-zero local fields and arbitrary ICC group factors. Every such bimodule is a summand of finite sums of models arising from finite-index subgroup isomorphisms and finite-dimensional projective representations. The classification also recovers the group, scalar cocycle and amplification scale up to the stated stable equivalence.", "scope_url": null }, { "id": "287", "title": "Isomorphism of the free group factors", "subject": "Operator algebras", "description": "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}.", "scope_url": "https://github.com/openai/math/blob/1581d4cbdd69a0faf34b2089614fe2a814316a6a/lean/docs/287.md" }, { "id": "288", "title": "Kadison's similarity conjecture", "subject": "Operator algebras", "description": "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.", "scope_url": "https://github.com/openai/math/blob/1581d4cbdd69a0faf34b2089614fe2a814316a6a/lean/docs/288.md" }, { "id": "289", "title": "Strong Kadison–Kastler stability and its spatial boundaries", "subject": "Operator algebras", "description": "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.", "scope_url": "https://github.com/openai/math/blob/1581d4cbdd69a0faf34b2089614fe2a814316a6a/lean/docs/289.md" }, { "id": "290", "title": "Relative bicentralizers and modular spectral recovery", "subject": "Operator algebras", "description": "Proves Connes' bicentralizer conjecture for every type III1 factor with separable predual and every faithful normal state. More generally, for every inclusion Nsubset M of von Neumann algebras with separable preduals admitting a faithful normal conditional expectation, constructs an amenable expected subalgebra Psubset N with P'cap c(M)=N'cap c(M), resolving the relative bicentralizer conjecture.", "scope_url": "https://github.com/openai/math/blob/1581d4cbdd69a0faf34b2089614fe2a814316a6a/lean/docs/290.md" }, { "id": "291", "title": "Cuntz comparison, nuclear dimension, and equivariant Jiang–Su stability", "subject": "Operator algebras", "description": "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.", "scope_url": "https://github.com/openai/math/blob/1581d4cbdd69a0faf34b2089614fe2a814316a6a/lean/docs/291.md" }, { "id": "292", "title": "Kirchberg's mathcal O_2 norm-ultrapower embedding problem", "subject": "Operator algebras", "description": "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.", "scope_url": "https://github.com/openai/math/blob/1581d4cbdd69a0faf34b2089614fe2a814316a6a/lean/docs/292.md" }, { "id": "293", "title": "Invariant projections, hyperinvariant subspaces, and transitive algebras", "subject": "Operator algebras", "description": "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.", "scope_url": "https://github.com/openai/math/blob/1581d4cbdd69a0faf34b2089614fe2a814316a6a/lean/docs/293.md" }, { "id": "294", "title": "Kaplansky's quasitrace conjecture and failure of tensor-product stable finiteness", "subject": "Operator algebras", "description": "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).", "scope_url": "https://github.com/openai/math/blob/1581d4cbdd69a0faf34b2089614fe2a814316a6a/lean/docs/294.md" }, { "id": "295", "title": "The Kadison–Ringrose cohomology conjecture", "subject": "Operator algebras", "description": "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.", "scope_url": "https://github.com/openai/math/blob/1581d4cbdd69a0faf34b2089614fe2a814316a6a/lean/docs/295.md" }, { "id": "296", "title": "The generator problem for finite factors", "subject": "Operator algebras", "description": "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 Psubset M of such factors, the unitaries u with M=W^*(P,u) form a dense Gδ subset in the trace 2-norm topology.", "scope_url": "https://github.com/openai/math/blob/1581d4cbdd69a0faf34b2089614fe2a814316a6a/lean/docs/296.md" }, { "id": "297", "title": "A ZFC counterexample to Naimark's problem", "subject": "Operator algebras", "description": "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.", "scope_url": "https://github.com/openai/math/blob/1581d4cbdd69a0faf34b2089614fe2a814316a6a/lean/docs/297.md" }, { "id": "298", "title": "Two notions of free entropy differ even when both are finite", "subject": "Operator algebras", "description": "Constructs a bounded self-adjoint tuple in a tracial von Neumann algebra whose microstates and nonmicrostates free entropies satisfy -inftylt chilt 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.", "scope_url": "https://github.com/openai/math/blob/1581d4cbdd69a0faf34b2089614fe2a814316a6a/lean/docs/298.md" }, { "id": "299", "title": "The Kirchberg–Rørdam character criterion and infinite tensor-power Jiang–Su stability", "subject": "Operator algebras", "description": "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.", "scope_url": "https://github.com/openai/math/blob/1581d4cbdd69a0faf34b2089614fe2a814316a6a/lean/docs/299.md" }, { "id": "300", "title": "Approximation and quadratic strong-operator paving", "subject": "Operator algebras", "description": "Proves that every self-adjoint element of a complex von Neumann algebra admits strong-operator paving relative to any maximal abelian subalgebra with O(varepsilon^{-2}) blocks. The norm bound holds after compression by a projection arbitrarily close to the identity in the strong topology, resolving the Popa–Vaes quadratic paving conjecture.", "scope_url": null }, { "id": "301", "title": "Trace cones and Razak–Jacelon stabilization", "subject": "Operator algebras", "description": "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.", "scope_url": "https://github.com/openai/math/blob/1581d4cbdd69a0faf34b2089614fe2a814316a6a/lean/docs/301.md" }, { "id": "302", "title": "Radius of comparison equals half the mean dimension", "subject": "Operator algebras", "description": "For every minimal homeomorphism h of an infinite compact metrizable space X, the radius of comparison of C(X)rtimes_hmathbb Z equals tfrac12mathrm{mdim}(X,h), including infinite values. Zero mean dimension is equivalent to the small boundary property, Jiang–Su stability and finite nuclear dimension; in this case nuclear dimension is at most one.", "scope_url": null }, { "id": "303", "title": "Weak pure infiniteness and Cuntz-algebra absorption", "subject": "Operator algebras", "description": "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.", "scope_url": "https://github.com/openai/math/blob/1581d4cbdd69a0faf34b2089614fe2a814316a6a/lean/docs/303.md" }, { "id": "304", "title": "The Hilbert–Smith conjecture in every dimension", "subject": "Topology", "description": "Every locally compact second-countable Hausdorff group acting faithfully and jointly continuously on a connected finite-dimensional topological manifold is a Lie group. This proves the Hilbert–Smith conjecture in all finite dimensions, for Hausdorff second-countable manifolds without boundary.", "scope_url": null }, { "id": "305", "title": "Four-dimensional disk embedding and Wall's conjecture", "subject": "Topology", "description": "The unrestricted four-dimensional disk-embedding conjecture fails: framed algebraic dual spheres do not suffice to obtain disjoint locally flat spanning disks. In particular, the free group F2 is not good in the sense of Freedman–Quinn. Also constructs a finitely presented integral Poincaré duality group of dimension four with a finite classifying space but no realization as the fundamental group of a closed aspherical topological four-manifold, disproving Wall's conjecture.", "scope_url": null }, { "id": "306", "title": "The purely cosmetic surgery conjecture", "subject": "Topology", "description": "Distinct Dehn surgery slopes on a nontrivial smooth knot in S3 never produce orientation-preservingly homeomorphic manifolds, proving the purely cosmetic surgery conjecture. The statement includes the meridional slope.", "scope_url": null }, { "id": "307", "title": "Failure of rational injectivity for maximal coarse assembly", "subject": "Topology", "description": "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.", "scope_url": "https://github.com/openai/math/blob/1581d4cbdd69a0faf34b2089614fe2a814316a6a/lean/docs/307.md" }, { "id": "308", "title": "Finite Smith–Toda complexes at every height", "subject": "Topology", "description": "For every n ≥ 0, constructs a finite Smith–Toda spectrum at a prime p depending on n, with Brown–Peterson homology BP_*/(p,v_1,ldots,v_n) and the canonical comodule structure. Thus Smith–Toda complexes exist at every height when the prime may vary; an explicit example realizes V(4) at p = 1009.", "scope_url": null }, { "id": "309", "title": "The Kervaire invariant problem at the prime three", "subject": "Topology", "description": "Resolves the odd-primary Kervaire invariant problem at the prime three: exactly the standard classes with indices 0, 2, and 3 survive in the mod-three Adams spectral sequence, in stems 10, 106, and 322. Each surviving detection coset contains an element of exact additive order three.", "scope_url": null }, { "id": "310", "title": "Quillen's conjecture in rational homology", "subject": "Topology", "description": "Proves the rational-homology form of Quillen's conjecture for every finite group and every prime. If the largest normal p-subgroup of G is trivial, the poset of nontrivial elementary abelian p-subgroups has nonzero augmented reduced rational homology and is therefore not contractible.", "scope_url": null }, { "id": "311", "title": "The Hovey–Strickland and Chai conjectures", "subject": "Topology", "description": "Proves Chai's invariant-ideal conjecture for Lubin–Tate deformation rings over finite residue fields, at every prime and positive height n. Through the implication of Barthel–Heard–Naumann, this proves the Hovey–Strickland conjecture: dualizable K(n)-local spectra have exactly n+2 thick tensor ideals, and their Balmer spectrum is a chain of n+1 points.", "scope_url": null }, { "id": "312", "title": "The Grothendieck homotopy hypothesis", "subject": "Topology", "description": "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.", "scope_url": "https://github.com/openai/math/blob/1581d4cbdd69a0faf34b2089614fe2a814316a6a/lean/docs/312.md" }, { "id": "313", "title": "Finite generation for the K(n)-local sphere", "subject": "Topology", "description": "Answers the degreewise finiteness question of Hovey and Hovey–Strickland: every homotopy group of the K(n)-local sphere is a finitely generated ℤp-module, for every prime, positive height and integer degree. The same conclusion holds after K(n)-localizing any finite p-local spectrum.", "scope_url": null }, { "id": "314", "title": "Cyclic length and chromatic fixed-point loss", "subject": "Topology", "description": "Determines the optimal chromatic loss from geometric H-fixed points to geometric G-fixed points for every subgroup H of a finite p-group G. At every nonnegative height, the loss equals the shortest subnormal-chain length from H to G with cyclic quotients. Each quotient counts once regardless of order, and finite spectra witness sharpness.", "scope_url": null }, { "id": "315", "title": "The four-dimensional Singer conjecture", "subject": "Topology", "description": "Proves that the L2-Betti numbers of the universal cover of every closed connected aspherical topological four-manifold vanish outside degree two. More generally, the same conclusion holds for every finite connected aspherical integral Poincaré complex of formal dimension four, proving the four-dimensional Singer conjecture in this wider class.", "scope_url": null }, { "id": "316", "title": "Curtis’s conjecture", "subject": "Topology", "description": "Proves Curtis’s conjecture: the positive-degree mod-two stable Hurewicz image of the sphere is spanned by the images of the Hopf-invariant-one classes η, ν, σ and the Kervaire-invariant-one classes that exist.", "scope_url": null }, { "id": "317", "title": "Thomason model structures in all strict higher dimensions", "subject": "Topology", "description": "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.", "scope_url": "https://github.com/openai/math/blob/1581d4cbdd69a0faf34b2089614fe2a814316a6a/lean/docs/317.md" }, { "id": "318", "title": "Chromatic splitting: filtrations and counterexamples", "subject": "Topology", "description": "Disproves strong chromatic splitting at height three for primes p ≥ 5, and weak splitting for the derived p-completed sphere at heights p (p ≥ 5) and p+1 (p ≥ 7). Nevertheless, for n ≥ 1 and pgt n+1, the overlap L_{n-1}L_{K(n)}S_p^wedge admits a 2^n-stage filtration by the predicted localized-sphere pieces. At height three and prime three, even finite assembly from such pieces fails in the category of E(2)-local modules over the derived completed sphere.", "scope_url": null }, { "id": "319", "title": "Counterexamples to finite generation at chromatic height two", "subject": "Topology", "description": "Refutes the Hahn–Wilson conjecture at chromatic height two. For every sufficiently large prime p, constructs a connective p-complete spectrum of exact fp-type two that cannot be built from completed mathrm{BP}langle2rangle by finitely many sums, shifts, cones and retracts. The examples nevertheless satisfy the finite and telescopic localization comparisons.", "scope_url": null }, { "id": "320", "title": "Nonhomeomorphic closed aspherical four-manifolds", "subject": "Topology", "description": "Constructs closed connected aspherical topological four-manifolds that are homotopy equivalent but not homeomorphic, with a common word-hyperbolic fundamental group. This disproves even the homeomorphism-existence formulation of the Borel conjecture in dimension four.", "scope_url": null }, { "id": "321", "title": "A counterexample to Wall's finite D(2) problem", "subject": "Topology", "description": "Constructs a finite connected three-dimensional CW complex whose universal cover has no integral homology above degree two and whose third cohomology vanishes for every local coefficient module, but which has no finite two-dimensional homotopy model. This disproves Wall's finite D(2) conjecture; the example has infinite fundamental group.", "scope_url": null }, { "id": "322", "title": "Tingley’s sphere-isometry problem", "subject": "Functional analysis", "description": "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.", "scope_url": "https://github.com/openai/math/blob/1581d4cbdd69a0faf34b2089614fe2a814316a6a/lean/docs/322.md" }, { "id": "323", "title": "Independence of the separable quotient problem", "subject": "Functional analysis", "description": "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.", "scope_url": "https://github.com/openai/math/blob/1581d4cbdd69a0faf34b2089614fe2a814316a6a/lean/docs/323.md" }, { "id": "324", "title": "Lipschitz equivalent Banach spaces need not be linearly isomorphic", "subject": "Functional analysis", "description": "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.", "scope_url": "https://github.com/openai/math/blob/1581d4cbdd69a0faf34b2089614fe2a814316a6a/lean/docs/324.md" }, { "id": "325", "title": "The complete Crouzeix conjecture", "subject": "Functional analysis", "description": "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]rVertle2sup_{zin 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.", "scope_url": "https://github.com/openai/math/blob/1581d4cbdd69a0faf34b2089614fe2a814316a6a/lean/docs/325.md" }, { "id": "326", "title": "The cotype–cotype conjecture under the approximation property", "subject": "Functional analysis", "description": "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.", "scope_url": "https://github.com/openai/math/blob/1581d4cbdd69a0faf34b2089614fe2a814316a6a/lean/docs/326.md" }, { "id": "327", "title": "Markov type characterizes superreflexivity", "subject": "Functional analysis", "description": "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.", "scope_url": "https://github.com/openai/math/blob/1581d4cbdd69a0faf34b2089614fe2a814316a6a/lean/docs/327.md" }, { "id": "328", "title": "Nonexpansive fixed points in reflexive Banach spaces", "subject": "Functional analysis", "description": "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.", "scope_url": "https://github.com/openai/math/blob/1581d4cbdd69a0faf34b2089614fe2a814316a6a/lean/docs/328.md" }, { "id": "329", "title": "A counterexample to metric-entropy duality", "subject": "Functional analysis", "description": "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.", "scope_url": "https://github.com/openai/math/blob/1581d4cbdd69a0faf34b2089614fe2a814316a6a/lean/docs/329.md" }, { "id": "330", "title": "A uniformly discrete counterexample to bounded approximation in Lipschitz-free spaces", "subject": "Functional analysis", "description": "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.", "scope_url": "https://github.com/openai/math/blob/1581d4cbdd69a0faf34b2089614fe2a814316a6a/lean/docs/330.md" }, { "id": "331", "title": "Reflexive midpoint convexity and diamond distortion", "subject": "Functional analysis", "description": "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.", "scope_url": "https://github.com/openai/math/blob/1581d4cbdd69a0faf34b2089614fe2a814316a6a/lean/docs/331.md" }, { "id": "332", "title": "Metric Markov cotype of ℓ1 and Hilbert-space Lipschitz extension", "subject": "Functional analysis", "description": "Proves that real ℓ1 has metric Markov cotype two, answering Mendel and Naor's question. Consequently, every Lipschitz map from an arbitrary subset of a real Hilbert space into ℓ1 extends to the whole space with a universal multiplicative loss in its Lipschitz constant, resolving Ball's extension problem for this target.", "scope_url": null }, { "id": "333", "title": "Smooth isometric immersions of surfaces into ℝ4", "subject": "Differential geometry", "description": "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.", "scope_url": "https://github.com/openai/math/blob/1581d4cbdd69a0faf34b2089614fe2a814316a6a/lean/docs/333.md" }, { "id": "334", "title": "A smooth surface metric with no local isometric immersion in ℝ3", "subject": "Differential geometry", "description": "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.", "scope_url": "https://github.com/openai/math/blob/1581d4cbdd69a0faf34b2089614fe2a814316a6a/lean/docs/334.md" }, { "id": "335", "title": "Gromov’s integral scalar-curvature bound for simplicial volume", "subject": "Differential geometry", "description": "Proves int_M(mathrm{Scal}_g^-)^{n/2},dV_gge a_nlVert MrVert for every closed connected oriented smooth n-manifold, n ≥ 3, and every smooth metric, with a_ngt 0 depending only on dimension. Here mathrm{Scal}_g^-=max{0,-mathrm{Scal}_g} and lVert MrVert is real simplicial volume. Also proves rational inessentiality under positive scalar curvature, resolving the Gromov–Lawson conjecture; every nonnegative-scalar-curvature metric on a closed aspherical manifold is flat.", "scope_url": null }, { "id": "336", "title": "Spectral scalar curvature, Urysohn width, and macroscopic dimension", "subject": "Differential geometry", "description": "Every complete connected smooth boundaryless n-manifold, n ≥ 3, satisfying -4Delta+mathrm{Scal}ge1 as a quadratic-form inequality admits a continuous map to a simplicial complex of dimension at most n-2 whose entire fibers have diameter bounded only by n in the original metric. This strengthens Gromov's width conclusion to spectral scalar curvature. Universal covers of closed positive-scalar-curvature manifolds also have continuous macroscopic dimension at most n-2 for every n ≥ 2.", "scope_url": null }, { "id": "337", "title": "Sharp Cartan–Hadamard isoperimetry and rigidity", "subject": "Differential geometry", "description": "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.", "scope_url": "https://github.com/openai/math/blob/1581d4cbdd69a0faf34b2089614fe2a814316a6a/lean/docs/337.md" }, { "id": "338", "title": "Yau's uniformization conjecture", "subject": "Differential geometry", "description": "Proves Yau's uniformization conjecture: every complete connected noncompact Kähler manifold with strictly positive holomorphic bisectional curvature is biholomorphic to ℂn.", "scope_url": null }, { "id": "339", "title": "Katok's entropy rigidity conjecture", "subject": "Differential geometry", "description": "Proves Katok's entropy rigidity conjecture for closed connected Riemannian manifolds of dimension at least three with strictly negative sectional curvature: normalized Liouville measure maximizes entropy for the geodesic flow if and only if the metric is locally symmetric.", "scope_url": null }, { "id": "340", "title": "A counterexample to the nearby Lagrangian conjecture", "subject": "Differential geometry", "description": "Disproves the unrestricted nearby Lagrangian conjecture. For some sufficiently large even N, constructs a closed exact embedded Lagrangian in T^*(S^9times S^{N-1}) that is diffeomorphic to the base but not Hamiltonian isotopic to its zero section.", "scope_url": null }, { "id": "341", "title": "Donaldson's hypersymplectic deformation conjecture", "subject": "Differential geometry", "description": "Proves Donaldson's hypersymplectic deformation conjecture in a cohomology-preserving form. Every positive triple of smooth closed two-forms on a closed connected oriented four-manifold, normalized by intomega_iwedgeomega_j=delta_{ij}, deforms through positive closed triples to a hyperkähler triple while preserving all three cohomology classes. Any positive triple can first be normalized by a constant linear change.", "scope_url": null }, { "id": "342", "title": "Donaldson's tamed-to-compatible conjecture", "subject": "Differential geometry", "description": "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.", "scope_url": "https://github.com/openai/math/blob/1581d4cbdd69a0faf34b2089614fe2a814316a6a/lean/docs/342.md" }, { "id": "343", "title": "Symplectic ball packing in higher dimensions", "subject": "Differential geometry", "description": "Resolves the Siegel–Yao conjecture for arbitrary capacities in every dimension 2nge6. 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^nlt R^n and R_i+R_jlt R for every distinct pair i, j.", "scope_url": "https://github.com/openai/math/blob/1581d4cbdd69a0faf34b2089614fe2a814316a6a/lean/docs/343.md" }, { "id": "344", "title": "The metric Blaschke conjecture", "subject": "Differential geometry", "description": "Proves the metric Blaschke conjecture: every closed connected Riemannian manifold of positive dimension whose injectivity radius equals its diameter is, up to scaling, a standard compact rank-one symmetric space.", "scope_url": null }, { "id": "345", "title": "Infinitely many closed geodesics on Riemannian spheres and closed three-manifolds", "subject": "Differential geometry", "description": "Proves that every smooth Riemannian metric on Sn, n ≥ 2, has infinitely many prime closed geodesics with pairwise distinct images. The same conclusion holds on every closed manifold admitting a finite smooth spherical cover and on every closed three-manifold, without orientability or nondegeneracy restrictions.", "scope_url": null }, { "id": "346", "title": "Sharp singular-set bounds for stationary integral varifolds", "subject": "Differential geometry", "description": "Proves that every stationary integral m-varifold in a Euclidean open set has singular set of Hausdorff dimension at most m-1, a sharp bound in every positive dimension and codimension. On round spheres, the family also proves almost-everywhere regularity: the singular set has zero m-dimensional Hausdorff measure.", "scope_url": null }, { "id": "347", "title": "Counterexamples to stable-Morse and strong Arnold fixed-point bounds", "subject": "Differential geometry", "description": "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.", "scope_url": "https://github.com/openai/math/blob/1581d4cbdd69a0faf34b2089614fe2a814316a6a/lean/docs/347.md" }, { "id": "348", "title": "Nonnegative-curvature Einstein classification and an L2 topological gap", "subject": "Differential geometry", "description": "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.", "scope_url": "https://github.com/openai/math/blob/1581d4cbdd69a0faf34b2089614fe2a814316a6a/lean/docs/348.md" }, { "id": "349", "title": "The Solomon–Yau least-volume conjecture", "subject": "Differential geometry", "description": "Proves the Solomon–Yau least-volume conjecture for minimal hypersurfaces of round spheres. For every m ≥ 2, a closed connected minimal immersion into the unit sphere S^{m+1} with non-totally-geodesic image has volume at least that of the smallest minimal Clifford product, counting covering multiplicity.", "scope_url": null }, { "id": "350", "title": "Yau’s nodal bounds: surfaces and higher dimensions", "subject": "Differential geometry", "description": "Proves the sharp Csqrtlambda 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_0gt 0, ruling out even arbitrarily small power losses.", "scope_url": "https://github.com/openai/math/blob/1581d4cbdd69a0faf34b2089614fe2a814316a6a/lean/docs/350.md" }, { "id": "351", "title": "Scalar curvature and finite-time Ricci-flow singularities", "subject": "Differential geometry", "description": "Proves that a smooth Ricci flow on a closed four-manifold extends past any finite time at which scalar curvature remains uniformly bounded. A higher-dimensional counterexample has bounded scalar curvature but unbounded full curvature at its finite maximal time, disproving the unrestricted scalar-curvature extension conjecture.", "scope_url": null }, { "id": "352", "title": "A finite-time singularity of Calabi flow", "subject": "Differential geometry", "description": "Disproves Chen's smooth long-time existence conjecture for Calabi flow by constructing a smooth U(10)-invariant Kähler metric on mathbb{CP}^{10} whose flow develops a finite-time singularity. The metric lies in the Fubini–Study class, so the failure occurs even in a class containing a constant-scalar-curvature metric.", "scope_url": null }, { "id": "353", "title": "Affine Bernstein rigidity through dimension nine and a smooth dimension-ten counterexample", "subject": "Differential geometry", "description": "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.", "scope_url": "https://github.com/openai/math/blob/1581d4cbdd69a0faf34b2089614fe2a814316a6a/lean/docs/353.md" }, { "id": "354", "title": "The isoperimetric profile of the cubic three-torus", "subject": "Differential geometry", "description": "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 4pi/81 and 1/pi, with exactly the adjacent two types minimizing at each transition.", "scope_url": "https://github.com/openai/math/blob/1581d4cbdd69a0faf34b2089614fe2a814316a6a/lean/docs/354.md" }, { "id": "355", "title": "Unique tangent flows at the first surface singularity", "subject": "Differential geometry", "description": "Proves the first-singular-time case of tangent-flow uniqueness for smooth compact connected embedded surfaces without boundary in ℝ3. At every singular point, all fixed-center backward tangent flows agree as area measures at every negative time in the original ambient coordinates, without mean-convexity or a prescribed tangent model.", "scope_url": null }, { "id": "356", "title": "Gigli’s characterization of Alexandrov curvature", "subject": "Differential geometry", "description": "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.", "scope_url": "https://github.com/openai/math/blob/1581d4cbdd69a0faf34b2089614fe2a814316a6a/lean/docs/356.md" }, { "id": "357", "title": "Bi-Lipschitz coordinates at every regular RCD point", "subject": "Differential geometry", "description": "Proves that every regular point of a noncollapsed mathop{mathrm{RCD}}nolimits (K,n) space, for K ∈ ℝ and integer n ≥ 2, has an open neighborhood bi-Lipschitz to an open subset of ℝn. Regularity requires all pointed tangents to be Euclidean, the reference measure is exactly mathcal H^n, and the chart compares ambient distances with a point-dependent finite constant.", "scope_url": null }, { "id": "358", "title": "A three-manifold without conjugate points or nonpositive curvature", "subject": "Differential geometry", "description": "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.", "scope_url": "https://github.com/openai/math/blob/1581d4cbdd69a0faf34b2089614fe2a814316a6a/lean/docs/358.md" }, { "id": "359", "title": "Negative Kähler curvature without bounded holomorphic coordinates", "subject": "Differential geometry", "description": "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.", "scope_url": "https://github.com/openai/math/blob/1581d4cbdd69a0faf34b2089614fe2a814316a6a/lean/docs/359.md" }, { "id": "360", "title": "Weak MTW curvature gives convexity and regular optimal transport", "subject": "Differential geometry", "description": "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.", "scope_url": "https://github.com/openai/math/blob/1581d4cbdd69a0faf34b2089614fe2a814316a6a/lean/docs/360.md" }, { "id": "361", "title": "Failure of integer-degree harmonic dimension comparison", "subject": "Differential geometry", "description": "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.", "scope_url": "https://github.com/openai/math/blob/1581d4cbdd69a0faf34b2089614fe2a814316a6a/lean/docs/361.md" }, { "id": "362", "title": "Global smoothness for relativistic Vlasov–Maxwell", "subject": "Partial differential equations", "description": "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.", "scope_url": "https://github.com/openai/math/blob/1581d4cbdd69a0faf34b2089614fe2a814316a6a/lean/docs/362.md" }, { "id": "363", "title": "Nonuniqueness with local conservation for the hard-sphere Boltzmann equation", "subject": "Partial differential equations", "description": "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.", "scope_url": "https://github.com/openai/math/blob/1581d4cbdd69a0faf34b2089614fe2a814316a6a/lean/docs/363.md" }, { "id": "364", "title": "Kinetic limits and fluctuations over the Boltzmann lifespan", "subject": "Partial differential equations", "description": "Derives the nonlinear Boltzmann equation from three-dimensional grand-canonical Newtonian gases throughout every regular kinetic interval with uniform Gaussian decay. Stable finite-range radial potentials may have attractive wells and a singular repulsive core; initial pair exclusion and spatially summable Gaussian density and gradient bounds are assumed. A companion gives finite-dimensional hard-sphere Gaussian fluctuations, centered at the exact microscopic expectation and governed by the linear fluctuating Boltzmann equation.", "scope_url": null }, { "id": "365", "title": "Joint metric and connection recovery from one boundary patch", "subject": "Partial differential equations", "description": "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.", "scope_url": "https://github.com/openai/math/blob/1581d4cbdd69a0faf34b2089614fe2a814316a6a/lean/docs/365.md" }, { "id": "366", "title": "The planar Mumford–Shah regularity conjecture and local weak-L4 gradient bounds", "subject": "Partial differential equations", "description": "Resolves the interior regularity conjecture for reduced absolute planar Mumford–Shah minimizers with bounded fidelity data. Locally, the closed discontinuity set is a C^{1,alpha} arc, a regular crack tip, or three arcs meeting at 120^circ; only finitely many global connected components meet any compact interior region.", "scope_url": null }, { "id": "367", "title": "The critical dimension for the one-phase Bernoulli problem", "subject": "Partial differential equations", "description": "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.", "scope_url": "https://github.com/openai/math/blob/1581d4cbdd69a0faf34b2089614fe2a814316a6a/lean/docs/367.md" }, { "id": "368", "title": "The three-dimensional Ball–Evans approximation problem", "subject": "Partial differential equations", "description": "Resolves the three-dimensional Ball–Evans approximation problem: every W^{1,p} homeomorphism between arbitrary bounded domains in ℝ3, for 1le plt infty, is a strong W^{1,p} limit of smooth diffeomorphisms onto the same target.", "scope_url": null }, { "id": "369", "title": "The hot spots conjecture for simply connected planar domains", "subject": "Partial differential equations", "description": "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.", "scope_url": "https://github.com/openai/math/blob/1581d4cbdd69a0faf34b2089614fe2a814316a6a/lean/docs/369.md" }, { "id": "370", "title": "The Lane–Emden and Hénon–Lane–Emden conjectures", "subject": "Partial differential equations", "description": "Resolves the subcritical Lane–Emden conjecture and its weighted Hénon extension. For n ≥ 2, p,qgt 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,Bgt -2.", "scope_url": "https://github.com/openai/math/blob/1581d4cbdd69a0faf34b2089614fe2a814316a6a/lean/docs/370.md" }, { "id": "371", "title": "Stable blowup for the defocusing Schrödinger equation", "subject": "Partial differential equations", "description": "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.", "scope_url": "https://github.com/openai/math/blob/1581d4cbdd69a0faf34b2089614fe2a814316a6a/lean/docs/371.md" }, { "id": "372", "title": "Global uniqueness in smooth isotropic elasticity", "subject": "Partial differential equations", "description": "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 3lambda+2mugt 0 on the closure. Neither analyticity, proximity to constant coefficients nor prior knowledge near the boundary is required.", "scope_url": "https://github.com/openai/math/blob/1581d4cbdd69a0faf34b2089614fe2a814316a6a/lean/docs/372.md" }, { "id": "373", "title": "Nonattainment of the three-marginal Coulomb Monge problem", "subject": "Partial differential equations", "description": "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.", "scope_url": null }, { "id": "374", "title": "Sharp one-third stability of Brenier maps", "subject": "Partial differential equations", "description": "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.", "scope_url": null }, { "id": "375", "title": "De Giorgi's conjecture in dimension eight", "subject": "Partial differential equations", "description": "Proves De Giorgi's conjecture at its sharp dimension-eight endpoint: every entire C2 solution u:mathbb R^8to(-1,1) of Delta u=u^3-u that is strictly increasing in one direction depends on only one linear coordinate. A stronger theorem classifies all stable entire solutions v:mathbb R^7to[-1,1] as constant wells or planar transitions, without an energy-growth assumption.", "scope_url": null }, { "id": "376", "title": "Universal computation in forced Navier–Stokes flows", "subject": "Partial differential equations", "description": "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.", "scope_url": null }, { "id": "377", "title": "Interior C^{1,alpha} regularity for infinity-harmonic functions", "subject": "Partial differential equations", "description": "Proves uniform interior C^{1,alpha_d} regularity for bounded infinity-harmonic functions in every dimension d ≥ 3, for a positive exponent depending only on dimension. The gradient's supremum norm and Hölder seminorm on the half unit ball are bounded by a dimension-dependent constant times the oscillation on the unit ball. The exponent is not explicit.", "scope_url": null } ], "papers": [ { "id": "001-01", "family_id": "001", "title": "Milne's rationality conjecture for abelian varieties", "subject": "Number theory", "date": "2026-10-07", "status": "claimed_proof", "formalization_catalogue": false, "family_scope_available": false, "paper_in_scope_note": false, "pdf_url": "https://raw.githubusercontent.com/openai/math/1581d4cbdd69a0faf34b2089614fe2a814316a6a/preprints/Milnes-rationality-conjecture-for-abelian-varieties-October-7-2026/paper.pdf", "source_url": "https://github.com/openai/math/tree/1581d4cbdd69a0faf34b2089614fe2a814316a6a/preprints/Milnes-rationality-conjecture-for-abelian-varieties-October-7-2026", "version_url": "https://github.com/openai/math/blob/1581d4cbdd69a0faf34b2089614fe2a814316a6a/preprints/Milnes-rationality-conjecture-for-abelian-varieties-October-7-2026/README.md", "scope_url": null, "independent_verification": false, "publisher": "OpenAI", "license": "Apache-2.0", "path": 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