id title coverage coverage_ref stateable key_mathlib comparator note 001 Milne’s rationality conjecture and algebraic specialization. NONE - NO - - Needs Hodge classes, crystalline/l-adic realizations of abelian varieties 002 The full BSD formula from low Selmer corank. PR #5542 NO WeierstrassCurve - Strong BSD needs Sha, regulator, Tamagawa, L-function continuation; main has weak BSD only 003 The quasi-Riemann hypothesis. NONE - YES DirichletCharacter.LFunction, riemannZeta QuasiRiemannHypothesis, DirichletSevenEighths, HeckeSevenEighths forall chi, s, 7/8 LFunction chi s != 0 004 Hilbert’s tenth problem over ℚ. PR #5391 YES ComputablePred, MvPolynomial, Primcodable - not ComputablePred (exists rational zero) on MvPolynomial (Fin n) Z 005 Irrationality of Catalan’s constant. MAIN FormalConjectures/Wikipedia/Irrational.lean YES Irrational, tsum Catalan Already stated as irrational_catalanConstant with OAI reference 006 Goldfeld’s conjecture: densities and mean analytic rank. MAIN FormalConjectures/Wikipedia/EllipticCurveRank.lean NO WeierstrassCurve, EllipticCurve.LFunction - Main states it with Mordell-Weil rank; analytic rank needs L(E,s) continuation 007 Ordinary two-point correlations and the corrected Elliott conjecture. NONE - YES ArithmeticFunction.liouville, Asymptotics.IsBigO OrdinaryTwoPointCorrelations, OrdinaryElliott sum_{n<=X} lambda(a n+b) lambda(c n+d) = O(X/(log X)^c) 008 The Deligne–Drinfeld conjecture. NONE - SMALL FreeLieAlgebra, LieAlgebra DeligneDrinfeld grt via pentagon/hexagon and Ihara bracket: sizable local defs over FreeLieAlgebra 009 Function-field reconstruction from Milnor K-theory and Galois data. NONE - NO - BogomolovPopInjectivity Milnor K-theory and pro-l Galois reconstruction missing 010 Unrestricted pro-modularity at the prime two. NONE - NO - - Completed Hecke algebras, Galois deformation, p-adic Hodge theory missing 011 Prime-factor statistics of $`p-1`$. MAIN FormalConjectures/ErdosProblems/821.lean SMALL Nat.primeFactorsList, Nat.totient - Only totient-preimage corollary stated; PD(1) law needs local stick-breaking def 012 Independent largest prime factors of consecutive integers. MAIN FormalConjectures/ErdosProblems/371.lean YES Nat.primeFactors, Filter.Tendsto JointDickman Already stated with OAI reference 013 Ostmann’s inverse Goldbach conjecture. MAIN FormalConjectures/ErdosProblems/431.lean YES Nat.Prime, Set.add OstmannPrimes, OstmannComplete Already stated with OAI reference 014 Restricted geometric Langlands, global Arthur enhancements, and generic Ramanujan. NONE - NO - - Geometric Langlands, automorphic representations missing 015 Torus-packet equidistribution in prime, quartic, and sextic degrees. NONE - NO - DukePrimeDegree Torus packets, homogeneous-space Haar equidistribution for orders missing 016 Zilber–Pink in abelian varieties and the Siegel threefold. NONE - NO - - Atypical subvarieties of abelian varieties missing; PR #6003 is modular Zilber-Pink only 017 The irrationality exponent of π is 2. MAIN FormalConjectures/Wikipedia/FlintCooksonHills.lean; PR #5340 YES Real.pi, LiouvilleWith PiExponent Flint-Hills in main; pi exponent statement in PR #5340 018 The Margulis–Platonov conjecture over global fields. NONE - NO - - Algebraic groups over global fields, anisotropic local groups missing 019 The local p-adic section conjecture and global consequences. NONE - NO - - Etale fundamental groups and sections missing 020 Squarefree quartics and power-free polynomial values. MAIN FormalConjectures/ErdosProblems/978.lean YES Squarefree, Polynomial.eval PowerFreeValues Already stated with OAI reference 021 A quadratic bound for Jacobsthal’s function. MAIN FormalConjectures/ErdosProblems/970.lean YES Nat.Coprime, Nat.primeFactors Jacobsthal, JacobsthalImproved Already stated with OAI reference 022 The weak inhomogeneous Duffin–Schaeffer conjecture. NONE - YES Nat.totient, MeasureTheory.volume, round - Inhomogeneous DS: divergence implies a.e. x has infinitely many q; Erdos 999 is homogeneous 023 Patterson's first moment for cubic Gauss sums. NONE - SMALL gaussSum, EisensteinInt PattersonFirstMoment Needs local cubic residue symbol on Z[omega] and primary primes 024 An asymptotic formula for the number of totients. MAIN FormalConjectures/ErdosProblems/416.lean YES Nat.totient, Filter.Tendsto TotientAsymptotic, TotientCompanionZero Already stated with OAI reference 025 Short Egyptian fractions. MAIN FormalConjectures/ErdosProblems/304.lean YES Finset.sum, Real.log ShortEgyptianFractions, EgyptianFractions Already stated with OAI reference 026 Positive lower density of large prime gaps. MAIN FormalConjectures/ErdosProblems/968.lean YES Nat.nth, Nat.Prime PrimeGaps Already stated with OAI reference 027 Potential integral density on curve character varieties. NONE - NO - CharacterVarietiesAllSeamsSupport? SL_r character varieties and integral points over number rings missing 028 Uniformly bounded components of Gaussian-prime graphs. MAIN FormalConjectures/ErdosProblems/952.lean YES GaussianInt, Prime GaussianMoat Already stated with OAI reference 029 Primitive roots for every admissible integer base. MAIN FormalConjectures/Wikipedia/ArtinPrimitiveRootsConjecture.lean YES orderOf, ZMod - artin_primitive_roots.parts.i covers infinitude 030 Modularity of elliptic curves over imaginary quadratic fields. NONE - NO WeierstrassCurve - Main modularity is over Q only; Bianchi modular forms missing 031 Uchida’s conjecture for open homomorphisms of Galois groups. NONE - NO - - Infinite Galois groups of number fields as topological groups; anabelian setup heavy 032 The rational Hodge conjecture for CM abelian varieties. PR #5538 NO - - Hodge conjecture PRs (#5538, #5585); CM abelian varieties and Hodge classes missing 033 Iitaka subadditivity, variation, and logarithmic additivity. NONE - NO - LogKodairaFiberNegative? Log Kodaira dimension, variation, Fujiki class C missing 034 Log abundance for compact Kähler spaces under logarithmic Iitaka subadditivity. NONE - NO - - Log canonical pairs, abundance, Kahler spaces missing 035 Log-canonical threefold abundance in numerical dimension one. NONE - NO - - Log canonical threefolds, nef/semiample divisors missing 036 Numerical semiampleness and generalized minimal models. NONE - NO - - Generalized pairs, minimal models, Bott-Chern cohomology missing 037 The ordinary-double-point volume gap. NONE - NO - - Normalized volume of klt singularities missing 038 Fujita’s freeness conjecture. NONE - NO - - Canonical bundles, global generation on smooth projective varieties missing 039 Nagata’s conjecture and maximal Seshadri constants. NONE - SMALL MvPolynomial, MvPolynomial.IsHomogeneous Nagata, MaximalSeshadriConstants Plane-curve form: local defs for multiplicity at point and very general points 040 Bloch’s conjecture for complex surfaces. NONE - NO - - Chow groups of zero-cycles and Albanese missing 041 Hyperkähler SYZ and projective-space bases. NONE - NO - - Hyperkahler manifolds, semiample line bundles missing 042 Oka classification for minimal compact complex surfaces: Kodaira dimension zero and class VII. NONE - NO - - Oka manifolds and compact complex surfaces missing 043 P = W for fixed-determinant SLn moduli spaces. NONE - NO - - Higgs moduli, perverse and weight filtrations missing 044 The equivariant cohomological Hikita conjecture. NONE - NO - - Nakajima varieties, Coulomb branches, equivariant cohomology missing 046 Shafarevich counterexamples in dimension two and with large fundamental group. NONE - NO - - Universal covers of projective varieties, holomorphic convexity missing 047 Zariski cancellation and affine fibrations over the complex numbers. MAIN FormalConjectures/Arxiv/2208.14736/ZariskiCancellation.lean YES MvPolynomial, Polynomial, AlgEquiv ComplexCancellation Already stated with OAI reference 048 A characteristic-zero counterexample to Lipman–Zariski. NONE - YES Derivation, Module.Free, IsRegularLocalRing - Normal f.g. C-domain dim 2, Der free rank 2, some localization not regular 049 A stable-coordinate counterexample in four variables. NONE - YES MvPolynomial, AlgEquiv, MvPolynomial.X StableCoordinateFour, AbhyankarSathaye exists f in C[x1..x4] not a coordinate, but coordinate in C[x1..x5] 050 A counterexample to Griffiths’ positivity conjecture. NONE - NO - - Ample vector bundles and Hermitian curvature missing 051 Kobayashi’s canonical-ampleness conjecture. NONE - NO - - Kahler manifolds, entire curves, ample canonical bundle missing 052 Tangent splittings and product decompositions. NONE - NO - KahlerSplitting, SplitTangentIntegrability Holomorphic tangent bundle splittings on Kahler manifolds missing 053 A counterexample to Pixton completeness in Chow. NONE - NO - - Moduli of stable curves and tautological rings missing 054 Irrational cubic fourfolds with Hodge-theoretic and categorical K3 associations. NONE - NO - - Rationality of cubic fourfolds, K3 categories missing 055 Gepner symmetry and large-volume stability on threefolds. NONE - NO - - Bridgeland stability conditions on derived categories missing 056 Termination of projective and Kähler fourfold minimal model programs. NONE - NO - - MMP flips and termination missing 057 Fundamental groups of special complex varieties and root orbifolds. NONE - NO - - Special Kahler manifolds (Campana) missing 058 Semialgebraic universal covers and bounded domains. NONE - NO - SymmetricDomains? Universal covers of projective varieties, bounded symmetric domains missing 059 Counterexamples to Zariski’s multiplicity conjecture. NONE - SMALL MvPowerSeries.order, Homeomorph, MvPolynomial - exists reduced f,g in C[x1..x4], germ homeomorphism of zero sets, orders 4 and 5 060 The Global Spherical Shell conjecture. NONE - NO - - Class VII surfaces, global spherical shells missing 062 Projective contact classification and the LeBrun–Salamon conjecture. NONE - NO - - Quaternionic-Kahler manifolds and contact Fano classification missing 063 The generalized Mukai conjecture. NONE - NO - - Fano manifolds, Picard number, pseudoindex missing 064 Topological triviality of μ-constant surface singularities. NONE - NO - - Milnor number families and topological triviality of germs missing 065 Virasoro constraints for complete intersections and projective-bundle towers. NONE - NO - - Gromov-Witten theory and Virasoro operators absent from Mathlib 066 Bounded klt complements for Fano contractions. NONE - NO - - klt singularities, log-canonical pairs, complements absent 067 The Campana–Peternell conjecture in dimension six. NONE - NO - - Fano varieties, nef tangent bundle, rational homogeneous spaces absent 068 Anticanonical nonvanishing in every dimension. NONE - NO - - canonical bundle, Hermitian metrics, curvature on varieties absent 069 Global quantum geometric Langlands at irrational level. NONE - NO - - geometric Langlands, twisted D-modules, derived categories absent 071 Koebe’s circle-domain conjecture. NONE - SMALL ConformalAt, DifferentiableOn, OnePoint KoebeCircleDomains state on domains in C: holomorphic bijection onto circle domain; define circle domain 072 Brennan's conjecture and the integral-means spectrum. MAIN FormalConjectures/Wikipedia/Brennanconjecture.lean YES DifferentiableOn, MeasureTheory.volume, deriv Brennan, BrennanSharp, StrictMeans Brennan conjecture and integral means spectrum already stated 073 The Falconer distance conjecture. MAIN FormalConjectures/Wikipedia/Falconer.lean YES dimH, MeasureTheory.volume, IsCompact FalconerAllDimensions, PlanarFalconer falconer_conjecture already stated for d >= 2 074 Kakeya in three and four dimensions. MAIN FormalConjectures/Wikipedia/Kakeya.lean YES dimH, EuclideanSpace - kakeya_set_conjecture n=4 covers 4D; 3D maximal operator bound not stated 075 The $`L\log L`$ Fourier-convergence conjecture. NONE - SMALL fourierCoeff, fourier, AddCircle.haarAddCircle FourierLLogL define L log L condition; ae convergence of symmetric partial sums 076 Real ultraflat Littlewood polynomials and unbounded binary merit factors. MAIN FormalConjectures/ErdosProblems/1150.lean YES Polynomial.coeff, Polynomial.eval, Metric.sphere AsymptoticallyMinimalLittlewood, LittlewoodFiniteFlatness erdos_1150 disproof; also PR #6991; Turyn merit factor not stated 077 Fourier restriction for positively curved surfaces. NONE - SMALL VectorFourier.fourierIntegral, hausdorffMeasure, eLpNorm - parametrize surface as graph; define extension operator and curvature condition 078 The three-dimensional Bochner–Riesz conjecture. NONE - SMALL Real.fourierIntegral, SchwartzMap, eLpNorm - define Bochner-Riesz multiplier on Schwartz functions; L^p bound 079 Local smoothing in three dimensions. NONE - SMALL SchwartzMap, Real.fourierIntegral, eLpNorm - define half-wave propagator and Bessel-potential Sobolev norm 080 The exact Sobolev endpoint for Schrödinger convergence. NONE - SMALL Real.fourierIntegral, SchwartzMap, MeasureTheory.ae - define H^s via Fourier and Schrodinger propagator; ae limit 081 Riesz transforms and rectifiability in higher codimension. NONE - SMALL hausdorffMeasure, LipschitzWith, MeasureTheory.Measure RieszQuantitative define ADR measure, truncated Riesz transform, uniform rectifiability 082 Annular variation and dyadic absolute bounds for the triangular Hilbert transform. NONE - SMALL MeasureTheory.integral, eLpNorm, MeasureTheory.Lp TriangularHilbert define truncated triangular Hilbert transform and r-variation 083 Hilbert transforms along Lipschitz directions. NONE - SMALL LipschitzWith, MeasureTheory.integral, eLpNorm LipschitzHilbert define truncated directional Hilbert transform along vector field 084 The geometric case of the Erdős similarity conjecture. MAIN FormalConjectures/ErdosProblems/120.lean YES MeasureTheory.volume, IsCompact - general similarity conjecture stated; geometric q^n case not a separate variant 085 Endpoint Sobolev regularity of centered disk averages. PR #7129 SMALL eLpNorm, MeasureTheory.integral DiskMaximal no weak derivative / W^{1,1} in Mathlib; needs local definition 086 An L3 bound for the trilinear Hilbert transform. NONE - SMALL MeasureTheory.integral, eLpNorm, MeasureTheory.Lp - define principal-value trilinear Hilbert transform; L3xL3xL3 to L1 bound 087 The Mahler conjectures, functional inequalities and polar-product symplectic width. PR #6977 SMALL ConvexBody, LinearMap.polar, MeasureTheory.volume MahlerConjecture, GeneralMahler, SymmetricMahlerEquality, SymmetricPolar define volume product and Hanner polytopes; Gromov width part needs symplectic 088 Sharp projection-body inequalities and a counterexample to simplex maximization. PR #7150 SMALL ConvexBody, orthogonalProjection, MeasureTheory.volume PettyProjectionVolume, ProjectionVolume, ProjectionCounterexample define projection body via projection volumes 089 Bounded-distortion L1 embeddings of planar and bounded-treewidth graphs. NONE - SMALL SimpleGraph, SimpleGraph.dist, MeasureTheory.Lp PlanarL1, BoundedTreewidthL1 treewidth and planarity not in Mathlib; local definitions needed 090 Triangular-lattice optimality, long-range Riesz and Coulomb energies, and spherical logarithmic energy. PR #7455 SMALL MeasureTheory.volume, Metric.sphere, Real.log TriangularEnergy PR covers Brauchart-Hardin-Saff log energy; lattice energy needs config defs 091 Logarithmic and Lp Brunn–Minkowski inequalities and the B-conjecture. PR #6978 SMALL ConvexBody, MeasureTheory.volume LogBrunnMinkowski define support function and log-Minkowski combination 092 The optimal order of convex-body covering density. NONE - SMALL IsZLattice, ConvexBody, MeasureTheory.volume CoveringDensity, SingleLatticeCovering define lattice covering density; Theta(n log n) bounds 093 Dimension-free logarithmic Sobolev inequality for subgaussian log-concave measures. NONE - SMALL MeasureTheory.Measure, Real.negMulLog, ConvexOn - define log-concave density, subgaussian marginals, entropy, LSI constant 094 Subpolynomial dimension reduction in Lp. NONE - YES MeasureTheory.Lp, PiLp, LipschitzWith SubpolynomialLp n-point subsets of L^p embed into l_p^d, d=n^o(1), distortion D 095 Hyperbolicity cones without semidefinite lifts. NONE - SMALL MvPolynomial.IsHomogeneous, Matrix.PosSemidef HyperbolicCones define hyperbolic polynomial, hyperbolicity cone, spectrahedral shadow 096 The Gaussian propeller conjecture in every dimension. PR #7002 SMALL ProbabilityTheory.stdGaussian, EuclideanSpace, MeasureTheory.integral GaussianPropeller sum of squared Gaussian moments over partitions <= 9/(8 pi) 097 The Euclidean Steinitz–Bergström bound. PR #6990 YES EuclideanSpace, Metric.closedBall, Finset.sum SteinitzBergstrom signs with partial sums bounded by C sqrt d 098 Compact counterexamples to bi-Lipschitz dimension reduction. PR #6998 SMALL LipschitzWith, NormedSpace, FiniteDimensional DoublingHilbert, CompactBanach metric doubling not defined in Mathlib; short local definition 099 The sharp exponential scale of edit-distance distortion. NONE - SMALL MeasureTheory.Lp, LipschitzWith EditDistance, BinaryEditLower no edit distance in pinned Mathlib; define it and l1 distortion 100 Cylinder coverings below the half-area bound. NONE - SMALL MeasureTheory.volume, convexHull, orthogonalProjection CylinderCovering define cylinders with triangular bases; tetrahedron covering 101 The sharp simplex conjecture for isotropic constants. NONE - SMALL ConvexBody, MeasureTheory.volume, Matrix.det - define isotropic constant; simplices uniquely maximize 102 The Unique Games Conjecture and optimal approximation thresholds. PR #5430 SMALL Turing.TM2ComputableInPolyTime UniqueGamesTheorem, OptimalMaxCut, VertexCover, MinUncut, DirectedFeedback repo has NP and IsPolyTime; needs gap problems and reductions 103 Exact derandomization of logarithmic space: $`\mathsf L=\mathsf{RL}=\mathsf{BPL}`$. NONE - NO - LogspaceEquality no space-bounded or randomized logspace machine model 104 Quasipolynomial algorithms for mean-payoff, stochastic and parity games. NONE - SMALL Turing.TM2ComputableInTime RandomizedMeanPayoff define mean-payoff games; quasipolynomial time bound via TM2 105 Perfect completeness for 2-to-1 games. PR #5430 SMALL Turing.TM2ComputableInPolyTime PerfectCompleteness PR states 2-to-1 games with perfect completeness 106 Hardness of coloring three-colorable graphs. NONE - SMALL SimpleGraph.Colorable, Turing.TM2ComputableInPolyTime - NP-hardness of c-coloring 3-colorable graphs via repo NP class 107 Matrix multiplication with exponent at most 9/4. PR #5316 SMALL TensorProduct, Matrix MatrixMultiplication PR defines tensor rank of matmul tensor; omega <= 9/4 108 A cubic permanent–determinant lower bound. NONE - SMALL Matrix.det, Matrix.permanent, MvPolynomial PermanentCubic define border determinantal complexity via closure 109 Integer multiplication below $`n\log n`$. NONE - SMALL Turing.TM2ComputableInTime - needs multitape TM model; Mathlib TM2 is stack-based 110 Optimal-order randomized k-server on arbitrary metrics. NONE - SMALL MetricSpace, PMF KServer, UniformKServer define randomized online k-server and competitive ratio 111 One-sample matroid prophet inequalities against an almighty adversary. NONE - SMALL Matroid, ProbabilityTheory.iIndepFun MatroidProphet define online sample-based policy and almighty adversary 112 Beyond the square-root exponent for depth-three circuits. NONE - SMALL Turing.TM2ComputableInPolyTime DepthThree define depth-3 OR-AND-OR circuits; PR #7121 is B2 circuits only 113 Approximate counting and entropy of perfect matchings. NONE - SMALL SimpleGraph, Real.negMulLog MatchingFPRAS, MatchingEntropy, MatchingEntropyBounds entropy conjecture is pure math; FPRAS needs randomized model 114 Approximate counting of common integer polymatroid bases. NONE - NO Matroid CommonBasesFPRAS needs oracle randomized machine model for FPRAS 115 Sampling and counting contingency tables with arbitrary margins. NONE - NO Matrix ContingencyTables needs randomized sampler model with expected time 116 Uniform black-box noncommutative identity testing across characteristics. NONE - SMALL FreeAlgebra, Matrix FormulaHitting, RationalHitting define noncommutative formulas; explicit hitting set construction 117 Uniform sparsest cut: hardness and semidefinite gaps. NONE - SMALL SimpleGraph, Turing.TM2ComputableInPolyTime UniformSparsestCut NP-hardness via repo NP; SDP gap part needs metric defs 118 Bin packing and unbounded configuration-LP gaps. NONE - SMALL Finset, Turing.TM2ComputableInPolyTime BinPackingGap MIRUP disproof is pure math: OPT - ceil(LP) unbounded 119 The Courtade–Kumar and Hellinger conjectures. MAIN FormalConjectures/Paper/CourtadeKumar.lean YES Real.negMulLog, Finset CourtadeKumar courtade_kumar already stated 120 Almost-linear-time exact matching and prescribed-degree factors in general graphs. NONE - NO SimpleGraph - almost-linear word-RAM time; no RAM model 121 Almost-linear approximation of edit distance. NONE - NO - EditApproximation almost-linear randomized time; no RAM model 122 Quantitative trace-reconstruction bounds with a uniform decoder. NONE - SMALL PMF, ProbabilityTheory TraceReconstruction lower bound on traces is pure probability; define deletion channel 124 Polynomial-time scheduling on three identical machines. NONE - SMALL Turing.TM2ComputableInPolyTime ThreeMachine define P3 prec unit-job scheduling; polytime optimal algorithm 125 The metric k-median approximation threshold and recovery. NONE - SMALL Turing.TM2ComputableInPolyTime, MetricSpace KMedianThreshold, KMedianRecovery, KMedianRefinedRecovery, MetricKMedian polytime (1+2/e+eps)-approx; threshold assumes P != NP 126 Exponential semidefinite complexity of perfect matching. NONE - SMALL Matrix.PosSemidef, convexHull MatchingPSD, MatchingAffineLift define psd rank of slack matrix; exponential lower bound 127 Average sensitivity of polynomial threshold functions. NONE - YES MvPolynomial.totalDegree, Fintype, SignType.sign GotsmanLinial average sensitivity of degree-d PTF <= 8 d sqrt n 128 A factor-two approximation for shortest common superstring. NONE - SMALL Turing.TM2ComputableInPolyTime, List.IsInfix Superstring polytime 2-approximation for shortest common superstring 129 Exponential state costs for two-way automata. NONE - SMALL NFA, DFA, Language TwoWayComplementation, TwoWayDeterminization Needs local 2NFA/2DFA defs; state-count lower bound family over growing alphabets 130 Exact Fourier transforms below $`n\log n`$. NONE - SMALL DFT via AddChar (ZMod n) ℂ, Polynomial ExactFourier Needs local straight-line arithmetic program model and operation count 131 Rapid mixing of graph switches for every degree sequence. NONE - SMALL SimpleGraph, ProbabilityTheory SwitchChain Needs local switch Markov chain, TV distance, mixing time definitions 132 A superquadratic separation of sensitivity and block sensitivity. MAIN FormalConjectures/Paper/StrongSensitivityConjecture.lean SMALL Finset, Fin n → Bool SensitivitySeparation Existing file states bs ≤ s^2; disproof is its negation 133 The computational complexity of Weisfeiler–Leman refinement. NONE - NO SimpleGraph WeisfeilerLeman, WLIdentification, VariableWL WL definable, but n^Ω(k) time bounds and EXPTIME-completeness need machine models 134 Generalized star height at most three. NONE - SMALL RegularExpression, Language.IsRegular, KStar GeneralizedStarHeight Local inductive generalized regex with complement; ∀ regular L, star height ≤ 3 135 Homogeneous depth-five lower bounds for iterated matrix multiplication. NONE - SMALL MvPolynomial, Matrix DepthFive Needs local homogeneous depth-five circuit model and size measure 136 A quasilinear PCP theorem for PPAD. NONE - NO - - PCP/PPAD reductions and complexity classes absent 137 One-tape time simulation in two-fifths-power space. NONE - NO Turing.TM2 OneTapeSpace Space-bounded simulation model absent; Mathlib only has time-bounded TM2 138 Subset Sum in $`O(2^{0.49n})`$ time. NONE - NO - - Randomized word-RAM running time model absent 139 Subpolynomial query complexity for log-concave sampling. NONE - NO MeasureTheory, gradient LogConcaveQuery Oracle query complexity of randomized samplers needs substantial algorithm model 140 Memory–sample lower bounds for noiseless Gaussian regression. NONE - NO gaussianReal NoiselessRegression Memory-bounded streaming learner model absent 141 Existential–universal real sentences in the counting hierarchy. NONE - NO - - Counting hierarchy and ETR complexity absent 142 Deterministic polynomial factorization over prime fields. NONE - SMALL Polynomial, ZMod, UniqueFactorizationMonoid - Use FC4M IsPolyTime with bitstring encoding; needs factorization output encoding 143 Hilbert's sixteenth problem: uniform bounds for limit cycles. NONE - SMALL MvPolynomial, HasDerivAt, IsIntegralCurve QuinticLienard Local defs of periodic orbit/limit cycle; ∀ degree n ∃ bound H(n) 144 Banach’s simple Lebesgue-spectrum problem. NONE - SMALL MeasurePreserving, Lp, HilbertBasis, ContDiff ThreeTorus Koopman iterates of one observable form orthonormal basis of L2_0(T^3) 145 Rokhlin’s multiple-mixing problem. NONE - SMALL MeasurePreserving, MeasurableEquiv, ProbabilityMeasure Rokhlin Local mixing/k-mixing defs; mixing → mixing of all orders 146 Positive metric entropy for the standard map. NONE - SMALL AddCircle, volume, MeasurePreserving StandardMapEntropy, StandardMapLyapunov Kolmogorov–Sinai entropy absent; needs local partition-entropy def 147 The near-boundary Birkhoff conjecture. NONE - NO Convex, ContDiff - Smooth billiard map, caustics, invariant-curve foliations all missing 148 The entropy-rate dimension formula for self-similar measures. NONE - SMALL dimH, Measure, negMulLog SelfSimilar, SelfSimilarCorollaries Local defs of measure Hausdorff dimension and random-walk entropy rate 149 Classwise permanence for weakly reversible mass-action systems. PR #7010 SMALL ODE, HasDerivAt MassAction PR states endotactic permanence (generalizes weakly reversible); needs CRN defs 150 Weak mixing of triangular billiards with an irrational angle. NONE - SMALL EuclideanSpace, MeasurePreserving IrrationalTriangleBilliard Needs local polygonal billiard flow and weak-mixing definition 151 A C1 counterexample to the entropy conjecture. NONE - SMALL coverEntropy, SingularHomology, ContMDiff C1EntropyCounterexample Topological entropy and singular homology exist; manifold self-map counterexample 152 Zero entropy does not guarantee a smooth positive-volume model. NONE - NO MeasurePreserving, ContMDiff - Needs KS entropy, smooth positive densities on arbitrary manifolds 153 Arithmetic classification and non-Pisot singularity for Bernoulli convolutions. NONE - SMALL Measure.map, Polynomial, IsAlgebraic - Core: Bernoulli convolution singular at reciprocals of quartic Salem numbers 154 Pointwise multiple ergodic averages for mixing transformations. NONE - SMALL MeasurePreserving, birkhoffAverage - Local mixing def; a.e. convergence of multiple averages to product of integrals 155 A counterexample to periodic tiling in dimension three. NONE - SMALL Set, ℤ^3 translates PeriodicTilingThree Local tiling/periodicity defs; ∃ finite tile in ℤ^3 with no periodic tiling 156 Borsuk's conjecture fails in dimension nine. MAIN FormalConjectures/Wikipedia/BorsukConjecture.lean YES EuclideanSpace, Metric.diam BorsukNine PR #7312 adds n ≥ 9 variant 157 Graph coloring, clique minors, and Colin de Verdière invariants. PR #5196, #6981 SMALL SimpleGraph.chromaticNumber, Subgraph HadwigerCounterexample, ListHadwiger Needs minors, fractional chromatic number; Colin de Verdière μ missing 158 The Euclidean plane cannot be colored with five colors. MAIN FormalConjectures/ErdosProblems/508.lean; FormalConjectures/Wikipedia/HadwigerNelson.lean YES SimpleGraph.chromaticNumber, EuclideanSpace, SimpleGraph/UnitDistance EuclideanFiveColor, PlaneColoring χ(ℝ²) ≥ 6 is a lower bound variant in 508 159 Erdős’s reciprocal-sum conjecture and quasipolynomial Szemerédi bounds. MAIN FormalConjectures/ErdosProblems/3.lean YES Summable, Set.IsAPOfLength ErdosReciprocal Divergent reciprocal sum implies arbitrarily long APs 160 Superexponential van der Waerden numbers. MAIN FormalConjectures/ErdosProblems/138.lean YES Finset, Nat QuantitativeVanDerWaerden W(k)^{1/k} → ∞ 161 Counterexamples to Sidorenko’s conjecture and the forcing conjecture. MAIN FormalConjectures/Wikipedia/SidorenkoConjecture.lean YES SimpleGraph.Copy, FC4M HomDensity SidorenkoCounterexample Forcing conjecture not stated 162 Counterexamples to Ryser’s covering conjecture. PR #6812 SMALL Finset, SetFamily RyserCovering Needs local r-partite hypergraph, cover number τ, matching ν 164 Hindman’s finite sums and products conjecture. MAIN FormalConjectures/ErdosProblems/172.lean YES Finset.sum, Finset.prod - Already marked solved with linked formal proof 165 The Harary–Hill and Zarankiewicz crossing-number formulas. NONE - SMALL SimpleGraph, Path, ContinuousMap CompleteCrossing, BipartiteCrossing Needs local plane drawing and crossing number; PR #4111 is draft placeholder 166 The higher-dimensional Erdős distinct-distances conjecture. MAIN FormalConjectures/ErdosProblems/1083.lean YES EuclideanSpace, Finset.image dist - 1083 states n^{2/d-o(1)}; result gives c_d n^{2/d} 167 Planar distinct distances and unit-distance bounds. MAIN FormalConjectures/ErdosProblems/604.lean; FormalConjectures/ErdosProblems/90.lean YES EuclideanSpace, dist PinnedDistances, PlanarUnitDistances PR #7340 links weak pinned proof; all-but-o(n) version stronger 168 Combinatorial invariance of Kazhdan–Lusztig polynomials. NONE - SMALL CoxeterSystem, CoxeterSystem.length KLInvariance Bruhat order and KL polynomials absent; definable via R-polynomial recursion 169 Shareshian–Wachs elementary positivity. NONE - SMALL MvPolynomial.esymm, SimpleGraph.Coloring ElementaryPositivity Local chromatic quasisymmetric function in finitely many variables 170 Sharp logarithmic exponents for off-diagonal Ramsey numbers. MAIN FormalConjectures/ErdosProblems/986.lean YES SimpleGraph.CliqueFree, indepNum SharpLogRamsey 986 is lower bound k^{s-1}/(log k)^c; sharp exponent is stronger 171 The hypercube Ramsey conjecture. MAIN FormalConjectures/ErdosProblems/181.lean YES SimpleGraph.Copy, FC4M Hypercube - R(Q_n) ≪ 2^n 172 Classification of finite Euclidean Ramsey configurations. NONE - SMALL EuclideanSpace, Isometry EuclideanRamsey, EuclideanRamseyTransitive Erdős 214 and PR #6980 are special cases; LRW conjecture not stated 173 Seymour’s second-neighborhood conjecture. PR #5188 YES Digraph SeymourSecondNeighborhood Oriented graph = Digraph with no loops or 2-cycles 174 Deterministic construction of strong thin spanning trees. NONE - SMALL SimpleGraph.IsEdgeConnected, IsTree StrongThinTree, AlgorithmicThinTrees Multigraphs need local def; algorithmic part harder 175 Talagrand’s expectation thresholds, discrete convexity, and graph decompositions. NONE - SMALL Finset, SetFamily TalagrandExpectationThreshold, TalagrandDiscreteConvexity GreensOpenProblems/54 is Gaussian convexity, different; needs p-biased measure 176 The second Kahn–Kalai conjecture with an edge-count bound. NONE - SMALL SimpleGraph.Copy, ProbabilityTheory SecondKahnKalai PR #4018 is first Kahn–Kalai; needs local G(n,p) measure 177 Bounded-degree coboundary expanders. NONE - SMALL ZMod 2, Finset CoboundaryExpanders Needs local abstract simplicial complex and F2 coboundary expansion 178 Deterministic nonbipartite Ramanujan graphs in every fixed degree. NONE - YES SimpleGraph.adjMatrix, Matrix.IsHermitian.eigenvalues, IsRegularOfDegree - Existence part stateable; poly-time construction part not 179 The circulant Hadamard and Barker-sequence conjectures. MAIN FormalConjectures/Arxiv/2402.13202/CirculantHadamard.lean; FormalConjectures/Arxiv/2104.00502/BarkerSequence.lean YES Matrix.circulant, Matrix CirculantHadamard, EvenBarker Circulant Hadamard only in orders 1, 4 180 Barnette’s Hamiltonian-cycle conjecture. PR #7149 SMALL SimpleGraph.IsHamiltonian, IsRegularOfDegree BarnetteHamiltonian Planarity absent in Mathlib; needs local def 181 The Erdős–Gallai cycle-decomposition conjecture. MAIN FormalConjectures/ErdosProblems/184.lean YES SimpleGraph, Walk.IsCycle CycleDecomposition O(n) cycles and edges partition 182 Power savings for intersective polynomial differences and prime arguments. NONE - YES Polynomial ℤ, ZMod, Finset.range - Intersective h; difference-free subsets of [N] have size O(N^{1-c_k}) 183 Power savings for planar halving lines and k-sets. PR #6979 YES EuclideanSpace, Collinear HalvingLines Halving lines O(n^{4/3-ε}) for points with no three collinear 184 Correspondence coloring with a fixed forbidden subgraph. NONE - YES SimpleGraph.CliqueFree, indepNum, maxDegree CliqueFreeLog AEKS core: K_r-free implies α ≥ c n log d / d; DP coloring needs local def 185 Counterexamples to infinite matroid intersection and packing/covering. NONE - YES Matroid, Matroid.Indep, Matroid.dual InfiniteMatroid, InfiniteMatroidCorollaries Mathlib matroids allow infinite ground sets; state intersection conjecture negation 186 Uniform influence and sharp thresholds for graph and hypergraph properties. NONE - SMALL SimpleGraph, Finset SharpThreshold Needs local p-biased measure and threshold width 187 Snaky in 21 Maker moves. NONE - SMALL ℤ × ℤ, Finset SnakyTwentyOne, SnakyCertificate, SnakyConditional Needs local Maker–Breaker game and winning strategy definitions 188 The sharp terminal leave in random triangle removal. NONE - SMALL SimpleGraph, ProbabilityTheory TriangleRemoval Needs local random triangle removal process 189 Cycle–clique Ramsey numbers. MAIN FormalConjectures/ErdosProblems/551.lean YES SimpleGraph.cycleGraph, CliqueFree CycleCliqueRamsey PR #6992 records full formula 190 Polynomial removal fails for ordered binary matrices. NONE - SMALL Matrix (Fin n) (Fin n) Bool MatrixRemoval Needs local ordered pattern copy density and removal distance 191 A power improvement in the Heilbronn triangle lower bound. MAIN FormalConjectures/ErdosProblems/507.lean YES EuclideanSpace, MeasureTheory.volume HeilbronnTriangle Lower bound n^{-2+c} for Heilbronn 192 Boolean functions violate the square-root degree bound by arbitrary factors. NONE - SMALL Finset.sum; Real.sqrt; Fintype SquareRootDegree Local defs of Boolean Fourier coefficient and degree; existential over C. PR 3807 is different 193 Serre’s intersection-multiplicity conjecture. MAIN FormalConjectures/Wikipedia/SerreMultiplicityConjectures.lean SMALL IsRegularLocalRing; Module.length; TensorProduct - intersectionMultiplicity_pos already stated; PR 7215 adds proof note 194 Lech’s multiplicity conjecture. NONE - SMALL Module.Flat; IsLocalHom; Module.length; ringKrullDim - Needs local Hilbert-Samuel multiplicity def; then e(R) ≤ e(S) for flat local maps 195 A counterexample to the small Cohen–Macaulay module conjecture. MAIN FormalConjectures/Wikipedia/SmallCohenMacaulayModules.lean SMALL IsAdicComplete; IsLocalRing.maximalIdeal; ringKrullDim - exists_isSmallCohenMacaulay already stated; PR 7214 adds counterexample note 196 A counterexample to Kaplansky’s zero-divisor conjecture. MAIN FormalConjectures/Wikipedia/Kaplansky.lean YES MonoidAlgebra; Monoid.IsTorsionFree; Group.IsFinitelyPresented TorsionFreeZeroDivisors zero_divisor_conjecture already marked solved, links OpenAI proof 197 A torsion-free group algebra that is not directly finite. MAIN FormalConjectures/Wikipedia/SurjunctiveGroup.lean YES MonoidAlgebra; IsDedekindFiniteMonoid; Group.IsFinitelyPresented KaplanskyDirectFiniteness; KaplanskyFinitelyPresented; OddKaplansky; GroupRingDeterminant Gottschalk stated; direct-finiteness conjecture itself not stated in repo 198 A counterexample to finitistic-dimension finiteness. NONE - YES projectiveDimension; HasProjectiveDimensionLE; ModuleCat; FiniteDimensional LittleFinitistic; FinitisticAsymmetry Exists f.d. ℂ-algebra with unbounded finite projective dimensions of f.d. modules 199 Counterexamples to Auslander–Reiten, Tachikawa and related homological conjectures. MAIN FormalConjectures/Paper/AuslanderReiten.lean YES CategoryTheory.Abelian.Ext; ModuleCat; Module.Projective AuslanderReiten; Tachikawa Also open PRs #7067 (Tachikawa), #5547 (Nakayama) 200 Eisenbud–Green–Harris and lex-plus-powers. MAIN FormalConjectures/Paper/EisenbudGreenHarris.lean YES MvPolynomial; Ideal.IsHomogeneous; Module.finrank - eisenbud_green_harris stated; lex-plus-powers Betti part not stated 201 A counterexample to Kurosh’s division-ring problem. NONE - YES DivisionRing; Subring.center; Algebra.IsAlgebraic; Module.rank - ∃ char-0 division ring, algebraic over center, 2-generated, infinite rank over center 202 The blockwise Alperin weight conjecture. NONE - NO - - Needs blocks, Brauer characters, weights of finite groups 203 Donovan's conjecture over fields and complete mixed-characteristic DVRs. NONE - NO - - Needs blocks, defect groups, Morita equivalence classes 204 Tensor saturation for even spin groups. NONE - NO - - Needs Spin(2n) representation theory and dominant weights 205 Saxl’s conjecture and universal tensor squares. NONE - SMALL FDRep; FDRep.character; CategoryTheory.Simple; Equiv.Perm Saxl; UniversalTensorSquares Universal-tensor-square variant stateable via FDRep; staircase needs Specht modules 206 Finite lattice representation and undecidability. NONE - SMALL Lattice; Fintype; OrderIso; Setoid FiniteCongruenceGraph(?) Needs local finite universal-algebra and congruence-lattice defs 207 The ℓ¹-Bass conjecture for all discrete groups. MAIN FormalConjectures/Arxiv/math.0703548/FarrellJonesConsequences.lean SMALL MonoidAlgebra; IsIdempotentElem; lp; Matrix.trace BassTrace; BassTorsionFree Bass (domains) and Kaplansky idempotent stated; ℓ¹ form needs local HS trace 208 Finite symmetric tensor categories and the Verlinde tower. NONE - NO - - Needs symmetric tensor categories and higher Verlinde categories 209 Integral counterexamples to Gersten’s conjecture. NONE - NO - - Needs higher algebraic K-theory of rings 210 Foulkes' conjecture for sixth powers and quadratic stabilization. NONE - SMALL SymmetricPower; Representation; LinearMap FoulkesHowe Define canonical Foulkes-Howe map; state surjectivity for b ≥ a(a-1) 211 The geometric phase diagram, diffusion, and spectra of random planar maps. NONE - NO - FKCRT Needs LQG, Brownian CRT, Liouville Brownian motion 212 Planar first-passage geometry and the absence of bigeodesics. NONE - SMALL Measure.infinitePi; SimpleGraph.Walk; IsProbabilityMeasure PlanarFirstPassage; GammaPassage Local FPP passage time and bigeodesic defs on ℤ²; a.s. no bigeodesic 213 Critical percolation on every quasi-transitive graph. NONE - SMALL Measure.infinitePi; SimpleGraph.Connected; SimpleGraph.LocallyFinite CriticalPercolation; CriticalZ3; CayleyPercolation(?) Local Bernoulli percolation, p_c, quasi-transitivity defs; θ(p_c)=0 214 The Benjamini–Schramm nonuniqueness conjecture. NONE - SMALL Measure.infinitePi; SimpleGraph.LocallyFinite; SimpleGraph.Connected BenjaminiSchramm Local percolation, p_u, nonamenability defs; p_c < p_u 215 Canonical $`O(3)`$ continuum limit and exact $`O(4)`$ mass asymptotics. NONE - NO - ClassicalON Continuum QFT limit and infinite-volume Gibbs states missing 216 Critical and near-critical XY scaling and BKT universality. NONE - NO - - Infinite-volume XY Gibbs states, BKT asymptotics missing 217 The low-temperature Sherrington–Kirkpatrick fluctuation law. NONE - SMALL gaussianReal; ProbabilityTheory.variance; Measure.pi - Finite-n SK log partition function concrete; Var ~ c n^{1/3} 218 Conformal universality for weakly interacting and random-bond Ising models. NONE - NO - - Needs SLE3 and Ising scaling limits 219 GOE bulk universality for regular graphs with weak Anderson disorder. NONE - NO - - Needs GOE point-process limits for random regular graphs 220 Directional zero–one laws beyond iid environments and iid ballisticity. NONE - SMALL ProbabilityTheory.Kernel.traj; Measure.infinitePi; Filter.Tendsto DirectionalBallisticity; DirectionalWalk Local RWRE quenched/annealed law; transience implies positive velocity 221 The Mézard–Parisi formula for diluted spin glasses. NONE - NO - DilutedSpin Mézard-Parisi hierarchical cavity functional too heavy 222 Perceptron free energies and microscopic jamming exponents. NONE - NO - PerceptronFreeEnergy Variational free-energy formulas too heavy 223 Random-cluster interfaces: critical, disordered, thermal, and natural-time scaling. NONE - NO - - Needs SLE/CLE 224 Critical and quenched near-critical universality for Poisson–Voronoi percolation. NONE - NO - - Needs Poisson-Voronoi percolation and conformal Cardy limit 225 Gaussian free field limits throughout the balanced six-vertex regime. NONE - NO - - Needs Gaussian free field convergence 226 The double-dimer loop ensemble converges to CLE4. NONE - NO - - Needs CLE4 227 Critical SK autocorrelation processes and dynamics across the temperature transition. NONE - SMALL Matrix.exp; gaussianReal; Fintype CriticalSK; CriticalSKMixing Mixing-time bounds stateable with local TV and heat-bath generator defs 228 Continuum phase transitions for radial pair potentials. NONE - NO - ContinuumTransition; RadialTransition Thermodynamic limit of continuum canonical free energy missing 229 Exact three- and four-state reconstruction thresholds and four-state tree capacity. NONE - SMALL Measure; Filter.Tendsto; Fintype ThreeStateSupercritical(?); ThreeStateTreeClauses(?) Local finite-depth broadcast on d-ary tree; reconstruction iff dλ²>1 230 Exact Hausdorff gauges for SLE. NONE - NO - SLELowerPositivity(?) Needs SLE 231 The free uniform spanning forest is a factor of IID. NONE - NO - FreeUniformSpanningForest; StronglyRayleighDPP Needs free uniform spanning forest and factor-of-iid machinery 232 Gaussian fields and interfaces for triangular-lattice Lipschitz heights. NONE - NO - TriangularGaussian(?) Needs GFF and SLE4 limits 233 The joint critical Ashkin–Teller current limit. NONE - NO - GaussianReplacement(?) Needs GFF and its local sets 234 All-temperature pressure of orthogonally invariant Ising spin glasses. NONE - NO - - Variational pressure formula too heavy 235 Limiting random SAT thresholds, sharp variance and computability. NONE - SMALL PMF; Finset.card; Filter.Tendsto SATComputability; SATSharpness; SATVariance; FixedClauseThreshold Local random k-SAT model; ∃ r_k sharp threshold limit 236 The exact factor-of-IID threshold for free Ising spins on trees. NONE - NO - FreeIsing Needs infinite-tree Ising Gibbs state and factor-of-iid defs 237 The three-quarter exponent for honeycomb self-avoiding walk. NONE - SMALL SimpleGraph.Walk.IsPath; Finset.card; Real.rpow HoneycombBridgeFiniteness(?); HoneycombBridgeMassSupport(?) Uniform n-step honeycomb SAW diameter n^{3/4+o(1)}; PR 4014 unrelated 238 Optimal logarithmic mixing of the Thorp shuffle. NONE - SMALL Equiv.Perm; PMF; Fin ThorpRouting; ThorpFirstReciprocal; ThorpRemaining; ThorpWeightedCompatibility Local Thorp shuffle and TV mixing time; t_mix = Θ(d) 239 Sharp singularity rates for symmetric random sign matrices. NONE - YES Matrix.det; Matrix.IsSymm; Finset.card; Filter.Tendsto - Count symmetric ±1 matrices with det 0; nth root → 1/2 240 Shelah's eventual categoricity and the prescribed-threshold obstruction. NONE - SMALL FirstOrder.Language.Structure; Cardinal; FirstOrder.Language.Embedding CHObstruction Needs local AEC definition and categoricity; then eventual categoricity 241 Rigidity of the Turing degrees. NONE - YES TuringDegree; TuringDegree.instPartialOrder; OrderIso DegreeRigidity ∀ f : TuringDegree ≃o TuringDegree, f = id 242 Single-fold Diophantine representations and undecidability under an at-most-one-solution promise. NONE - SMALL Dioph; REPred; MvPolynomial SingleFold Local single-fold Diophantine representation def; every r.e. set has one 243 Separating choiceless counting from polynomial time and witnessed choice. NONE - NO - ChoicelessPolynomialTime; WitnessedChoice Needs CPT+C logic formalism 244 The Partition Principle does not imply Choice. NONE - NO - PartitionPrinciple Needs ZF models/symmetric extensions 245 Weak normalization implies strong normalization in pure type systems. NONE - SMALL Relation.ReflTransGen; Acc; WellFounded TypeSystemNormalization Local PTS syntax, β-reduction, typing; WN → SN 246 Cannon's conjecture. NONE - NO - CannonGeometricAction Gromov boundary missing (FCForMathlib has hyperbolic groups only) 247 An infinite finitely presented residually finite 2-group and a finitely presented nil algebra. PR 7412 YES Group.IsFinitelyPresented; ResiduallyFinite; IsOfFinOrder; Infinite PeriodicGroup ∃ infinite f.p. residually finite group, every element order 2^k 248 Thompson's group F is nonamenable. NONE - SMALL PresentedGroup; Group.FG; MeasureTheory FoelnerFilter ThompsonNonamenability Local Thompson F presentation and Følner amenability def; ¬amenable 249 A finitely generated Eilenberg–Ganea counterexample. NONE - NO - EilenbergGanea Needs geometric dimension (classifying spaces) 250 Boone–Higman embeddings with higher finiteness. NONE - SMALL IsSimpleGroup; Group.IsFinitelyPresented; ComputablePred BooneHigman; UniversalFInfinity Local decidable-word-problem def; iff embeds in f.p. simple group 251 Amenability, unitarizability, and strong Ulam stability. NONE - SMALL ContinuousLinearMap; InnerProductSpace; unitary Dixmier; DixmierAllDiscrete Local amenability and uniformly bounded rep defs; unitarizable iff amenable 252 A torsion-free hyperbolic group that is neither residually finite nor linear over any field. MAIN FormalConjectures/Other/HyperbolicGroupsResiduallyFinite.lean YES ResiduallyFinite; Monoid.IsTorsionFree; (FCForMathlib IsHyperbolic) TorsionFreeHyperbolic hyperbolic_residuallyFinite stated; non-linearity part not stated 253 An infinite finitely presented simple amenable group. NONE - SMALL IsSimpleGroup; Group.IsFinitelyPresented; Infinite; FoelnerFilter SimpleAmenable Needs local group amenability def 254 Classifying spaces and geometric obstructions for Artin groups. NONE - SMALL CoxeterMatrix, PresentedGroup, Subgroup.closure ArtinParabolicIntersections, ArtinCAT0 Parabolic intersection stateable via Artin presentation; K(pi,1)/CAT(0) parts NO 255 Quasi-isometric recognition of virtually polycyclic groups. NONE - SMALL Group.FG, IsSolvable, Subgroup.FiniteIndex PolycyclicRecognition Needs local polycyclic and word-metric quasi-isometry definitions 256 Nonsingular systems of equations over arbitrary groups. NONE - YES Monoid.Coprod, Subgroup.normalClosure, FreeGroup Kervaire Kervaire: G nontrivial implies (G * Z)/<> nontrivial; Howie needs exponent sums 257 A hyperbolic group without a geometric CAT(0) action. NONE - NO - HyperbolicObstruction Needs Gromov hyperbolic groups, CAT(0) spaces, geometric actions 258 Gersten’s conjecture and virtual compact specialness of one-relator groups. NONE - NO PresentedGroup - Needs word-hyperbolicity and virtual compact specialness 259 A group without fixed price. NONE - NO - - Needs cost of pmp equivalence relations 260 Spacetime Penrose inequalities: enclosing area, charge, rotation, and anti-de Sitter extensions. NONE - NO - - Needs Lorentzian initial data, ADM mass, trapped surfaces 261 Localization and delocalization in the Anderson model. NONE - NO - PlanarAndersonSpectrum Needs random Schroedinger operators and spectral type decomposition 262 Sharp finite-matrix Lieb–Thirring inequalities and all equality cases. NONE - NO - LiebThirring Needs matrix Schroedinger operators on L2(R) and their eigenvalue sums 263 The ionization and generalized ionization conjectures. NONE - NO - CoulombIonization, CoulombRadii Needs many-body Coulomb Hamiltonians and ground-state binding 264 Strong cosmic censorship near two-ended Kerr data. NONE - NO - - Needs Einstein vacuum evolution and extension theory 265 Area laws and tensor networks for two-dimensional gapped systems. NONE - NO - - Needs local Hamiltonians, entanglement entropy, PEPS on lattices 266 Exactly three mutually unbiased bases in dimension six. MAIN FormalConjectures/OpenQuantumProblems/13.lean YES Matrix.unitaryGroup, Complex.abs MUBSix mutuallyUnbiasedBases_dim6 states max MUB count in dimension 6 267 Positive-temperature Bose–Einstein condensation and exact quantum depletion. NONE - NO - HardSphere Needs canonical Gibbs states and thermodynamic limits for hard spheres 268 The spin-one Haldane gap. NONE - SMALL Matrix.IsHermitian.eigenvalues, Matrix.kroneckerMap - Finite spin-1 Heisenberg ring Hamiltonian; gap bounded below uniformly in L 269 Uniform Laughlin gap and stability under bounded scalar disorder. NONE - NO - LaughlinGap Needs lowest Landau level on sphere and V1 pseudopotential setup 270 Threshold and positive-energy bound states of the BFSS matrix model. NONE - NO - - Needs BFSS supersymmetric matrix quantum mechanics on L2 271 Bloch's law, its lattice correction, and the spherical magnetization law. NONE - NO - Heisenberg Needs quantum Gibbs states, thermodynamic limit, magnetization asymptotics 272 Entanglement without distillable secret key. NONE - SMALL Matrix.kroneckerMap, Matrix.PosSemidef DimensionTenPair PPT-square disproof stateable via Choi matrices; distillable key part NO 273 The entropy photon-number inequality. NONE - NO - EntropyPhotonNumber Needs bosonic Fock space, beam splitters, von Neumann entropy 274 Parity is not in QAC0. NONE - SMALL Matrix.unitaryGroup, Matrix.kroneckerMap QACParity Needs local QAC0 circuit model definitions on qubit tensor powers 275 QMA-hardness of continuum Coulomb energy. NONE - NO - ContinuumCoulombHardness Needs QMA and polynomial-time reductions 276 Classical capacity of generalized amplitude damping. NONE - NO - AmplitudeDamping Needs quantum channel capacity and Holevo quantities 277 Threshold repetition for entangled games. NONE - SMALL Matrix.PosSemidef, Matrix.kroneckerMap EntangledGames Needs local nonlocal game, entangled value, k-fold repetition definitions 278 Failure of Kohn–Sham ensemble representation. NONE - NO - CoulombCounterexample Needs Coulomb ground-state densities and Kohn-Sham potentials 279 Exact quantum factoring over a fixed finite gate set. NONE - NO - ExactQuantumFactoring Needs uniform quantum circuit families and polynomial-time complexity 280 Unitary vertex operator algebras and conformal nets. NONE - NO - VertexAlgebraNet Needs vertex operator algebras and conformal nets 281 QAOA attains the SK optimum in the thermodynamic-first limit. NONE - NO - - Needs SK model limits and QAOA energy asymptotics 282 From scale symmetry to local conformal symmetry in four-dimensional QFT. NONE - NO - - Needs algebraic QFT local nets and stress tensors 283 Polynomial-time unitary synthesis from a Boolean oracle. NONE - NO - - Needs uniform quantum oracle circuits and diamond norm 284 The optimal quartic separation between randomized and quantum queries. NONE - NO - - Needs randomized and quantum query complexity 285 Counterexamples to Baum–Connes and Kadison–Kaplansky. NONE - NO - - Needs Baum-Connes assembly and reduced group C*-algebras 286 Rigidity and arithmetic of lattice von Neumann algebras. NONE - NO - - Needs group von Neumann algebras and bimodule theory 287 Isomorphism of the free group factors. NONE - NO - InterpolatedFactors Needs free group factors L(F_n) 288 Kadison's similarity conjecture. NONE - YES CStarAlgebra, ContinuousLinearMap, StarAlgHom, Units KadisonSimilarity Bounded unital hom A -> B(H) is similar to a star-homomorphism 289 Strong Kadison–Kastler stability and its spatial boundaries. NONE - YES VonNeumannAlgebra, Metric.hausdorffDist, unitary StrongKadisonKastler Close vN algebras on H are conjugate by unitary near identity 290 Relative bicentralizers and modular spectral recovery. NONE - NO - - Needs type III1 factors, bicentralizers, core c(M) 291 Cuntz comparison, nuclear dimension, and equivariant Jiang–Su stability. NONE - NO - - Needs Jiang-Su algebra, nuclear dimension, strict comparison 292 Kirchberg's $`\mathcal O_2`$ norm-ultrapower embedding problem. NONE - NO - NuclearUltrapower Needs full group C*-algebras, norm ultrapowers, O_2 293 Invariant projections, hyperinvariant subspaces, and transitive algebras. NONE - YES spectralRadius, Submodule, Commute, VonNeumannAlgebra HyperinvariantSubspaces Exists quasinilpotent T with no nontrivial closed hyperinvariant subspace 294 Kaplansky's quasitrace conjecture and failure of tensor-product stable finiteness. NONE - NO - KaplanskyQuasitrace, KaplanskyStableFiniteness Needs 2-quasitraces and minimal tensor products of C*-algebras 295 The Kadison–Ringrose cohomology conjecture. NONE - SMALL VonNeumannAlgebra, ContinuousMultilinearMap KadisonRingrose Define bounded Hochschild cochains and coboundary; higher cohomology vanishes 296 The generator problem for finite factors. NONE - SMALL VonNeumannAlgebra, Set.centralizer FactorGeneration, RelativeGeneration Needs II1 factor, normal trace, separable predual local definitions 297 A ZFC counterexample to Naimark's problem. NONE - SMALL CStarAlgebra, StarAlgHom, ContinuousLinearMap Naimark Define irreducible reps and unitary equivalence; existence of counterexample 298 Two notions of free entropy differ even when both are finite. NONE - NO - FiniteEntropySeparation Needs Voiculescu microstates and non-microstates free entropy 299 The Kirchberg–Rørdam character criterion and infinite tensor-power Jiang–Su stability. NONE - NO - CharacterCriterion Needs Jiang-Su algebra, central sequence algebras, ultrapowers 300 Approximation and quadratic strong-operator paving. NONE - NO VonNeumannAlgebra - Needs masas, conditional expectations, SOT paving framework 301 Trace cones and Razak–Jacelon stabilization. NONE - NO - - Needs Razak-Jacelon algebra and trace-cone classification 302 Radius of comparison equals half the mean dimension. NONE - NO - - Needs radius of comparison and mean dimension 303 Weak pure infiniteness and Cuntz-algebra absorption. NONE - NO - ExactInfiniteness, IndividualStrongInfiniteness Needs pure infiniteness, Cuntz comparison, O_infinity absorption 304 The Hilbert–Smith conjecture in every dimension. MAIN FormalConjectures/HilbertProblems/5.lean YES IsTopologicalGroup, LocallyCompactSpace, ChartedSpace, LieGroup - hilbert_smith_conjecture; PR #7043 adds second-countable variants 305 Four-dimensional disk embedding and Wall's conjecture. NONE - NO - - Needs Freedman-Quinn good groups and Poincare duality groups 306 The purely cosmetic surgery conjecture. NONE - NO - - Needs knots in S3 and Dehn surgery 307 Failure of rational injectivity for maximal coarse assembly. NONE - NO - CoarseAssembly Needs Roe algebras and coarse assembly maps 308 Finite Smith–Toda complexes at every height. NONE - NO - - Needs spectra and Brown-Peterson homology 309 The Kervaire invariant problem at the prime three. NONE - NO - - Needs Adams spectral sequence and stable stems 310 Quillen's conjecture in rational homology. NONE - SMALL nerve, alternatingFaceMapComplex, Sylow - Order complex rational homology of elementary abelian p-subgroup poset 311 The Hovey–Strickland and Chai conjectures. NONE - NO - - Needs Lubin-Tate rings, K(n)-local spectra, Balmer spectrum 312 The Grothendieck homotopy hypothesis. NONE - NO - - Needs Grothendieck coherators and infinity-groupoid homotopy theory 313 Finite generation for the $`K(n)`$-local sphere. NONE - NO - - Needs K(n)-local sphere and its homotopy groups 314 Cyclic length and chromatic fixed-point loss. NONE - NO - - Needs equivariant spectra, geometric fixed points, chromatic height 315 The four-dimensional Singer conjecture. NONE - NO - - Needs L2-Betti numbers and aspherical topological 4-manifolds 316 Curtis’s conjecture. NONE - NO - - Stable homotopy groups of spheres, Hurewicz map to H_*(HF2) absent 317 Thomason model structures in all strict higher dimensions. NONE - NO - ThomasonModelStructures Model categories and strict n-categories with Thomason structures absent 318 Chromatic splitting: filtrations and counterexamples. NONE - NO - - Spectra, K(n)-localization, chromatic homotopy absent 319 Counterexamples to finite generation at chromatic height two. NONE - NO - - Spectra, BP<2>, fp-type absent 320 Nonhomeomorphic closed aspherical four-manifolds. NONE - YES HomotopyGroup; ContinuousMap.HomotopyEquiv; ChartedSpace; Homeomorph - Core: exist closed aspherical top. 4-manifolds, homotopy equivalent, not homeomorphic 321 A counterexample to Wall's finite D(2) problem. NONE - NO - - Needs CW complexes plus cohomology with local coefficients 322 Tingley’s sphere-isometry problem. PR #6988 YES Metric.sphere; IsometryEquiv; LinearIsometryEquiv; NormedSpace TingleySphereIsometry Surjective sphere isometry extends to linear isometry 323 Independence of the separable quotient problem. PR #7001 YES Cardinal.continuum; SeparableSpace; NormedSpace; Submodule SeparableQuotientNegative CH half stateable; consistency/independence half needs set-theory metatheory 324 Lipschitz equivalent Banach spaces need not be linearly isomorphic. PR #6999 YES LipschitzWith; ContinuousLinearEquiv; SeparableSpace LipschitzEquivalence Exist separable Banach X,Y bi-Lipschitz but not linearly isomorphic 325 The complete Crouzeix conjecture. PR #5247 SMALL InnerProductSpace; ContinuousLinearMap; Matrix; Polynomial.aeval CompleteCrouzeix; CrouzeixHilbert; DirectCrouzeix; HilbertCrouzeix; StructuralCrouzeix Define numerical range, matrix-valued polynomial calculus; PR states scalar Crouzeix 326 The cotype–cotype conjecture under the approximation property. PR #7153 SMALL NormedSpace; ContinuousLinearMap; StrongDual Cotype Define Rademacher type/cotype, K-convexity, approximation property 327 Markov type characterizes superreflexivity. NONE - SMALL UniformConvexSpace; NormedSpace; Matrix MarkovType Define Markov type via stationary reversible chains; renorming to uniformly convex 328 Nonexpansive fixed points in reflexive Banach spaces. PR #6989 YES NormedSpace.inclusionInDoubleDual; LipschitzWith; Convex; Bornology.IsBounded ReflexiveFixedPoints Nonexpansive self-map of closed bounded convex set in reflexive space has fixed point 329 A counterexample to metric-entropy duality. PR #7152 SMALL Metric.coveringNumber; polar; Convex MetricEntropyDuality Define translate-covering number N(K,L); negate Pietsch duality 330 A uniformly discrete counterexample to bounded approximation in Lipschitz-free spaces. NONE - NO - DiscreteLipschitzFree Lipschitz-free space construction absent 331 Reflexive midpoint convexity and diamond distortion. NONE - SMALL NormedSpace.inclusionInDoubleDual; SimpleGraph; LipschitzWith DiamondDistortion; MidpointLenses Define asymptotic (midpoint) uniform convexity moduli, diamond graphs, distortion 332 Metric Markov cotype of ℓ1 and Hilbert-space Lipschitz extension. NONE - YES lp; LipschitzWith; InnerProductSpace; LipschitzWith.extend_real LipschitzHilbert Core: Lipschitz maps from subsets of Hilbert into l1 extend with constant C 333 Smooth isometric immersions of surfaces into ℝ4. PR #7211 SMALL Bundle.RiemannianMetric; IsRiemannianManifold; mfderiv SurfaceImmersion Closed Riemannian surface isometrically immerses in R^4; PR covers orientable 334 A smooth surface metric with no local isometric immersion in ℝ3. NONE - SMALL ContDiff; fderiv; Matrix.PosDef IsometricImmersion Smooth PD matrix field g on (-1,1)^2; no local f with Df^T Df = g 335 Gromov’s integral scalar-curvature bound for simplicial volume. NONE - NO - - Scalar curvature, simplicial volume absent 336 Spectral scalar curvature, Urysohn width, and macroscopic dimension. NONE - NO - - Scalar curvature, Urysohn width, macroscopic dimension absent 337 Sharp Cartan–Hadamard isoperimetry and rigidity. NONE - NO - SharpCAT0Filling Sectional curvature, finite-perimeter sets, integral currents absent 338 Yau's uniformization conjecture. NONE - NO - - Kähler manifolds, bisectional curvature absent 339 Katok's entropy rigidity conjecture. NONE - NO - - Sectional curvature, geodesic flow entropy, Liouville measure absent 340 A counterexample to the nearby Lagrangian conjecture. NONE - NO - - Cotangent bundles symplectic structure, Hamiltonian isotopy absent 341 Donaldson's hypersymplectic deformation conjecture. NONE - NO - - Differential forms on manifolds, de Rham cohomology absent 342 Donaldson's tamed-to-compatible conjecture. NONE - NO - TamingCompatibility Almost complex structures, symplectic forms on manifolds absent 343 Symplectic ball packing in higher dimensions. NONE - SMALL EuclideanSpace; ContDiff; fderiv; Matrix.J BallPacking; BallPackingNecessity Define standard symplectic form and symplectic embeddings of balls in R^2n 344 The metric Blaschke conjecture. NONE - NO - - Injectivity radius, CROSS classification absent 345 Infinitely many closed geodesics on Riemannian spheres and closed three-manifolds. NONE - NO - - Closed geodesics on Riemannian manifolds absent 346 Sharp singular-set bounds for stationary integral varifolds. NONE - NO - - Varifolds, first variation absent 347 Counterexamples to stable-Morse and strong Arnold fixed-point bounds. NONE - NO - ArnoldCounterexample Hamiltonian diffeomorphisms on Kähler manifolds absent 348 Nonnegative-curvature Einstein classification and an L2 topological gap. NONE - NO - EinsteinFour Einstein metrics, sectional curvature absent 349 The Solomon–Yau least-volume conjecture. NONE - NO - - Minimal immersions in spheres, Riemannian volume absent 350 Yau’s nodal bounds: surfaces and higher dimensions. NONE - NO - NodalLength; SmoothYau; YauCounterexample Laplace–Beltrami eigenfunctions on closed manifolds absent 351 Scalar curvature and finite-time Ricci-flow singularities. NONE - NO - - Ricci flow absent 352 A finite-time singularity of Calabi flow. NONE - NO - - Calabi flow, Kähler metrics on CP^n absent 353 Affine Bernstein rigidity through dimension nine and a smooth dimension-ten counterexample. NONE - SMALL ContDiff; fderiv; Matrix.det; ConvexOn AffineBernstein Entire convex graphs solving affine-maximal 4th-order PDE are quadratic, n<=9 354 The isoperimetric profile of the cubic three-torus. NONE - NO - CubicTorus; ThreeTorus Finite-perimeter sets on flat torus absent 355 Unique tangent flows at the first surface singularity. NONE - NO - - Mean curvature flow, tangent flows absent 356 Gigli’s characterization of Alexandrov curvature. NONE - NO - - RCD spaces, Alexandrov curvature absent 357 Bi-Lipschitz coordinates at every regular RCD point. NONE - NO - - RCD spaces, tangent cones absent 358 A three-manifold without conjugate points or nonpositive curvature. NONE - NO - ConjugatePoints Conjugate points, sectional curvature absent 359 Negative Kähler curvature without bounded holomorphic coordinates. NONE - NO - PinchedKahler Complete Kähler metrics, holomorphic sectional curvature absent 360 Weak MTW curvature gives convexity and regular optimal transport. NONE - NO - WeakMTWGlobalSupport MTW tensor, cut locus, Riemannian OT absent 361 Failure of integer-degree harmonic dimension comparison. NONE - SMALL laplacian; ContDiff; Matrix; FiniteDimensional.finrank HarmonicGrowth Metric on R^3 as matrix field; define Ricci, Laplace–Beltrami in coordinates 362 Global smoothness for relativistic Vlasov–Maxwell. NONE - SMALL fderiv; ContDiff; EuclideanSpace; HasCompactSupport VlasovMaxwell Classical PDE system statable with smooth functions, like Millennium/NavierStokes 363 Nonuniqueness with local conservation for the hard-sphere Boltzmann equation. NONE - NO - BoltzmannNonuniqueness Entropy solutions of Boltzmann equation, L^1 weak theory absent 364 Kinetic limits and fluctuations over the Boltzmann lifespan. NONE - NO - - Particle-system kinetic limits, BBGKY absent 365 Joint metric and connection recovery from one boundary patch. NONE - NO - Conductivity Weak elliptic solutions, Dirichlet-to-Neumann maps on bundles absent 366 The planar Mumford–Shah regularity conjecture and local weak-L4 gradient bounds. NONE - NO - - SBV, Mumford–Shah functional absent 367 The critical dimension for the one-phase Bernoulli problem. NONE - NO - BernoulliNonflatInSeven Sobolev H^1 minimizers of Alt–Caffarelli energy absent 368 The three-dimensional Ball–Evans approximation problem. NONE - NO - - Sobolev spaces W^{1,p} absent 369 The hot spots conjecture for simply connected planar domains. NONE - SMALL laplacian; ContDiff; IsSimplyConnected; IsMaxOn HotSpots Classical Neumann eigenfunctions on smooth planar domains; define normal derivative 370 The Lane–Emden and Hénon–Lane–Emden conjectures. NONE - YES InnerProductSpace.laplacian; ContDiffOn; EuclideanSpace HenonEmden No positive entire classical solution of the Hénon–Lane–Emden system 371 Stable blowup for the defocusing Schrödinger equation. NONE - NO - DefocusingNLS Sobolev H^k on torus, NLS well-posedness absent 372 Global uniqueness in smooth isotropic elasticity. NONE - SMALL fderiv; ContDiff; EuclideanSpace ElasticityUniqueness Cauchy data of smooth Lamé solutions; define smooth domain, normal 373 Nonattainment of the three-marginal Coulomb Monge problem. NONE - SMALL MeasureTheory.Measure.map; ProbabilityMeasure; lintegral CoulombCounterexample Define multi-marginal couplings, Monge/Kantorovich costs; Monge infimum not attained 374 Sharp one-third stability of Brenier maps. NONE - SMALL MeasureTheory.Measure.map; ConvexOn; gradient Brenier Define W2 via couplings, Brenier map; Hölder-1/3 stability, sharp 375 De Giorgi's conjecture in dimension eight. MAIN FormalConjectures/Paper/DeGiorgi.lean YES InnerProductSpace.laplacian; ContDiff; EuclideanSpace - DeGiorgi_le_eight / dimension-8 case already stated 376 Universal computation in forced Navier–Stokes flows. NONE - NO - ForcedNavierStokesComputation Forced NS solutions plus Turing compilation; too heavy 377 Interior $`C^{1,\alpha}`$ regularity for infinity-harmonic functions. PR #7446 SMALL ContDiff; fderiv; HolderWith - Define viscosity infinity-harmonic functions; C^{1,alpha} interior estimate