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372 results

001
Milne’s rationality conjecture and algebraic specialization
Proves Milne's rationality conjecture for abelian varieties over $Q$ 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.
To be verifiedNumber theory
002
The full BSD formula from low Selmer corank
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.
To be verified Updated versionNumber theory
003
The quasi-Riemann hypothesis
Proves that every Dirichlet L-function, including $ζ (s)$ , is zero-free in $ℜ s > 7/8$ , resolving the quasi-Riemann hypothesis. The same half-plane is zero-free for every finite-order Hecke L-function over $Q (- 3)$ . A companion gives a different proof of the zero-free half-plane $ℜ s > 11/12$ .
Verified by LeanNumber theory
004
Hilbert’s tenth problem over ℚ
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.
To be verifiedNumber theory
005
Irrationality of Catalan’s constant
Proves that Catalan's constant $G = \sum_{j \geq 0} (- 1)^{j} / (2 j + 1)^{2}$ is irrational.
Verified by LeanNumber theory
006
Goldfeld’s conjecture: densities and mean analytic rank
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.
To be verifiedNumber theory
007
Ordinary two-point correlations and the corrected Elliott conjecture
Proves the ordinary two-point Chowla conjecture, with a bound $O (X / (lo g X)^{c})$ 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 $n^{i t}$ for $∣ t ∣ \leq X$ .
Verified by Lean Reasoning summaryNumber theory
008
The Deligne–Drinfeld conjecture
Proves that the rational Grothendieck–Teichmüller Lie algebra, with the Ihara bracket, is freely generated by one element in each odd weight 3, 5, 7, …, resolving the Deligne–Drinfeld conjecture.
Verified by LeanNumber theory
009
Function-field reconstruction from Milnor K-theory and Galois data
Reconstructs function fields of transcendence degree at least two over algebraically closed constants from $K_{1}^{M} / ℓ$ , $K_{2}^{M} / ℓ$ , 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.
Verified by LeanNumber theory
010
Unrestricted pro-modularity at the prime two
Every continuous odd absolutely irreducible two-dimensional 2-adic representation of $G_{Q}$ 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.
To be verifiedNumber theory
011
Prime-factor statistics of $p - 1$
Proves that the normalized ordered logarithms of the prime factors of $p - 1$ , counted with multiplicity, converge jointly to the Poisson–Dirichlet law $PD (1)$ as p ranges uniformly over primes up to x and $x \to \infty$ . This resolves the Ford–Konyagin–Luca conjecture. It also proves that infinitely many integers n have more than $n^{1 - ε}$ totient preimages, for every ε > 0.
To be verifiedNumber theory
012
Independent largest prime factors of consecutive integers
Resolves the Erdős–Pomerance joint Dickman conjecture: the logarithmic sizes of the largest prime factors of n and $n + 1$ are asymptotically independent in ordinary natural density. In particular, the integers satisfying $P^{+} (n) < P^{+} (n + 1)$ have density 1/2.
Verified by LeanNumber theory
013
Ostmann’s inverse Goldbach conjecture
Proves that no finite modification of the primes can be written as $A + B$ with $A, B \subseteq Z_{\geq 0}$ each containing at least two elements. This resolves Ostmann's inverse Goldbach conjecture on additive indecomposability.
Verified by LeanNumber theory
014
Restricted geometric Langlands, global Arthur enhancements, and generic Ramanujan
Proves the restricted geometric Langlands equivalence for connected reductive groups on smooth projective connected curves over $F_{q}$ 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 $p ∤ ∣ W_{G} ∣$ . 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.
To be verifiedNumber theory
015
Torus-packet equidistribution in prime, quartic, and sextic degrees
Proves Haar equidistribution without escape of mass for complete volume-weighted torus packets from totally real fields: arbitrary lattices and prescribed local types in fixed prime degree at least five, arbitrary-order Picard packets in primitive quartic fields, and maximal-order ideal-class packets in primitive sextic fields. Here primitive means having no proper intermediate field; the relevant order or field discriminant tends to infinity.
Verified by LeanNumber theory
016
Zilber–Pink in abelian varieties and the Siegel threefold
Proves the abelian Zilber–Pink conjecture over $Q$ : 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 $A_{2}$ for Hodge-generic curves defined over $Q$ , without boundary or reduction assumptions.
To be verifiedNumber theory
017
The irrationality exponent of π is 2
Proves that the irrationality exponent of π is exactly 2: for every ε > 0 and all sufficiently large denominators q, every rational $p / q$ satisfies $∣ π - p / q ∣ \geq q^{- 2 - ε}$ . This also proves convergence of the Flint–Hills series $\sum_{n \geq 1} 1/ (n^{3} sin^{2} n)$ , with angles in radians.
Verified by Lean Reasoning summaryNumber theory
018
The Margulis–Platonov conjecture over global fields
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.
To be verifiedNumber theory
019
The local p-adic section conjecture and global consequences
Proves that rational points on every smooth proper geometrically connected curve of genus at least two over a finite extension of ℚ_p 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.
To be verifiedNumber theory
020
Squarefree quartics and power-free polynomial values
Proves the squarefree-values conjecture for irreducible integer quartics with no fixed prime-square divisor: squarefree values on positive integers have the predicted positive Euler-product density. More generally, establishes the $(d - 2)$ -power-free density for irreducible integer polynomials of degrees four through eight under the necessary local condition; together with Browning's higher-degree theorem, this covers every d ≥ 4.
Verified by LeanNumber theory
021
A quadratic bound for Jacobsthal’s function
Answers Jacobsthal's quadratic-bound question: every interval of $C k^{2}$ 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.
Verified by LeanNumber theory
022
The weak inhomogeneous Duffin–Schaeffer conjecture
Proves that for every real shift γ and finite-valued $ψ : N \to [0, \infty)$ , divergence of $\sum_{q} ϕ (q) ψ (q) / q$ implies $∥ q x - γ ∥ < ψ (q)$ 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.
To be verifiedNumber theory
023
Patterson's first moment for cubic Gauss sums
Proves unconditionally the all-primary-prime form of Patterson’s first-moment asymptotic: normalized cubic Gauss sums over primary Eisenstein primes of norm at most X, including both conjugates, have an explicit positive main term of order $X^{5/6} / lo g X$ . Every fixed nonzero prime-angle Fourier mode has smaller order.
Verified by LeanNumber theory
024
An asymptotic formula for the number of totients
Gives an asymptotic equivalent for the number $V (x)$ of distinct totient values up to x, with a positive bounded phase-dependent factor determined by convergent arithmetic approximations. In particular, $V (c x) / V (x) \to c$ for every fixed c > 0, answering Erdős and Hall’s scaling question.
Verified by LeanNumber theory
025
Short Egyptian fractions
Every rational $a / b$ with $1 \leq a < b$ is a sum of $O (lo g lo g b)$ distinct positive unit fractions. The worst-case minimum number of terms has the same order, resolving Erdős’s conjecture on short Egyptian fractions.
Verified by LeanNumber theory
026
Positive lower density of large prime gaps
For every fixed C > 0, a positive proportion of consecutive prime gaps exceed $C lo g p_{n}$ , 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.
Verified by LeanNumber theory
027
Potential integral density on curve character varieties
Resolves the determinant-one curve case of Litt's integral-density question. For every smooth connected complex algebraic curve and every rank, integral points become Zariski dense in every component of its SL_r character variety over the full ring of integers of one number field. Prescribed quasi-unipotent boundary conjugacy classes are allowed, including nonsemisimple classes.
Verified by LeanNumber theory
028
Uniformly bounded components of Gaussian-prime graphs
Proves the Gaussian moat conjecture: no infinite walk through distinct Gaussian primes can have uniformly bounded steps. More strongly, for every distance bound D, the graph joining Gaussian primes at distance at most D has uniformly bounded finite component sizes, depending only on D, including primes on the coordinate axes.
Verified by LeanNumber theory
029
Primitive roots for every admissible integer base
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 $c_{a} x / (lo g x)^{2}$ primes in every sufficiently large interval $(x, 2 x)$ have primitive root a, with $c_{a} > 0$ .
To be verifiedNumber theory
030
Modularity of elliptic curves over imaginary quadratic fields
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.
To be verifiedNumber theory
031
Uchida’s conjecture for open homomorphisms of Galois groups
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.
To be verifiedNumber theory
032
The rational Hodge conjecture for CM abelian varieties
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.
To be verifiedAlgebraic and complex geometry
033
Iitaka subadditivity, variation, and logarithmic additivity
Proves Campana's orbifold Iitaka subadditivity conjecture for smooth Fujiki-class- $C$ manifolds with rational simple-normal-crossing boundaries. For projective fibrations $f : U \to V$ of smooth complex quasi-projective varieties with connected fibers, general fiber F, and $κ ˉ (V) \geq 0$ , proves Popa's inequality $κ ˉ (U) \geq κ (F) + max {κ ˉ (V), Var (f)}$ , where variation measures the whole geometric generic fiber.
Verified by LeanAlgebraic and complex geometry
034
Log abundance for compact Kähler spaces under logarithmic Iitaka subadditivity
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.
To be verified Updated versionAlgebraic and complex geometry
035
Log-canonical threefold abundance in numerical dimension one
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.
To be verifiedAlgebraic and complex geometry
036
Numerical semiampleness and generalized minimal models
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.
To be verifiedAlgebraic and complex geometry
037
The ordinary-double-point volume gap
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.
To be verifiedAlgebraic and complex geometry
038
Fujita’s freeness conjecture
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 $K_{X} + m L$ is globally generated for every integer $m \geq n + 1$ .
To be verifiedAlgebraic and complex geometry
039
Nagata’s conjecture and maximal Seshadri constants
Proves Nagata's strict inequality $\sum_{i} m_{i} < d r$ for every nonzero effective plane curve of degree d through r ≥ 10 very general complex points, with arbitrary multiplicities m_i. It also proves maximal multipoint Seshadri constants $(L^{n} / r)^{1/ n}$ for every smooth polarized projective variety of dimension n ≥ 2 and all sufficiently large r: at very general points over ℂ, and at the geometric generic tuple over any algebraically closed field of positive characteristic.
Verified by LeanAlgebraic and complex geometry
040
Bloch’s conjecture for complex surfaces
Proves Bloch's conjecture: for every smooth connected projective complex surface S with $p_{g} (S) = 0$ , the Albanese map $CH_{0} (S)^{0} \to Alb (S) (C)$ 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.
To be verifiedAlgebraic and complex geometry
041
Hyperkähler SYZ and projective-space bases
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.
To be verifiedAlgebraic and complex geometry
042
Oka classification for minimal compact complex surfaces: Kodaira dimension zero and class VII
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.
To be verifiedAlgebraic and complex geometry
043
P = W for fixed-determinant SL_n moduli spaces
Proves $P_{k} = W_{2 k} = W_{2 k + 1}$ 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.
To be verifiedAlgebraic and complex geometry
044
The equivariant cohomological Hikita conjecture
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.
To be verifiedAlgebraic and complex geometry
046
Shafarevich counterexamples in dimension two and with large fundamental group
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.
To be verifiedAlgebraic and complex geometry
047
Zariski cancellation and affine fibrations over the complex numbers
Constructs an integral complex affine fourfold $X \neq ≅ A^{4}$ with $X \times A^{1} ≅ A^{5}$ , disproving affine-space cancellation over ℂ in dimension four. It also disproves the Dolgachev–Weisfeiler affine-fibration conjecture: smooth surjections $X \to A^{1}$ and $A^{5} \to A^{2}$ have every residue-field fiber isomorphic to affine three-space but are not Zariski-locally trivial.
Verified by LeanAlgebraic and complex geometry
048
A characteristic-zero counterexample to Lipman–Zariski
Constructs a singular normal affine complex surface with free rank-two tangent sheaf, disproving the characteristic-zero Lipman–Zariski conjecture.
To be verifiedAlgebraic and complex geometry
049
A stable-coordinate counterexample in four variables
Constructs a polynomial in four complex variables that is not a coordinate but becomes one after adjoining a single variable, disproving the Stable Coordinate conjecture in four variables. Every fiber is affine three-space, yet none of its embeddings is rectifiable, also disproving the Abhyankar–Sathaye conjecture even when all fibers are affine spaces.
Verified by LeanAlgebraic and complex geometry
050
A counterexample to Griffiths’ positivity conjecture
Constructs ample rank-two bundles on $P^{1} \times P^{1}$ with no smooth Hermitian metric of strictly Griffiths-positive curvature, disproving Griffiths' positivity conjecture already on the quadric surface.
Verified by LeanAlgebraic and complex geometry
051
Kobayashi’s canonical-ampleness conjecture
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.
To be verifiedAlgebraic and complex geometry
052
Tangent splittings and product decompositions
A splitting of the tangent bundle of a compact Kähler manifold into two integrable holomorphic subbundles induces a compatible product decomposition of its universal cover, proving the two-summand form of Beauville's splitting conjecture. On smooth rationally connected projective manifolds, both summands are automatically integrable, establishing Höring's conjecture and the corresponding product decomposition.
Verified by LeanAlgebraic and complex geometry
053
A counterexample to Pixton completeness in Chow
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.
To be verifiedAlgebraic and complex geometry
054
Irrational cubic fourfolds with Hodge-theoretic and categorical K3 associations
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.
To be verifiedAlgebraic and complex geometry
055
Gepner symmetry and large-volume stability on threefolds
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.
To be verifiedAlgebraic and complex geometry
056
Termination of projective and Kähler fourfold minimal model programs
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.
To be verifiedAlgebraic and complex geometry
057
Fundamental groups of special complex varieties and root orbifolds
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.
To be verifiedAlgebraic and complex geometry
058
Semialgebraic universal covers and bounded domains
Proves the Kollár–Pardon conjecture: the semialgebraic universal covers of connected normal projective complex varieties are exactly products $D \times C^{m} \times F$ , with D bounded symmetric and F simply connected, normal, and projective. A universal cover is quasi-projective exactly when the bounded symmetric factor is absent. In particular, a smooth projective variety covered by ℂⁿ has a finite étale cover by an abelian variety.
Verified by LeanAlgebraic and complex geometry
059
Counterexamples to Zariski’s multiplicity conjecture
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.
To be verifiedAlgebraic and complex geometry
060
The Global Spherical Shell conjecture
Every connected minimal compact complex surface of class VII with $b_{2} > 0$ 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 $C^{2} ∖ {0}$ whose complement is connected.
To be verifiedAlgebraic and complex geometry
062
Projective contact classification and the LeBrun–Salamon conjecture
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 $b_{2} \geq 2$ have underlying manifold $P (T^{*} Z)$ for a smooth projective variety Z.
To be verifiedAlgebraic and complex geometry
063
The generalized Mukai conjecture
Proves the generalized Mukai conjecture: every positive-dimensional smooth complex projective Fano manifold of dimension n, Picard number ρ and pseudoindex ι satisfies $ρ (ι - 1) \leq n$ , with equality exactly for $(P^{ι - 1})^{ρ}$ . Here the pseudoindex is the least anticanonical degree of a rational curve.
To be verifiedAlgebraic and complex geometry
064
Topological triviality of μ-constant surface singularities
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.
To be verifiedAlgebraic and complex geometry
065
Virasoro constraints for complete intersections and projective-bundle towers
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.
To be verifiedAlgebraic and complex geometry
066
Bounded klt complements for Fano contractions
Proves the finite-rational-coefficient form of Shokurov’s bounded-klt-complement conjecture for ϵ-lc complex Fano-type pairs with nef anti-log-canonical divisor. For ϵ-lc Fano contractions over any algebraically closed characteristic-zero field, it gives klt complements near every base point, with index bounded only by dimension and positive rational ϵ.
Verified by LeanAlgebraic and complex geometry
067
The Campana–Peternell conjecture in dimension six
Proves the Campana–Peternell conjecture in complex dimension six: every smooth connected complex projective Fano sixfold with nef tangent bundle is rational homogeneous.
To be verifiedAlgebraic and complex geometry
068
Anticanonical nonvanishing in every dimension
If X is a smooth connected complex projective variety and $- K_{X}$ admits a smooth Hermitian metric with nonnegative curvature, then $H^{0} (X, - m K_{X}) \neq = 0$ for some m > 0. Thus smooth semipositivity forces a nonzero section of a positive tensor power of the anticanonical bundle in every dimension.
To be verifiedAlgebraic and complex geometry
069
Global quantum geometric Langlands at irrational level
Proves the unramified de Rham quantum geometric Langlands equivalence for every connected simple complex algebraic group on every smooth projective connected complex curve, at every shifted level $c \in C ∖ Q$ . It identifies the full derived categories of twisted D-modules for the group and its Langlands dual, retaining all global forms and connected components.
To be verifiedAlgebraic and complex geometry
071
Koebe’s circle-domain conjecture
Resolves the existence part of Koebe's circle-domain conjecture: every domain in the Riemann sphere is conformally equivalent to a domain whose complementary components are round disks or points. It also proves that circle domains with conformally removable boundary are rigid, meaning every conformal equivalence to another circle domain is Möbius, establishing this direction of the He–Schramm conjecture.
Verified by LeanReal and complex analysis
072
Brennan's conjecture and the integral-means spectrum
Proves Brennan's conjecture: for every conformal bijection ϕ from a simply connected plane domain onto the disk, $∣ ϕ^{'} ∣^{s}$ is area-integrable for $4/3 < s < 4$ . The sharp universal integral-means identity is $B_{S} (t) = ∣ t ∣ - 1$ for t ≤ −2. A strict bound $B_{b} (- 1) < 1/4$ for bounded univalent functions disproves Kraetzer's prediction at that parameter.
Verified by LeanReal and complex analysis
073
The Falconer distance conjecture
Resolves the Falconer distance conjecture in every dimension d ≥ 2: every compact set $E \subset R^{d}$ with Hausdorff dimension greater than $d /2$ determines a set of Euclidean distances of positive Lebesgue measure.
Verified by LeanReal and complex analysis
074
Kakeya in three and four dimensions
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 $L^{3} (R^{3})$ to $L^{3} (S^{2})$ with norm $O_{ε} (δ^{- ε})$ for every ε > 0. In four dimensions, every set containing a unit segment in every direction has Hausdorff dimension four.
To be verifiedReal and complex analysis
075
The $L lo g L$ Fourier-convergence conjecture
Proves that the ordinary symmetric Fourier partial sums of every complex-valued function in $L lo g L (T)$ converge almost everywhere along the full sequence. This resolves the classical sufficiency conjecture at the $L lo g L$ scale.
Verified by LeanReal and complex analysis
076
Real ultraflat Littlewood polynomials and unbounded binary merit factors
Constructs polynomials with N consecutive coefficients in ${- 1, 1}$ whose modulus is $(1 + o (1)) N$ 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.
Verified by LeanReal and complex analysis
077
Fourier restriction for positively curved surfaces
Proves the diagonal Fourier extension conjecture for positively curved surfaces in three dimensions. For every compact smooth positively curved surface $Σ \subset R^{3}$ , including surfaces with boundary, the extension operator is bounded from $L^{p} (Σ)$ to $L^{p} (R^{3})$ for every p > 3.
To be verifiedReal and complex analysis
078
The three-dimensional Bochner–Riesz conjecture
Resolves the three-dimensional Bochner–Riesz conjecture in its strict range: the Bochner–Riesz multipliers of order δ are bounded on $L^{p} (R^{3})$ for every $1 \leq p \leq \infty$ whenever $δ > max {3∣1/ p - 1/2∣ - 1/2, 0}$ .
To be verifiedReal and complex analysis
079
Local smoothing in three dimensions
Resolves Sogge's local smoothing conjecture for the Euclidean wave equation in three spatial dimensions. The estimate holds throughout the full strict range $2 < p < \infty$ , with Sobolev regularity above $max {0, 1 - 3/ p}$ . In particular, the critical L³ estimate holds with every positive Sobolev loss.
To be verifiedReal and complex analysis
080
The exact Sobolev endpoint for Schrödinger convergence
Proves almost-everywhere convergence $e^{i t Δ} f \to f$ as $t ↓ 0$ for every $f \in H^{n / (2 (n + 1))} (R^{n})$ and every dimension n ≥ 2. This attains the sharp Sobolev equality case of Carleson's Schrödinger convergence problem, including the planar endpoint H^1/3.
To be verifiedReal and complex analysis
081
Riesz transforms and rectifiability in higher codimension
Resolves the remaining higher-codimension Riesz-transform rectifiability problem: for d ≥ 4 and $2 \leq n \leq d - 2$ , an n-Ahlfors–David regular Radon measure on ℝ^d is uniformly n-rectifiable whenever its n-dimensional Riesz transform is uniformly L²-bounded over all positive hard truncations. The rectifiability bounds depend only on dimension, regularity and operator bounds.
Verified by LeanReal and complex analysis
082
Annular variation and dyadic absolute bounds for the triangular Hilbert transform
Proves maximal and annular r-variation bounds, for every r > 2, from complex $L^{3} (R^{2}) \times L^{3} (R^{2})$ to $L^{3/2} (R^{2})$ . The maximal estimate controls both hard truncation endpoints and gives almost-everywhere and norm convergence. Pairing with a third input settles the triangular Hilbert transform estimate at the symmetric $L^{3} \times L^{3} \times L^{3}$ point.
Verified by LeanReal and complex analysis
083
Hilbert transforms along Lipschitz directions
Proves a uniform strong L² bound for the planar Hilbert transform along any Lipschitz unit vector field, at integration lengths bounded by an absolute multiple of its reciprocal Lipschitz constant. The estimate is uniform over inner truncations and yields an L²-bounded principal-value operator, establishing Stein's weak-type conjecture at this short scale.
Verified by LeanReal and complex analysis
084
The geometric case of the Erdős similarity conjecture
For every fixed $q \in (0, 1)$ , constructs compact subsets of $[0, 1]$ with measure arbitrarily close to one containing no translated and nontrivially dilated copy of ${q^{n} : n \geq 1}$ , with dilations of either sign. This resolves the geometric-progression case of the Erdős similarity conjecture for every ratio.
Verified by LeanReal and complex analysis
085
Endpoint Sobolev regularity of centered disk averages
Resolves the planar centered-disk case of the Hajłasz–Onninen maximal-function regularity problem. For every real $f \in W^{1, 1} (R^{2})$ , the centered disk maximal function satisfies $∥\nabla M f ∥_{1} \leq C ∥\nabla f ∥_{1}$ with an absolute constant. It belongs locally to $W^{1, 1}$ and has a globally integrable weak gradient.
Verified by LeanReal and complex analysis
086
An L³ bound for the trilinear Hilbert transform
Proves that the principal-value trilinear Hilbert transform with shifts $x - t$ , $x - 2 t$ , $x - 3 t$ is bounded from $L^{3} (R)^{3}$ to $L^{1} (R)$ . This resolves the L³ exponent case of the standard conjecture for slopes 1, 2, 3.
To be verifiedReal and complex analysis
087
The Mahler conjectures, functional inequalities and polar-product symplectic width
Resolves the symmetric and nonsymmetric geometric Mahler conjectures in every dimension, with Hanner polytopes and simplices as the respective volume-product minimizers and all equality cases classified. The corresponding sharp functional Mahler inequalities also hold. For n ≥ 2, every symmetric polar product $K \times K^{\circ}$ in dimension $2 n$ has Gromov width 4.
Verified by Lean Reasoning summaryConvex and metric geometry
088
Sharp projection-body inequalities and a counterexample to simplex maximization
Proves Petty's projection-volume conjecture in the remaining dimensions n ≥ 4: ellipsoids uniquely minimize projection-body volume at fixed body volume. Also establishes the full Lutwak–Petty projection inequalities. In contrast, products of simplices exceed Brannen's proposed simplex maximum for normalized projection-body volume by an exponential factor in every sufficiently large dimension.
Verified by LeanConvex and metric geometry
089
Bounded-distortion L₁ embeddings of planar and bounded-treewidth graphs
Resolves the planar and bounded-treewidth cases of the Gupta–Newman–Rabinovich–Sinclair conjecture. Shortest-path metrics of finite connected graphs with arbitrary positive edge lengths embed into real L₁ with universal distortion for planar graphs, and distortion depending only on treewidth for bounded-treewidth graphs. The corresponding multicommodity flow–cut gaps are uniformly bounded.
Verified by LeanConvex and metric geometry
090
Triangular-lattice optimality, long-range Riesz and Coulomb energies, and spherical logarithmic energy
Proves that the triangular lattice minimizes the lower limit of energy per particle for every nonnegative completely monotone potential of squared distance among locally finite planar configurations of centered-disk density one. It also minimizes unit-background renormalized Riesz energies for $0 < s < 2$ and Coulomb energy, resolving Sandier–Serfaty and the two-dimensional Brauchart–Hardin–Saff conjecture on the linear term of optimal spherical logarithmic energy.
Verified by LeanConvex and metric geometry
091
Logarithmic and L_p Brunn–Minkowski inequalities and the B-conjecture
Proves the logarithmic Brunn–Minkowski inequality for origin-symmetric convex bodies in every dimension, and the scalar-dilation B-conjecture for all even log-concave Radon measures. For Lebesgue volume it also proves the additive L_p Brunn–Minkowski inequality for full-dimensional origin-symmetric convex bodies throughout $0 < p < 1$ .
Verified by LeanConvex and metric geometry
092
The optimal order of convex-body covering density
Determines the optimal worst-case covering density as $Θ (n lo g n)$ , for both lattice and unrestricted translative coverings. Every convex body in ℝⁿ, n ≥ 2, admits a lattice covering of density at most $C n lo g n$ ; centrally symmetric examples in every sufficiently large dimension require at least $c n lo g n$ even without the lattice restriction, for absolute $c, C > 0$ .
Verified by LeanConvex and metric geometry
093
Dimension-free logarithmic Sobolev inequality for subgaussian log-concave measures
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.
To be verifiedConvex and metric geometry
094
Subpolynomial dimension reduction in L_p
For every fixed $1 < p < \infty$ and distortion D > 1, every n-point subset of real L_p embeds into $ℓ_{p}^{d}$ 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 $Θ (n^{2})$ when p ≠ 2.
Verified by LeanConvex and metric geometry
095
Hyperbolicity cones without semidefinite lifts
Disproves the Projected Lax conjecture: some hyperbolicity cones are not spectrahedral shadows. The examples admit no exact finite affine semidefinite lift, regardless of the number of auxiliary variables or the real coefficients used. This also disproves the generalized Lax conjecture that every hyperbolicity cone is spectrahedral.
Verified by LeanConvex and metric geometry
096
The Gaussian propeller conjecture in every dimension
Proves that the sum of squared Gaussian first moments of any finite measurable partition is at most $9/ (8 π)$ . In dimension at least two, three planar sectors of angle $2 π /3$ , extended orthogonally, attain the bound. Combined with the separate Unique Games theorem, this proves NP-hardness of improving the loss factor $(8 π /9) (1 - 1/ k)$ for identity-target kernel clustering with fixed k ≥ 3 on rational centered positive semidefinite inputs.
Verified by LeanConvex and metric geometry
097
The Euclidean Steinitz–Bergström bound
Proves that any finite sequence in the Euclidean unit ball of ℝ^d admits signs keeping every partial sum within $C d$ , 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 $S_{2} (d) = Θ (d)$ .
Verified by LeanConvex and metric geometry
098
Compact counterexamples to bi-Lipschitz dimension reduction
Every infinite-dimensional real Banach space contains a compact doubling set that admits no bi-Lipschitz embedding into any finite-dimensional normed space. The doubling constant is universal. This answers the Lang–Plaut problem negatively, even for compact subsets of Hilbert space.
Verified by LeanConvex and metric geometry
099
The sharp exponential scale of edit-distance distortion
Determines the least distortion of embedding edit distance on words of length at most d into real ℓ₁: it is $exp (Θ (lo g d lo g lo g d))$ . 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.
Verified by LeanConvex and metric geometry
100
Cylinder coverings below the half-area bound
Covers the entire closed regular tetrahedron by finitely many cylinders with compact triangular perpendicular bases whose total area is less than half its smallest orthogonal projection area. This disproves Bang's half-area cylinder-covering bound and the stronger directionwise normalized conjecture in dimension three.
Verified by LeanConvex and metric geometry
101
The sharp simplex conjecture for isotropic constants
Proves that simplices uniquely maximize the isotropic constant among convex bodies in every dimension, resolving the strong isotropic constant conjecture. Also establishes the sharp entropy lower bound for log-concave probability densities, with equality precisely for invertible affine images of products of one-sided exponential laws.
To be verifiedConvex and metric geometry
102
The Unique Games Conjecture and optimal approximation thresholds
Proves Khot's Unique Games Conjecture. Independent direct reductions also establish NP-hardness, on unweighted graphs, of approximation beyond the Goemans–Williamson ratio for Max-Cut, below factor two for Vertex Cover, and within any fixed constant factor for Min-UnCut and directed feedback vertex set. These direct proofs use established PCP and Label Cover hardness results.
Verified by LeanTheoretical computer science
103
Exact derandomization of logarithmic space: $L = RL = BPL$
Proves $L = RL = BPL$ , 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.
Verified by LeanTheoretical computer science
104
Quasipolynomial algorithms for mean-payoff, stochastic and parity games
Gives deterministic algorithms using $2^{O ((l o g (L + 2))^{2})}$ 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.
Verified by LeanTheoretical computer science
105
Perfect completeness for 2-to-1 games
Proves Khot's 2-to-1 Games Conjecture with perfect completeness: for every fixed rational $δ \in (0, 1)$ , it is NP-hard to distinguish satisfiable games from games whose optimum is at most δ, on explicit unweighted instances. The alphabet depends only on δ, and every right-hand label has exactly two preimages under each constraint map.
Verified by LeanTheoretical computer science
106
Hardness of coloring three-colorable graphs
It is NP-hard to color a three-colorable graph using any fixed number c ≥ 3 of colors. More strongly, for every fixed $0 < δ < 1/3$ , 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 $δ n$ otherwise, where n is the number of vertices.
Verified by LeanTheoretical computer science
107
Matrix multiplication with exponent at most 9/4
Proves $ω \leq 9/4$ over ℂ, giving $O_{ε} (n^{9/4 + ε})$ arithmetic operations for square matrix multiplication. In characteristic zero, some inner dimension n^a with a > 0.465 permits $n^{2 + o (1)}$ rectangular multiplication. Further square bounds give ω < 2.258 outside finitely many positive characteristics and ω < 2.371054886006746 over every fixed field.
Verified by LeanTheoretical computer science
108
A cubic permanent–determinant lower bound
Proves an $Ω (n^{3})$ lower bound for the border determinantal complexity of the $n \times n$ permanent over ℂ. Even coefficientwise limits of determinants of affine-linear matrices require matrix size at least $c n^{3}$ , for an absolute c > 0 and all sufficiently large n; the same bound therefore holds for exact representations.
Verified by LeanTheoretical computer science
109
Integer multiplication below $n lo g n$
Multiplies two n-bit integers exactly at every input length in deterministic worst-case time $O (n (lo g n)^{1 - κ})$ , with $κ = 2^{- 182}$ , on one fixed finite-alphabet Turing machine with finitely many one-dimensional tapes. This disproves the Schönhage–Strassen $n lo g n$ optimality conjecture in the ordinary multitape bit model.
To be verifiedTheoretical computer science
110
Optimal-order randomized k-server on arbitrary metrics
Establishes a randomized competitive ratio $O (lo g^{2} (k + 1))$ 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 $lo g (t + 1)$ at request t, with a finite instance-dependent additive movement constant.
Verified by LeanTheoretical computer science
111
One-sample matroid prophet inequalities against an almighty adversary
For every finite matroid known in advance, gives a distribution-independent online rule using one independent sample per element and earning a universal constant fraction of the expected offline optimum. Values are independent and nonnegative, with finite expected optimum. The guarantee holds even when the arrival-order adversary sees all samples, values, and the rule's entire random seed; no polynomial-time implementation is asserted.
Verified by LeanTheoretical computer science
112
Beyond the square-root exponent for depth-three circuits
Constructs a single language in deterministic polynomial time whose n-bit membership function requires $2^{ω (n)}$ 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.
Verified by LeanTheoretical computer science
113
Approximate counting and entropy of perfect matchings
Gives a fully polynomial randomized approximation scheme for counting perfect matchings in arbitrary finite simple graphs, with exact detection of zero counts. Also proves the perfect-matching entropy conjecture of Anari, Oveis Gharan, and Vinzant, bounding the maximum entropy of a matching law at every feasible edge-marginal vector in a loopless labelled multigraph, including boundary points.
Verified by LeanTheoretical computer science
114
Approximate counting of common integer polymatroid bases
Gives a fully polynomial randomized approximation scheme for counting common integer bases of two integral polymatroids of equal total rank, supplied by exact rank-value oracles. Capacities are binary-encoded, each integer vector counts once, and oracle calls and bit operations outside the oracles are polynomial on every execution. For matroids presented by independence oracles, the results also cover common independent sets of prescribed, unrestricted, or maximum cardinality, even when the ranks differ.
Verified by LeanTheoretical computer science
115
Sampling and counting contingency tables with arbitrary margins
For nonnegative integer matrices with prescribed row and column sums, gives exact uniform sampling in expected polynomial bit time and almost-uniform sampling in worst-case polynomial bit time. The dimensions and binary-encoded margins are unrestricted. Also gives a fully polynomial randomized approximation scheme for counting such tables with arbitrary individual cell bounds, including structural zeros, with polynomial cost on every execution.
Verified by LeanTheoretical computer science
116
Uniform black-box noncommutative identity testing across characteristics
For each characteristic, constructs in deterministic polynomial bit time a polynomial-dimensional matrix tuple detecting every nonzero division-free noncommutative formula of bounded size over any field of that characteristic. Rational formulas over ℚ also admit polynomial-size hitting lists whenever they have a defined rational-matrix evaluation.
Verified by LeanTheoretical computer science
117
Uniform sparsest cut: hardness and semidefinite gaps
Proves that approximating Uniform Sparsest Cut within any fixed constant factor is NP-hard, even with nonnegative rational capacities and unit demands. The Goemans–Linial semidefinite relaxation also has integrality gaps of order at least $lo g n / (lo g lo g n)^{3}$ , approaching the square-root-logarithmic upper bound.
Verified by LeanTheoretical computer science
118
Bin packing and unbounded configuration-LP gaps
Disproves the modified integer round-up conjecture of Scheithauer and Terno: the integral bin-packing optimum can exceed its configuration linear-programming value by an arbitrarily large additive constant. Approximating the optimum within any fixed additive constant is also NP-hard, even when every item exceeds 1/6 and each bin holds at most five items.
Verified by LeanTheoretical computer science
119
The Courtade–Kumar and Hellinger conjectures
Proves the Courtade–Kumar conjecture: among Boolean functions of independent uniform bits, a single coordinate retains the most mutual information after independent bit-flip noise. A stronger theorem treats randomized binary summaries at fixed initial information. The Hellinger conjecture is also proved for every Boolean output bias and noise correlation.
Verified by LeanTheoretical computer science
120
Almost-linear-time exact matching and prescribed-degree factors in general graphs
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.
To be verifiedTheoretical computer science
121
Almost-linear approximation of edit distance
For every fixed rational $ε \in (0, 1)$ , gives a randomized $(1 + ε)$ 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.
Verified by LeanTheoretical computer science
122
Quantitative trace-reconstruction bounds with a uniform decoder
At every fixed deletion probability in $(0, 1)$ , reconstructing an arbitrary length-n binary string requires $n^{Ω (l o g l o g n)}$ 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 $n^{- ε}$ for fixed ε > 0, both bounds become polynomial in the input and parameter encoding.
Verified by LeanTheoretical computer science
124
Polynomial-time scheduling on three identical machines
Resolves the three-processor unit-job scheduling problem of Garey and Johnson: a deterministic polynomial-time algorithm minimizes makespan for nonpreemptive unit-length jobs with arbitrary precedence constraints on three identical parallel machines. For an explicitly given precedence graph, it decides deadline feasibility exactly and constructs a feasible schedule.
Verified by LeanTheoretical computer science
125
The metric k-median approximation threshold and recovery
Gives a deterministic polynomial-time $(1 + 2/ e + ε)$ -approximation for finite rational metric k-median with specified candidate facilities, for every fixed ε > 0. Assuming $P \neq = N P$ , the optimal infimum approximation factor is $1 + 2/ e$ .
Verified by LeanTheoretical computer science
126
Exponential semidefinite complexity of perfect matching
Proves that every exact semidefinite lift of the perfect matching polytope has exponential size, answering Rothvoss's polynomial-size lift question negatively. The bound holds even for the positive semidefinite rank of its odd-cut slack matrix after any fixed shift $0 < ρ < 1$ , allowing arbitrary real positive semidefinite factors.
Verified by LeanTheoretical computer science
127
Average sensitivity of polynomial threshold functions
Proves that a degree-at-most-d polynomial threshold function on the uniform n-dimensional Boolean cube has average sensitivity at most $8 d n$ , uniformly for $1 \leq d \leq n$ . Average sensitivity counts expected output changes under single-bit flips. This establishes the asymptotic Gotsman–Linial conjecture, allowing polynomial zeros with $sign (0) = 1$ .
Verified by LeanTheoretical computer science
128
A factor-two approximation for shortest common superstring
Gives a deterministic polynomial-time algorithm constructing a common superstring of length at most twice the optimum for every finite family of explicitly represented strings. The running time is polynomial in the full encoded input length, including symbol labels.
Verified by LeanTheoretical computer science
129
Exponential state costs for two-way automata
Proves exponential lower bounds both for complementing two-way nondeterministic finite automata and for simulating one-way nondeterministic automata by two-way deterministic ones. The latter resolves the Sakoda–Sipser state-succinctness conjecture over growing finite alphabets; both results rule out polynomial state bounds independent of alphabet size.
Verified by LeanTheoretical computer science
130
Exact Fourier transforms below $n lo g n$
Gives a deterministic length-n discrete Fourier transform algorithm using $O (n (lo g n)^{1 - δ})$ operations for every n, with explicit $δ = 1 0^{- 13}$ . The model uses exact complex arithmetic, unrestricted coefficients and a supplied root of unity, and counts scalar preparation and logarithmic-word indexing.
Verified by LeanTheoretical computer science
131
Rapid mixing of graph switches for every degree sequence
Resolves the simple-undirected Kannan–Tetali–Vempala conjecture: the lazy edge-switch chain mixes in $O (n^{8})$ time for every graphical labeled degree sequence. The same degree-constrained graphs can also be sampled exactly uniformly by an almost-surely terminating algorithm with expected polynomial bit running time.
Verified by LeanTheoretical computer science
132
A superquadratic separation of sensitivity and block sensitivity
Constructs total Boolean functions with block sensitivity $bs (f) \geq s (f)^{α}$ 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.
Verified by LeanTheoretical computer science
133
The computational complexity of Weisfeiler–Leman refinement
Proves unconditional $n^{Ω (k)}$ 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.
Verified by LeanTheoretical computer science
134
Generalized star height at most three
Every regular language over a finite alphabet has a generalized regular expression with at most three nested Kleene stars, allowing union, concatenation and complement over the same alphabet. This establishes an absolute bound independent of automaton size, resolving the uniform-boundedness version of the generalized star-height problem.
Verified by LeanTheoretical computer science
135
Homogeneous depth-five lower bounds for iterated matrix multiplication
Over every characteristic-zero field, the $(1, 1)$ entry of a product of n independent $n \times n$ variable matrices requires $n^{Θ (n)}$ gates in homogeneous depth-five sum–product circuits. This sharp bound allows shared gates and bottom linear forms involving all variables.
Verified by LeanTheoretical computer science
136
A quasilinear PCP theorem for PPAD
Resolves the quasilinear PCP-for-PPAD conjecture. An End-of-Line instance of length N reduces to numerical circuit constraints of total length $N (lo g N)^{O (1)}$ 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.
To be verifiedTheoretical computer science
137
One-tape time simulation in two-fifths-power space
Determines the halting and finite-control outcome of a fixed deterministic one-writable-tape machine up to time T using $O (T^{2/5} lo g^{C} (T + 2))$ 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.
Verified by LeanTheoretical computer science
138
Subset Sum in $O (2^{0.49 n})$ time
Gives a uniform randomized classical algorithm for worst-case Subset Sum in ordinary $O (2^{0.49 n})$ 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.
To be verifiedTheoretical computer science
139
Subpolynomial query complexity for log-concave sampling
For C² potentials with a supplied minimizer and $I ⪯ \nabla^{2} V ⪯ 2 I$ , proves that sampling within total variation 1/10 requires only $C_{ε} d^{ε}$ 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.
Verified by LeanTheoretical computer science
140
Memory–sample lower bounds for noiseless Gaussian regression
For fixed A > 0, a one-pass learner with $A d^{2}$ persistent bits needs $Ω_{A} (d lo g (1/ ϵ))$ noiseless Gaussian samples to recover a unit vector to angular error $0 < ϵ \leq 1/10$ 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.
Verified by LeanTheoretical computer science
141
Existential–universal real sentences in the counting hierarchy
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.
To be verifiedTheoretical computer science
142
Deterministic polynomial factorization over prime fields
Gives a uniform deterministic algorithm that completely factors every nonzero dense degree-n polynomial over a prime field 𝔽_p, including multiplicities, in bit complexity polynomial in $(n + 1) lo g p$ . The prime is supplied in binary. No randomness, integer-factorization or primitive-root oracle, or GRH assumption is required.
To be verifiedTheoretical computer science
143
Hilbert's sixteenth problem: uniform bounds for limit cycles
Resolves the uniform boundedness assertion in Hilbert's sixteenth problem: the number of isolated periodic orbits of a real planar polynomial vector field is bounded by a finite constant depending only on its degree. For classical quintic Liénard systems, the exact maximum is two limit cycles.
Verified by LeanDynamical systems and ergodic theory
144
Banach’s simple Lebesgue-spectrum problem
Resolves the probability-preserving form of Banach's simple Lebesgue-spectrum problem within smooth dynamics. A smooth volume-preserving diffeomorphism of the standard-volume three-torus has simple Lebesgue spectrum on its entire complex mean-zero L² space: the bilateral iterates of one real observable form an orthonormal basis of that space.
Verified by LeanDynamical systems and ergodic theory
145
Rokhlin’s multiple-mixing problem
Proves that every invertible mixing probability-preserving transformation is mixing of all finite orders, resolving Rokhlin's multiple-mixing problem for a single transformation. Correlations among any finite collection of measurable sets converge to the product of their measures whenever all pairwise time separations diverge.
Verified by LeanDynamical systems and ergodic theory
146
Positive metric entropy for the standard map
Proves that the standard sine map on the two-dimensional torus has positive metric entropy with respect to area for every sufficiently large positive parameter. This establishes Sinai's positive-parameter-measure conjecture for the original family, with the stronger conclusion of a full parameter tail.
Verified by LeanDynamical systems and ergodic theory
147
The near-boundary Birkhoff conjecture
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.
To be verifiedDynamical systems and ergodic theory
148
The entropy-rate dimension formula for self-similar measures
For every self-similar measure on the line generated by finitely many contracting similarities, proves $dim_{H} μ = min {1, h_{RW} / χ}$ , where $h_{RW}$ 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.
Verified by LeanDynamical systems and ergodic theory
149
Classwise permanence for weakly reversible mass-action systems
Proves the permanence conjecture for every finite weakly reversible mass-action system with fixed positive rate constants. Every positive stoichiometric compatibility class, even an unbounded one, has a common compact convex forward-invariant absorbing set. All positive trajectories in that class therefore eventually share positive lower and finite upper concentration bounds.
Verified by LeanDynamical systems and ergodic theory
150
Weak mixing of triangular billiards with an irrational angle
Proves that the billiard flow in every nondegenerate Euclidean triangle with at least one angle irrational relative to π is weakly mixing for normalized area times uniform direction. This strengthens ergodicity on the entire irrational-angle class, with no genericity or Diophantine restrictions.
Verified by LeanDynamical systems and ergodic theory
151
A C¹ counterexample to the entropy conjecture
Constructs a noninvertible C¹ self-map of a compact smooth manifold with zero topological entropy but eigenvalue 2 on second homology. This disproves the homological entropy lower bound for general C¹ self-maps: homological growth need not force positive orbit complexity.
Verified by LeanDynamical systems and ergodic theory
152
Zero entropy does not guarantee a smooth positive-volume model
Constructs a zero-entropy ergodic invertible transformation of a standard nonatomic probability space that is not measurably conjugate to any C^∞ diffeomorphism preserving a strictly positive smooth probability density on a compact finite-dimensional manifold. One example rules out every finite dimension.
Verified by LeanDynamical systems and ergodic theory
153
Arithmetic classification and non-Pisot singularity for Bernoulli convolutions
Classifies singular and absolutely continuous unbiased Bernoulli convolutions for every $λ \in (0, 1)$ 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.
To be verifiedDynamical systems and ergodic theory
154
Pointwise multiple ergodic averages for mixing transformations
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.
To be verifiedDynamical systems and ergodic theory
155
A counterexample to periodic tiling in dimension three
Constructs a finite translational tile in ℤ³ that tiles space but admits no fully periodic tiling, disproving the periodic tiling conjecture in the smallest possible lattice dimension. Its unit-cube thickening gives the same counterexample in ℝ³, even with arbitrary real translation vectors.
Verified by LeanCombinatorics
156
Borsuk's conjecture fails in dimension nine
Constructs a compact subset of ℝ⁹ that cannot be covered by ten sets of strictly smaller diameter, disproving Borsuk's covering assertion already in dimension nine. The example consists of rank-one orthogonal projectors onto lines in ℝ⁴, with the Frobenius metric.
Verified by LeanCombinatorics
157
Graph coloring, clique minors, and Colin de Verdière invariants
Disproves Hadwiger's conjecture even for fractional coloring: arbitrarily large finite simple graphs with independence number at most two satisfy $χ_{f} (G) > h (G)$ , where $h (G)$ is the largest clique-minor order. Also disproves the fractional Colin de Verdière chromatic bound $χ_{f} (G) \leq μ (G) + 1$ . In the positive direction, every finite nonempty graph satisfies $χ_{list} (G) \leq C h (G)$ for a universal constant C.
Verified by LeanCombinatorics
158
The Euclidean plane cannot be colored with five colors
Proves that every five-coloring of the Euclidean plane has a monochromatic pair at distance one, with no restriction on the color classes. This advances the Hadwiger–Nelson problem: together with the classical seven-coloring, only six and seven remain possible chromatic numbers of the plane.
Verified by LeanCombinatorics
159
Erdős’s reciprocal-sum conjecture and quasipolynomial Szemerédi bounds
Proves Erdős's conjecture that every set of positive integers with divergent reciprocal sum contains arithmetic progressions of every finite length. Quantitatively, for each fixed k ≥ 3, every subset of ${1, \dots, N}$ with no nonconstant k-term progression has size at most $C_{k} N exp [- c_{k} (lo g N)^{ε_{k}}]$ , with positive constants depending only on k.
Verified by Lean Reasoning summaryCombinatorics
160
Superexponential van der Waerden numbers
Resolves Erdős's superexponential-growth question for van der Waerden numbers. If $W_{r} (k)$ is the least interval length forcing a monochromatic k-term progression in every r-coloring, then $W_{r} (k) > k^{c k ⌊ l o g_{2} r ⌋}$ for an absolute c > 0, all r ≥ 2 and sufficiently large k, uniformly in r. In particular, $W_{r} (k)^{1/ k} \to \infty$ for each fixed r.
Verified by LeanCombinatorics
161
Counterexamples to Sidorenko’s conjecture and the forcing conjecture
Disproves Sidorenko's conjecture with a connected bipartite pattern on 35 vertices and 66 edges that occurs less frequently than in a random graph of the same edge density. The same pattern disproves the forcing conjecture of Skokan and Thoma: matching its density and the edge density of a constant graphon need not force quasirandomness.
Verified by LeanCombinatorics
162
Counterexamples to Ryser’s covering conjecture
Disproves Ryser's covering conjecture by constructing intersecting $(q + 1)$ -partite, $(q + 1)$ -uniform hypergraphs with covering number $q + 1$ , rather than the predicted bound q, for every sufficiently large prime q. A separate construction over extension fields also disproves Gyárfás's monochromatic tree-cover conjecture.
Verified by LeanCombinatorics
164
Hindman’s finite sums and products conjecture
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.
To be verifiedCombinatorics
165
The Harary–Hill and Zarankiewicz crossing-number formulas
Resolves the Harary–Hill conjecture and Turán's brickyard problem in the Zarankiewicz formulation, determining the crossing numbers of every complete and complete bipartite graph. The result proves the optimality of the classical drawings among all plane drawings with continuous edge arcs.
Verified by LeanCombinatorics
166
The higher-dimensional Erdős distinct-distances conjecture
For every fixed d ≥ 3, any n ≥ 2 distinct points in ℝ^d determine at least $c_{d} n^{2/ d}$ distinct distances, with $c_{d} > 0$ depending only on dimension. This matches the integer-grid order and resolves the higher-dimensional Erdős distinct-distances conjecture with a constant-factor bound.
To be verifiedCombinatorics
167
Planar distinct distances and unit-distance bounds
Proves the weak pinned Erdős distance conjecture: for every fixed ε > 0, all but $o (n)$ points of any n-point planar set determine at least $n^{1 - ε}$ distinct nonzero distances. A complementary theorem bounds the number of unit-distance pairs by $O (n^{4/3 - δ})$ for an absolute δ > 0.
Verified by LeanCombinatorics
168
Combinatorial invariance of Kazhdan–Lusztig polynomials
Resolves the full combinatorial invariance conjecture: isomorphic Bruhat intervals in arbitrary Coxeter systems have identical equal-parameter Kazhdan–Lusztig polynomials. Thus the abstract order of the interval determines the polynomial, even across different Coxeter systems.
Verified by LeanCombinatorics
169
Shareshian–Wachs elementary positivity
Resolves the elementary-positivity part of the Shareshian–Wachs conjecture: the chromatic quasisymmetric function of every natural unit interval graph has elementary-basis coefficients in $N [q]$ . The coefficients count explicitly described permutations, giving a combinatorial explanation of positivity.
Verified by LeanCombinatorics
170
Sharp logarithmic exponents for off-diagonal Ramsey numbers
For every fixed integer s ≥ 5, proves $r (s, t) = t^{s - 1} / (lo g t)^{s - 2 + o (1)}$ as $t \to \infty$ , determining the logarithmic exponent and matching the classical upper bound at that scale. Here $r (s, t)$ is the least number of vertices forcing an s-clique or a t-vertex independent set.
Verified by LeanCombinatorics
171
The hypercube Ramsey conjecture
Resolves the Burr–Erdős hypercube Ramsey conjecture: the two-color Ramsey number of the n-dimensional cube is $Θ (2^{n})$ . 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.
To be verifiedCombinatorics
172
Classification of finite Euclidean Ramsey configurations
Classifies finite point configurations that occur monochromatically, at their original scale, in every finite coloring of sufficiently high-dimensional Euclidean space. The characterization is an algebraic condition over the coordinate field. It also disproves the Leader–Russell–Walters conjecture that every such configuration is a subset of a finite transitive set.
Verified by LeanCombinatorics
173
Seymour’s second-neighborhood conjecture
Proves Seymour's second-neighborhood conjecture: every nonempty finite oriented graph has a vertex with at least as many vertices at directed distance exactly two as at directed distance one. Oriented graphs may be arbitrary apart from the exclusion of loops and oppositely directed edge pairs.
Verified by LeanCombinatorics
174
Deterministic construction of strong thin spanning trees
Resolves the strong thin-tree conjecture constructively. Every finite loopless k-edge-connected multigraph on at least two vertices has a spanning tree containing at most a universal $C / k$ fraction of the edges of every cut. Such a tree can be found deterministically in polynomial time, even with binary-encoded parallel-edge multiplicities.
Verified by LeanCombinatorics
175
Talagrand’s expectation thresholds, discrete convexity, and graph decompositions
Proves that integral and fractional expectation thresholds differ by at most a universal factor, and resolves Talagrand's discrete-convexity conjecture. An application proves the Ascoli–He–Park–Talagrand graph-decomposition conjecture: every graph's edges split into a universally bounded number of fixed pieces, each with containment threshold at most a universal constant times the original graph's integral expectation threshold. The pieces' embeddings need not agree on shared vertices.
Verified by LeanCombinatorics
176
The second Kahn–Kalai conjecture with an edge-count bound
Proves the second Kahn–Kalai conjecture: for every finite simple graph H with h ≥ 1 edges and at most n vertices, its appearance threshold in $G (n, p)$ is at most $C p_{E} (n, H) (1 + lo g_{2} h)$ , with universal C. Here $p_{E}$ is the least density at which every subgraph of H has expected copy count at least 1/2.
Verified by LeanCombinatorics
177
Bounded-degree coboundary expanders
Constructs arbitrarily large finite d-dimensional simplicial complexes, for every d ≥ 3, with uniformly bounded vertex degrees and uniform 𝔽₂ coboundary expansion in every degree below d. Together with the known graph and two-dimensional cases, this establishes the existence of such expanders in every positive dimension.
Verified by LeanCombinatorics
178
Deterministic nonbipartite Ramanujan graphs in every fixed degree
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 $- 2 d - 1$ and $2 d - 1$ . A deterministic algorithm outputs the full adjacency list in polynomial bit time, with exponent depending on d.
To be verifiedCombinatorics
179
The circulant Hadamard and Barker-sequence conjectures
Proves that real circulant Hadamard matrices exist exactly in orders 1 and 4, resolving the circulant Hadamard conjecture. Together with classical Barker-sequence results, this shows that binary sequences whose nontrivial aperiodic autocorrelations have magnitude at most 1 exist at lengths n > 1 exactly when $n \in {2, 3, 4, 5, 7, 11, 13}$ .
Verified by LeanCombinatorics
180
Barnette’s Hamiltonian-cycle conjecture
Proves that every finite simple cubic bipartite planar 3-vertex-connected graph has a Hamiltonian cycle, resolving Barnette's conjecture. Equivalently, every three-edge path in a finite simple cubic 3-vertex-connected bipartite Pfaffian graph lies in a Hamiltonian cycle.
Verified by LeanCombinatorics
181
The Erdős–Gallai cycle-decomposition conjecture
Proves that the edges of every finite simple undirected graph on n vertices can be partitioned into at most $C n$ 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.
Verified by LeanCombinatorics
182
Power savings for intersective polynomial differences and prime arguments
For every fixed intersective integer polynomial h of degree k ≥ 2 with positive leading coefficient, proves that a subset of ${1, \dots, N}$ avoiding nonzero values $h (1), h (2), \dots$ as differences has size $O_{h} (N^{1 - c_{k}})$ , with $c_{k} > 0$ 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.
Verified by LeanCombinatorics
183
Power savings for planar halving lines and k-sets
Improves the planar halving-line bound to $O (n^{4/3 - ε})$ for sets with no three collinear and an absolute ε > 0. More generally, an n-point set with no three collinear has $O (n (k + 1)^{1/3 - ε_{0}})$ strictly separable k-subsets for $1 \leq k \leq n /2$ , with an absolute $ε_{0} > 0$ . The constants and positive exponents are nonquantitative.
Verified by LeanCombinatorics
184
Correspondence coloring with a fixed forbidden subgraph
Proves the Alon–Krivelevich–Sudakov coloring conjecture in correspondence-coloring form: graphs avoiding any fixed subgraph F need $O_{F} (Δ/ lo g Δ)$ colors when their maximum degree Δ is sufficiently large. Also proves the Ajtai–Erdős–Komlós–Szemerédi independence conjecture: for fixed r ≥ 4, every n-vertex K_r-free graph of average degree d ≥ 2 has an independent set of size $Ω_{r} (n lo g d / d)$ .
Verified by LeanCombinatorics
185
Counterexamples to infinite matroid intersection and packing/covering
Disproves the unrestricted infinite matroid intersection and packing/covering conjectures in ZFC, using two self-dual partitional matroids on a countably infinite ground set. The same examples answer Joó’s partitional-matroid question negatively. They are neither finitary nor cofinitary, so Nash-Williams’ original finitary conjecture remains outside the result.
Verified by LeanCombinatorics
186
Uniform influence and sharp thresholds for graph and hypergraph properties
Proves the Friedgut–Kalai threshold-width conjectures for graphs and fixed-uniformity hypergraphs. For fixed $0 < ε < 1/2$ , every nontrivial increasing relabeling-invariant property crosses from probability ε to $1 - ε$ within width $O ((lo g n)^{- 2})$ for graphs and $O_{r} ((lo g n)^{- r / (r - 1)})$ for r-uniform hypergraphs, r ≥ 3. The hypergraph influence bound also applies to nonmonotone properties.
Verified by LeanCombinatorics
187
Snaky in 21 Maker moves
Settles the Snaky achievement problem: Maker can force the six-cell Snaky shape within 21 of its own moves on the initially empty infinite square board. Maker moves first, each player claims one free cell per turn, and translations, rotations and reflections count as wins.
Verified by LeanCombinatorics
188
The sharp terminal leave in random triangle removal
Starting from the complete graph on n vertices, repeatedly delete a uniformly chosen remaining triangle. The terminal edge count is asymptotic to $n^{3/2} / (22)$ , with mean-square convergence after normalization by n^3/2. This proves the triangle case of the Joos–Kühn sharp-constant conjecture.
Verified by LeanCombinatorics
189
Cycle–clique Ramsey numbers
Proves the Erdős–Faudree–Rousseau–Schelp conjecture: $R (C_{m}, K_{n}) = (m - 1) (n - 1) + 1$ for every $m \geq n \geq 3$ , 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.
Verified by LeanCombinatorics
190
Polynomial removal fails for ordered binary matrices
Disproves polynomial ordered binary matrix removal with one fixed $66 \times 66$ 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.
Verified by LeanCombinatorics
191
A power improvement in the Heilbronn triangle lower bound
For every sufficiently large n, constructs n points in the unit square such that every triangle has area at least $n^{- 2 + c}$ for one absolute c > 0. This disproves the conjectured almost-n⁻² upper bound in Heilbronn's triangle problem, which asks how large the smallest determined triangle can be.
Verified by LeanCombinatorics
192
Boolean functions violate the square-root degree bound by arbitrary factors
Disproves the proposed square-root bound relating a Boolean function's linear Fourier coefficients to its polynomial degree. For every C > 0, there is a sign-valued Boolean function f with $\sum_{i} f ({i}) > C de g (f)$ . Thus its total signed correlation with individual input bits can exceed the proposed bound by an arbitrary factor.
Verified by LeanCombinatorics
193
Serre’s intersection-multiplicity conjecture
Proves strict positivity of Serre's intersection multiplicity $χ^{R} (M, N)$ for nonzero finitely generated modules over any regular local ring, provided $M \otimes_{R} N$ has finite length and $dim M + dim N = dim R$ . This resolves the positivity conjecture, including ramified mixed characteristic.
To be verifiedAlgebra
194
Lech’s multiplicity conjecture
Proves $e (R) \leq e (S)$ 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.
Verified by LeanAlgebra
195
A counterexample to the small Cohen–Macaulay module conjecture
Constructs a three-dimensional complete Noetherian normal local domain over ℂ with no nonzero finitely generated maximal Cohen–Macaulay module. A three-dimensional local domain essentially of finite type over ℂ has the same property, disproving the domain form of the small Cohen–Macaulay module conjecture.
Verified by LeanAlgebra
196
A counterexample to Kaplansky’s zero-divisor conjecture
Constructs a finitely presented torsion-free group G whose group algebra $F_{2} [G]$ has nonzero zero divisors, disproving Kaplansky's zero-divisor conjecture. The group has a finite two-dimensional classifying space.
Verified by LeanAlgebra
197
A torsion-free group algebra that is not directly finite
Constructs a finitely presented torsion-free nonsofic group whose group algebra over 𝔽₂ is not directly finite, disproving Kaplansky's conjecture even without torsion. Companion examples give injective nonsurjective cellular automata on all configurations, refuting Gottschalk's surjunctivity conjecture. Another counterexample is an integral group-ring matrix, invertible over the rational group ring, with Fuglede–Kadison determinant strictly between zero and one, disproving the unrestricted Determinant Conjecture.
Verified by Lean Reasoning summaryAlgebra
198
A counterexample to finitistic-dimension finiteness
Constructs a finite-dimensional complex algebra whose finite-dimensional modules have unbounded finite projective dimensions. This disproves the little finitistic-dimension conjecture.
Verified by LeanAlgebra
199
Counterexamples to Auslander–Reiten, Tachikawa and related homological conjectures
Constructs finite-dimensional algebras over a characteristic-two rational-function field that disprove the Auslander–Reiten and Gorenstein-projective conjectures, and Tachikawa's second conjecture. An associated endomorphism algebra also disproves the classical, generalized and strong Nakayama conjectures, the Auslander–Gorenstein conjecture, and the Wakamatsu tilting conjecture. The counterexamples persist under every extension of the base field.
Verified by LeanAlgebra
200
Eisenbud–Green–Harris and lex-plus-powers
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.
To be verifiedAlgebra
201
A counterexample to Kurosh’s division-ring problem
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.
To be verifiedAlgebra
202
The blockwise Alperin weight conjecture
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.
To be verifiedAlgebra
203
Donovan's conjecture over fields and complete mixed-characteristic DVRs
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.
To be verifiedAlgebra
204
Tensor saturation for even spin groups
Proves saturation factor one for $Spin (2 n)$ , 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.
To be verifiedAlgebra
205
Saxl’s conjecture and universal tensor squares
Proves Saxl's conjecture: the tensor square of every staircase representation contains every irreducible complex representation of the corresponding symmetric group. More generally, every S_n with $n \in / {2, 4, 9}$ has an irreducible representation whose tensor square contains all irreducibles.
Verified by LeanAlgebra
206
Finite lattice representation and undecidability
Some finite lattices are not congruence lattices of any finite algebra, answering the finite lattice representation problem negatively. Moreover, no algorithm decides whether a finite lattice has such a representation, or whether it is a full subgroup interval of a finite group.
Verified by LeanAlgebra
207
The ℓ¹-Bass conjecture for all discrete groups
Proves the ℓ¹-Bass conjecture for every discrete group: Hattori–Stallings traces of idempotent matrices over $ℓ^{1} (G)$ 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.
Verified by LeanAlgebra
208
Finite symmetric tensor categories and the Verlinde tower
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 $Ver_{p^{n}}$ . The level may depend on the category, and the theorem includes characteristic two, resolving the finite case of the Benson–Etingof–Ostrik conjecture.
To be verifiedAlgebra
209
Integral counterexamples to Gersten’s conjecture
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.
To be verifiedAlgebra
210
Foulkes' conjecture for sixth powers and quadratic stabilization
Proves the sixth case of Foulkes’ conjecture: $Sym^{6} (Sym^{b} V)$ embeds equivariantly in $Sym^{b} (Sym^{6} V)$ for every b ≥ 6 and finite-dimensional complex V. More generally, the canonical multiplication map $Sym^{b} (Sym^{a} V) \to Sym^{a} (Sym^{b} V)$ is surjective for a ≥ 2 and $b \geq a (a - 1)$ , giving dimension-independent quadratic stabilization.
Verified by LeanAlgebra
211
The geometric phase diagram, diffusion, and spectra of random planar maps
Critical Fortuin–Kasteleyn planar maps converge to Liouville quantum gravity spheres for $0 < q \leq 4$ 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.
Verified by LeanProbability and statistical mechanics
212
Planar first-passage geometry and the absence of bigeodesics
Proves that planar first-passage percolation has no doubly infinite geodesic for iid nonnegative nonatomic edge weights when the minimum of four weights has finite second moment. For exponential weights, the limit shape is strictly convex with C¹ boundary. Differentiability also holds for every Gamma law with positive shape and rate.
Verified by LeanProbability and statistical mechanics
213
Critical percolation on every quasi-transitive graph
Resolves the Benjamini–Schramm criticality conjecture for bond percolation on every infinite connected locally finite quasi-transitive graph with $p_{c} < 1$ : 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.
Verified by LeanProbability and statistical mechanics
214
The Benjamini–Schramm nonuniqueness conjecture
Proves $p_{c} < p_{u}$ 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.
Verified by LeanProbability and statistical mechanics
215
Canonical $O (3)$ continuum limit and exact $O (4)$ mass asymptotics
Constructs the canonical continuum limit of the two-dimensional nearest-neighbor $O (3)$ model: a non-Gaussian local relativistic theory with a unique vacuum and a positive mass gap. For the square-lattice $O (4)$ model, determines the exact leading asymptotic of the full transfer gap, $m_{lat} (β) \sim 32 e^{π /4 - 1/2} β e^{- π β}$ . The family also proves exponential spin-correlation decay for two-dimensional nearest-neighbor $O (n)$ models with n ≥ 3 at every positive temperature.
Verified by LeanProbability and statistical mechanics
216
Critical and near-critical XY scaling and BKT universality
For the square-lattice nearest-neighbor cosine XY model, proves critical axis correlations $C_{β_{c}} (r) \sim A r^{- 1/4} (lo g r)^{1/8}$ and the Berezinskii–Kosterlitz–Thouless essential singularity $β_{c} - β lo g ξ (β) \to B$ , with $A, B > 0$ 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.
To be verifiedProbability and statistical mechanics
217
The low-temperature Sherrington–Kirkpatrick fluctuation law
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 $c_{β} n^{1/3}$ , with $c_{β} > 0$ , confirming the predicted n^1/6 standard-deviation scale. Exact centering and standardization give full-sequence convergence to a uniquely characterized nondegenerate law.
To be verifiedProbability and statistical mechanics
218
Conformal universality for weakly interacting and random-bond Ising models
Weak finite-range square-symmetric even multispin perturbations of the square-lattice Ising model preserve critical bulk spin and energy limits; weak square-symmetric contour interactions also yield chordal SLE₃ interface limits. With sufficiently weak iid bond disorder of any fixed bounded nondegenerate mean-zero law, critical spin interfaces converge to the same law in probability over environments.
Verified by LeanProbability and statistical mechanics
219
GOE bulk universality for regular graphs with weak Anderson disorder
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.
To be verifiedProbability and statistical mechanics
220
Directional zero–one laws beyond iid environments and iid ballisticity
On ℤ^d, d ≥ 3, directional escape has probability zero or one for iid strictly elliptic nearest-neighbor environments, and for stationary ergodic finite-range-dependent environments under uniform ellipticity. In iid uniformly elliptic environments with d ≥ 2, almost-sure directional transience implies a deterministic limiting velocity with positive projection in that direction, resolving the ballisticity conjecture.
Verified by LeanProbability and statistical mechanics
221
The Mézard–Parisi formula for diluted spin glasses
Proves the Mézard–Parisi hierarchical cavity formula for Poisson-diluted even-arity Ising models satisfying the Panchenko–Talagrand factorization and positivity assumptions, with only first-moment integrability. The limiting free energy equals the infimum over finite-depth hierarchical trial laws. This includes the Viana–Bray model, symmetric diluted even-spin models, and weighted soft even-K satisfiability.
Verified by Lean Reasoning summaryProbability and statistical mechanics
222
Perceptron free energies and microscopic jamming exponents
Determines finite-temperature variational free energies for Gaussian Ising perceptrons with bounded Borel log-potentials and Gaussian spherical perceptrons with bounded continuous potentials, at every positive pattern density. A spherical extension treats bi-orthogonally invariant disorder with compact limiting singular-value distributions and no outliers. At margin −1, the quadratic-penalty spherical model has a sharp feasibility threshold and limiting gap and force laws, with system size, zero temperature, and critical density taken in that order.
Verified by LeanProbability and statistical mechanics
223
Random-cluster interfaces: critical, disordered, thermal, and natural-time scaling
Proves chordal SLE_κ limits for critical square-lattice random-cluster interfaces for $0 < q \leq 4$ , with $κ = 4 π / arccos (- q /2)$ : bounded Jordan domains are allowed for q ≥ 1, and smooth Jordan domains for q < 1, under the stated marked-boundary approximations. For $1 \leq q \leq 4$ , complete nested plane loops converge to CLE_κ.
To be verifiedProbability and statistical mechanics
224
Critical and quenched near-critical universality for Poisson–Voronoi percolation
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.
To be verifiedProbability and statistical mechanics
225
Gaussian free field limits throughout the balanced six-vertex regime
The balanced square-lattice six-vertex height field with $a = b = 1$ and $0 < c \leq 2$ 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 $- (2 π)^{- 1} lo g ∣ x - y ∣$ , the exact variance multiplier is $1/ arcsin (c /2)$ .
To be verifiedProbability and statistical mechanics
226
The double-dimer loop ensemble converges to CLE₄
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.
To be verifiedProbability and statistical mechanics
227
Critical SK autocorrelation processes and dynamics across the temperature transition
For zero-field Gaussian SK heat-bath dynamics with rate-one updates per spin, proves worst-start cutoff on the $lo g n$ scale for fixed $0 \leq β < 1$ , 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.
Verified by LeanProbability and statistical mechanics
228
Continuum phase transitions for radial pair potentials
Constructs stable distance-dependent pair interactions for three-dimensional classical particles with a first-order phase transition: the canonical free energy has a derivative jump at one inverse temperature throughout an open density interval. One potential has a divergent repulsive core; another is bounded and continuous with an integrable power-law tail, realizing the type of transition sought in Simon's continuum problem.
Verified by LeanProbability and statistical mechanics
229
Exact three- and four-state reconstruction thresholds and four-state tree capacity
Proves the exact reconstruction threshold $d λ^{2} > 1$ , 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.
Verified by LeanProbability and statistical mechanics
230
Exact Hausdorff gauges for SLE
Resolves Schramm’s Hausdorff-measure question for chordal SLE_κ, $0 < κ < 8$ . The explicit gauge $r^{d} (lo g lo g (1/ r))^{(2 - d) /2}$ , $d = 1 + κ /8$ , gives almost surely positive finite measure to every trace segment $γ ([s, t])$ with $0 < s < t < \infty$ , and finite expected measure to the trace in every bounded disk.
Verified by LeanProbability and statistical mechanics
231
The free uniform spanning forest is a factor of IID
On every infinite connected locally finite simple unweighted graph, the free uniform spanning forest is a factor of independent vertex labels, by one isomorphism-equivariant rule using no root. Translation-invariant strongly Rayleigh binary processes on every countable group, including invariant determinantal processes with Hermitian positive-contraction kernels, are also factors of IID.
Verified by LeanProbability and statistical mechanics
232
Gaussian fields and interfaces for triangular-lattice Lipschitz heights
Proves Gaussian free field limits on bounded smooth simply connected domains for triangular-lattice height models: uniform odd heights with increments $0, \pm 2$ and two-arc boundary values $\pm 1$ , and zero-boundary integer Lipschitz heights weighted by fixed $x \in [1/ 2, 1]$ . 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.
To be verifiedProbability and statistical mechanics
233
The joint critical Ashkin–Teller current limit
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.
To be verifiedProbability and statistical mechanics
234
All-temperature pressure of orthogonally invariant Ising spin glasses
Gives an exact variational formula for the limiting pressure of orthogonally invariant Ising spin glasses at every fixed temperature, both almost surely and in expectation. The coupling matrix is a Haar-random rotation of a deterministic spectrum converging to a compactly supported law, with extreme eigenvalues converging to its support edges. The zero-field ground-state energy follows as temperature tends to zero.
Verified by LeanProbability and statistical mechanics
235
Limiting random SAT thresholds, sharp variance and computability
For random k-SAT with independent uniformly signed proper clauses sampled with replacement, proves finite positive limiting thresholds and hitting-time variance $Θ_{k} (n)$ 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.
Verified by LeanProbability and statistical mechanics
236
The exact factor-of-IID threshold for free Ising spins on trees
Determines when the free zero-field ferromagnetic Ising state on the infinite d-regular tree is a factor of independent vertex labels: exactly when $tanh β \leq (d - 1)^{- 1/2}$ , 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.
Verified by LeanProbability and statistical mechanics
237
The three-quarter exponent for honeycomb self-avoiding walk
Proves the diameter form of Nienhuis's predicted three-quarter exponent: a uniformly chosen n-step self-avoiding walk on the honeycomb lattice has diameter $n^{3/4 + o (1)}$ . Its local mass and covering numbers have exponent 4/3. These estimates hold at every sufficiently large fixed length, simultaneously across scales, with arbitrarily high polynomial probability.
Verified by LeanProbability and statistical mechanics
238
Optimal logarithmic mixing of the Thorp shuffle
Proves that the Thorp shuffle randomizes $N = 2^{d}$ labeled cards in $Θ (lo g N)$ 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.
Verified by LeanProbability and statistical mechanics
239
Sharp singularity rates for symmetric random sign matrices
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 $p \in (0, 1) ∖ {1/2}$ , the rate is $(p^{2} + (1 - p)^{2} + o (1))^{n}$ . In the biased case, agreeing rows attain this rate.
To be verifiedProbability and statistical mechanics
240
Shelah's eventual categoricity and the prescribed-threshold obstruction
Proves Shelah's eventual categoricity conjecture in ZFC: for each bound on the Löwenheim–Skolem number, a uniform threshold makes categoricity of an abstract elementary class in one cardinal above that threshold imply categoricity throughout the same tail. Categoricity means uniqueness up to isomorphism at a given cardinality. Under the continuum hypothesis, a proposed specific Hanf threshold need not suffice.
Verified by LeanMathematical logic
241
Rigidity of the Turing degrees
Every order automorphism of the Turing degrees is the identity, resolving their rigidity problem. Thus no nontrivial relabeling of degrees preserves the ordering by relative computability.
Verified by LeanMathematical logic
242
Single-fold Diophantine representations and undecidability under an at-most-one-solution promise
Every recursively enumerable set of tuples of natural numbers has a Diophantine representation with exactly one auxiliary solution for each member and none for nonmembers. This proves the single-fold conjecture and hence the finite-fold conjecture. Diophantine solvability over the nonnegative integers remains undecidable even with an at-most-one-solution promise.
Verified by LeanMathematical logic
243
Separating choiceless counting from polynomial time and witnessed choice
Confirms the Blass–Gurevich–Shelah noncapture conjecture: consistency of a linear system over 𝔽₃ defines a polynomial-time query on unordered finite structures that choiceless polynomial time with counting cannot express. A separate result shows that adding witnessed symmetric choice strictly increases expressive power. Both separations hold for the full counting formalism, allowing hereditarily finite sets of arbitrary finite rank.
Verified by LeanMathematical logic
244
The Partition Principle does not imply Choice
Assuming ZF is consistent, constructs a model in which every surjective image of a set injects into that set, yet the axiom of choice fails. Choice for ordinal-indexed families still holds. From any countable transitive model of ZFC, a separate construction gives a transitive symmetric extension with these properties and no new countable sequences of ground-model elements.
Verified by LeanMathematical logic
245
Weak normalization implies strong normalization in pure type systems
Proves that weak normalization implies strong normalization for every pure type system: if every legal expression in every valid context has a β-normal form, every β-reduction sequence terminates. This resolves the β-Barendregt–Geuvers–Klop conjecture, including nonfunctional rules and open contexts.
Verified by LeanMathematical logic
246
Cannon's conjecture
Every word-hyperbolic group with boundary homeomorphic to S² admits a proper cocompact isometric action on hyperbolic three-space with finite kernel, proving Cannon's conjecture. Every torsion-free such group is therefore the fundamental group of a closed hyperbolic three-manifold.
Verified by LeanGroup theory
247
An infinite finitely presented residually finite 2-group and a finitely presented nil algebra
Constructs an infinite finitely presented residually finite group whose elements all have finite 2-power order, answering the finitely presented Burnside problem negatively even in this class. The construction also yields an infinite-dimensional finitely presented nil associative 𝔽₂-algebra and a finitely presented infinite-dimensional algebraic unitization, giving negative answers to the corresponding nilpotence and Kurosh finiteness questions.
Verified by LeanGroup theory
248
Thompson's group F is nonamenable
Proves that Thompson's group F, the group of dyadic piecewise linear homeomorphisms of the interval, is nonamenable, resolving its longstanding amenability problem.
Verified by LeanGroup theory
249
A finitely generated Eilenberg–Ganea counterexample
Constructs a finitely generated residually finite group with integral cohomological dimension two and geometric dimension three, disproving the Eilenberg–Ganea conjecture. It has no two-dimensional classifying space, even with infinitely many cells.
Verified by LeanGroup theory
250
Boone–Higman embeddings with higher finiteness
A finitely generated group has decidable word problem exactly when it embeds in a finitely presented simple group, proving the Boone–Higman conjecture. The target can have type F_∞: a classifying space with finitely many cells in each dimension. A single group of type F_∞ can also contain every finitely presented group.
Verified by LeanGroup theory
251
Amenability, unitarizability, and strong Ulam stability
Resolves Dixmier's problem for all discrete groups: amenability is equivalent to every uniformly bounded Hilbert-space representation being similar to a unitary representation. For countable discrete groups, amenability is also equivalent to strong Ulam stability: sufficiently accurate unitary approximate representations are uniformly close in operator norm to genuine representations on the same, possibly infinite-dimensional, Hilbert space.
Verified by LeanGroup theory
252
A torsion-free hyperbolic group that is neither residually finite nor linear over any field
Constructs a torsion-free word-hyperbolic group that is not residually finite, answering the residual-finiteness question negatively. One fixed nonidentity element is killed by every finite-dimensional linear representation over every commutative field, so the group is not linear over any such field.
Verified by LeanGroup theory
253
An infinite finitely presented simple amenable group
Constructs an infinite finitely presented simple amenable group, answering the longstanding question of whether these properties can occur simultaneously.
Verified by LeanGroup theory
254
Classifying spaces and geometric obstructions for Artin groups
The Salvetti complex of every finite-rank Artin group is aspherical, proving the Artin $K (π, 1)$ 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.
Verified by LeanGroup theory
255
Quasi-isometric recognition of virtually polycyclic groups
Proves that every finitely generated group quasi-isometric to a finitely generated virtually polycyclic group is virtually polycyclic, resolving the Eskin–Fisher–Whyte lattice-recognition conjecture. Equivalently, a group quasi-isometric to a lattice in a connected simply connected solvable Lie group is virtually a uniform lattice in some such Lie group, possibly a different one.
Verified by LeanGroup theory
256
Nonsingular systems of equations over arbitrary groups
Proves that every finite system of equations over an arbitrary group whose exponent-sum matrix has full row rank over ℚ has a simultaneous solution in an overgroup, resolving Howie's conjecture. The coefficient group embeds in the presented quotient. A companion proves Kervaire's conjecture: adjoining one generator and one relation cannot trivialize a nontrivial group.
Verified by LeanGroup theory
257
A hyperbolic group without a geometric CAT(0) action
Answers negatively whether every word-hyperbolic group is a CAT(0) group. Constructs one with a finite classifying space but no proper cocompact isometric action on any proper complete $CAT (0)$ space, in any dimension.
Verified by LeanGroup theory
258
Gersten’s conjecture and virtual compact specialness of one-relator groups
Proves Gersten's conjecture: every finitely generated one-relator group containing no Baumslag–Solitar subgroup $BS (m, n)$ , with $m, n \neq = 0$ , is word-hyperbolic. It also proves that every word-hyperbolic one-relator group is virtually compact special.
To be verifiedGroup theory
259
A group without fixed price
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.
To be verifiedGroup theory
260
Spacetime Penrose inequalities: enclosing area, charge, rotation, and anti-de Sitter extensions
Proves the sharp enclosing-area spacetime Penrose inequality for smooth one-ended asymptotically flat initial data in every spatial dimension n ≥ 3, under dominant energy, weak future trapping, positive enclosing area, and the stated decay assumptions. It bounds invariant ADM mass below using minimum enclosing area, with equality rigidity under additional horizon hypotheses. Charged upper-area bounds treat dyonic three-dimensional data; the higher-dimensional purely electric extension uses the matched neutral theorem.
Verified by LeanMathematical physics
261
Localization and delocalization in the Anderson model
Resolves the predicted spectral contrast for the lattice Anderson model with independent uniform site potentials. In dimension two, every positive disorder strength gives almost surely pure-point spectrum. In every fixed dimension d ≥ 3, sufficiently weak positive disorder gives purely absolutely continuous spectrum on a fixed open interval with nonzero spectral weight.
Verified by LeanMathematical physics
262
Sharp finite-matrix Lieb–Thirring inequalities and all equality cases
Proves the sharp one-dimensional Lieb–Thirring inequality for $1/2 < γ < 3/2$ and arbitrary finite-matrix potentials W ≥ 0 with $\int tr (W^{γ + 1/2}) < \infty$ : 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.
Verified by LeanMathematical physics
263
The ionization and generalized ionization conjectures
For the full nonrelativistic Coulomb model with two electron spin states, proves that a molecule with M fixed nuclei of charges at least one and total charge Z strictly binds at most $Z + C M$ electrons. Neutral-atom first ionization energies and radii containing all but an expected half-electron have universal positive upper and lower bounds. The energy cost of removing m electrons has Thomas–Fermi asymptotics as $m \to \infty$ and $Z / m \to \infty$ ; neutral-atom outer radii have the corresponding iterated-limit asymptotics, taking $Z \to \infty$ first.
Verified by LeanMathematical physics
264
Strong cosmic censorship near two-ended Kerr data
Proves local strong cosmic censorship near each fixed rotating subextremal Kerr bridge. A dense G_δ 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.
To be verifiedMathematical physics
265
Area laws and tensor networks for two-dimensional gapped systems
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 $L \times L$ 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.
To be verifiedMathematical physics
266
Exactly three mutually unbiased bases in dimension six
Proves $N (6) = 3$ , resolving Zauner's dimension-six mutually unbiased bases conjecture: three such bases exist in ℂ⁶, but four cannot. The exclusion is a complete certified computation under the stated binary64 arithmetic and compiler conditions. An independent companion proves the Matolcsi–Ruzsa–Weiner Fourier-vanishing conjecture for order-six complex Hadamard matrices outside Tao's cubic equivalence class.
Verified by LeanMathematical physics
267
Positive-temperature Bose–Einstein condensation and exact quantum depletion
Proves Bose–Einstein condensation for the exact canonical Gibbs state of the three-dimensional hard-sphere gas: each fixed exclusion distance and sufficiently small fixed density admit a strictly positive temperature, independent of volume, with positive condensate fraction in the thermodynamic limit. At zero temperature, proves the Bogoliubov leading quantum-depletion law for hard spheres and fixed bounded nonnegative radial finite-range potentials of positive scattering length, taking the thermodynamic limit before the dilute limit.
Verified by LeanMathematical physics
268
The spin-one Haldane gap
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.
To be verifiedMathematical physics
269
Uniform Laughlin gap and stability under bounded scalar disorder
Proves the fermionic Laughlin spectral-gap conjecture for the full V₁ interaction at filling 1/3 on the round sphere. The unique ground state remains uniformly gapped under sufficiently weak bounded real scalar one-body potentials projected to the lowest Landau level. Both the gap and disorder threshold are uniform over all sufficiently large particle numbers and all normalized potential profiles.
Verified by LeanMathematical physics
270
Threshold and positive-energy bound states of the BFSS matrix model
Proves that the undeformed relative $SU (N)$ BFSS model has exactly one normalizable zero-energy state for every finite N ≥ 2, resolving the threshold-bound-state conjecture. For $SU (2)$ , 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.
To be verifiedMathematical physics
271
Bloch's law, its lattice correction, and the spherical magnetization law
Proves Bloch's T^3/2 law with its exact coefficient for three-dimensional quantum Heisenberg ferromagnets at every positive quantum spin, allowing nonnegative symmetric finite-range couplings whose support generates ℤ³. The thermodynamic limit precedes the zero-field derivative and low-temperature limit. The family also proves spontaneous magnetization for nearest-neighbor models in every dimension d ≥ 3 and determines the first lattice correction for three-dimensional nearest-neighbor couplings.
Verified by Lean Reasoning summaryMathematical physics
272
Entanglement without distillable secret key
Constructs an entangled state on $C^{10} \otimes C^{10}$ 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 $M_{21} (C)$ whose square is not entanglement breaking disproves Christandl's PPT-square conjecture.
Verified by LeanMathematical physics
273
The entropy photon-number inequality
Proves the entropy photon-number inequality for beam-splitter mixing of two independent finite-energy bosonic inputs in any finite number of modes: the output's entropy photon number is at least the transmissivity-weighted average of the inputs'. Arbitrary entanglement within each input is allowed, and product thermal inputs attain equality even when their entropies differ.
Verified by LeanMathematical physics
274
Parity is not in QAC⁰
Resolves Moore's parity conjecture in the measured-output model: constant-depth quantum circuits with arbitrary one-qubit gates, unbounded-arity Toffoli gates and polynomially many total qubits cannot compute parity with any fixed positive worst-case advantage. Ancillas start in zero, one output qubit is measured, and all other registers may be discarded. Xu–Li's reductions give the same bounded-error obstruction for strict majority.
Verified by LeanMathematical physics
275
QMA-hardness of continuum Coulomb energy
Proves QMA-hardness of approximating the electronic Coulomb energy infimum in three dimensions, minimizing over the full spinful fermionic continuum space. Deterministic polynomial-time reductions work even with only unit-charge nuclei at distinct rational positions, polynomially many electrons and an energy-threshold separation of at least one.
Verified by LeanMathematical physics
276
Classical capacity of generalized amplitude damping
Determines the unassisted classical capacity of every qubit generalized amplitude-damping channel, including all damping and thermal parameters. An explicit one-variable optimization gives the capacity, attained by independent two-state signal ensembles with collective decoding. Holevo capacity, minimum output entropy and regularized classical capacity are additive when tensoring with any finite-dimensional quantum channel.
Verified by LeanMathematical physics
277
Threshold repetition for entangled games
Proves exponential threshold repetition for every finite two-player one-round game: if its entangled value is v < 1, the probability of winning at least a fraction $v + δ$ of k independent repetitions decays exponentially in k, for $0 < δ < 1 - v$ . Arbitrary joint finite-dimensional entangled strategies and correlated question distributions are allowed.
Verified by LeanMathematical physics
278
Failure of Kohn–Sham ensemble representation
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 $L^{3/2} (R^{3}) + L^{\infty} (R^{3})$ . This disproves Kohn–Sham ensemble representability for that potential class; the required nuclear charge is specified nonnumerically.
To be verifiedMathematical physics
279
Exact quantum factoring over a fixed finite gate set
Gives a polynomial-time uniform quantum circuit family that outputs the complete prime factorization of every integer with probability one. Both gate count and qubit count are polynomial in the input length, and one fixed finite gate set suffices.
Verified by LeanMathematical physics
280
Unitary vertex operator algebras and conformal nets
Proves the strongly rational case of the strong-locality conjecture: every simple unitary strongly rational complex vertex operator algebra generates a completely rational conformal net. Its simple modules are unitarizable, and its representation category agrees with the net’s finite-index sectors as a braided unitary tensor category.
Verified by LeanMathematical physics
281
QAOA attains the SK optimum in the thermodynamic-first limit
Proves that QAOA approaches the ground-state energy of the Gaussian zero-field Sherrington–Kirkpatrick model when system size tends to infinity before circuit depth. For every accuracy, finite depth and deterministic angles independent of size and disorder achieve the required limiting expected energy per spin. This also yields leading-order optimal expected MaxCut values on large-degree random regular graphs, with size tending to infinity before degree.
Verified by LeanMathematical physics
282
From scale symmetry to local conformal symmetry in four-dimensional QFT
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.
To be verifiedMathematical physics
283
Polynomial-time unitary synthesis from a Boolean oracle
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.
To be verifiedMathematical physics
284
The optimal quartic separation between randomized and quantum queries
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.
To be verifiedMathematical physics
285
Counterexamples to Baum–Connes and Kadison–Kaplansky
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.
To be verifiedOperator algebras
286
Rigidity and arithmetic of lattice von Neumann algebras
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.
To be verifiedOperator algebras
287
Isomorphism of the free group factors
Resolves the free group factor isomorphism problem: $L (F_{2}) ≅ L (F_{3})$ , and hence all interpolated free group factors, including $L (F_{\infty})$ , are isomorphic. Their common factor has fundamental group $R_{> 0}$ .
Verified by Lean Reasoning summaryOperator algebras
288
Kadison's similarity conjecture
Proves Kadison's similarity conjecture: every bounded complex-linear unital algebra homomorphism from a unital complex C^∗-algebra to operators on a Hilbert space becomes a $*$ -homomorphism after conjugation by a bounded invertible operator.
Verified by LeanOperator algebras
289
Strong Kadison–Kastler stability and its spatial boundaries
Proves that sufficiently close unital von Neumann algebras on the same Hilbert space are conjugate by a unitary arbitrarily close to the identity, with a universal tolerance in the operator-norm distance between unit balls. Counterexamples show that near-identity conjugacy fails for one-sided near inclusions, and that arbitrarily close norm-separable C^∗-algebras need not be ambiently unitarily conjugate.
Verified by LeanOperator algebras
290
Relative bicentralizers and modular spectral recovery
Proves Connes' bicentralizer conjecture for every type III₁ factor with separable predual and every faithful normal state. More generally, for every inclusion $N \subset M$ of von Neumann algebras with separable preduals admitting a faithful normal conditional expectation, constructs an amenable expected subalgebra $P \subset N$ with $P^{'} \cap c (M) = N^{'} \cap c (M)$ , resolving the relative bicentralizer conjecture.
Verified by LeanOperator algebras
291
Cuntz comparison, nuclear dimension, and equivariant Jiang–Su stability
Proves equivariant Jiang–Su stability for every countable discrete amenable group action on a simple separable unital infinite-dimensional nuclear stably finite Jiang–Su-stable C^∗-algebra, resolving this case of Szabó's conjecture without restrictions on trace dynamics. The family also proves the unital Toms–Winter conjecture, equating strict comparison, finite nuclear dimension and Jiang–Su stability in the simple separable unital infinite-dimensional nuclear setting.
Verified by LeanOperator algebras
292
Kirchberg's $O_{2}$ norm-ultrapower embedding problem
Constructs an explicit separable unital full group C^∗-algebra that cannot embed unitally into the norm ultrapower of any fixed nonzero unital nuclear C^∗-algebra, for any free ultrafilter on the natural numbers. Taking the target to be $O_{2}$ answers Kirchberg's norm-ultrapower embedding problem negatively.
Verified by LeanOperator algebras
293
Invariant projections, hyperinvariant subspaces, and transitive algebras
Constructs a nonzero norm-quasinilpotent operator on every infinite-dimensional separable complex Hilbert space with no nonzero proper closed subspace invariant under every commuting operator. The construction also gives operators with no nontrivial invariant projection in the hyperfinite type II₁ factor.
Verified by LeanOperator algebras
294
Kaplansky's quasitrace conjecture and failure of tensor-product stable finiteness
Disproves Kaplansky's quasitrace conjecture by constructing a separable unital complex C^∗-algebra admitting normalized 2-quasitraces, all of which are nonadditive. As a consequence, two unital simple stably finite C^∗-algebras can have a properly infinite minimal tensor product, with one factor $C_{r}^{*} (F_{2})$ .
Verified by LeanOperator algebras
295
The Kadison–Ringrose cohomology conjecture
Proves that every bounded Hochschild cocycle of degree at least two on a complex von Neumann algebra, with coefficients in the algebra itself, has a bounded primitive. Equivalently, all higher bounded Hochschild cohomology groups vanish, resolving the Kadison–Ringrose conjecture.
Verified by LeanOperator algebras
296
The generator problem for finite factors
Proves that every type II₁ factor with separable predual is generated by a single operator, equivalently by two self-adjoint operators, resolving the generator problem. More strongly, for every irreducible inclusion $P \subset M$ of such factors, the unitaries u with $M = W^{*} (P, u)$ form a dense G_δ subset in the trace 2-norm topology.
Verified by LeanOperator algebras
297
A ZFC counterexample to Naimark's problem
Gives an alternative to [Tanaka's ZFC construction](https://arxiv.org/abs/2609.26930v1) of a unital infinite-dimensional simple complex C^∗-algebra with a faithful tracial state and exactly one nonzero irreducible representation up to unitary equivalence. Thus the unrestricted compact-operator characterization fails without additional set-theoretic assumptions; the counterexample is nonseparable.
Verified by LeanOperator algebras
298
Two notions of free entropy differ even when both are finite
Constructs a bounded self-adjoint tuple in a tracial von Neumann algebra whose microstates and nonmicrostates free entropies satisfy $- \infty < χ < χ^{*} < \infty$ . This answers Voiculescu’s finite-entropy equality question negatively: the matrix-approximation and free-Fisher-information definitions differ even when both are finite.
Verified by LeanOperator algebras
299
The Kirchberg–Rørdam character criterion and infinite tensor-power Jiang–Su stability
A nonzero unital separable complex C^∗-algebra is Jiang–Su stable exactly when its norm central-sequence algebra has no characters, for every free ultrafilter. This answers the Kirchberg–Rørdam character question. Also, the infinite minimal tensor power of every such algebra without characters is Jiang–Su stable, answering the Dadarlat–Toms question.
Verified by LeanOperator algebras
300
Approximation and quadratic strong-operator paving
Proves that every self-adjoint element of a complex von Neumann algebra admits strong-operator paving relative to any maximal abelian subalgebra with $O (ε^{- 2})$ 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.
To be verifiedOperator algebras
301
Trace cones and Razak–Jacelon stabilization
Classifies separable nuclear complex C^∗-algebras after tensoring with the Razak–Jacelon algebra and the compact operators, using their full topological cones of extended lower-semicontinuous tracial weights. This answers Robert’s trace-cone question, including algebras with arbitrary ideal structure and both finite and infinite subquotients.
Verified by LeanOperator algebras
302
Radius of comparison equals half the mean dimension
For every minimal homeomorphism h of an infinite compact metrizable space X, the radius of comparison of $C (X) ⋊_{h} Z$ equals $\frac{1}{2} mdim (X, h)$ , 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.
To be verifiedOperator algebras
303
Weak pure infiniteness and Cuntz-algebra absorption
Resolves the ordinary-to-strong pure-infiniteness question of Kirchberg and Rørdam for complex C^∗-algebras. For exact algebras, proper infiniteness of one fixed finite amplification of every positive element also suffices. Consequently, every separable nuclear algebra with this property absorbs $O_{\infty}$ , without unitality or simplicity assumptions.
Verified by LeanOperator algebras
304
The Hilbert–Smith conjecture in every dimension
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.
To be verifiedTopology
305
Four-dimensional disk embedding and Wall's conjecture
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.
To be verifiedTopology
306
The purely cosmetic surgery conjecture
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.
To be verifiedTopology
307
Failure of rational injectivity for maximal coarse assembly
Constructs a uniformly discrete bounded-geometry space whose maximal coarse assembly map is not rationally injective. The example is a coarse disjoint union of finite connected graphs of uniformly bounded degree, with an infinite-order kernel class. A companion gives the analogous failure for reduced coarse assembly, disproving the rational coarse Novikov conjecture.
Verified by LeanTopology
308
Finite Smith–Toda complexes at every height
For every n ≥ 0, constructs a finite Smith–Toda spectrum at a prime p depending on n, with Brown–Peterson homology $B P_{*} / (p, v_{1}, \dots, v_{n})$ 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.
To be verifiedTopology
309
The Kervaire invariant problem at the prime three
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.
To be verifiedTopology
310
Quillen's conjecture in rational homology
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.
To be verifiedTopology
311
The Hovey–Strickland and Chai conjectures
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.
To be verifiedTopology
312
The Grothendieck homotopy hypothesis
Proves the Grothendieck homotopy hypothesis for ∞-groupoids associated with every Grothendieck coherator in the Ara–Henry convention: these algebraic objects recover the homotopy theory of spaces.
Verified by LeanTopology
313
Finite generation for the $K (n)$ -local sphere
Answers the degreewise finiteness question of Hovey and Hovey–Strickland: every homotopy group of the $K (n)$ -local sphere is a finitely generated ℤ_p-module, for every prime, positive height and integer degree. The same conclusion holds after $K (n)$ -localizing any finite p-local spectrum.
To be verifiedTopology
314
Cyclic length and chromatic fixed-point loss
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.
To be verifiedTopology
315
The four-dimensional Singer conjecture
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.
To be verifiedTopology
316
Curtis’s conjecture
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.
To be verifiedTopology
317
Thomason model structures in all strict higher dimensions
Resolves the Ara–Maltsiniotis conjecture: for every n ≥ 1 and n = ω, small strict globular n-categories admit proper combinatorial Thomason model structures Quillen equivalent to simplicial sets. Thus strict higher categories model the homotopy theory of spaces in every stated dimension.
Verified by LeanTopology
318
Chromatic splitting: filtrations and counterexamples
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 $p > n + 1$ , the overlap $L_{n - 1} L_{K (n)} S_{p}^{\land}$ 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.
To be verifiedTopology
319
Counterexamples to finite generation at chromatic height two
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 $BP ⟨ 2 ⟩$ by finitely many sums, shifts, cones and retracts. The examples nevertheless satisfy the finite and telescopic localization comparisons.
To be verifiedTopology
320
Nonhomeomorphic closed aspherical four-manifolds
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.
To be verifiedTopology
321
A counterexample to Wall's finite D(2) problem
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.
To be verifiedTopology
322
Tingley’s sphere-isometry problem
Resolves Tingley's problem: every surjective isometry between the unit spheres of nonzero real Banach spaces extends uniquely to a surjective real-linear isometry of the whole spaces. No dimension restriction is imposed, so the metric geometry of the unit sphere determines the Banach space up to linear isometry.
Verified by LeanFunctional analysis
323
Independence of the separable quotient problem
Establishes, relative to the consistency of a measurable cardinal, that the separable quotient problem is independent of ZFC. The assertion that every infinite-dimensional Banach space has a separable infinite-dimensional quotient can hold for all real and complex Banach spaces, whereas the continuum hypothesis yields counterexamples over both fields.
Verified by LeanFunctional analysis
324
Lipschitz equivalent Banach spaces need not be linearly isomorphic
Constructs separable real Banach spaces that are globally bi-Lipschitz equivalent but not linearly isomorphic, resolving the separable Lipschitz-isomorphism problem negatively. Thus even the complete metric structure up to bi-Lipschitz equivalence does not determine a separable Banach space's linear isomorphism class.
Verified by LeanFunctional analysis
325
The complete Crouzeix conjecture
Resolves the complete Crouzeix conjecture: for every bounded operator A on a complex Hilbert space and every finite matrix-valued polynomial P, one has $∥ P [A]∥ \leq 2 sup_{z \in W (A)} ∥ P (z)∥$ , where $W (A)$ is the numerical range. The constant 2 is sharp, independent of the matrix size, and valid in infinite dimensions.
Verified by LeanFunctional analysis
326
The cotype–cotype conjecture under the approximation property
Resolves the cotype–cotype conjecture for real Banach spaces with the approximation property. Such a nonzero space is K-convex if and only if both it and its dual have finite Rademacher cotype, with possibly different exponents. Equivalently, these cotype assumptions force nontrivial Rademacher type.
Verified by LeanFunctional analysis
327
Markov type characterizes superreflexivity
Proves that every real Banach space with Markov type p for some p > 1 admits an equivalent uniformly convex norm, answering Naor's renorming question. Together with the known converse, this characterizes superreflexivity by nontrivial Markov type.
Verified by LeanFunctional analysis
328
Nonexpansive fixed points in reflexive Banach spaces
Resolves Kirk's reflexive-space fixed-point problem: every nonexpansive selfmap of a nonempty closed bounded convex subset of a real reflexive Banach space has a fixed point. The result uses the original norm, without assuming uniform convexity.
Verified by LeanFunctional analysis
329
A counterexample to metric-entropy duality
Disproves Pietsch's dimension-free duality conjecture for metric entropy. Origin-symmetric convex bodies violate every proposed choice of universal constants in the conjectured comparison between covering numbers and those of the polar bodies, even when the covering body is a cube.
Verified by LeanFunctional analysis
330
A uniformly discrete counterexample to bounded approximation in Lipschitz-free spaces
Constructs a countable uniformly discrete metric space whose real Lipschitz-free Banach space has the approximation property but not the bounded approximation property, answering Kalton's question negatively. Finite-rank operators approximate the identity on every compact set, but their norms cannot share a finite bound.
Verified by LeanFunctional analysis
331
Reflexive midpoint convexity and diamond distortion
Constructs a real reflexive Banach space with an asymptotically midpoint uniformly convex norm but no equivalent asymptotically uniformly convex norm, extending Baudier's separation to reflexive spaces. In the same space, depth-k countably branching diamonds require distortion at least $1 + k /12$ , so midpoint uniform convexity does not force uniformly bounded diamond distortion even under reflexivity.
Verified by LeanFunctional analysis
332
Metric Markov cotype of ℓ₁ and Hilbert-space Lipschitz extension
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.
To be verifiedFunctional analysis
333
Smooth isometric immersions of surfaces into ℝ⁴
Every closed smooth Riemannian surface admits a smooth isometric immersion into ℝ⁴, resolving the closed-surface form of the four-dimensional isometric-immersion problem. This includes nonorientable surfaces and metrics of arbitrary Gaussian curvature.
Verified by LeanDifferential geometry
334
A smooth surface metric with no local isometric immersion in ℝ³
Constructs a smooth positive-definite metric on $(- 1, 1)^{2}$ for which no neighborhood of the origin admits a smooth isometric immersion into ℝ³. This answers the unrestricted smooth local isometric realization problem for surfaces negatively, even after shrinking the neighborhood.
Verified by LeanDifferential geometry
335
Gromov’s integral scalar-curvature bound for simplicial volume
Proves $\int_{M} (Scal_{g}^{-})^{n /2} d V_{g} \geq a_{n} ∥ M ∥$ for every closed connected oriented smooth n-manifold, n ≥ 3, and every smooth metric, with $a_{n} > 0$ depending only on dimension. Here $Scal_{g}^{-} = max {0, - Scal_{g}}$ and $∥ M ∥$ 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.
To be verifiedDifferential geometry
336
Spectral scalar curvature, Urysohn width, and macroscopic dimension
Every complete connected smooth boundaryless n-manifold, n ≥ 3, satisfying $- 4Δ + Scal \geq 1$ 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.
To be verifiedDifferential geometry
337
Sharp Cartan–Hadamard isoperimetry and rigidity
Proves generalized Cartan–Hadamard isoperimetry in every dimension: in a complete simply connected manifold with sectional curvature at most κ ≤ 0, every finite-volume finite-perimeter set satisfies the sharp comparison with the equal-volume model ball. Bounded positive-volume equality regions for κ = 0 are Euclidean balls. Also proves sharp Euclidean filling bounds for compactly supported integral n-cycles, n ≥ 2, in arbitrary proper CAT $(0)$ spaces.
Verified by LeanDifferential geometry
338
Yau's uniformization conjecture
Proves Yau's uniformization conjecture: every complete connected noncompact Kähler manifold with strictly positive holomorphic bisectional curvature is biholomorphic to ℂⁿ.
To be verifiedDifferential geometry
339
Katok's entropy rigidity conjecture
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.
To be verifiedDifferential geometry
340
A counterexample to the nearby Lagrangian conjecture
Disproves the unrestricted nearby Lagrangian conjecture. For some sufficiently large even N, constructs a closed exact embedded Lagrangian in $T^{*} (S^{9} \times S^{N - 1})$ that is diffeomorphic to the base but not Hamiltonian isotopic to its zero section.
To be verifiedDifferential geometry
341
Donaldson's hypersymplectic deformation conjecture
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 $\int ω_{i} \land ω_{j} = δ_{ij}$ , 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.
To be verifiedDifferential geometry
342
Donaldson's tamed-to-compatible conjecture
Proves Donaldson's tamed-to-compatible conjecture: every smooth almost complex structure on a closed four-manifold that is tamed by a symplectic form admits a compatible symplectic form. The almost complex structure stays fixed; the form's cohomology class may change.
Verified by LeanDifferential geometry
343
Symplectic ball packing in higher dimensions
Resolves the Siegel–Yao conjecture for arbitrary capacities in every dimension $2 n \geq 6$ . Finitely many closed symplectic balls of capacities $R_{1}, \dots, R_{k}$ embed disjointly into an open ball of capacity R exactly when $\sum_{i} R_{i}^{n} < R^{n}$ and $R_{i} + R_{j} < R$ for every distinct pair i, j.
Verified by LeanDifferential geometry
344
The metric Blaschke conjecture
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.
To be verifiedDifferential geometry
345
Infinitely many closed geodesics on Riemannian spheres and closed three-manifolds
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.
To be verifiedDifferential geometry
346
Sharp singular-set bounds for stationary integral varifolds
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.
To be verifiedDifferential geometry
347
Counterexamples to stable-Morse and strong Arnold fixed-point bounds
Disproves stable-Morse lower bounds for nondegenerate Hamiltonian fixed points: on simply connected closed Kähler manifolds of real dimension 22, the deficit below the stable Morse number is unbounded. A separate Hamiltonian diffeomorphism of the complex quadric threefold has exactly three fixed points, fewer than the four critical points required of every smooth function.
Verified by LeanDifferential geometry
348
Nonnegative-curvature Einstein classification and an L² topological gap
Classifies closed connected Einstein four-manifolds with positive Einstein constant and nonnegative sectional curvature: up to scaling, their universal Riemannian covers are the round S⁴, Fubini–Study $CP^{2}$ , 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.
Verified by LeanDifferential geometry
349
The Solomon–Yau least-volume conjecture
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.
To be verifiedDifferential geometry
350
Yau’s nodal bounds: surfaces and higher dimensions
Proves the sharp $C λ$ 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 $λ^{1/2 + ε_{0}}$ for some fixed $ε_{0} > 0$ , ruling out even arbitrarily small power losses.
Verified by LeanDifferential geometry
351
Scalar curvature and finite-time Ricci-flow singularities
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.
To be verifiedDifferential geometry
352
A finite-time singularity of Calabi flow
Disproves Chen's smooth long-time existence conjecture for Calabi flow by constructing a smooth $U (10)$ -invariant Kähler metric on $CP^{10}$ 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.
To be verifiedDifferential geometry
353
Affine Bernstein rigidity through dimension nine and a smooth dimension-ten counterexample
Proves that every smooth locally uniformly convex affine-maximal graph of dimension three through nine, complete for its induced Euclidean metric, is an elliptic paraboloid. A smooth entire nonquadratic example in dimension ten makes this range sharp. In dimensions three through nine, the paraboloid classification also holds for connected open locally uniformly convex affine-maximal hypersurfaces complete for the affine Berwald–Blaschke metric.
Verified by LeanDifferential geometry
354
The isoperimetric profile of the cubic three-torus
Determines the isoperimetric profile of the unit cubic flat three-torus and classifies every finite-perimeter minimizer: balls, circular tubes around shortest closed geodesics, coordinate slabs, and their complements. The transition volumes are $4 π /81$ and $1/ π$ , with exactly the adjacent two types minimizing at each transition.
Verified by LeanDifferential geometry
355
Unique tangent flows at the first surface singularity
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.
To be verifiedDifferential geometry
356
Gigli’s characterization of Alexandrov curvature
Proves Gigli's conjecture: in every integer dimension n ≥ 2, Alexandrov curvature at least κ is characterized by the full-support $RCD ((n - 1) κ, n)$ condition with reference measure $H^{n}$ and distributional sectional curvature at least κ in the original global test classes. The RCD condition is unreduced.
Verified by LeanDifferential geometry
357
Bi-Lipschitz coordinates at every regular RCD point
Proves that every regular point of a noncollapsed $RCD (K, n)$ 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 $H^{n}$ , and the chart compares ambient distances with a point-dependent finite constant.
To be verifiedDifferential geometry
358
A three-manifold without conjugate points or nonpositive curvature
Constructs a closed connected orientable smooth three-manifold that admits a metric without conjugate points but no metric of nonpositive sectional curvature. This answers negatively, already in dimension three, whether the first metric-existence property implies the second.
Verified by LeanDifferential geometry
359
Negative Kähler curvature without bounded holomorphic coordinates
Constructs a contractible domain in ℂ³ with a complete negatively pinched Kähler metric but no bounded holomorphic coordinates, disproving bounded-domain uniformization in this setting. A higher-dimensional example has sectional curvature at most −1 and only constant bounded holomorphic functions; its curvature is not bounded below.
Verified by LeanDifferential geometry
360
Weak MTW curvature gives convexity and regular optimal transport
On every closed connected Riemannian manifold of dimension at least two satisfying weak Ma–Trudinger–Wang curvature, all tangent injectivity domains are convex, resolving Villani’s conjecture in this setting. For squared-distance transport between measurable probability densities bounded above and away from zero, the optimal map and its inverse are Hölder continuous.
Verified by LeanDifferential geometry
361
Failure of integer-degree harmonic dimension comparison
Disproves Yau's proposed Euclidean dimension bound for harmonic functions of integer growth on manifolds with nonnegative Ricci curvature. For every sufficiently large integer k, a complete smooth metric on ℝ³ has at least $(k + 2)^{2}$ independent harmonic functions of growth at most k, exceeding the Euclidean count $(k + 1)^{2}$ . The metric may depend on k.
Verified by LeanDifferential geometry
362
Global smoothness for relativistic Vlasov–Maxwell
Proves large-data global existence and uniqueness for the three-dimensional, one-species relativistic Vlasov–Maxwell system. Smooth admissible initial data may be arbitrary provided the particle density is compactly supported and the electromagnetic fields have finite energy and bounded derivatives of every order; the solution remains smooth on every finite time interval.
Verified by Lean Reasoning summaryPartial differential equations
363
Nonuniqueness with local conservation for the hard-sphere Boltzmann equation
Constructs two distinct global entropy solutions of the three-dimensional periodic hard-sphere Boltzmann equation from the same nonnegative initial density, with bounded velocity support and finite mass, energy and absolute entropy. Both are strongly continuous in L¹ and satisfy exact local conservation of mass, momentum and kinetic energy.
Verified by LeanPartial differential equations
364
Kinetic limits and fluctuations over the Boltzmann lifespan
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.
To be verifiedPartial differential equations
365
Joint metric and connection recovery from one boundary patch
Zero-frequency measurements on any nonempty open boundary patch determine a smooth metric and smooth unitary connection on a trivial Hermitian rank-two bundle over a compact connected manifold of dimension at least three, up to diffeomorphism and gauge fixed on that patch. Inputs and observations use the same patch. In contrast, distinct uniformly positive bounded measurable scalar conductivities on a three-dimensional ball can have identical full-boundary data.
Verified by LeanPartial differential equations
366
The planar Mumford–Shah regularity conjecture and local weak-L⁴ gradient bounds
Resolves the interior regularity conjecture for reduced absolute planar Mumford–Shah minimizers with bounded fidelity data. Locally, the closed discontinuity set is a $C^{1, α}$ arc, a regular crack tip, or three arcs meeting at $12 0^{\circ}$ ; only finitely many global connected components meet any compact interior region.
To be verifiedPartial differential equations
367
The critical dimension for the one-phase Bernoulli problem
Establishes seven as the first dimension admitting a nonflat, one-homogeneous global minimizer of the one-phase Bernoulli energy. Consequently, minimizing free boundaries are smooth through dimension six, and their singular sets have dimension at most $n - 7$ in higher dimensions.
Verified by LeanPartial differential equations
368
The three-dimensional Ball–Evans approximation problem
Resolves the three-dimensional Ball–Evans approximation problem: every $W^{1, p}$ homeomorphism between arbitrary bounded domains in ℝ³, for $1 \leq p < \infty$ , is a strong $W^{1, p}$ limit of smooth diffeomorphisms onto the same target.
To be verifiedPartial differential equations
369
The hot spots conjecture for simply connected planar domains
Proves a strict form of Burdzy's simply connected hot spots conjecture. On every smooth bounded simply connected planar domain, each nonzero eigenfunction for the first positive Neumann eigenvalue has no interior critical point, so all global extrema lie on the boundary. Eigenvalue multiplicity is allowed.
Verified by LeanPartial differential equations
370
The Lane–Emden and Hénon–Lane–Emden conjectures
Resolves the subcritical Lane–Emden conjecture and its weighted Hénon extension. For n ≥ 2, $p, q > 0$ and real A, B, the system $- Δ u = ∣ x ∣^{A} v^{p}$ , $- Δ v = ∣ x ∣^{B} u^{q}$ has no positive entire solution when $(n + A) / (p + 1) + (n + B) / (q + 1) > n - 2$ , with solutions continuous at the origin and classical elsewhere. No symmetry or growth assumption is needed. Known radial existence gives the exact existence criterion for n ≥ 3 and $A, B > - 2$ .
Verified by LeanPartial differential equations
371
Stable blowup for the defocusing Schrödinger equation
For a sufficiently large odd nonlinearity power, constructs a nonempty open set of initial data in $H^{k} (T^{12})$ , 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.
Verified by LeanPartial differential equations
372
Global uniqueness in smooth isotropic elasticity
Proves that full static boundary displacement-to-traction data determine both smooth real Lamé moduli on every bounded connected smooth domain in ℝ³, provided μ > 0 and $3 λ + 2 μ > 0$ on the closure. Neither analyticity, proximity to constant coefficients nor prior knowledge near the boundary is required.
Verified by LeanPartial differential equations
373
Nonattainment of the three-marginal Coulomb Monge problem
An optimal three-particle Coulomb configuration need not be a deterministic function of the first particle, even for smooth identical spatial densities. The Monge and Kantorovich infima agree, but the Monge infimum is not attained. The same phenomenon occurs for every inverse-power Riesz exponent in each dimension at least two, with a suitable density in each case.
Verified by LeanPartial differential equations
374
Sharp one-third stability of Brenier maps
For uniform source measure ρ on a compact convex body with interior in dimension at least two, quadratic optimal transport maps satisfy $∥ T_{μ} - T_{ν} ∥_{L^{2} (ρ)} \leq C W_{2} (μ, ν)^{1/3}$ uniformly over targets in a fixed compact set. The exponent is sharp, even for three-atom targets, disproving Letrouit's conjectured square-root bound.
Verified by LeanPartial differential equations
375
De Giorgi's conjecture in dimension eight
Proves De Giorgi's conjecture at its sharp dimension-eight endpoint: every entire C² solution $u : R^{8} \to (- 1, 1)$ of $Δ u = u^{3} - u$ that is strictly increasing in one direction depends on only one linear coordinate. A stronger theorem classifies all stable entire solutions $v : R^{7} \to [- 1, 1]$ as constant wells or planar transitions, without an energy-growth assumption.
To be verifiedPartial differential equations
376
Universal computation in forced Navier–Stokes flows
Constructs viscous incompressible flows starting from rest on a fixed flat three-dimensional domain that perform universal computation under smooth external forcing. A terminating compiler turns a Turing machine and input into a finite program for the force, so a designated particle reaches a fixed region exactly when the machine halts. The viscosity is fixed, positive and computable.
Verified by Lean Updated versionPartial differential equations
377
Interior $C^{1, α}$ regularity for infinity-harmonic functions
Proves uniform interior $C^{1, α_{d}}$ 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.
To be verifiedPartial differential equations

Explorer

Milne’s rationality conjecture and algebraic specialization

The full BSD formula from low Selmer corank

The quasi-Riemann hypothesis

Hilbert’s tenth problem over ℚ

Irrationality of Catalan’s constant

Goldfeld’s conjecture: densities and mean analytic rank

Ordinary two-point correlations and the corrected Elliott conjecture

The Deligne–Drinfeld conjecture

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

Unrestricted pro-modularity at the prime two

Prime-factor statistics of p−1

Independent largest prime factors of consecutive integers

Ostmann’s inverse Goldbach conjecture

Restricted geometric Langlands, global Arthur enhancements, and generic Ramanujan

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

Zilber–Pink in abelian varieties and the Siegel threefold

The irrationality exponent of π is 2

The Margulis–Platonov conjecture over global fields

The local p-adic section conjecture and global consequences

Squarefree quartics and power-free polynomial values

A quadratic bound for Jacobsthal’s function

The weak inhomogeneous Duffin–Schaeffer conjecture

Patterson's first moment for cubic Gauss sums

An asymptotic formula for the number of totients

Short Egyptian fractions

Positive lower density of large prime gaps

Potential integral density on curve character varieties

Uniformly bounded components of Gaussian-prime graphs

Primitive roots for every admissible integer base

Modularity of elliptic curves over imaginary quadratic fields

Uchida’s conjecture for open homomorphisms of Galois groups

The rational Hodge conjecture for CM abelian varieties

Iitaka subadditivity, variation, and logarithmic additivity

Log abundance for compact Kähler spaces under logarithmic Iitaka subadditivity

Log-canonical threefold abundance in numerical dimension one

Numerical semiampleness and generalized minimal models

The ordinary-double-point volume gap

Fujita’s freeness conjecture

Nagata’s conjecture and maximal Seshadri constants

Bloch’s conjecture for complex surfaces

Hyperkähler SYZ and projective-space bases

Oka classification for minimal compact complex surfaces: Kodaira dimension zero and class VII

P = W for fixed-determinant SLn moduli spaces

The equivariant cohomological Hikita conjecture

Shafarevich counterexamples in dimension two and with large fundamental group

Zariski cancellation and affine fibrations over the complex numbers

A characteristic-zero counterexample to Lipman–Zariski

A stable-coordinate counterexample in four variables

A counterexample to Griffiths’ positivity conjecture

Kobayashi’s canonical-ampleness conjecture

Tangent splittings and product decompositions

A counterexample to Pixton completeness in Chow

Irrational cubic fourfolds with Hodge-theoretic and categorical K3 associations

Gepner symmetry and large-volume stability on threefolds

Termination of projective and Kähler fourfold minimal model programs

Fundamental groups of special complex varieties and root orbifolds

Semialgebraic universal covers and bounded domains

Counterexamples to Zariski’s multiplicity conjecture

The Global Spherical Shell conjecture

Projective contact classification and the LeBrun–Salamon conjecture

The generalized Mukai conjecture

Topological triviality of μ-constant surface singularities

Virasoro constraints for complete intersections and projective-bundle towers

Bounded klt complements for Fano contractions

The Campana–Peternell conjecture in dimension six

Anticanonical nonvanishing in every dimension

Global quantum geometric Langlands at irrational level

Koebe’s circle-domain conjecture

Brennan's conjecture and the integral-means spectrum

The Falconer distance conjecture

Kakeya in three and four dimensions

The LlogL Fourier-convergence conjecture

Real ultraflat Littlewood polynomials and unbounded binary merit factors

Fourier restriction for positively curved surfaces

The three-dimensional Bochner–Riesz conjecture

Local smoothing in three dimensions

The exact Sobolev endpoint for Schrödinger convergence

Riesz transforms and rectifiability in higher codimension

Annular variation and dyadic absolute bounds for the triangular Hilbert transform

Prime-factor statistics of $p - 1$

P = W for fixed-determinant SL_n moduli spaces

The $L lo g L$ Fourier-convergence conjecture

An L³ bound for the trilinear Hilbert transform

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

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

Subpolynomial dimension reduction in L_p

Exact derandomization of logarithmic space: $L = RL = BPL$

Integer multiplication below $n lo g n$

Exact Fourier transforms below $n lo g n$

Subset Sum in $O (2^{0.49 n})$ time

A C¹ counterexample to the entropy conjecture