{"status": {"release": "openai-math", "upstream_repo": "https://github.com/openai/math", "upstream_commit": "adc7f1241b42e322a6451854ab7e4b4c146bf78a", "generated": "2026-10-08T23:07:21Z", "review_status": "independent third-party review, pre-referee", "families_total": 372, "families_with_lean": 235, "families_without_lean": 137, "challenges": 405, "verdicts": {"full": 144, "partial": 60, "weaker-statement": 19, "supporting-only": 12}, "non_full": 91, "built_locally": ["AsymptoticallyMinimalLittlewood", "BorsukNine", "CannonGeometricAction", "CirculantHadamard", "CoboundaryExpanders", "Conductivity", "ContingencyTables", "DimensionTenChannel", "DimensionTenPair", "DirichletSevenEighths", "ElasticityUniqueness", "EntangledGames", "ErdosReciprocal", "EuclideanFiveColor", "EvenBarker", "ExactFourier", "GeneralizedStarHeight", "GotsmanLinial", "InterpolatedFactors", "KServer", "ListHadwiger", "LittlewoodFiniteFlatness", "MahlerConjecture", "MassAction", "MatchingAffineLift", 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"ThreeStateTreeClauses", "ThreeTorus", "TingleySphereIsometry", "TorsionFreeHyperbolic", "TorsionFreeZeroDivisors", "TraceReconstruction", "TreeEdit", "TriangleFace", "TriangleRemoval", "TriangularCovering", "TriangularEnergy", "TriangularGaussian", "TriangularHilbert", "TruffetCounterexample", "TwoWayComplementation", "TwoWayDeterminization", "TypeSystemNormalization", "UniformCommutator", "UniformGamma", "UniformKServer", "UniformSparsestCut", "UniqueGamesTheorem", "UniversalFInfinity", "VariableWL", "VelocityHemisphere", "VertexAlgebraNet", "VertexCover", "VlasovMaxwell", "WLIdentification", "WeakHessian", "WeakMTWGlobalSupport", "WeightedSweepMoments", "WeisfeilerLeman", "WitnessedChoice", "YauCounterexample"], "comparator_sandboxed_pass": ["AbhyankarSathaye", "AffineBernstein", "AlgorithmicThinTrees", "AmplitudeDamping", "ArnoldCounterexample", "ArtinCAT0", "ArtinParabolicIntersections", "AsymptoticallyMinimalLittlewood", "AtomicGaussian", "AuslanderReiten", 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"CoulombCounterexample", "CoulombIonization", "CoulombRadii", "CourtadeKumar", "CoveringDensity", "CriticalPercolation", "CriticalSK", "CriticalSKMixing", "CriticalStripMass", "CriticalZ3", "CrouzeixHilbert", "CycleDecomposition", "CylinderCovering", "DaugavetModuli", "DegreeRigidity", "DeligneDrinfeld", "DepthFive", "DepthThree", "DiamondDistortion", "DilutedSpin", "DimensionTenChannel", "DimensionTenPair", "DirectCrouzeix", "DirectionalBallisticity", "DirectionalWalk", "DiscreteLipschitzFree", "DiskMaximal", "Dixmier", "DixmierAllDiscrete", "DoublingHilbert", "DuttaDomain", "DyadicAvoidance", "EditDistance", "EgyptianFractions", "EilenbergGanea", "ElasticityUniqueness", "ElementaryPositivity", "EntangledGames", "EntropyPhotonNumber", "EuclideanFiveColor", "EuclideanRamsey", "EuclideanRamseyCircle", "EuclideanRamseyNine", "EuclideanRamseyQuadratic", "EuclideanRamseySpherical", "EuclideanRamseyTransitive", "EvenBarker", "ExactFourier", "ExactQuantumFactoring", "FKCRT", "FactorGeneration", "FillingCoefficient", "FiniteCircle", "FiniteCongruenceGraph", "FiniteEntropySeparation", "FiniteFactor", "FinitisticAsymmetry", "FixedClauseThreshold", "ForcedNavierStokesComputation", "ForestSpace", "FormulaHitting", "FoulkesHowe", "FourRowPermanent", "FreeIsing", "FreeUniformSpanningForest", "GammaPassage", "GaussianFiniteEntropy", "GaussianInformation", "GaussianMoat", "GaussianPropeller", "GaussianReplacement", "GeneralMahler", "GeneralizedStarHeight", "GotsmanLinial", "GrahamSpherical", "GrothendieckElementaryExpansion", "GroupRingDeterminant", "HadamardCubeFiber", "HalvingLines", "HardSphere", "HarmonicArtin", "HarmonicGrowth", "HeilbronnTriangle", "Heisenberg", "HenonEmden", "HilbertCrouzeix", "HoneycombBridgeFiniteness", "HoneycombFreeEnergy", "HotSpots", "HyperbolicCones", "HyperbolicObstruction", "HyperinvariantSubspaces", "IndependentProducts", "IndependentSets", "IndividualStrongInfiniteness", "InfiniteMatroid", "InfiniteMatroidCorollaries", 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"SKFullSupport", "SKHighTemperature", "SKRatio", "SKValue", "SLELowerPositivity", "Saxl", "SecondKahnKalai", "SelfSimilar", "SelfSimilarCorollaries", "SensitivitySeparation", "SeparableQuotientNegative", "SeymourSecondNeighborhood", "SharpCAT0Filling", "SharpLogRamsey", "SharpThreshold", "ShortEgyptianFractions", "SidorenkoCounterexample", "SignedFiniteBand", "SignedSweepMoment", "SimpleAmenable", "SimpleOvergroups", "SingleFold", "SingleLatticeCovering", "SingletonLoopMatching", "SmoothInitialForm", "SmoothObstruction", "SmoothYau", "SnakyCertificate", "SnakyConditional", "SoftChannel204", "SolenoidalSheetPrograms", "SphericalField", "SpinAngle", "SplitTangentIntegrability", "SquareDifference", "SquareRootDegree", "StandardMapComponents", "StandardMapEntropy", "StandardMapLyapunov", "SteinitzBergstrom", "StrictMeans", "StrongThinTree", "StronglyRayleighDPP", "StructuralCrouzeix", "SubpolynomialLp", "SubsphereCurrent", "Superstring", "SwitchChain", "SymmetricDomains", "SymmetricMahlerEquality", "SymmetricPolar", "Tachikawa", "TalagrandDiscreteConvexity", "TalagrandExpectationThreshold", "TamingCompatibility", "ThomasonModelStructures", "ThompsonNonamenability", "ThorpFirstReciprocal", "ThorpRemaining", "ThorpWeightedCompatibility", "ThreeMachine", "ThreeStateSupercritical", "ThreeStateTreeClauses", "ThreeTorus", "TingleySphereIsometry", "TorsionFreeHyperbolic", "TorsionFreeZeroDivisors", "TraceReconstruction", "TreeEdit", "TriangleFace", "TriangleRemoval", "TriangularCovering", "TriangularEnergy", "TriangularGaussian", "TriangularHilbert", "TruffetCounterexample", "TwoWayComplementation", "TwoWayDeterminization", "TypeSystemNormalization", "UniformCommutator", "UniformGamma", "UniformKServer", "UniformSparsestCut", "UniversalFInfinity", "VariableWL", "VelocityHemisphere", "VertexAlgebraNet", "VertexCover", "VlasovMaxwell", "WLIdentification", "WeakHessian", "WeakMTWGlobalSupport", "WeightedSweepMoments", "WeisfeilerLeman", "WitnessedChoice", "YauCounterexample"], "comparator_running": [], "build_in_progress": ["ContinuumCoulombHardness"]}, "families": [{"family": "003", "title": "The quasi-Riemann hypothesis", "subject": "Number theory", "headline": "Proves that every Dirichlet $L$-function, including $\\zeta(s)$, is zero-free in $\\Re s>7/8$, resolving the quasi-Riemann hypothesis. The same half-plane is zero-free for every finite-order Hecke $L$-function over $\\mathbb Q(\\sqrt{-3})$. A companion gives a different proof of the zero-free half-plane $\\Re s>11/12$.", "verdict": "full", "challenges": ["QuasiRiemannHypothesis", "DirichletSevenEighths", "HeckeSevenEighths", "SiegelZeros"], "review_note": "No gap for zeta/Dirichlet L (Mathlib objects). Hecke 7/8 clause rests on a custom WeakFEPair-defined L-function (`LFunction := continuedLattice χ s / 6`) whose identification with the Dirichlet series is not in the statement; 11/12 companion not stated separately (implied).", "definitions_to_check": ["Hecke part: `def LFunction (χ : Character) (s : ℂ) : ℂ := continuedLattice χ s / 6` with `continuedLattice χ = HeckeTheta.latticeL (coefficients χ)` (custom theta/WeakFEPair construction; custom `structure Character` with `modulus`, `residue : MulChar (O ⧸ modulus) ℂ`, `unit_trivial`, `period`); no lemma linking it to Σ χ(a) N(a)^{-s}"], "external_packages": ["PrimeNumberTheoremAnd", "RellichKondrachov"], "cone_lines_max": 486525, "machine_check": "comparator-pass", "lab_scope_note": "The quasi-Riemann hypothesis asks for a fixed zero-free half-plane $\\Re s>\\theta$ with $\\theta<1$. The formalization gives $\\theta=7/8$ for the Riemann zeta function and every Dirichlet $L$-function, uniformly over all positive moduli and all characters. It also establishes the same bound for finite-order Hecke $L$-functions over $\\mathbb Q(\\sqrt{-3})$.\n\nThe principal-character poles at $s=1$ are excluded in the Dirichlet and Hecke statements. The paper's later applications are not included.\n\nThe formalized result gives a uniform logarithmic exclusion region for Landau–Siegel zeros. There is one constant $c>0$ such that every primitive nonprincipal real Dirichlet character of conductor $q\\ge3$ and every real zero $0<\\beta<1$ of its $L$-function satisfy $1-\\beta\\ge c/\\log q$.\n\nBoth character parities are included. No explicit value of $c$ is given. This excludes real zeros in $1-c/\\log q<\\beta<1$, but does not rule out real zeros elsewhere in $(0,1)$.", "lab_docs_url": "https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/docs/003.md", "overview_entry": "family 003 in overview.pdf / CONTENTS.md", "papers": [{"dir": "The-Quasi-Riemann-Hypothesis-September-30-2026", "title": "The Quasi-Riemann Hypothesis: A Zero-Free Half-Plane $`\\Re s\\gt 7/8`$", "url": "https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-Quasi-Riemann-Hypothesis-September-30-2026"}, {"dir": "Uniform-exclusion-of-Landau-Siegel-zeros-October-1-2026", "title": "Uniform exclusion of Landau–Siegel zeros", "url": "https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Uniform-exclusion-of-Landau-Siegel-zeros-October-1-2026"}], "audit_part": 1, "second_reader": "", "referee_verdict": ""}, {"family": "005", "title": "Irrationality of Catalan's constant", "subject": "Number theory", "headline": "Proves that Catalan's constant $G=\\sum_{j\\ge0}(-1)^j/(2j+1)^2$ is irrational.", "verdict": "full", "challenges": ["Catalan"], "review_note": "None; statement is the headline verbatim with Mathlib `Irrational`/`tsum`.", "definitions_to_check": [], "external_packages": ["PrimeNumberTheoremAnd"], "cone_lines_max": 574902, "machine_check": "none", "lab_scope_note": "The formalization proves that Catalan's constant $G=\\sum_{j=0}^{\\infty}(-1)^j/(2j+1)^2$ is irrational. This is the mathematical assertion in the paper's title.", "lab_docs_url": "https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/docs/005.md", "overview_entry": "family 005 in overview.pdf / CONTENTS.md", "papers": [{"dir": "Catalans-constant-is-irrational-September-24-2026", "title": "Catalan's constant is irrational", "url": "https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Catalans-constant-is-irrational-September-24-2026"}], "audit_part": 1, "second_reader": "", "referee_verdict": ""}, {"family": "007", "title": "Two-point Chowla and the corrected binary Elliott conjecture", "subject": "Number theory", "headline": "Proves the ordinary two-point Chowla conjecture, with a bound $O(X/(\\log X)^c)$ for Liouville correlation sums along fixed nonproportional affine forms, where $c>0$ is absolute. More generally, proves the binary corrected Elliott conjecture for complex multiplicative functions bounded by one when one factor is uniformly nonpretentious against each fixed Dirichlet character times $n^{it}$ for $|t|\\le X$.", "verdict": "full", "challenges": ["OrdinaryElliott", "OrdinaryTwoPointCorrelations"], "review_note": "None found; affine forms take natural-number coefficients (a₁,a₂>0, b₁,b₂≥0), which loses nothing up to re-indexing; custom non-pretentiousness definition checked against the standard one.", "definitions_to_check": ["`def UniformlyNonpretentious (f : ℕ → ℂ) : Prop := ∀ (q : ℕ), 0 < q → ∀ (χ : DirichletCharacter ℂ q) (K : ℝ), ∀ᶠ N : ℕ in atTop, ∀ t : ℝ, |t| ≤ (N : ℝ) → K ≤ squaredDistance f (characterTwist χ t) N` (custom but matches the standard definition; two files use different but equivalent encodings)"], "external_packages": ["PrimeNumberTheoremAnd", "StrongPNT"], "cone_lines_max": 153427, "machine_check": "comparator-sandboxed-pass", "lab_scope_note": "The formalization proves ordinary two-point cancellation for multiplicative functions. For the Liouville function on every fixed pair of nonproportional affine forms, the correlation sum up to $X$ is $O(X/(\\log X)^c)$ for an absolute $c>0$, with the implied constant depending on the forms.\n\nFor two one-bounded multiplicative functions, if at least one is uniformly nonpretentious against all Dirichlet-character twists with frequency $|t|\\le N$, then their shifted and nonproportional affine correlation sums divided by $N$ tend to zero. These are ordinary averages, with no logarithmic averaging.", "lab_docs_url": "https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/docs/007.md", "overview_entry": "family 007 in overview.pdf / CONTENTS.md", "papers": [{"dir": "Ordinary-two-point-correlations-of-multiplicative-functions-September-24-2026", "title": "Ordinary two-point correlations of multiplicative functions", "url": "https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Ordinary-two-point-correlations-of-multiplicative-functions-September-24-2026"}], "audit_part": 1, "second_reader": "", "referee_verdict": ""}, {"family": "008", "title": "The Deligne--Drinfeld conjecture", "subject": "Number theory", "headline": "Proves that the rational Grothendieck--Teichm\\\"uller Lie algebra, with the Ihara bracket, is freely generated by one element in each odd weight $3,5,7,\\ldots$, resolving the Deligne--Drinfeld conjecture.", "verdict": "full", "challenges": ["DeligneDrinfeld"], "review_note": "None for the headline; W is the graded (polynomial) solution space; the weight-completion clause is formal (coordinatewise on weight pieces).", "definitions_to_check": ["`instance : TopologicalSpace L := ⊥` and `instance : TopologicalSpace OddFree := ⊥` (discrete topologies, so the `Continuous` clauses on weight-completions are essentially formal); custom `Tangent` Lie algebra used to define `ihara` (checked: it is the dual-numbers construction giving the Ihara derivation)"], "external_packages": [], "cone_lines_max": 29733, "machine_check": "comparator-sandboxed-pass", "lab_scope_note": "The Deligne–Drinfeld conjecture predicts that the rational Grothendieck–Teichmüller Lie algebra is freely generated by one element in each odd weight $3,5,7,\\ldots$. The formalization establishes this for the rational solution space of antisymmetry, the three-term equation, and the four-strand pentagon with the Ihara bracket. It also gives the corresponding continuous isomorphism after completion by weight.\n\nThe regularized-transport proposition and graph-complex consequences are not included.", "lab_docs_url": "https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/docs/008.md", "overview_entry": "family 008 in overview.pdf / CONTENTS.md", "papers": [{"dir": "The-Deligne-Drinfeld-conjecture-September-23-2026", "title": "The Deligne-Drinfeld conjecture", "url": "https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-Deligne-Drinfeld-conjecture-September-23-2026"}], "audit_part": 1, "second_reader": "", "referee_verdict": ""}, {"family": "009", "title": "Bogomolov--Pop and Milnor $K$-theoretic reconstruction", "subject": "Number theory", "headline": "Reconstructs function fields of transcendence degree at least two over algebraically closed constants from $K^{\\mathrm M}_1/\\ell$, $K^{\\mathrm M}_2/\\ell$, and their product. These data recover the perfect closure and constants when $\\ell$ 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-$\\ell$ Galois data away from the characteristic.", "verdict": "partial", "challenges": ["BogomolovPopInjectivity"], "review_note": "Only injectivity half of Bogomolov-Pop reconstruction (Galois data side); no existence/surjectivity and no Milnor K-theoretic reconstruction or perfect-closure/constants recovery statement.", "definitions_to_check": ["`def ScalarStep ... ∃ u : (PadicInt ell)ˣ, ∀ x, Tendsto (fun n => (φ.1 x) ^ PadicInt.appr (u : PadicInt ell) n) atTop (nhds (χ.1 x))` (ℓ-adic unit scaling encoded via `appr` approximants; custom but plausible); `abelianLift := Quotient.out x` (choice-based section used in `bracket`, harmless)"], "external_packages": [], "cone_lines_max": 3339, "machine_check": "comparator-sandboxed-pass", "lab_scope_note": "The Bogomolov–Pop reconstruction conjecture concerns recovering a function field and its constants from pro-$\\ell$ abelian-by-central Galois data. The linked formalization proves injectivity of the reconstruction correspondence for function fields of transcendence degree at least two over algebraically closed fields of characteristic different from the prime $\\ell$.\n\nIf two isomorphisms of perfect closures preserving the constant fields induce the same Galois-data isomorphism modulo $\\ell$-adic unit scaling, then they agree modulo Frobenius. The selected statement proves this uniqueness; existence of a field isomorphism for every admissible Galois-data isomorphism is outside it.", "lab_docs_url": "https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/docs/009.md", "overview_entry": "family 009 in overview.pdf / CONTENTS.md", "papers": [{"dir": "The-Bogomolov-Pop-reconstruction-theorem-September-23-2026", "title": "The Bogomolov-Pop reconstruction theorem", "url": "https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-Bogomolov-Pop-reconstruction-theorem-September-23-2026"}], "audit_part": 1, "second_reader": "", "referee_verdict": ""}, {"family": "012", "title": "Independent largest prime factors of consecutive integers", "subject": "Number theory", "headline": "Resolves the Erd\\H{o}s--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)= 5 only; the primitive quartic (Picard packets) and primitive sextic (maximal-order ideal-class packets) results of the summary are not stated.", "definitions_to_check": ["Whole packet-measure apparatus is custom (`packetOrbits`, `orbitBasepoint := Classical.epsilon ...`, `orbitalHaar := if h : HasAddFundamentalDomain ... then ... else 0`, `latticePoint := if h : ∃ g, ... then ⟦h.choose⟧ else ⟦1⟧`): junk `else` branches are unreachable for genuine torus packets, but the encoding of 'complete volume-weighted packet' is not checkable against a standard Mathlib notion"], "external_packages": ["ClassFieldTheory"], "cone_lines_max": 130452, "machine_check": "none", "lab_scope_note": "The formalization proves equidistribution of volume-weighted torus packets for totally real number fields of every fixed prime degree at least five. For any sequence of full lattices whose multiplier-order discriminants tend to infinity, the packet measures converge weakly to Haar probability measure and form a tight family, so no mass escapes. Arbitrary local homothety types are allowed, and the fields may vary along the sequence.", "lab_docs_url": "https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/docs/015.md", "overview_entry": "family 015 in overview.pdf / CONTENTS.md", "papers": [{"dir": "Equidistribution-of-Prime-Degree-Torus-Packets-with-Arbitrary-Local-Type-September-24-2026", "title": "Equidistribution of Prime-Degree Torus Packets with Arbitrary Local Type", "url": "https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Equidistribution-of-Prime-Degree-Torus-Packets-with-Arbitrary-Local-Type-September-24-2026"}], "audit_part": 1, "second_reader": "", "referee_verdict": ""}, {"family": "017", "title": "The irrationality exponent of $\\pi$ is $2$", "subject": "Number theory", "headline": "Proves that the irrationality exponent of $\\pi$ is exactly $2$: for every $\\varepsilon>0$ and all sufficiently large denominators $q$, every rational $p/q$ satisfies $|\\pi-p/q|\\ge q^{-2-\\varepsilon}$. This also proves convergence of the Flint--Hills series $\\sum_{n\\ge1}1/(n^3\\sin^2 n)$, with angles in radians.", "verdict": "full", "challenges": ["PiExponent"], "review_note": "Headline (irrationality exponent of pi = 2, both forms) stated with Mathlib objects; the Flint-Hills series convergence corollary is not stated.", "definitions_to_check": [], "external_packages": [], "cone_lines_max": 94948, "machine_check": "comparator-sandboxed-pass", "lab_scope_note": "The formalization proves that the irrationality exponent of $\\pi$ is exactly two. For every $\\nu>2$, all sufficiently large positive denominators $q$ satisfy $|\\pi-p/q|\\ge q^{-\\nu}$ for every integer numerator $p$. It also states the exact supremum characterization using infinitely many rational approximations.\n\nThe paper's convergence consequence for the Flint–Hills series is outside this selected statement.", "lab_docs_url": "https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/docs/017.md", "overview_entry": "family 017 in overview.pdf / CONTENTS.md", "papers": [{"dir": "The-irrationality-exponent-of-pi-is-2-September-24-2026", "title": "The irrationality exponent of pi is 2", "url": "https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-irrationality-exponent-of-pi-is-2-September-24-2026"}], "audit_part": 1, "second_reader": "", "referee_verdict": ""}, {"family": "020", "title": "Squarefree quartics and power-free polynomial values", "subject": "Number theory", "headline": "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 $d\\ge4$.", "verdict": "full", "challenges": ["PowerFreeValues"], "review_note": "None; statement is for every d >= 4 (stronger than the new range 4..8 alone).", "definitions_to_check": [], "external_packages": [], "cone_lines_max": 23269, "machine_check": "comparator-sandboxed-pass", "lab_scope_note": "The formalized result proves positive-density power-free values for every integer polynomial $f$ irreducible over $\\mathbb Q$ of degree $d\\ge4$. Put $k=d-2$ and assume no prime $k$th power divides every value of $f$. Then the number of positive integers $n\\le X$ for which $f(n)$ is $k$-free is $c_fX+o_f(X)$, where $c_f>0$ is the convergent product of local factors. Negative values are allowed and zero is excluded. No monicity, primitivity, or coefficient-height restriction is imposed.", "lab_docs_url": "https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/docs/020.md", "overview_entry": "family 020 in overview.pdf / CONTENTS.md", "papers": [{"dir": "Squarefree-values-of-quartics-and-power-free-values-of-polynomials-September-24-2026", "title": "Squarefree values of quartics and power-free values of polynomials", "url": "https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Squarefree-values-of-quartics-and-power-free-values-of-polynomials-September-24-2026"}], "audit_part": 1, "second_reader": "", "referee_verdict": ""}, {"family": "021", "title": "A quadratic bound for Jacobsthal's function", "subject": "Number theory", "headline": "Answers Jacobsthal's quadratic-bound question: every interval of $Ck^2$ consecutive integers contains an integer coprime to any prescribed positive integer with at most $k$ distinct prime divisors, for an absolute constant $C$. The bound is uniform over prime sets and interval positions and removes the classical logarithmic loss.", "verdict": "full", "challenges": ["Jacobsthal", "JacobsthalImproved"], "review_note": "None; both the quadratic and the (log log 3k)^-2 improved bound are stated with elementary definitions.", "definitions_to_check": [], "external_packages": ["AbsorptionCutoff", "PrimeNumberTheoremAnd", "StrongPNT"], "cone_lines_max": 152494, "machine_check": "none", "lab_scope_note": "Let $h(k)$ be the least interval length that guarantees an integer coprime to any prescribed positive modulus with at most $k$ distinct prime factors. The formalization proves the paper's strengthened Jacobsthal bound $h(k)\\le Ck^2/(\\log\\log(3k))^2$ for one absolute $C>0$ and every $k\\ge1$. Intervals may begin at any signed integer. The earlier quadratic bound is also retained as a separate statement; the optimal order of growth is not determined.", "lab_docs_url": "https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/docs/021.md", "overview_entry": "family 021 in overview.pdf / CONTENTS.md", "papers": [{"dir": "A-quadratic-bound-for-Jacobsthals-function-September-25-2026", "title": "A quadratic bound for Jacobsthal's function", "url": "https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-quadratic-bound-for-Jacobsthals-function-September-25-2026"}], "audit_part": 1, "second_reader": "", "referee_verdict": ""}, {"family": "023", "title": "Patterson’s first moment for cubic Gauss sums", "subject": "Number theory", "headline": "Proves unconditionally the all-primary-prime form of Patterson’s first-moment asymptotic: normalized cubic Gauss sums over primary Eisenstein primes of norm at most $X$, including both conjugates, have an explicit positive main term of order $X^{5/6}/\\log X$. Every fixed nonzero prime-angle Fourier mode has smaller order.", "verdict": "full", "challenges": ["PattersonFirstMoment"], "review_note": "None; custom Gauss-sum/cubic-symbol definitions match the paper's displayed definition (checked in paper.tex).", "definitions_to_check": ["Gauss sum, cubic residue symbol and primary-prime predicate are all custom (`cubicSymbolAtPrime` uses `Classical.choose` and a fallback value 0; `residueRepresentative := Classical.choose ...`; `attribute [local instance] Classical.propDecidable`), but they agree with the paper's definition and the summand is independent of the representative"], "external_packages": ["PrimeNumberTheoremAnd", "RellichKondrachov", "StrongPNT"], "cone_lines_max": 306361, "machine_check": "none", "lab_scope_note": "Patterson's first-moment conjecture concerns the average of normalized cubic Gauss sums over primary Eisenstein primes. The formalization proves the sharp-cutoff asymptotic with main term $\\frac65c_*X^{5/6}/\\log X$, where $c_*=(2\\pi)^{2/3}/(3\\Gamma(2/3))$. It also proves the corresponding angular comparison for every integer Fourier mode and cancellation of order $o(X^{5/6}/\\log X)$ for each fixed nonzero mode. All primary Eisenstein primes are included, without an extra conjectural premise.", "lab_docs_url": "https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/docs/023.md", "overview_entry": "family 023 in overview.pdf / CONTENTS.md", "papers": [{"dir": "An-unconditional-first-moment-for-cubic-Gauss-sums-September-25-2026", "title": "An unconditional first moment for cubic Gauss sums", "url": "https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/An-unconditional-first-moment-for-cubic-Gauss-sums-September-25-2026"}], "audit_part": 1, "second_reader": "", "referee_verdict": ""}, {"family": "024", "title": "An asymptotic formula for the number of totients", "subject": "Number theory", "headline": "Gives an asymptotic equivalent for the number $V(x)$ of distinct totient values up to $x$, with a positive bounded phase-dependent factor determined by convergent arithmetic approximations. In particular, $V(cx)/V(x)\\to c$ for every fixed $c>0$, answering Erd\\H{o}s and Hall’s scaling question.", "verdict": "full", "challenges": ["TotientAsymptotic", "TotientCompanionZero"], "review_note": "None for the headline (V(cx)/V(x) -> c is stated in terms of V alone). The explicit main term rests on a large custom apparatus that cannot be checked against a standard object from the statement.", "definitions_to_check": ["Main term built from custom `rho := sInf RenewalRootSet`, `lam`, `gamma`, `g` (renewal recursion), `AH` (uses `∑ᶠ` finsums over `d` and over `Finset (TailDatum H)`, whose junk value 0 on infinite support is not excluded by the statement) and `A f s := limUnder atTop (fun H => AH H f s)` (junk value if no limit; the statement separately asserts `TendstoUniformlyOn` so it is not vacuous)"], "external_packages": ["PrimeNumberTheoremAnd"], "cone_lines_max": 73021, "machine_check": "none", "lab_scope_note": "Let $V(x)$ count the distinct values of Euler's totient function up to $x$. The formalization constructs the paper's explicit positive main term from finite arithmetic approximants and proves that their ratio tends to one. In particular, $V(cx)/V(x)\\to c$ for every fixed $c>0$, answering the Erdős–Hall regular-variation question.\n\nThe formalization also gives asymptotics for totients $v\\le x$ whose least preimage $\\ell(v)=\\min\\{n\\ge1:\\varphi(n)=v\\}$ lies between $kx$ and $(k+1)x$. The associated coefficient is positive under the stated existence condition; when no totient $d$ satisfies $kd<\\ell(d)$, the count and coefficient are identically zero. The cases $k=1,2$ have positive coefficients.", "lab_docs_url": "https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/docs/024.md", "overview_entry": "family 024 in overview.pdf / CONTENTS.md", "papers": [{"dir": "An-asymptotic-formula-for-the-number-of-totients-September-25-2026", "title": "An asymptotic formula for the number of totients", "url": "https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/An-asymptotic-formula-for-the-number-of-totients-September-25-2026"}], "audit_part": 1, "second_reader": "", "referee_verdict": ""}, {"family": "025", "title": "Erd\\H{o}s’s short Egyptian-fraction conjecture", "subject": "Number theory", "headline": "Every rational $a/b$ with $1\\le a0$; padding can preserve that denominator. If $v(k)$ is the least integer at least $2$ absent from all exact $k$-term expansions of $1$, then eventually $e^{e^{k/600}}\\le v(k)\\le1+k^{2^{k-1}}$, and $\\liminf_{k\\to\\infty}\\log\\log v(k)/k\\ge\\log2/257$.", "lab_docs_url": "https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/docs/025.md", "overview_entry": "family 025 in overview.pdf / CONTENTS.md", "papers": [{"dir": "Short-Egyptian-fractions-September-25-2026", "title": "Short Egyptian fractions", "url": "https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Short-Egyptian-fractions-September-25-2026"}], "audit_part": 1, "second_reader": "", "referee_verdict": ""}, {"family": "026", "title": "Positive lower density of large prime gaps", "subject": "Number theory", "headline": "For every fixed $C>0$, a positive proportion of consecutive prime gaps exceed $C\\log p_n$, throughout every sufficiently large initial segment of the primes. The proportion may depend on $C$. Consequently, the indices where $p_n/n$ increases have positive lower density, answering Erd\\H{o}s and Prachar.", "verdict": "full", "challenges": ["PrimeGaps"], "review_note": "None; challenge states the large-gap theorem and the ratio corollary. Lower density is a custom sSup-defined liminf (checked).", "definitions_to_check": ["`def lowerAsymptoticDensity (A : Set ℕ) : ℝ := sSup {d : ℝ | ∃ N0 : ℕ, ∀ N : ℕ, N0 ≤ N → d ≤ (initialCount A N : ℝ) / (N : ℝ)}` (custom liminf-style density; correct since the set is nonempty and bounded by 1)"], "external_packages": [], "cone_lines_max": 22533, "machine_check": "comparator-sandboxed-pass", "lab_scope_note": "The formalized supplement proves that the indices $n$ for which $p_n/n= 0 branch.", "definitions_to_check": ["Entire framework is custom scheme theory: `SmoothProjectiveVariety`, `ReducedSNCBoundary`, `kodaira := ⨆ (k : ℕ) (_ : E.HasImageDimensionAtLeast k), (k : WithBot ℕ∞)` (Iitaka dimension via algebraically independent ratios of log pluriforms), `VeryGenerally`; the conclusion quantifies `∀ F : FiberModel ...`, so it is vacuous if no `FiberModel` can exist (non-vacuity is not visible from the statement)"], "external_packages": [], "cone_lines_max": 3902, "machine_check": "comparator-sandboxed-pass", "lab_scope_note": "The paper studies logarithmic Kodaira additivity for connected-fiber morphisms of smooth projective reduced simple-normal-crossing pairs that are smooth on all boundary strata away from the base boundary. The linked formalization proves the negative-fiber branch: for a very general base point, if the logarithmic Kodaira dimension of the fiber is $-\\infty$, then the total logarithmic Kodaira dimension equals the sum of the base and fiber dimensions and is $-\\infty$. Every positive-degree logarithmic pluriform section on the total space then vanishes.\n\nThe finite-dimension and negative-base branches of the paper's additivity theorem are outside this selected statement.", "lab_docs_url": "https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/docs/033.md", "overview_entry": "family 033 in overview.pdf / CONTENTS.md", "papers": [{"dir": "The-reverse-logarithmic-Kodaira-inequality-and-additivity-September-26-2026", "title": "The reverse logarithmic Kodaira inequality and additivity", "url": "https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-reverse-logarithmic-Kodaira-inequality-and-additivity-September-26-2026"}], "audit_part": 1, "second_reader": "", "referee_verdict": ""}, {"family": "039", "title": "Nagata's conjecture and maximal Seshadri constants", "subject": "Algebraic and complex geometry", "headline": "Proves Nagata's strict inequality $\\sum_i m_i= 10, very general, arbitrary multiplicities) fully stated; maximal multipoint Seshadri constants only for smooth projective surfaces over C, not (L^n/r)^{1/n} in dimension n >= 2 and not over algebraically closed fields of positive characteristic.", "definitions_to_check": ["Seshadri side is custom scheme theory: `selfIntersection S L := χ(L^2) - 2χ(L) + χ(O)` (= L^2 by Riemann-Roch, checked), `curveDegree := χ(L|_C) - χ(O_C)`, `cohomology := Abelian.Ext (structureSheaf X) M n` with `cohomologyDimension := Module.finrank ℂ` (junk 0 if infinite-dimensional), `IsBlowup` by universal property, `curveMultiplicity` via `idealOrder` of the stalk kernel; Nagata side defines the curve equation by `Classical.choose` of a sorry'd existence theorem (`curveEquation`) that is itself a checked theorem_name"], "external_packages": [], "cone_lines_max": 56552, "machine_check": "comparator-sandboxed-pass", "lab_scope_note": "Nagata's conjecture asserts that a nonzero effective plane curve of degree $d$, with multiplicities at least $m_i$ at $r\\ge10$ very general points, satisfies $\\sum_i m_i= 4) and commuting LNDs; 'every fibre is A^3' not stated.", "definitions_to_check": [], "external_packages": [], "cone_lines_max": 3240, "machine_check": "comparator-sandboxed-pass", "lab_scope_note": "The Abhyankar–Sathaye conjecture predicts that a polynomial defining an affine-space quotient must be an ambient coordinate. For every $n\\ge4$, the formalized counterexample gives $F\\in\\mathbb C[x_1,\\ldots,x_n]$ with quotient $\\mathbb C[x_1,\\ldots,x_n]/(F)\\cong\\mathbb C[y_1,\\ldots,y_{n-1}]$, although $F$ is not a coordinate.\n\nA companion gives $n-1$ commuting, locally nilpotent derivations, linearly independent over the polynomial ring, whose common kernel is generated by the construction's explicit polynomial and contains no ambient coordinate. The extra $n-4$ derivations are ordinary partial derivatives in the added variables. The three-variable case is not covered.", "lab_docs_url": "https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/docs/049.md", "overview_entry": "family 049 in overview.pdf / CONTENTS.md", "papers": [{"dir": "An-explicit-noncoordinate-polynomial-with-affine-three-space-zero-fibre-September-24-2026", "title": "An explicit noncoordinate polynomial with affine three-space zero fibre", "url": "https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/An-explicit-noncoordinate-polynomial-with-affine-three-space-zero-fibre-September-24-2026"}], "audit_part": 1, "second_reader": "", "referee_verdict": ""}, {"family": "050", "title": "A counterexample to Griffiths' positivity conjecture", "subject": "Algebraic and complex geometry", "headline": "Constructs ample rank-two bundles on $\\mathbb P^1\\times\\mathbb P^1$ with no smooth Hermitian metric of strictly Griffiths-positive curvature, disproving Griffiths' positivity conjecture already on the quadric surface.", "verdict": "full", "challenges": ["QuadricBundles"], "review_note": "Headline stated; the 'very-ampleness for all admissible even exponents' result is not (docs). All notions (algebraic bundle on P1xP1, ampleness, Hermitian metric, Griffiths positivity) are bespoke, with no bridge to Mathlib objects.", "definitions_to_check": ["Bespoke `AlgebraicRankTwoBundle` (4 charts by polynomial principal opens + regular transition matrices), `IsAmple V := ∃ n > 0, ∃ N, ∃ s : Fin N → SymmetricSection V n, GivesClosedProjectiveEmbedding s` (a 4-clause custom 'closed embedding' predicate), `SmoothHermitianMetric` on the dual bundle (`dualMatrix`), `StrictlyGriffithsPositive` via `metricMixedHessian`/`holomorphicMetricDerivative`; a mistake in any of these would silently change what is proved, and the statement is vacuous-prone since it ends in `¬ ∃ h`"], "external_packages": [], "cone_lines_max": 12256, "machine_check": "comparator-sandboxed-pass", "lab_scope_note": "Griffiths' positivity conjecture predicts that every ample holomorphic vector bundle on a smooth complex projective variety admits a smooth Hermitian metric with strictly Griffiths-positive curvature. The formalized counterexample starts with a rank-two algebraic bundle $G$ on $\\mathbb P^1\\times\\mathbb P^1$. Its coordinatewise power pullbacks, tensored with $\\mathcal O(1,1)$, are ample for every positive power but admit no such metric for all sufficiently large powers.\n\nThe separate very-ampleness result for all admissible even exponents is not included.", "lab_docs_url": "https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/docs/050.md", "overview_entry": "family 050 in overview.pdf / CONTENTS.md", "papers": [{"dir": "ample-rank-two-bundles-on-the-quadric-surface-without-griffiths-positive-metrics-September-24-2026", "title": "Ample rank-two bundles on the quadric surface without Griffiths-positive metrics", "url": "https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/ample-rank-two-bundles-on-the-quadric-surface-without-griffiths-positive-metrics-September-24-2026"}], "audit_part": 1, "second_reader": "", "referee_verdict": ""}, {"family": "052", "title": "Tangent-bundle splittings and universal covers", "subject": "Algebraic and complex geometry", "headline": "A splitting of the tangent bundle of a compact K\\\"ahler 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 H\\\"oring's conjecture and the corresponding product decomposition.", "verdict": "full", "challenges": ["KahlerSplitting", "SplitTangentIntegrability"], "review_note": "Both headline theorems are stated; the 'corresponding product decomposition' for rationally connected manifolds (composition of the two, needing a Kahler metric from the embedding) is not a single statement. Complex manifold, Kahler metric, integrability and rational connectedness are all bespoke encodings.", "definitions_to_check": ["Bespoke `ComplexManifold`/`KahlerMetric` (closedness of the fundamental form via a cyclic derivative sum in charts), `Integrable` (Frobenius condition through chartwise Lie brackets of holomorphic local sections), `OrdinaryUniversalCover`, `CompatibleProduct`, `ProjectiveEmbedding` (injective holomorphic immersion into P^N) and `RationallyConnected` (dense Zariski-open set of pairs joined by a `RationalCurve`); plausible but unchecked against Mathlib/standard notions"], "external_packages": [], "cone_lines_max": 17281, "machine_check": "comparator-sandboxed-pass", "lab_scope_note": "The splitting question asks whether a holomorphic decomposition of the tangent bundle comes from a product decomposition of the universal cover. The formalized result gives an affirmative answer for a compact connected Kähler manifold whose tangent bundle splits into two positive-rank, integrable holomorphic subbundles. The universal cover is a product of connected simply connected complex manifolds of the prescribed dimensions, and the differential identifies the two tangent factors with the original summands.\n\nBoth integrability assumptions are required. Automatic integrability and the paper's additional corollaries are not included.\n\nThe formalization proves integrability of both summands in a holomorphic splitting of the tangent bundle of a rationally connected projective manifold. More precisely, for a compact connected complex manifold of dimension at least two with a projective embedding witnessing rational connectedness, each positive-rank summand of a holomorphic tangent-bundle splitting is integrable. The paper's compatible product decomposition is a separate consequence, outside this selected statement.", "lab_docs_url": "https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/docs/052.md", "overview_entry": "family 052 in overview.pdf / CONTENTS.md", "papers": [{"dir": "Universal-cover-splitting-for-compact-Kahler-manifolds-September-23-2026", "title": "Universal-cover splitting for compact Kähler manifolds", "url": "https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Universal-cover-splitting-for-compact-Kahler-manifolds-September-23-2026"}, {"dir": "Integrability-of-split-tangent-bundles-on-rationally-connected-manifolds-September-23-2026", "title": "Integrability of split tangent bundles on rationally connected manifolds", "url": "https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Integrability-of-split-tangent-bundles-on-rationally-connected-manifolds-September-23-2026"}], "audit_part": 1, "second_reader": "", "referee_verdict": ""}, {"family": "058", "title": "The Koll\\'ar--Pardon universal-cover conjecture", "subject": "Algebraic and complex geometry", "headline": "Proves the Koll\\'ar--Pardon conjecture: the semialgebraic universal covers of connected normal projective complex varieties are exactly products $D\\times\\mathbb C^m\\times F$, with $D$ bounded symmetric and $F$ simply connected, normal, and projective. A universal cover is quasi-projective exactly when the bounded symmetric factor is absent. In particular, a smooth projective variety covered by $\\mathbb C^n$ has a finite \\textnormal{étale} cover by an abelian variety.", "verdict": "supporting-only", "challenges": ["SymmetricDomains"], "review_note": "Only the bounded-domain symmetry lemma (companion paper); the Kollar-Pardon classification of semialgebraic universal covers D x C^m x F, the quasi-projective criterion and the 'covered by C^n => finite etale abelian cover' corollary are not stated.", "definitions_to_check": ["Custom `IsSemialgebraic` (inductive closure of `p = 0`, `0 < p`, complement, union), `HolomorphicOnSubset`, `Biholomorph`, `IsSmooth`, `IsBoundedSymmetricDomain` (open, connected, bounded, every point an isolated fixed point of a holomorphic involution); plausible, standard-looking"], "external_packages": [], "cone_lines_max": 48720, "machine_check": "comparator-sandboxed-pass", "lab_scope_note": "The formalized result answers the paper's symmetry question affirmatively. A nonempty connected bounded semialgebraic subset, relatively open in a complex affine algebraic set, is smooth and biholomorphic to a bounded symmetric domain whenever a discrete group acts properly and holomorphically with compact quotient.\n\nThe action need not be free, and the ambient algebraic set may be singular.", "lab_docs_url": "https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/docs/058.md", "overview_entry": "family 058 in overview.pdf / CONTENTS.md", "papers": [{"dir": "Symmetry-of-semialgebraic-bounded-domains-with-compact-quotient-September-24-2026", "title": "Symmetry of semialgebraic bounded domains with compact quotient", "url": "https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Symmetry-of-semialgebraic-bounded-domains-with-compact-quotient-September-24-2026"}], "audit_part": 1, "second_reader": "", "referee_verdict": ""}, {"family": "071", "title": "Koebe's circle-domain conjecture and circle-domain rigidity", "subject": "Real and complex analysis", "headline": "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 M\\\"obius, establishing this direction of the He--Schramm conjecture.", "verdict": "full", "challenges": ["KoebeCircleDomains"], "review_note": "None; both the existence part of Koebe's conjecture and removability => rigidity are stated for arbitrary domains. Conformality is encoded chartwise on OnePoint C (custom).", "definitions_to_check": ["Custom conformality on the sphere: `IsConformalAt f p := ContinuousAt f p ∧ ∃ d ≠ 0, HasDerivAt (fun z => chart (f p) (f (chartSymm p z))) d (chart p p)` (hand-rolled charts at ∞ via `sphereInv`; equals holomorphic-with-nonzero-derivative; fine)"], "external_packages": ["Schoenflies"], "cone_lines_max": 272799, "machine_check": "none", "lab_scope_note": "The formalization proves the removability-to-rigidity direction of the He–Schramm conjecture. If a circle domain has conformally removable boundary, then every conformal equivalence from it to another circle domain agrees on the source with a Möbius transformation. There is no bound on the number of complementary components.\n\nThe same Comparator file also states Koebe's circle-domain theorem, which gives a circle-domain representative for every domain in the Riemann sphere.\n\nKoebe's circle-domain conjecture asks whether every domain in the Riemann sphere is conformally equivalent to a circle domain, whose complementary components are round disks or points. The formalization proves this for every domain. It also proves rigidity under a separate hypothesis: a conformal equivalence between circle domains is a Möbius transformation when the source boundary is conformally removable.", "lab_docs_url": "https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/docs/071.md", "overview_entry": "family 071 in overview.pdf / CONTENTS.md", "papers": [{"dir": "Removable-Boundaries-and-Rigidity-of-Circle-Domains-September-23-2026", "title": "Removable Boundaries and Rigidity of Circle Domains", "url": "https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Removable-Boundaries-and-Rigidity-of-Circle-Domains-September-23-2026"}, {"dir": "Koebes-Circle-Domain-Conjecture-September-23-2026", "title": "Koebe's Circle-Domain Conjecture", "url": "https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Koebes-Circle-Domain-Conjecture-September-23-2026"}], "audit_part": 1, "second_reader": "", "referee_verdict": ""}, {"family": "072", "title": "Brennan's conjecture and a counterexample to Kraetzer's prediction", "subject": "Real and complex analysis", "headline": "Proves Brennan's conjecture: for every conformal bijection $\\phi$ from a simply connected plane domain onto the disk, $|\\phi'|^s$ is area-integrable for $4/30$ and the spectrum value $B_{\\mathcal S}(-2)=1$. The Koebe map and its inverse give divergence at all four boundary exponents.\n\nKraetzer's proposed integral-means spectrum predicts $B_b(-1)=1/4$ for bounded univalent maps. The formalized result proves $B_b(-1)<1/4$, contradicting that prediction.\n\nThe underlying estimate gives constants $0<\\varepsilon<1/4$ and $C<\\infty$ such that the normalized circular mean of $|f'|^{-1}$ is at most $C(1-r)^{-1/4+\\varepsilon}$ for every normalized univalent disk map and $1/2\\le r<1$. The bound requires neither bounded image nor boundary regularity, and no numerical value of $\\varepsilon$ is specified.", "lab_docs_url": "https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/docs/072.md", "overview_entry": "family 072 in overview.pdf / CONTENTS.md", "papers": [{"dir": "Brennans-conjecture-and-sharp-inverse-square-integral-means-September-24-2026", "title": "Brennan's conjecture and sharp inverse-square integral means", "url": "https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Brennans-conjecture-and-sharp-inverse-square-integral-means-September-24-2026"}, {"dir": "A-strict-inverse-first-power-bound-for-univalent-functions-September-24-2026", "title": "A strict inverse-first-power bound for univalent functions", "url": "https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-strict-inverse-first-power-bound-for-univalent-functions-September-24-2026"}], "audit_part": 1, "second_reader": "", "referee_verdict": ""}, {"family": "073", "title": "The Falconer distance conjecture", "subject": "Real and complex analysis", "headline": "Resolves the Falconer distance conjecture in every dimension $d\\ge2$: every compact set $E\\subset\\mathbb R^d$ with Hausdorff dimension greater than $d/2$ determines a set of Euclidean distances of positive Lebesgue measure.", "verdict": "full", "challenges": ["PlanarFalconer", "FalconerAllDimensions"], "review_note": "None.", "definitions_to_check": [], "external_packages": [], "cone_lines_max": 185097, "machine_check": "comparator-sandboxed-pass", "lab_scope_note": "The formalization proves the Falconer distance conjecture in every dimension. For every integer $d\\ge2$ and compact set $E\\subset\\mathbb R^d$ with Hausdorff dimension greater than $d/2$, the set $\\{\\lVert x-y\\rVert:x,y\\in E\\}$ has positive Lebesgue measure. The linked statements include both this all-dimensional result and the earlier planar case.", "lab_docs_url": "https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/docs/073.md", "overview_entry": "family 073 in overview.pdf / CONTENTS.md", "papers": [{"dir": "The-Falconer-distance-conjecture-in-all-dimensions-September-23-2026", "title": "The Falconer distance conjecture in all dimensions", "url": "https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-Falconer-distance-conjecture-in-all-dimensions-September-23-2026"}], "audit_part": 1, "second_reader": "", "referee_verdict": ""}, {"family": "076", "title": "Ultraflat real Littlewood polynomials", "subject": "Real and complex analysis", "headline": "Constructs polynomials with $N$ consecutive coefficients in $\\{-1,1\\}$ whose modulus is $(1+o(1))\\sqrt N$ uniformly on the entire unit circle, for every sufficiently large integer length $N$. Thus real Littlewood polynomials are ultraflat, including at the real endpoints. Their binary merit factors tend to infinity, disproving Turyn's bounded-merit-factor conjecture.", "verdict": "weaker-statement", "challenges": ["AsymptoticallyMinimalLittlewood", "LittlewoodFiniteFlatness"], "review_note": "Only max-modulus <= (1+eta)sqrt(N) and L^p (finite p) flatness are stated; uniform ultraflatness (two-sided, sup norm), merit factor -> infinity and the disproof of Turyn's conjecture are not. Two of the three index papers (Oct 5) are not linked in docs.", "definitions_to_check": [], "external_packages": [], "cone_lines_max": 9712, "machine_check": "comparator-sandboxed-pass", "lab_scope_note": "The formalization gives real Littlewood polynomials of every sufficiently large length whose maximum modulus on the unit circle is at most $(1+\\eta)\\sqrt N$ for any fixed $\\eta>0$. Thus the smallest possible maximum is asymptotically minimal through all integer lengths.\n\nA further statement chooses one family of real sign polynomials for all lengths such that, for every fixed finite $p>0$, the $L^p$ mean of $\\bigl||P_N(z)|/\\sqrt N-1\\bigr|$ on the unit circle tends to zero. The results are existential and do not give an effective convergence rate or a signing algorithm.", "lab_docs_url": "https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/docs/076.md", "overview_entry": "family 076 in overview.pdf / CONTENTS.md", "papers": [{"dir": "Asymptotically-minimal-maxima-of-real-Littlewood-polynomials-September-23-2026", "title": "Asymptotically minimal maxima of real Littlewood polynomials", "url": "https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Asymptotically-minimal-maxima-of-real-Littlewood-polynomials-September-23-2026"}], "audit_part": 1, "second_reader": "", "referee_verdict": ""}, {"family": "081", "title": "The David--Semmes Riesz-transform problem in higher codimension", "subject": "Real and complex analysis", "headline": "Resolves the remaining higher-codimension Riesz-transform rectifiability problem: for $d\\ge4$ and $2\\le n\\le d-2$, an $n$-Ahlfors--David regular Radon measure on $\\mathbb R^d$ is uniformly $n$-rectifiable whenever its $n$-dimensional Riesz transform is uniformly $L^2$-bounded over all positive hard truncations. The rectifiability bounds depend only on dimension, regularity and operator bounds.", "verdict": "full", "challenges": ["RieszQuantitative"], "review_note": "None for the headline; uniform n-rectifiability is encoded as big pieces of Lipschitz images (standard equivalent form). The variation / principal-value corollary is not stated.", "definitions_to_check": ["`ADRegularWithConstant`, `RieszL2BoundedWithConstant` (real-valued f, vector-valued hard truncation `truncated`), `BallImageConclusion` are custom but standard in form; `AdmissibleRadius` uses `Metric.ediam μ.support`"], "external_packages": ["FixedPointTheorems"], "cone_lines_max": 65638, "machine_check": "none", "lab_scope_note": "The formalized result proves that bounded Riesz transforms force quantitative uniform rectifiability in higher codimension. For $d\\ge4$ and $2\\le n\\le d-2$, an $n$-Ahlfors–David regular measure whose positive hard truncations have one uniform $L^2$ operator bound has big pieces of Lipschitz images. The mass fraction and Lipschitz constant depend only on the dimensions and the stated regularity and operator bounds, and work at every support point and admissible radius, including unbounded support. The variation and principal-value corollary is not included.", "lab_docs_url": "https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/docs/081.md", "overview_entry": "family 081 in overview.pdf / CONTENTS.md", "papers": [{"dir": "Riesz-transforms-and-uniform-rectifiability-in-higher-codimension-September-24-2026", "title": "Riesz transforms and uniform rectifiability in higher codimension", "url": "https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Riesz-transforms-and-uniform-rectifiability-in-higher-codimension-September-24-2026"}], "audit_part": 1, "second_reader": "", "referee_verdict": ""}, {"family": "082", "title": "Maximal and variational bounds for the triangular Hilbert transform", "subject": "Real and complex analysis", "headline": "Proves maximal and annular $r$-variation bounds, for every $r>2$, from complex $L^3(\\mathbb R^2)\\times L^3(\\mathbb R^2)$ to $L^{3/2}(\\mathbb R^2)$. The maximal estimate controls both hard truncation endpoints and gives almost-everywhere and norm convergence. Pairing with a third input settles the triangular Hilbert transform estimate at the symmetric $L^3\\times L^3\\times L^3$ point.", "verdict": "partial", "challenges": ["TriangularHilbert"], "review_note": "Maximal L^3 x L^3 -> L^{3/2} bound only; annular r-variation (r > 2), a.e./norm convergence and the trilinear symmetric-point estimate are not stated.", "definitions_to_check": ["`maximal F G z := if GoodPoint F G z then ⨆ E : Endpoints, ENNReal.ofReal ‖truncation F G E z‖ else 0` (junk value 0 off the good set; the statement itself asserts `∀ᵐ z, GoodPoint F G z`)"], "external_packages": [], "cone_lines_max": 10861, "machine_check": "comparator-sandboxed-pass", "lab_scope_note": "The formalized result proves the symmetric maximal estimate for the triangular Hilbert transform: for arbitrary complex $F,G\\in L^3(\\mathbb R^2)$, the $L^{3/2}$ norm of the supremum over all finite hard-truncation intervals is at most $C\\|F\\|_3\\|G\\|_3$ for one absolute constant $C$. It also establishes a common full-measure set on which the truncated integrals are defined and almost-everywhere measurability of the maximal output.", "lab_docs_url": "https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/docs/082.md", "overview_entry": "family 082 in overview.pdf / CONTENTS.md", "papers": [{"dir": "The-maximal-triangular-Hilbert-transform-at-the-symmetric-point-September-24-2026", "title": "The maximal triangular Hilbert transform at the symmetric point", "url": "https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-maximal-triangular-Hilbert-transform-at-the-symmetric-point-September-24-2026"}], "audit_part": 1, "second_reader": "", "referee_verdict": ""}, {"family": "083", "title": "Stein's conjecture for Hilbert transforms along Lipschitz directions", "subject": "Real and complex analysis", "headline": "Proves a uniform strong $L^2$ 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 $L^2$-bounded principal-value operator, establishing Stein's weak-type conjecture at this short scale.", "verdict": "full", "challenges": ["LipschitzHilbert"], "review_note": "None; statement is exactly the short-scale uniform bound plus principal-value/weak-type consequences (not the full-scale Stein conjecture).", "definitions_to_check": ["`smoothPrincipalValue` extends the symmetric quotient at 0 by `Function.update ... 0 (deriv g 0)` (custom, correct for Schwartz f); file temporarily removes/re-adds the instance `instCommCStarAlgebraComplex` to fix scalar-multiplication elaboration (harmless hack)"], "external_packages": [], "cone_lines_max": 185580, "machine_check": "none", "lab_scope_note": "The formalization proves uniform $L^2$ bounds for short Hilbert transforms along Lipschitz unit vector fields in the plane, including fields depending on both coordinates. For Lipschitz constant $K>0$, every hard truncation with $0<\\varepsilon\\le R\\le1/(10^6K)$ has one universal $L^2$ bound on Schwartz functions, independent of the inner cutoff.\n\nAt a fixed short scale for $1$-Lipschitz fields, the formalization also gives principal-value and weak-$(2,2)$ estimates and bounded extensions to all of $L^2$. These results establish the stated short-scale form of Stein's question.", "lab_docs_url": "https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/docs/083.md", "overview_entry": "family 083 in overview.pdf / CONTENTS.md", "papers": [{"dir": "A-uniform-Hilbert-transform-estimate-for-Lipschitz-directions-September-25-2026", "title": "A uniform Hilbert transform estimate for Lipschitz directions", "url": "https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-uniform-Hilbert-transform-estimate-for-Lipschitz-directions-September-25-2026"}], "audit_part": 1, "second_reader": "", "referee_verdict": ""}, {"family": "084", "title": "The geometric case of the Erd\\H{o}s similarity conjecture", "subject": "Real and complex analysis", "headline": "For every fixed $q\\in(0,1)$, constructs compact subsets of $[0,1]$ with measure arbitrarily close to one containing no translated and nontrivially dilated copy of $\\{q^n:n\\ge1\\}$, with dilations of either sign. This resolves the geometric-progression case of the Erd\\H{o}s similarity conjecture for every ratio.", "verdict": "partial", "challenges": ["DyadicAvoidance"], "review_note": "Only q = 1/2 (dyadic) is stated; the geometric-progression case for every ratio q in (0,1) is not.", "definitions_to_check": [], "external_packages": [], "cone_lines_max": 8455, "machine_check": "comparator-sandboxed-pass", "lab_scope_note": "The formalization proves the dyadic case of the Erdős similarity conjecture. For every $0<\\eta<1$, it constructs a compact set $E\\subset[0,1]$ of measure greater than $1-\\eta$ that contains no affine copy of $\\{2^{-n}:n\\ge1\\}$. Explicitly, for every translation $x$ and every nonzero real dilation $s$, some point $x+s2^{-n}$ lies outside $E$. Both signs of $s$ are included.", "lab_docs_url": "https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/docs/084.md", "overview_entry": "family 084 in overview.pdf / CONTENTS.md", "papers": [{"dir": "The-dyadic-case-of-the-Erdos-similarity-conjecture-September-25-2026", "title": "The dyadic case of the Erdős similarity conjecture", "url": "https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-dyadic-case-of-the-Erdos-similarity-conjecture-September-25-2026"}], "audit_part": 1, "second_reader": "", "referee_verdict": ""}, {"family": "085", "title": "Endpoint regularity of the planar centered maximal function", "subject": "Real and complex analysis", "headline": "Resolves the planar centered-disk case of the Haj{\\l}asz--Onninen maximal-function regularity problem. For every real $f\\in W^{1,1}(\\mathbb R^2)$, the centered disk maximal function satisfies $\\|\\nabla Mf\\|_1\\le C\\|\\nabla f\\|_1$ with an absolute constant. It belongs locally to $W^{1,1}$ and has a globally integrable weak gradient.", "verdict": "full", "challenges": ["SignedFiniteBand", "DiskMaximal"], "review_note": "None; the BV extension is not claimed (docs). Weak gradients are encoded by hand (`HasWeakGradient` against C_c^∞ test functions), which is the standard definition.", "definitions_to_check": [], "external_packages": [], "cone_lines_max": 19397, "machine_check": "comparator-sandboxed-pass", "lab_scope_note": "The centered disk maximal operator takes the supremum of averages of $|f|$ over disks centered at each point. The formalization proves that every real $f\\in W^{1,1}(\\mathbb R^2)$ has a maximal function that is finite almost everywhere, belongs to $W^{1,1}_{\\mathrm{loc}}$, and has a globally integrable weak gradient satisfying $\\|\\nabla Mf\\|_1\\le C\\|\\nabla f\\|_1$ for one absolute constant $C$.\n\nThe earlier signed finite-band estimate for smooth compactly supported functions is also retained. The general endpoint statement covers the Sobolev setting of the paper; a BV extension is outside these statements.", "lab_docs_url": "https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/docs/085.md", "overview_entry": "family 085 in overview.pdf / CONTENTS.md", "papers": [{"dir": "An-Endpoint-Gradient-Bound-for-the-Centered-Disk-Maximal-Operator-September-26-2026", "title": "An Endpoint Gradient Bound for the Centered Disk Maximal Operator", "url": "https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/An-Endpoint-Gradient-Bound-for-the-Centered-Disk-Maximal-Operator-September-26-2026"}], "audit_part": 1, "second_reader": "", "referee_verdict": ""}, {"family": "087", "title": "The Mahler conjectures and symplectic width", "subject": "Convex and metric geometry", "headline": "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 $n\\ge2$, every symmetric polar product $K\\times K^\\circ$ in dimension $2n$ has Gromov width $4$.", "verdict": "full", "challenges": ["MahlerConjecture", "SymmetricMahlerEquality", "GeneralMahler", "SymmetricPolar"], "review_note": "Symmetric and general Mahler with all equality cases and the Gromov width 4 are stated; the sharp functional Mahler inequalities of the summary are not.", "definitions_to_check": ["Custom but standard encodings: `coordinatePolar` (polar w.r.t. the dot product), `IsHanner`/`hannerJoin`/`hannerProduct`, `volumeProduct := sInf (... '' interior K)`, `HasSymplecticEmbedding` (C^∞ on U, embedding, `omega0` pullback condition) and `gromovWidth := sSup {c | HasSymplecticEmbedding (capacityBall n c) U}` with capacity π r^2"], "external_packages": [], "cone_lines_max": 40205, "machine_check": "comparator-sandboxed-pass", "lab_scope_note": "The symmetric Mahler conjecture predicts $|K||K^\\circ|\\ge4^n/n!$ for every origin-symmetric convex body $K\\subset\\mathbb R^n$. The formalization establishes this for every $n\\ge1$ and characterizes equality exactly by invertible linear images of Hanner bodies, built from intervals using Cartesian products and convex-hull joins.\n\nThe nonsymmetric Mahler conjecture and the functional inequalities are not included.\n\nThe general Mahler conjecture gives a sharp lower bound for the volume product of a convex body and its polar. For every $n\\ge1$ and compact convex body $K\\subset\\mathbb R^n$ with nonempty interior, the formalization proves\n$\\inf_{z\\in\\mathrm{int}\\,K}|K|\\,|(K-z)^\\circ|\\ge (n+1)^{n+1}/(n!)^2$.\nEquality holds exactly when $K$ is a simplex. The paper's functional inequality is outside this selected statement.\n\nFor an origin-symmetric convex body $K\\subset\\mathbb R^n$, $n\\ge2$, the formalized result determines the symplectic ball capacity of $\\mathrm{int}\\,K\\times\\mathrm{int}\\,K^\\circ$: its Gromov width is $4$, and every ball of capacity $0= 4, with ellipsoid equality case) fully stated; the product-of-simplices counterexample is stated only in dimension 20 (ratio 1.015...), not the exponential factor in every sufficiently large dimension; the full Lutwak-Petty inequality family is not stated.", "definitions_to_check": ["Projection body defined by hand: `projectionBody K := {x | ∀ u, ‖u‖ = 1 → ⟪u, x⟫ ≤ shadowVolume K u}` with `shadowVolume` the `volume` of the orthogonal projection on the subtype `(span {u})ᗮ` (three files use slightly different encodings: `volume` on the subtype vs `Measure.euclideanHausdorffMeasure (d - 1)`), and `simplexConstant d := (d+1) * d^d / d!` in `ProjectionVolume` is an asserted value for the simplex, not computed in the statement"], "external_packages": [], "cone_lines_max": 13670, "machine_check": "comparator-sandboxed-pass", "lab_scope_note": "Petty's projection-volume conjecture predicts\n\n$\\displaystyle \\frac{|\\Pi K|}{|K|^{n-1}}\\ge \\kappa_{n-1}^{n}\\kappa_n^{2-n},$\n\nwith equality exactly for ellipsoids; here $\\kappa_j$ is the volume of the Euclidean unit ball in dimension $j$. The formalization establishes this for every convex body $K\\subset\\mathbb R^n$ and every $n\\ge4$.\n\nOther inequalities in the projection-body family are not included.\n\nBrannen's simplex-maximization conjecture predicts that a simplex maximizes normalized projection-body volume $|\\Pi K|/|K|^{n-1}$ in dimension $n$. The formalized counterexample is the product of two ten-dimensional simplices. Its normalized projection volume, divided by that of a twenty-dimensional simplex, is\n\n$\\displaystyle \\frac{22{,}355{,}476}{22{,}020{,}096}>1.$\n\nThus the proposed maximum fails in dimension $20$. The paper's exponential-factor result for every sufficiently large dimension is not included.", "lab_docs_url": "https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/docs/088.md", "overview_entry": "family 088 in overview.pdf / CONTENTS.md", "papers": [{"dir": "Pettys-projection-volume-conjecture-in-dimensions-at-least-four-September-24-2026", "title": "Petty’s projection-volume conjecture in dimensions at least four", "url": "https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Pettys-projection-volume-conjecture-in-dimensions-at-least-four-September-24-2026"}, {"dir": "A-product-counterexample-to-the-simplex-maximum-for-projection-body-volume-September-24-2026", "title": "A product counterexample to the simplex maximum for projection-body volume", "url": "https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-product-counterexample-to-the-simplex-maximum-for-projection-body-volume-September-24-2026"}], "audit_part": 1, "second_reader": "", "referee_verdict": ""}, {"family": "089", "title": "Bounded-distortion $L_1$ embeddings of planar and bounded-treewidth graphs", "subject": "Convex and metric geometry", "headline": "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 $L_1$ 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.", "verdict": "full", "challenges": ["PlanarL1", "BoundedTreewidthL1"], "review_note": "Both L1-embedding theorems are stated; the flow-cut-gap corollary is not. Planarity is a hand-rolled topological drawing predicate.", "definitions_to_check": ["`IsPlanar` is a custom 'drawing by simple plane arcs' predicate (injective points, `arc u v h : C(unitInterval, ℝ × ℝ)` per oriented edge, interiors of arcs of distinct unoriented edges disjoint); `HasTreeDecomposition` and `graphDistance := sInf (range walkLength)` are also hand-rolled; both read as standard"], "external_packages": [], "cone_lines_max": 33806, "machine_check": "comparator-sandboxed-pass", "lab_scope_note": "The planar case of the Gupta–Newman–Rabinovich–Sinclair embedding problem asks for a universal distortion bound in $L_1$. The formalized result gives one constant $C$ for every finite connected planar graph with arbitrary positive real edge lengths: its weighted shortest-path metric embeds into real $L_1$ with distortion at most $C$. The constant is independent of the number of vertices and the ratios between edge lengths; singleton graphs are included.\n\nThe formalized result gives a uniform $L_1$ embedding bound for each bounded-treewidth class. For every bag-size bound $k\\ge2$, there is a constant $C(k)$ such that the weighted shortest-path metric of every nonempty finite connected graph with a tree decomposition of bag size at most $k$ embeds into finite-dimensional real $L_1$ with distortion at most $C(k)$. Edge lengths may be arbitrary positive real numbers, and the constant is independent of the graph and decomposition size.", "lab_docs_url": "https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/docs/089.md", "overview_entry": "family 089 in overview.pdf / CONTENTS.md", "papers": [{"dir": "Planar-Graph-Metrics-Embed-into-L1-with-Constant-Distortion-September-23-2026", "title": "Planar Graph Metrics Embed into L1 with Constant Distortion", "url": "https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Planar-Graph-Metrics-Embed-into-L1-with-Constant-Distortion-September-23-2026"}, {"dir": "L1-Embeddings-of-Graphs-of-Bounded-Treewidth-September-23-2026", "title": "L1 Embeddings of Graphs of Bounded Treewidth", "url": "https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/L1-Embeddings-of-Graphs-of-Bounded-Treewidth-September-23-2026"}], "audit_part": 1, "second_reader": "", "referee_verdict": ""}, {"family": "090", "title": "Universal optimality of the triangular lattice", "subject": "Convex and metric geometry", "headline": "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 $00$.", "verdict": "full", "challenges": ["SingleLatticeCovering", "CoveringDensity"], "review_note": "None; the lower bound for centrally symmetric bodies is stated for the translative supremum too (hence for the lattice one).", "definitions_to_check": ["Hand-rolled density notions: `euclideanVolume` (pushforward of product Lebesgue measure), `euclideanCovolume`, `thetaL`, `thetaT` with `upperCenterIntensity := limsup #(X ∩ cube R) / (2R)^n`; standard in form"], "external_packages": [], "cone_lines_max": 40097, "machine_check": "comparator-sandboxed-pass", "lab_scope_note": "The formalized result gives an absolute constant $C>0$ such that every convex body $K\\subset\\mathbb R^n$, $n\\ge2$, admits a covering by translates along one full-rank lattice $L$ with density $|K|/\\mathrm{covol}(L)\\le Cn\\log n$. The covering is exact, and no symmetry, boundary regularity, or volume normalization is assumed.\n\nThe formalization determines the order of the largest covering density in dimension $n$. For all sufficiently large $n$, the suprema of translative covering density and lattice covering density are each bounded above and below by absolute positive multiples of $n\\log n$, both over all convex bodies and over centrally symmetric convex bodies.", "lab_docs_url": "https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/docs/092.md", "overview_entry": "family 092 in overview.pdf / CONTENTS.md", "papers": [{"dir": "A-single-lattice-covering-bound-of-order-n-log-n-September-23-2026", "title": "A single-lattice covering bound of order n log n", "url": "https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-single-lattice-covering-bound-of-order-n-log-n-September-23-2026"}, {"dir": "Translative-covering-densities-of-order-n-log-n-September-23-2026", "title": "Translative covering densities of order n log n", "url": "https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Translative-covering-densities-of-order-n-log-n-September-23-2026"}], "audit_part": 1, "second_reader": "", "referee_verdict": ""}, {"family": "094", "title": "Subpolynomial dimension reduction in $L_p$", "subject": "Convex and metric geometry", "headline": "For every fixed $11$, every $n$-point subset of real $L_p$ embeds into $\\ell_p^d$ with distortion at most $D$ and dimension $d=n^{o(1)}$, answering Naor's sublinear-dimension question for $p\\ne2$. In contrast, exact embeddings require worst-case dimension $\\Theta(n^2)$ when $p\\ne2$.", "verdict": "full", "challenges": ["SubpolynomialLp"], "review_note": "None; both the subpolynomial upper bound and the quadratic exact-embedding bounds are stated for every p > 1, p != 2.", "definitions_to_check": ["`dimension p n D := sInf {d | GoodDimension p n D d}` is a natural-number infimum (junk 0 if empty); the statement itself asserts `GoodDimension ... (dimension ...)`, ruling that out"], "external_packages": [], "cone_lines_max": 7844, "machine_check": "comparator-sandboxed-pass", "lab_scope_note": "For $p>1$, $p\\ne2$, the formalization bounds the least dimension needed to embed every $n$-point subset of real $L_p$ with distortion $D$. For each fixed $D>1$, there is a constant $C=C(p,D)$ such that this dimension lies between $\\log n/\\log(1+2D)$ and $\\exp(C(\\log n)^{\\gamma(p)})$ for every $n\\ge2$, where $\\gamma(p)=2-p$ for $p<2$ and $1-2/p$ for $p>2$. At distortion $1$ and $n\\ge9$, it lies between $\\lfloor(n-1)/4\\rfloor^2$ and $\\binom n2$. The least dimension is attained, and its logarithm divided by $\\log n$ tends to $0$ for $D>1$ and to $2$ for $D=1$.", "lab_docs_url": "https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/docs/094.md", "overview_entry": "family 094 in overview.pdf / CONTENTS.md", "papers": [{"dir": "Subpolynomial-dimension-reduction-in-Lp-September-23-2026", "title": "Subpolynomial dimension reduction in Lp", "url": "https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Subpolynomial-dimension-reduction-in-Lp-September-23-2026"}], "audit_part": 1, "second_reader": "", "referee_verdict": ""}, {"family": "095", "title": "Hyperbolicity cones without semidefinite lifts", "subject": "Convex and metric geometry", "headline": "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.", "verdict": "weaker-statement", "challenges": ["HyperbolicCones"], "review_note": "Only the generalized Lax disproof (cone is not spectrahedral, no auxiliary variables); the Projected Lax disproof (not a spectrahedral shadow, no lift at all) is not stated, and the companion paper shows the formalized cone does have an exact semidefinite lift.", "definitions_to_check": ["`cone := {x | ∀ t : ℂ, (linePolynomial x).aeval t = 0 → t.im = 0 ∧ 0 ≤ t.re}` (closed hyperbolicity cone defined via roots of `t ↦ p(t e - x)`, standard) and the polynomial itself via `matrixValue`/`phi`/`adjugate`; only homogeneous pencils `L : Ambient →ₗ Sym N` are excluded"], "external_packages": [], "cone_lines_max": 5649, "machine_check": "comparator-sandboxed-pass", "lab_scope_note": "The generalized Lax conjecture predicts that every hyperbolicity cone is spectrahedral. The formalized counterexample is an explicit homogeneous polynomial of degree $20$ in $23$ real variables. It is hyperbolic, but its closed hyperbolicity cone cannot be represented as the positive-semidefinite region of any finite real symmetric linear matrix pencil, in any positive matrix size.", "lab_docs_url": "https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/docs/095.md", "overview_entry": "family 095 in overview.pdf / CONTENTS.md", "papers": [{"dir": "A-Nonspectrahedral-Hyperbolicity-Cone-September-24-2026", "title": "A nonspectrahedral hyperbolicity cone", "url": "https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-Nonspectrahedral-Hyperbolicity-Cone-September-24-2026"}], "audit_part": 1, "second_reader": "", "referee_verdict": ""}, {"family": "096", "title": "The Gaussian propeller conjecture", "subject": "Convex and metric geometry", "headline": "Proves that the sum of squared Gaussian first moments of any finite measurable partition is at most $9/(8\\pi)$. In dimension at least two, three planar sectors of angle $2\\pi/3$, extended orthogonally, attain the bound. Combined with the separate Unique Games theorem, this proves NP-hardness of improving the loss factor $(8\\pi/9)(1-1/k)$ for identity-target kernel clustering with fixed $k\\ge3$ on rational centered positive semidefinite inputs.", "verdict": "full", "challenges": ["GaussianPropeller"], "review_note": "Sharp 9/(8 pi) bound and attainment are stated; the NP-hardness of improving the kernel-clustering loss factor (a consequence combined with the Unique Games theorem) is not.", "definitions_to_check": [], "external_packages": [], "cone_lines_max": 11622, "machine_check": "comparator-sandboxed-pass", "lab_scope_note": "The Gaussian propeller problem asks how large the sum of squared Gaussian first moments can be over a partition. The formalized result proves the sharp bound $9/(8\\pi)$ for every positive dimension and every positive number of labelled cells, allowing empty cells and arbitrary masses. In dimension at least two with at least three cells, three planar sectors of angle $120^\\circ$, extended in orthogonal directions, attain equality. The separate Gaussian-maxima and kernel-clustering consequences are not included.", "lab_docs_url": "https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/docs/096.md", "overview_entry": "family 096 in overview.pdf / CONTENTS.md", "papers": [{"dir": "The-Gaussian-Propeller-Bound-in-Every-Dimension-September-24-2026", "title": "The Gaussian propeller bound in every dimension", "url": "https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-Gaussian-Propeller-Bound-in-Every-Dimension-September-24-2026"}], "audit_part": 1, "second_reader": "", "referee_verdict": ""}, {"family": "097", "title": "The Euclidean Steinitz--Bergstr\\\"om conjecture", "subject": "Convex and metric geometry", "headline": "Proves that any finite sequence in the Euclidean unit ball of $\\mathbb R^d$ admits signs keeping every partial sum within $C\\sqrt d$, independently of length. Consequently every zero-sum family can be reordered with the same bound on unsigned partial sums. A matching lower bound gives the optimal order $S_2(d)=\\Theta(\\sqrt d)$.", "verdict": "full", "challenges": ["SteinitzBergstrom"], "review_note": "Signed-prefix and zero-sum reordering bounds with an absolute constant are stated; the matching Omega(sqrt d) lower bound (optimal order) is not.", "definitions_to_check": [], "external_packages": [], "cone_lines_max": 15082, "machine_check": "comparator-sandboxed-pass", "lab_scope_note": "The formalized result gives the Euclidean Steinitz–Bergström bound with one absolute constant $C$. For every finite family of vectors in the unit ball of $\\mathbb R^d$, signs can be chosen so that every prefix in the prescribed order has norm at most $C\\sqrt d$. When the vector sum is zero, a permutation makes every unsigned prefix satisfy the same bound. Repeated and zero vectors are included. The result is existential and does not supply an online algorithm.", "lab_docs_url": "https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/docs/097.md", "overview_entry": "family 097 in overview.pdf / CONTENTS.md", "papers": [{"dir": "The-Euclidean-Steinitz-Bergstrom-theorem-September-24-2026", "title": "The Euclidean Steinitz–Bergström theorem", "url": "https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-Euclidean-Steinitz-Bergstrom-theorem-September-24-2026"}], "audit_part": 1, "second_reader": "", "referee_verdict": ""}, {"family": "098", "title": "A negative answer to the Lang--Plaut problem", "subject": "Convex and metric geometry", "headline": "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.", "verdict": "full", "challenges": ["DoublingHilbert", "CompactBanach"], "review_note": "None.", "definitions_to_check": [], "external_packages": [], "cone_lines_max": 4664, "machine_check": "comparator-sandboxed-pass", "lab_scope_note": "The formalized result gives a doubling subset of real $\\ell_2$, with doubling constant at most $76800$, that admits no bi-Lipschitz embedding into any finite-dimensional Euclidean space at any finite distortion. A companion gives one universal doubling constant such that every infinite-dimensional real Banach space contains a compact doubling subset with no bi-Lipschitz embedding into any finite-dimensional normed space.", "lab_docs_url": "https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/docs/098.md", "overview_entry": "family 098 in overview.pdf / CONTENTS.md", "papers": [{"dir": "A-doubling-Hilbert-subset-with-no-finite-dimensional-bi-Lipschitz-embedding-September-25-2026", "title": "A doubling Hilbert subset with no finite-dimensional bi-Lipschitz embedding", "url": "https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-doubling-Hilbert-subset-with-no-finite-dimensional-bi-Lipschitz-embedding-September-25-2026"}], "audit_part": 1, "second_reader": "", "referee_verdict": ""}, {"family": "099", "title": "The sharp distortion of edit distance into $\\ell_1$", "subject": "Convex and metric geometry", "headline": "Determines the least distortion of embedding edit distance on words of length at most $d$ into real $\\ell_1$: it is $\\exp(\\Theta(\\sqrt{\\log d\\,\\log\\log d}))$. Insertions, deletions and substitutions have unit cost. The constants are uniform over all finite alphabets with at least two symbols, even when the alphabet grows with $d$; binary words already force the lower bound.", "verdict": "full", "challenges": ["EditDistance", "FiniteCircle", "BinaryEditLower", "TreeEdit"], "review_note": "None.", "definitions_to_check": ["Hand-rolled `distortion ρ f := sSup {‖f x - f y‖/ρ x y} * sSup {ρ x y/‖f x - f y‖}` and `leastDistortion := sInf (range distortion)` over injective maps into `lp 1` (standard distortion; junk values only for degenerate sets with fewer than two points)"], "external_packages": [], "cone_lines_max": 25424, "machine_check": "comparator-sandboxed-pass", "lab_scope_note": "The formalization determines the exponential scale of the least $\\ell_1$ distortion of unit-cost edit distance on strings of length at most $d$. For every sufficiently large $d$, uniformly over finite alphabets with at least two symbols, the distortion lies between $\\exp(c\\sqrt{\\log d\\,\\log\\log d})$ and $\\exp(C\\sqrt{\\log d\\,\\log\\log d})$ for absolute constants $c,C>0$. The lower bound has a witness consisting of binary strings of one common length, and the same scale controls the supremum over finite alphabets.\n\nThe formalization gives two finite-circle constructions of binary strings whose least $\\ell_1$ distortion is at least $\\exp(c\\sqrt{\\log d\\,\\log\\log d})$ for all sufficiently large length caps $d$. The witnesses use strings of one common length and include the binary coding transfers with their stated uniform bounds.\n\nIt also gives a finite histogram embedding with distortion at most $\\exp(C\\sqrt{\\log d\\,\\log\\log d})$ for all strings of length at most $d$, uniformly over finite alphabets, including the empty string. Thus the lower and upper exponential scales agree up to absolute constants.\n\nThe formalization gives two lower-bound constructions for the $\\ell_1$ distortion of ordinary edit distance on binary words. For every sufficiently large length cap $d$, each construction supplies a finite set of at least two binary words of one common length at most $d$ whose least $\\ell_1$ distortion is at least $\\exp(c\\sqrt{\\log d\\,\\log\\log d})$ for an absolute $c>0$.\n\nThe selected statements are the binary lower bounds. The paper's constant-distortion binary conversion and the companion upper embedding theorem are outside them.", "lab_docs_url": "https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/docs/099.md", "overview_entry": "family 099 in overview.pdf / CONTENTS.md", "papers": [{"dir": "Edit-Distance-in-l1-Matching-Bounds-up-to-Constants-in-the-Exponent-September-27-2026", "title": "Edit Distance in l1: Matching Bounds up to Constants in the Exponent", "url": "https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Edit-Distance-in-l1-Matching-Bounds-up-to-Constants-in-the-Exponent-September-27-2026"}, {"dir": "Finite-Circle-Obstructions-Binary-Codes-and-Histogram-Embeddings-for-Edit-Distance-September-27-2026", "title": "Finite-Circle Obstructions, Binary Codes, and Histogram Embeddings for Edit Distance", "url": "https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Finite-Circle-Obstructions-Binary-Codes-and-Histogram-Embeddings-for-Edit-Distance-September-27-2026"}, {"dir": "Tree-Constructions-for-the-l1-Distortion-of-Binary-Edit-Distance-September-27-2026", "title": "Tree Constructions for the l1 Distortion of Binary Edit Distance", "url": "https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Tree-Constructions-for-the-l1-Distortion-of-Binary-Edit-Distance-September-27-2026"}], "audit_part": 1, "second_reader": "", "referee_verdict": ""}, {"family": "100", "title": "A counterexample to Bang's cylinder-covering bound", "subject": "Convex and metric geometry", "headline": "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.", "verdict": "full", "challenges": ["CylinderCovering", "TriangularCovering", "RuledCovering"], "review_note": "None for the headline; the aggregate file additionally states ruled-set and radial-sweep approximation results.", "definitions_to_check": ["`CylinderCovering.lean` contains ten `run_cmd Lean.modifyEnv fun env => Lean.Meta.auxLemmasExt.setState env {}` lines (environment-resetting metaprogramming inside a challenge file; a memory/performance hack that does not change statements)"], "external_packages": [], "cone_lines_max": 70623, "machine_check": "comparator-sandboxed-pass", "lab_scope_note": "The half-area cylinder-covering conjecture predicts that a cylinder cover of a convex body has total perpendicular-base area at least half its smallest projection area. The formalization constructs finite covers of a regular tetrahedron by cylinders with compact triangular bases whose total area is strictly below that bound. For the explicit small parameter $\\varepsilon$, the normalized area is $1/2-(13/6000)\\varepsilon^2+O(\\varepsilon^4)$, giving the strict saving.\n\nIt also gives a counterexample to the directionwise normalized half-bound. The broader Comparator file includes the companion ruled-set and radial-sweep approximation results.\n\nThe formalization approximates compact ruled families of line segments by finite cylinder covers with perpendicular-base area at most the integral projection cost plus any prescribed positive error. The selected results cover the stated $C^1$ hyperbolic differential condition, square-zero and nilpotent cases, physical rescalings, logarithmic and product-cell estimates, and singular cases. Compact label sets need not have regular boundary.\n\nThe differential and segment-length hypotheses remain part of each result. The aggregate Comparator file also contains the companion tetrahedron counterexamples and radial-sweep estimates.\n\nThe formalization proves that every regular tetrahedron has a finite cylinder cover whose total perpendicular-base area is strictly less than half its minimum projection area. It supplies covers with parallelogram bases and also proves the directionwise normalized cost is below one half. These are counterexamples to the two cylinder-covering bounds studied in the paper.\n\nThe linked aggregate includes the accompanying explicit angular covers and finite approximation theorems for ruled sets and radial sweeps. The affine extension of the directionwise conclusion to every nondegenerate tetrahedron is outside the selected regular-tetrahedron statements.\n\nThe formalization proves finite triangular-cylinder approximation for radially aligned segment sweeps. For every tagged partition of the parameter interval, it constructs one cylinder per interval with the prescribed radial direction, and the total perpendicular-base area converges to the weighted parameter area as the mesh tends to zero. It also covers the corresponding actual swept set under the stated geometric hypotheses.\n\nThe same aggregate includes finite cylinder covers of regular tetrahedra below half the minimum projection area. These are the radial approximation and covering consequences relevant to the paper.", "lab_docs_url": "https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/docs/100.md", "overview_entry": "family 100 in overview.pdf / CONTENTS.md", "papers": [{"dir": "Finite-angular-cylinder-covers-below-the-half-area-bound-September-27-2026", "title": "Finite angular cylinder covers below the half-area bound", "url": "https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Finite-angular-cylinder-covers-below-the-half-area-bound-September-27-2026"}, {"dir": "Finite-cylinder-approximation-of-ruled-sets-September-27-2026", "title": "Finite cylinder approximation of ruled sets", "url": "https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Finite-cylinder-approximation-of-ruled-sets-September-27-2026"}, {"dir": "Slope-field-perturbations-of-the-two-cylinder-covering-September-27-2026", "title": "Slope-field perturbations of the two-cylinder covering", "url": "https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Slope-field-perturbations-of-the-two-cylinder-covering-September-27-2026"}, {"dir": "Finite-triangular-approximation-of-radial-sweeps-September-27-2026", "title": "Finite triangular approximation of radial sweeps", "url": "https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Finite-triangular-approximation-of-radial-sweeps-September-27-2026"}], "audit_part": 1, "second_reader": "", "referee_verdict": ""}, {"family": "102", "title": "The Unique Games Conjecture and optimal approximation thresholds", "subject": "Theoretical computer science", "headline": "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.", "verdict": "full", "challenges": ["UniqueGamesTheorem", "OptimalMaxCut", "VertexCover", "MinUncut", "DirectedFeedback"], "review_note": "None found; all five hardness statements are formalized as polynomial-time (Mathlib TM2, finite stack alphabets) Karp gap reductions from binary 3SAT (or an arbitrary NP verifier language), which is the standard NP-hardness encoding. Cone is ~190k lines and uses 'Std'/'Lean' (core) only; the OptimalMaxCut axiom_files flag is a comment-only false positive.", "definitions_to_check": ["Complexity notions are Mathlib's `Turing.TM2ComputableInPolyTime` (note: Mathlib's `FinTM2` only forces the input stack alphabet to be finite, so every challenge adds `finiteAlphabet : ∀ k, Finite (computation.tm.Γ k)`), outputs are measured in explicit unary/binary encodings (`gameBits` writes every number in unary); `alphaGW` is defined as an `sInf` over ρ ∈ [-1,1) (equals the GW constant, checked); `InNP` is the certificate definition via `NPVerifier` with a polynomial `witnessBound`"], "external_packages": [], "cone_lines_max": 188149, "machine_check": "comparator-sandboxed-pass", "lab_scope_note": "The Unique Games conjecture asks for hardness of distinguishing nearly satisfiable unique games from games of very small value. The formalized result gives, for every fixed $0<\\varepsilon,\\delta<1/2$, a deterministic polynomial-time reduction from binary 3SAT to nonempty unweighted simple bipartite unique games. Satisfiable inputs have value at least $1-\\varepsilon$ and unsatisfiable inputs have value at most $\\delta$. The alphabet is a fixed $\\mathbb F_2^s$ depending only on the errors, and every constraint is a translation.\n\nThe formalized result establishes hardness of approximating Max-Cut beyond the Goemans–Williamson constant $\\alpha_{\\mathrm{GW}}$. For every fixed $\\alpha_{\\mathrm{GW}}<\\alpha\\le1$, it gives a deterministic polynomial-time reduction from binary 3SAT to a strictly separated Max-Cut gap on finite simple unweighted graphs. The reduction includes the explicit positive integer scaling used in the gap statement.\n\nThe formalized result gives the factor-two hardness threshold for Vertex Cover. For each integer $m\\ge4$, a polynomial-time reduction from binary 3SAT produces simple unweighted graphs with cover density below $1/2+1/m$ on satisfiable inputs and above $1-1/m$ on unsatisfiable inputs. Consequently, any polynomial-time approximation with a fixed factor $1\\le\\alpha<2$ would give a polynomial-time decision algorithm for 3SAT. No assumption that $P\\ne NP$ is built into the statement.\n\nMin-UnCut minimizes the number of edges left uncut by a bipartition. The formalization constructs a deterministic reduction from encoded $3$-SAT formulas to finite simple unweighted Min-UnCut instances with arbitrarily large fixed multiplicative gaps. For every integer $K\\ge2$, satisfiable formulas give optimum at most the output threshold, while unsatisfiable formulas give optimum strictly greater than $K$ times that threshold. Runtime and output length are polynomial for each fixed $K$, establishing hardness for every fixed approximation factor greater than one.\n\nA directed feedback vertex set meets every directed cycle. The formalization proves hardness of approximating the minimum such set within any fixed constant factor. For every real factor $A\\ge1$ and every language in NP, it constructs a polynomial-time gap reduction to unweighted directed graphs with the stated completeness and soundness separation. Thus a fixed-factor polynomial-time approximation would imply a polynomial-time algorithm for every NP language.", "lab_docs_url": "https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/docs/102.md", "overview_entry": "family 102 in overview.pdf / CONTENTS.md", "papers": [{"dir": "The-Unique-Games-Theorem-September-23-2026", "title": "The Unique Games Theorem", "url": "https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-Unique-Games-Theorem-September-23-2026"}, {"dir": "A-Direct-Proof-of-Optimal-Max-Cut-Hardness-September-23-2026", "title": "A Direct Proof of Optimal Max-Cut Hardness", "url": "https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-Direct-Proof-of-Optimal-Max-Cut-Hardness-September-23-2026"}, {"dir": "The-Factor-Two-Hardness-Threshold-for-Vertex-Cover-September-23-2026", "title": "The Factor-Two Hardness Threshold for Vertex Cover", "url": "https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-Factor-Two-Hardness-Threshold-for-Vertex-Cover-September-23-2026"}, {"dir": "Constant-factor-hardness-of-Min-UnCut-September-23-2026", "title": "Constant-factor hardness of Min-UnCut", "url": "https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Constant-factor-hardness-of-Min-UnCut-September-23-2026"}, {"dir": "Constant-factor-hardness-of-directed-feedback-vertex-set-September-23-2026", "title": "Constant-factor hardness of directed feedback vertex set", "url": "https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Constant-factor-hardness-of-directed-feedback-vertex-set-September-23-2026"}], "audit_part": 1, "second_reader": "", "referee_verdict": ""}, {"family": "104", "title": "Quasipolynomial algorithms for mean-payoff, stochastic and parity games", "subject": "Theoretical computer science", "headline": "Gives deterministic algorithms using $2^{O((\\log(L+2))^2)}$ bit operations, for complete binary input length $L$, for ordinary mean-payoff games and two separate extensions. They compute exact values and optimal positional strategies in ordinary games, the nonnegative expectation-of-liminf value set in turn-based stochastic games, and the winning set for nonnegative liminf mean payoff conjoined with parity. Signed rewards, rational chance probabilities, and parity priorities are unrestricted and binary-encoded.", "verdict": "weaker-statement", "challenges": ["TruffetCounterexample", "RandomizedMeanPayoff"], "review_note": "Only a randomized (error 1/8) quasipolynomial algorithm for the winning set; the deterministic algorithm, exact values/optimal positional strategies, turn-based stochastic games and mean-payoff-parity games (two of four index papers) are not stated. TruffetCounterexample is a side negative result.", "definitions_to_check": ["Custom 3-tape randomized Turing machine (`structure Machine`, `step`, `run M code random`, one random bit per step, alphabet `Option Bool`, heads in ℤ) with `T` existentially quantified per input and `time = number of steps`; standard in form but NOT Mathlib's TM model, so 'quasipolynomial bit time' rests on this hand-rolled definition. Mean payoff is `liminf` of running averages and strategies are history-dependent (`Strategy G := List (Fin G.m) → ...`), both standard"], "external_packages": [], "cone_lines_max": 27355, "machine_check": "comparator-sandboxed-pass", "lab_scope_note": "The formalization checks that a specified two-step execution of Truffet's elimination procedure terminates with a feasible but nonoptimal output. It is a finite counterexample to that proposed optimization step.\n\nThe formalized result gives one randomized algorithm for the complete zero-threshold winning set of every finite mean-payoff game with signed binary weights. It succeeds with probability at least $7/8$ and runs in quasipolynomial bit time on every random tape. Self-loops, parallel edges, and history-dependent strategies are allowed, and winning means a nonnegative liminf average payoff.", "lab_docs_url": "https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/docs/104.md", "overview_entry": "family 104 in overview.pdf / CONTENTS.md", "papers": [{"dir": "Deterministic-quasipolynomial-time-mean-payoff-games-September-25-2026", "title": "Deterministic quasipolynomial-time mean-payoff games", "url": "https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Deterministic-quasipolynomial-time-mean-payoff-games-September-25-2026"}, {"dir": "Randomized-quasipolynomial-time-mean-payoff-games-September-25-2026", "title": "Randomized quasipolynomial-time mean-payoff games", "url": "https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Randomized-quasipolynomial-time-mean-payoff-games-September-25-2026"}], "audit_part": 1, "second_reader": "", "referee_verdict": ""}, {"family": "105", "title": "The 2-to-1 Games Conjecture with perfect completeness", "subject": "Theoretical computer science", "headline": "Proves Khot's 2-to-1 Games Conjecture with perfect completeness: for every fixed rational $\\delta\\in(0,1)$, it is NP-hard to distinguish satisfiable games from games whose optimum is at most $\\delta$, on explicit unweighted instances. The alphabet depends only on $\\delta$, and every right-hand label has exactly two preimages under each constraint map.", "verdict": "full", "challenges": ["PerfectCompleteness"], "review_note": "None; NP-hardness is encoded as a polynomial-time Karp gap reduction from binary 3SAT (Mathlib TM2 model, finite stack alphabets).", "definitions_to_check": ["Same Mathlib `TM2ComputableInPolyTime` + `FiniteAlphabet` encoding as family 102; instances are multigraphs (edge list with repeats), outputs written with unary `encodeWord`"], "external_packages": [], "cone_lines_max": 257263, "machine_check": "comparator-pass", "lab_scope_note": "The formalized result establishes perfect-completeness hardness for 2-to-1 games. For every fixed rational $0<\\delta<1$, a deterministic polynomial-time reduction from binary 3SAT produces a nonempty unweighted game of value exactly $1$ on satisfiable inputs and at most $\\delta$ on unsatisfiable inputs. Each constraint projection has exactly two preimages for each output label. The alphabet depends only on $\\delta$, and the runtime is measured in the original input bit length.", "lab_docs_url": "https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/docs/105.md", "overview_entry": "family 105 in overview.pdf / CONTENTS.md", "papers": [{"dir": "Perfect-completeness-for-2-to-1-games-September-23-2026", "title": "Perfect completeness for 2-to-1 games", "url": "https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Perfect-completeness-for-2-to-1-games-September-23-2026"}], "audit_part": 1, "second_reader": "", "referee_verdict": ""}, {"family": "106", "title": "Hardness of coloring three-colorable graphs", "subject": "Theoretical computer science", "headline": "It is NP-hard to color a three-colorable graph using any fixed number $c\\ge3$ of colors. More strongly, for every fixed $0<\\delta<1/3$, a deterministic polynomial-time reduction from $3$SAT produces simple unweighted graphs that are three-colorable in the satisfiable case and have no independent set of size $\\delta n$ otherwise, where $n$ is the number of vertices.", "verdict": "full", "challenges": ["IndependentSets"], "review_note": "Independent-set gap reduction stated for every 00.465$ permits $n^{2+o(1)}$ rectangular multiplication. Further square bounds give $\\omega<2.258$ outside finitely many positive characteristics and $\\omega<2.371054886006746$ over every fixed field.", "verdict": "full", "challenges": ["MatrixMultiplication", "MatrixFields"], "review_note": "ω(C) <= 9/4, α > 0.465 and ω(F) < 2.371054886006746 for every field are stated; 'ω < 2.258 outside finitely many positive characteristics' is not. Exponents are sInf of uniform-constant, non-uniform straight-line program families (standard).", "definitions_to_check": ["`omega F := sInf {τ | AdmissibleExponent F τ}` is only meaningful if the set is nonempty and bounded below (true: τ = 3 admissible, and cost ≥ n² forces τ ≥ 2; junk value 0 otherwise); `complexAlpha := sSup {k ∈ [0,1] | rectangularOmega ℂ k = 2}`; the first-conjunct-only reading of `omega_lt_source_constant` is padded with a pure numeric inequality `2.371054886006746 < 2.371056`"], "external_packages": ["FixedPointTheorems"], "cone_lines_max": 181751, "machine_check": "comparator-pass", "lab_scope_note": "The formalized results bound the complex matrix-multiplication exponent by $\\omega(\\mathbb C)\\le9/4$, the dual exponent by $\\alpha>0.465$, and the rectangular exponent at aspect ratio $0.709$ by $\\omega(\\mathbb C;1,0.709,1)<2.092$. The dual exponent is the supremum of rectangular aspect ratios attainable with exponent $2$. The arithmetic model counts additions, subtractions, and multiplications in finite division-free programs, with arbitrary positive exponent slack. The square bound implies the paper's weaker $2.258$ headline bound.\n\nThe formalized result gives the unconditional bound $\\omega(F)<2.371054886006746$ for every field $F$, including finite fields and fields of positive characteristic. The exponent counts additions, subtractions, and multiplications in finite division-free arithmetic programs, with arbitrary positive exponent slack. No numerical inequalities remain as hypotheses. The result concerns arithmetic complexity, rather than bit complexity or practical crossover sizes.", "lab_docs_url": "https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/docs/107.md", "overview_entry": "family 107 in overview.pdf / CONTENTS.md", "papers": [{"dir": "Complex-Matrix-Multiplication-Below-2.258-and-Rectangular-Bounds-September-24-2026", "title": "Complex Matrix Multiplication Below 2.258 and Rectangular Bounds", "url": "https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Complex-Matrix-Multiplication-Below-2.258-and-Rectangular-Bounds-September-24-2026"}, {"dir": "Staggered-extraction-for-exact-matrix-multiplication-over-every-field-September-24-2026", "title": "Staggered extraction for exact matrix multiplication over every field", "url": "https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Staggered-extraction-for-exact-matrix-multiplication-over-every-field-September-24-2026"}], "audit_part": 1, "second_reader": "", "referee_verdict": ""}, {"family": "108", "title": "A cubic permanent--determinant lower bound", "subject": "Theoretical computer science", "headline": "Proves an $\\Omega(n^3)$ lower bound for the border determinantal complexity of the $n\\times n$ permanent over $\\mathbb C$. Even coefficientwise limits of determinants of affine-linear matrices require matrix size at least $cn^3$, for an absolute $c>0$ and all sufficiently large $n$; the same bound therefore holds for exact representations.", "verdict": "full", "challenges": ["PermanentCubic", "SmoothInitialForm"], "review_note": "None; the algebraic-branching-program consequences are not stated (docs). Border complexity is encoded as sequential coefficientwise limits (equivalent to Zariski closure here).", "definitions_to_check": [], "external_packages": [], "cone_lines_max": 20739, "machine_check": "comparator-sandboxed-pass", "lab_scope_note": "The formalization proves a cubic lower bound for both exact and border determinantal representations of the complex $m\\times m$ permanent. For $m\\ge1408$, every affine-linear determinant representation of size $n$, including coefficientwise limits, satisfies $n\\ge m^3/(5529600e)$.\n\nA general supporting theorem applies to a polynomial in $d\\ge2$ variables whose first nonzero homogeneous Taylor term has degree $r\\ge2$ and no nonzero singular zero. Its exact and border determinantal sizes are at least $(r-1)(d-1)/(4e)$. The paper's algebraic-branching-program consequences are outside these statements.", "lab_docs_url": "https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/docs/108.md", "overview_entry": "family 108 in overview.pdf / CONTENTS.md", "papers": [{"dir": "A-cubic-lower-bound-for-border-determinantal-complexity-of-the-permanent-September-24-2026", "title": "A cubic lower bound for border determinantal complexity of the permanent", "url": "https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-cubic-lower-bound-for-border-determinantal-complexity-of-the-permanent-September-24-2026"}], "audit_part": 1, "second_reader": "", "referee_verdict": ""}, {"family": "110", "title": "Optimal-order randomized $k$-server on arbitrary metrics", "subject": "Theoretical computer science", "headline": "Establishes a randomized competitive ratio $O(\\log^2(k+1))$ for $k$-server on every metric space, matching the worst-case lower-bound order. One policy serves every finite oblivious request sequence, including on infinite unbounded metrics. On finite rational metrics, a uniform implementation has polynomial preprocessing and per-request bit cost in the input length and $\\log(t+1)$ at request $t$, with a finite instance-dependent additive movement constant.", "verdict": "full", "challenges": ["KServer", "UniformKServer"], "review_note": "Both the arbitrary-metric O(log^2 (k+1)) bound and the uniform polynomial-time implementation on finite rational metrics (2 <= k < n) are stated; the matching lower bound and any polynomial bound on the additive constant are not (docs: constant not claimed polynomially bounded).", "definitions_to_check": ["Hand-rolled bit-operation model for the uniform claim: `BitMachine`/`BitAction`/`MachineState` (read-only input tape, `tapes` work tapes over `Option Bool`, `emit`, `yield`), run for exactly `requestBudget c e L t := c * (L + Nat.clog 2 (t+1) + 1)^e` coin flips per request with a deterministic `boot` of `polynomialBudget` steps; not Mathlib's TM model. KServer's `definition_names` list 9 definitions that the solution must reproduce exactly"], "external_packages": [], "cone_lines_max": 50260, "machine_check": "comparator-sandboxed-pass", "lab_scope_note": "The formalization proves an $O((\\log(k+1))^2)$ competitive ratio for randomized $k$-server against oblivious finite request sequences. For every $k\\ge2$, every metric space containing at least $k+1$ points, and every initial configuration, one policy works for all request sequences, including in infinite and unbounded spaces.\n\nThe bound permits a configuration-dependent additive constant, which is zero when the initial server positions are distinct. The universal multiplicative constant is independent of the metric and $k$.\n\nThe formalization constructs one uniform randomized bit algorithm for $k$-server on finite rational metrics, for $2\\le k0$. The language and its polynomial-time algorithm are fixed before $A$ is chosen, while circuits may vary with the input length. Gates have unrestricted finite fan-in and sharing, with free input negations and constants. The result does not assert a fixed exponent $n^{1/2+\\varepsilon}$ or a lower bound for unrestricted depth.", "lab_docs_url": "https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/docs/112.md", "overview_entry": "family 112 in overview.pdf / CONTENTS.md", "papers": [{"dir": "Beyond-the-Square-Root-Exponent-for-Depth-Three-Boolean-Circuits-September-23-2026", "title": "Beyond the Square-Root Exponent for Depth-Three Boolean Circuits", "url": "https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Beyond-the-Square-Root-Exponent-for-Depth-Three-Boolean-Circuits-September-23-2026"}], "audit_part": 1, "second_reader": "", "referee_verdict": ""}, {"family": "113", "title": "Approximate counting and the perfect-matching entropy conjecture", "subject": "Theoretical computer science", "headline": "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.", "verdict": "full", "challenges": ["MatchingFPRAS", "MatchingEntropy", "BinaryMatching", "SingletonLoopMatching", "MatchingEntropyBounds", "TriangleFace"], "review_note": "None for the FPRAS and the pointwise entropy bounds; the sharp face-dimension bound is not stated (TriangleFace is only the triangle-expansion face correspondence).", "definitions_to_check": ["Randomized machine is hand-rolled: `RandomMachine` (single finite transition table on `Turing.Tape (Fin 8)`, one fair coin per tick, tick = one write/move via `Turing.TM0.Stmt`), input encoded sparsely with binary `encodeNat`; `maxMatchingEntropy := sSup (entropy '' feasibleLaws y)` (nonempty and bounded on the matching polytope)"], "external_packages": [], "cone_lines_max": 66246, "machine_check": "comparator-sandboxed-pass", "lab_scope_note": "The formalization gives a fully polynomial randomized approximation scheme for counting perfect matchings in every finite simple undirected graph. For rational $0<\\varepsilon<1$ and $0<\\delta<1/2$, it returns a nonnegative rational estimate with relative error at most $\\varepsilon$ with probability at least $1-\\delta$. If the graph has no perfect matching, every execution returns zero.\n\nThe algorithm is a fixed finite-alphabet randomized machine. Its worst-case running time is polynomial in the binary input length, $\\varepsilon^{-1}$, and $\\log(\\delta^{-1})$, including on unsuccessful random tapes.\n\nFor feasible edge marginals $x$ of perfect matchings in a loopless multigraph on $2m$ vertices, let $F(x)=-\\sum_e x_e\\log x_e$, $B(x)=-\\sum_e(1-x_e)\\log(1-x_e)$, and let $H(x)$ be the maximum entropy of a matching law with those marginals. The formalization proves $F(x)-(2-2/m)B(x)\\le H(x)\\le F(x)$ for $m\\ge1$, including boundary points of the polytope. It retains the earlier coefficient-eight entropy bound and the deterministic counting approximations with factors $512^n$ for loopless graphs and $2^{18n}$ with singleton loops.\n\nA further statement identifies the exact face obtained by expanding a degree-three vertex into a triangle and shows that minimal faces are carried to minimal faces by the expansion. The paper's sharp global face-dimension bound is not part of that selected statement.", "lab_docs_url": "https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/docs/113.md", "overview_entry": "family 113 in overview.pdf / CONTENTS.md", "papers": [{"dir": "A-Fully-Polynomial-Randomized-Approximation-Scheme-for-Perfect-Matchings-in-General-Graphs-September-23-2026", "title": "A Fully Polynomial Randomized Approximation Scheme for Perfect Matchings in General Graphs", "url": "https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-Fully-Polynomial-Randomized-Approximation-Scheme-for-Perfect-Matchings-in-General-Graphs-September-23-2026"}, {"dir": "Entropy-and-Face-Dimension-of-the-Perfect-Matching-Polytope-September-23-2026", "title": "Entropy and Face Dimension of the Perfect-Matching Polytope", "url": "https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Entropy-and-Face-Dimension-of-the-Perfect-Matching-Polytope-September-23-2026"}], "audit_part": 1, "second_reader": "", "referee_verdict": ""}, {"family": "114", "title": "Approximate counting of common integer polymatroid bases", "subject": "Theoretical computer science", "headline": "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.", "verdict": "partial", "challenges": ["CommonBasesFPRAS"], "review_note": "Matroid special case only (independence oracles, equal ranks); the integer-polymatroid FPRAS with binary capacities and rank-value oracles, and the variants for common independent sets of other cardinalities/unequal ranks, are not stated.", "definitions_to_check": ["Hand-rolled oracle register machine: `Instruction` = push/pop/coin/query/halt over `Fin (k+8)` bit-list registers with counters `oracleCalls`/`bitOperations`; a `query` on a register of length ≠ n silently outputs 0 and stops; `ExactOracles` says the oracle answers independence of the characteristic vector exactly; `inputLength` counts n and r in binary while the resource bound is polynomial in n itself"], "external_packages": [], "cone_lines_max": 26126, "machine_check": "comparator-sandboxed-pass", "lab_scope_note": "The formalized result gives a fully polynomial randomized approximation scheme for the number of common bases of two equal-rank matroids on an enumerated finite ground set, using independence oracles. For rational $0<\\varepsilon,\\delta<1$, the nonnegative rational output has relative error at most $\\varepsilon$ with probability at least $1-\\delta$. A zero count produces zero on every execution. Oracle calls and bit operations have polynomial bounds on every random tape in the input size, $\\varepsilon^{-1}$, and $\\log(\\delta^{-1})$.", "lab_docs_url": "https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/docs/114.md", "overview_entry": "family 114 in overview.pdf / CONTENTS.md", "papers": [{"dir": "Approximate-counting-of-common-bases-of-two-matroids-September-23-2026", "title": "Approximate counting of common bases of two matroids", "url": "https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Approximate-counting-of-common-bases-of-two-matroids-September-23-2026"}], "audit_part": 1, "second_reader": "", "referee_verdict": ""}, {"family": "115", "title": "Sampling and counting contingency tables with arbitrary margins", "subject": "Theoretical computer science", "headline": "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.", "verdict": "full", "challenges": ["ContingencyTables"], "review_note": "None material; counting statement assumes equal row/column totals (unequal totals, which also admit no table, are not required to output 0).", "definitions_to_check": ["Machine model is the hand-rolled `RandomMachine` (single-tape Post-Turing table over `Fin 8`, one fair coin per tick) shared with the matching FPRAS file; 'expected polynomial time' is `tsum (1 - haltMass t)`, 'uniform' is `tableMass → 1/card (Table r c)`, 'almost uniform' is total variation `(∑ |tableMass - 1/card|)/2`; `OutputsTable` uses an explicit `@Fin.instInhabited 8 alphabetSize_neZero` instance hack"], "external_packages": [], "cone_lines_max": 191813, "machine_check": "comparator-pass", "lab_scope_note": "The formalization gives an exact uniform sampler for nonnegative integer contingency tables with arbitrary prescribed row and column sums having equal totals. It terminates almost surely, every halted output is feasible, and each table has exactly the uniform limiting probability. Expected bit complexity is polynomial in the dimensions and the binary length of the margins.\n\nIt also gives a sampler with a fixed polynomial time bound on every execution whose total-variation error is at most $2^{-k}$ for requested precision $k\\ge1$. The same Comparator file includes the companion approximation scheme for counting tables with individual cell bounds.\n\nThe formalization gives a fully polynomial randomized approximation scheme for counting nonnegative integer matrices with prescribed row sums, column sums, and individual entry bounds. Both dimensions may vary, the margins and bounds are binary encoded, and zero entry bounds are allowed. For rational $0<\\varepsilon,\\delta<1$, the algorithm returns a nonnegative estimate with relative error at most $\\varepsilon$ with probability at least $1-\\delta$, and returns zero on every execution when no table exists.\n\nThe running time is polynomial in the encoded input size, $\\varepsilon^{-1}$, and $\\log(\\delta^{-1})$ on every random tape. The same Comparator file also contains the companion sampling results for tables without cell bounds.", "lab_docs_url": "https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/docs/115.md", "overview_entry": "family 115 in overview.pdf / CONTENTS.md", "papers": [{"dir": "Exact-Uniform-Sampling-of-Contingency-Tables-with-Arbitrary-Margins-September-24-2026", "title": "Exact Uniform Sampling of Contingency Tables with Arbitrary Margins", "url": "https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Exact-Uniform-Sampling-of-Contingency-Tables-with-Arbitrary-Margins-September-24-2026"}, {"dir": "An-FPRAS-for-Cell-Bounded-Contingency-Tables-September-24-2026", "title": "An FPRAS for Cell-Bounded Contingency Tables", "url": "https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/An-FPRAS-for-Cell-Bounded-Contingency-Tables-September-24-2026"}], "audit_part": 1, "second_reader": "", "referee_verdict": ""}, {"family": "116", "title": "Uniform identity testing for noncommutative formulas", "subject": "Theoretical computer science", "headline": "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 $\\mathbb Q$ also admit polynomial-size hitting lists whenever they have a defined rational-matrix evaluation.", "verdict": "partial", "challenges": ["FormulaHitting", "RationalHitting"], "review_note": "Characteristic-zero existence of one hitting tuple (no complexity claim) and the poly-time rational hitting lists are stated; positive-characteristic constructions and the polynomial-bit-time bound for the single tuple are not.", "definitions_to_check": ["`Hits` quantifies over `Admissible`/`Nonzero` formulas via a hand-rolled relational `Eval` (inverse requires an actual two-sided inverse of the evaluated operand); machine is Mathlib `Turing.TM0.Machine (Fin 4) (Fin (m+1))` with unary input `n,1,s` (time polynomial in n + s, not in a binary length)"], "external_packages": [], "cone_lines_max": 15304, "machine_check": "comparator-sandboxed-pass", "lab_scope_note": "The formalization gives one explicit tuple of rational matrices that simultaneously detects every nonzero division-free noncommutative formula with at most $s$ gates in $n$ variables, for $n,s\\ge1$. Evaluation at that tuple is a nonzero matrix over every characteristic-zero field.\n\nThe selected theorem states this universal hitting property. The paper's deterministic polynomial bit-construction bound and matrix-dimension bound $O(ns^2)$ are not separately asserted in it.\n\nThe formalized result constructs polynomial-size hitting lists for noncommutative rational formulas with rational constants, addition, multiplication, and inverse gates. Given the number of variables and a formula-size bound, one deterministic polynomial-time machine outputs rational matrix tuples of a common positive dimension. Every admissible nonzero formula within the size bound evaluates to an invertible matrix on some listed tuple. The matrix dimension, complete binary output length, and running time are all polynomially bounded.", "lab_docs_url": "https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/docs/116.md", "overview_entry": "family 116 in overview.pdf / CONTENTS.md", "papers": [{"dir": "One-Rational-Matrix-Hitting-Point-for-Noncommutative-Formulas-September-24-2026", "title": "One Rational Matrix Hitting Point for Noncommutative Formulas", "url": "https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/One-Rational-Matrix-Hitting-Point-for-Noncommutative-Formulas-September-24-2026"}, {"dir": "Polynomial-Hitting-Lists-for-Noncommutative-Rational-Formulas-September-24-2026", "title": "Polynomial Hitting Lists for Noncommutative Rational Formulas", "url": "https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Polynomial-Hitting-Lists-for-Noncommutative-Rational-Formulas-September-24-2026"}], "audit_part": 1, "second_reader": "", "referee_verdict": ""}, {"family": "117", "title": "Uniform sparsest cut: hardness and semidefinite gaps", "subject": "Theoretical computer science", "headline": "Proves that approximating Uniform Sparsest Cut within any fixed constant factor is NP-hard, even with nonnegative rational capacities and unit demands. The Goemans--Linial semidefinite relaxation also has integrality gaps of order at least $\\sqrt{\\log n}/(\\log\\log n)^3$, approaching the square-root-logarithmic upper bound.", "verdict": "partial", "challenges": ["UniformSparsestCut"], "review_note": "Only the Goemans-Linial integrality gap (real capacities, along a sequence of n); the NP-hardness of constant-factor approximation (the summary's first claim) is not stated.", "definitions_to_check": ["`NegativeType d` is `∃ x : Fin n → EuclideanSpace ℝ (Fin n), d i j = ‖x i - x j‖^2` plus triangle inequalities (n dimensions suffice, standard); `OPT`/`glValue` are `sInf`s over sets that must be nonempty and bounded below (true)"], "external_packages": [], "cone_lines_max": 9934, "machine_check": "comparator-sandboxed-pass", "lab_scope_note": "The formalized result gives integrality gaps for the Goemans–Linial relaxation of uniform sparsest cut. Along a sequence of instance sizes $n\\to\\infty$, the ratio of the integral optimum to the positive relaxation optimum is at least $c\\sqrt{\\log n}/(\\log\\log n)^3$ for one absolute $c>0$. Every unordered pair has unit demand, and the relaxation uses squared Euclidean distances satisfying the triangle inequalities. The result is an existential sequence, not a bound for every size or a computational hardness theorem.", "lab_docs_url": "https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/docs/117.md", "overview_entry": "family 117 in overview.pdf / CONTENTS.md", "papers": [{"dir": "Near-square-root-logarithmic-integrality-gaps-for-uniform-sparsest-cut-September-24-2026", "title": "Near-square-root logarithmic integrality gaps for uniform sparsest cut", "url": "https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Near-square-root-logarithmic-integrality-gaps-for-uniform-sparsest-cut-September-24-2026"}], "audit_part": 1, "second_reader": "", "referee_verdict": ""}, {"family": "118", "title": "Unbounded bin-packing gaps and the modified integer round-up conjecture", "subject": "Theoretical computer science", "headline": "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.", "verdict": "full", "challenges": ["BinPackingGap"], "review_note": "None; P/NP are defined by Mathlib TM2 poly-time deciders and the certificate definition (`InNP`).", "definitions_to_check": ["`PEqualsNP := ClassP = ClassNP` with `ClassP` via `TM2ComputableInPolyTime` deciders (+ finite alphabet) and `ClassNP` via `CookLevin.InNP` certificate verifiers; LP values are `sInf` over fractional covers (nonempty, bounded below)"], "external_packages": [], "cone_lines_max": 158527, "machine_check": "none", "lab_scope_note": "The formalized results rule out a universal additive bound for the configuration linear program in bin packing. For every integer $c\\ge0$, there are an integer $B$ and a rational instance with $5B$ items whose individual-copy and size-type configuration-LP values both equal $B$, but whose integral optimum is greater than $B+c$. Distinguishing a packing in $B$ bins from the absence of one in $B+c$ bins is NP-hard for each fixed $c$. A deterministic polynomial-time algorithm with a fixed absolute additive allowance exists exactly when $P=NP$; under that equality, an optimal algorithm is constructed.", "lab_docs_url": "https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/docs/118.md", "overview_entry": "family 118 in overview.pdf / CONTENTS.md", "papers": [{"dir": "Additive-hardness-and-unbounded-configuration-gaps-in-bin-packing-September-24-2026", "title": "Additive hardness and unbounded configuration gaps in bin packing", "url": "https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Additive-hardness-and-unbounded-configuration-gaps-in-bin-packing-September-24-2026"}], "audit_part": 1, "second_reader": "", "referee_verdict": ""}, {"family": "119", "title": "The Courtade--Kumar and Hellinger conjectures", "subject": "Theoretical computer science", "headline": "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.", "verdict": "partial", "challenges": ["CourtadeKumar", "SoftChannel204"], "review_note": "Courtade-Kumar (with attainment) and the soft/biased-Boolean mutual-information contraction are stated; the Hellinger conjecture is not.", "definitions_to_check": ["Information-theoretic quantities are re-implemented by hand (`mutualInformation` with an `if 0 < joint` guard and natural logs divided by `ell := Real.log 2`, `psi`, `H`, `psiInv := Function.invFunOn psi (Set.Icc 0 1)`, `production`, `hybrid`), not Mathlib objects; the SoftChannel statement also bundles seven properties (FullMain)"], "external_packages": [], "cone_lines_max": 11590, "machine_check": "comparator-sandboxed-pass", "lab_scope_note": "The formalization proves sharp contraction of the information carried by a binary channel under independent symmetric noise on a uniform discrete cube. At each fixed initial information level, a noisy coordinate channel attains the maximum retained information. It also proves the refined Boolean bound that accounts for output bias, with strict improvement for nonconstant biased outputs at nonzero noise correlation, and the selected mean-dependent entropy-production inequality.\n\nFor Boolean functions this includes the Courtade–Kumar inequality $I(f(X);Y)\\le1-h_2(\\varepsilon)$ in bits at crossover probability $0\\le\\varepsilon\\le1/2$, where $h_2$ is binary entropy. A coordinate and its complement attain equality.", "lab_docs_url": "https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/docs/119.md", "overview_entry": "family 119 in overview.pdf / CONTENTS.md", "papers": [{"dir": "Sharp-binary-information-contraction-on-the-discrete-cube-September-24-2026", "title": "Sharp binary-information contraction on the discrete cube", "url": "https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Sharp-binary-information-contraction-on-the-discrete-cube-September-24-2026"}], "audit_part": 1, "second_reader": "", "referee_verdict": ""}, {"family": "121", "title": "Almost-linear expected-time approximation of edit distance", "subject": "Theoretical computer science", "headline": "For every fixed rational $\\varepsilon\\in(0,1)$, gives a randomized $(1+\\varepsilon)$ approximation to unit-cost edit distance in worst-case expected time $N^{1+o(1)}$, with success probability at least $2/3$. The strings have total length $N$ and polynomially bounded integer symbols. This is an asymptotic guarantee at fixed accuracy.", "verdict": "full", "challenges": ["EditApproximation"], "review_note": "Statement matches the summary (2/3 success, (1+eps) estimate, expected work (N+2)^(1+eta), poly space), but the 'expected time' is a hand-instrumented work counter in a 19k-line bespoke model with no link to a RAM/TM semantics; it should be treated as unaudited for the complexity claim (the correctness/probability part is meaningful). Not audited line by line.", "definitions_to_check": ["Bespoke cost model: `integerBinaryWorkRawBand := computedFullOutputWork ...` built from `...WithWork` functions with hard-coded charges (e.g. `def trimBitWordWithWork : List Bool → List Bool × ℕ | bit :: bits => ... (…, tail.2 + 3)`, `queryInsertWithWork ... (…, test.2 + 2)`) and `inductive BitQuery | read (key) (next) | charge (work : ℕ) (next)`; 'random' is a `Measure` (`computedFullLaw`, `rejectionRuntimeArrayLaw`), not random bits; the integer-symbol coding `integerSymbolCode` and `BinaryFraction` for ε are custom; the file also carries `attribute [local instance]` plumbing and ~1,200 declarations that the theorem statement silently depends on"], "external_packages": [], "cone_lines_max": 137267, "machine_check": "none", "lab_scope_note": "The formalization gives a randomized $(1+\\varepsilon)$-approximation for unit-cost edit distance between explicitly stored integer strings, for every fixed rational $0<\\varepsilon<1$. Its estimate is at least the true distance and at most $(1+\\varepsilon)$ times it with probability at least $2/3$; equal strings return zero on every execution.\n\nFor total length at most $N$ and integer symbols bounded by a fixed polynomial in $N$, expected work is at most $(N+2)^{1+\\eta}$ for every fixed $\\eta>0$ and all sufficiently large $N$. Space is polynomially bounded on every execution.", "lab_docs_url": "https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/docs/121.md", "overview_entry": "family 121 in overview.pdf / CONTENTS.md", "papers": [{"dir": "An-Almost-Linear-Approximation-Scheme-for-Edit-Distance-September-24-2026", "title": "An Almost-Linear Approximation Scheme for Edit Distance", "url": "https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/An-Almost-Linear-Approximation-Scheme-for-Edit-Distance-September-24-2026"}], "audit_part": 1, "second_reader": "", "referee_verdict": ""}, {"family": "122", "title": "Superpolynomial lower bounds and quasipolynomial reconstruction from deletion traces", "subject": "Theoretical computer science", "headline": "At every fixed deletion probability in $(0,1)$, reconstructing an arbitrary length-$n$ binary string requires $n^{\\Omega(\\log\\log n)}$ independent traces, ruling out polynomial-sample reconstruction. A uniform decoder achieves quasipolynomial sample and running-time bounds for known fixed rational retention probabilities. When the deletion probability is at most $n^{-\\varepsilon}$ for fixed $\\varepsilon>0$, both bounds become polynomial in the input and parameter encoding.", "verdict": "partial", "challenges": ["TraceReconstruction"], "review_note": "Only the lower bounds (superpolynomial and n^{c log(q^3 log n)}) are stated; the quasipolynomial uniform decoder and the polynomial small-deletion-probability regime are not.", "definitions_to_check": ["`sampleComplexity n q s := sInf {m | ∃ estimator A, IsEstimator A ∧ ∀ x, s ≤ reconstructionSuccess q A x}` in ℝ≥0∞ (estimator = arbitrary map from m traces to a probability vector over words, success averaged exactly over deletion masks; standard)"], "external_packages": [], "cone_lines_max": 33984, "machine_check": "comparator-sandboxed-pass", "lab_scope_note": "The formalization proves superpolynomial sample lower bounds for exact worst-case reconstruction of binary words from independent deletion traces, with unrestricted computation and any fixed positive success probability. For every fixed deletion probability $q\\in(0,1)$, sample complexity grows faster than every fixed power of the word length; the minimum total-variation distance between two one-trace laws decays faster than every inverse power.\n\nMore quantitatively, along sequences with $q^3\\log n\\to\\infty$, the required number of samples is at least $n^{c\\log(q^3\\log n)}$ eventually for every fixed $00$. Assuming $P\\ne NP$, the optimal infimum approximation factor is $1+2/e$.", "verdict": "full", "challenges": ["KMedianRecovery", "KMedianRefinedRecovery", "KMedianThreshold", "MetricKMedian"], "review_note": "None; the hardness half is proven inside the `threshold` theorem under `PNeNP` (as the summary says 'assuming P != NP').", "definitions_to_check": ["Complexity classes are hand-defined from Mathlib `TM2ComputableInPolyTime` with a `FinitePolyTime` finite-alphabet clause: `P`, `NP` (certificate form with `encodePair`) and `PNeNP := P ≠ NP`; `approximationFactors` is a set of reals whose `sInf` must be nonempty/bounded below (true: ≥ 1)"], "external_packages": [], "cone_lines_max": 72493, "machine_check": "comparator-sandboxed-pass", "lab_scope_note": "The formalization gives an exact-budget recovery algorithm for metric $k$-median on the stated polynomially bounded integral metrics. Suppose a supplied anchor represents all but logarithmically many comparison clusters by distinct proxies, with total proxy cost at most the comparison cost plus a sufficiently small relative error. The algorithm always opens at most $k$ facilities and, with any requested confidence, attains a $(1+2/e+\\varepsilon)$ factor relative to those comparison centers in polynomial time.\n\nThe linked final approximation result applies to arbitrary finite rational metrics. For one absolute $\\sigma>0$, a polynomial-time fair-bit algorithm always returns a feasible solution, has expected cost at most $(2-\\sigma)\\mathrm{OPT}$, and attains the same factor with arbitrarily high polynomial confidence.\n\nThe formalization gives, for every fixed $\\varepsilon>0$, a deterministic polynomial-time $(1+2/e+\\varepsilon)$-approximation for metric $k$-median on finite rational metrics with specified candidate facilities. Every output is a nonempty subset of the candidate facilities and opens at most $k$ of them.\n\nAssuming $P\\ne NP$, it also proves that the infimum of all polynomial-time approximation factors in this model is exactly $1+2/e$.", "lab_docs_url": "https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/docs/125.md", "overview_entry": "family 125 in overview.pdf / CONTENTS.md", "papers": [{"dir": "Single-Exponential-Recovery-and-Bounded-Price-Strictness-for-Metric-k-Median-September-24-2026", "title": "Single-exponential recovery and bounded-price strictness for metric k-median", "url": "https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Single-Exponential-Recovery-and-Bounded-Price-Strictness-for-Metric-k-Median-September-24-2026"}, {"dir": "The-Approximation-Threshold-for-Metric-k-Median-September-24-2026", "title": "The approximation threshold for metric k-median", "url": "https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-Approximation-Threshold-for-Metric-k-Median-September-24-2026"}], "audit_part": 1, "second_reader": "", "referee_verdict": ""}, {"family": "126", "title": "Exponential semidefinite complexity of perfect matching", "subject": "Theoretical computer science", "headline": "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 $0<\\rho<1$, allowing arbitrary real positive semidefinite factors.", "verdict": "weaker-statement", "challenges": ["MatchingAffineLift", "MatchingPSD"], "review_note": "Superpolynomial (n^C for every C) lower bounds for exact affine SDP lifts and the unshifted PSD rank only; exponential size and the fixed-shift (00$, the PSD rank of the selected perfect-matching slack matrix exceeds $n^C$ for all sufficiently large even $n$. Every exact affine semidefinite lift of the perfect-matching polytope likewise requires matrix size greater than $n^C$.\n\nThese selected statements give superpolynomial growth. They do not state the paper's exponential bound for every fixed positive shift.", "lab_docs_url": "https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/docs/126.md", "overview_entry": "family 126 in overview.pdf / CONTENTS.md", "papers": [{"dir": "Exponential-PSD-rank-of-positively-shifted-matching-matrices-October-5-2026", "title": "Exponential PSD rank of positively shifted matching matrices", "url": "https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Exponential-PSD-rank-of-positively-shifted-matching-matrices-October-5-2026"}], "audit_part": 1, "second_reader": "", "referee_verdict": ""}, {"family": "127", "title": "The asymptotic Gotsman--Linial conjecture", "subject": "Theoretical computer science", "headline": "Proves that a degree-at-most-$d$ polynomial threshold function on the uniform $n$-dimensional Boolean cube has average sensitivity at most $8d\\sqrt n$, uniformly for $1\\le d\\le n$. Average sensitivity counts expected output changes under single-bit flips. This establishes the asymptotic Gotsman--Linial conjecture, allowing polynomial zeros with $\\operatorname{sign}(0)=1$.", "verdict": "full", "challenges": ["GotsmanLinial"], "review_note": "None; the exact-extremizer form of the Gotsman-Linial conjecture is not claimed.", "definitions_to_check": [], "external_packages": [], "cone_lines_max": 4022, "machine_check": "comparator-sandboxed-pass", "lab_scope_note": "The formalized result proves the average-sensitivity bound $8d\\sqrt n$ for every Boolean threshold function defined by a real multilinear polynomial of degree at most $d$ on the $n$-dimensional cube. The sign convention assigns value $1$ at zero. This establishes the stated asymptotic bound, rather than the separate conjecture identifying exact extremizers.", "lab_docs_url": "https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/docs/127.md", "overview_entry": "family 127 in overview.pdf / CONTENTS.md", "papers": [{"dir": "Average-Sensitivity-of-Polynomial-Threshold-Functions-September-25-2026", "title": "Average sensitivity of polynomial threshold functions", "url": "https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Average-Sensitivity-of-Polynomial-Threshold-Functions-September-25-2026"}], "audit_part": 1, "second_reader": "", "referee_verdict": ""}, {"family": "128", "title": "A factor-two approximation for shortest common superstring", "subject": "Theoretical computer science", "headline": "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.", "verdict": "full", "challenges": ["Superstring"], "review_note": "None.", "definitions_to_check": ["Polynomial time via Mathlib `TM2ComputableInPolyTime` plus a finite-alphabet clause (standard for this repo); symbols are arbitrary bit lists so the alphabet is unbounded, as the summary intends"], "external_packages": [], "cone_lines_max": 20361, "machine_check": "comparator-sandboxed-pass", "lab_scope_note": "The shortest common superstring problem asks for the shortest string containing each input string as a contiguous substring. The formalized result gives one deterministic polynomial-time algorithm whose output is a common superstring of length at most twice the unrestricted optimum. It applies to every finite list of explicitly encoded strings, with running time measured in input bits and the approximation ratio measured in symbol length.", "lab_docs_url": "https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/docs/128.md", "overview_entry": "family 128 in overview.pdf / CONTENTS.md", "papers": [{"dir": "A-Polynomial-Time-2-Approximation-for-Shortest-Common-Superstring-September-24-2026", "title": "A Polynomial-Time 2-Approximation for Shortest Common Superstring", "url": "https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-Polynomial-Time-2-Approximation-for-Shortest-Common-Superstring-September-24-2026"}], "audit_part": 1, "second_reader": "", "referee_verdict": ""}, {"family": "129", "title": "Exponential state costs for two-way automata", "subject": "Theoretical computer science", "headline": "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.", "verdict": "full", "challenges": ["TwoWayComplementation", "TwoWayDeterminization", "OneWayLiveness"], "review_note": "None; alphabet is `SetRel (Fin (n-2)) (Fin (n-2))` (resp. `BRel (Fin h)`), so no alphabet-independent polynomial bound, exactly as the summary says.", "definitions_to_check": ["Hand-rolled 2-way automata (`TwoNFA`, `DMachine`, `NMachine`) with endmarkers, stay moves and acceptance by an existential finite run (`FiniteRun positive`), a convention that differs from textbook halting-acceptance (documented in the docs and handled via the `positive` flag); `((n - 4) / 127)` is natural-number floor division in the exponent"], "external_packages": [], "cone_lines_max": 8477, "machine_check": "comparator-sandboxed-pass", "lab_scope_note": "The paper asks how many states are needed to complement or determinize two-way nondeterministic finite automata. The formalization gives, for every $n\\ge4$, an explicit $n$-state automaton whose complement requires at least $\\tfrac12 2^{\\lfloor(n-4)/127\\rfloor}-1$ states. For the same family of relational languages and every $n\\ge131$, every equivalent deterministic two-way automaton requires at least $\\tfrac12 2^{\\lfloor(n-4)/127\\rfloor}$ states.\n\nThe alphabet grows with $n$, so neither complementation nor determinization has a polynomial state bound uniform over alphabets. The model has two endmarkers, left, right, and stay moves, and acceptance by a finite run. The separate results about one-way liveness and the Sakoda–Sipser problem are outside this scope.\n\nThe formalization gives a family of one-way nondeterministic automata with $h+3$ states whose languages require exponentially many states for two-way deterministic recognition. For every $h\\ge2$, a deterministic recognizer with $s$ states satisfies $2^{\\lfloor(h-2)/31\\rfloor}\\le4(s+2)^2$ under both acceptance conventions considered in the paper; one convention has the sharper factor $4(s+1)^2$. The alphabet may grow with $h$, so no alphabet-independent polynomial simulation bound holds.", "lab_docs_url": "https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/docs/129.md", "overview_entry": "family 129 in overview.pdf / CONTENTS.md", "papers": [{"dir": "An-exponential-state-lower-bound-for-two-way-nondeterministic-complementation-September-25-2026", "title": "An exponential state lower bound for two-way nondeterministic complementation", "url": "https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/An-exponential-state-lower-bound-for-two-way-nondeterministic-complementation-September-25-2026"}, {"dir": "An-exponential-two-way-deterministic-state-lower-bound-for-one-way-liveness-September-25-2026", "title": "An exponential two-way deterministic state lower bound for one-way liveness", "url": "https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/An-exponential-two-way-deterministic-state-lower-bound-for-one-way-liveness-September-25-2026"}], "audit_part": 1, "second_reader": "", "referee_verdict": ""}, {"family": "130", "title": "Fourier transforms below $n\\log n$", "subject": "Theoretical computer science", "headline": "Gives a deterministic length-$n$ discrete Fourier transform algorithm using $O(n(\\log n)^{1-\\delta})$ operations for every $n$, with explicit $\\delta=10^{-13}$. The model uses exact complex arithmetic, unrestricted coefficients and a supplied root of unity, and counts scalar preparation and logarithmic-word indexing.", "verdict": "weaker-statement", "challenges": ["ExactFourier"], "review_note": "Existence of exact DFT circuits with < c n log2 n gates for infinitely many n (every c > 0) only; no O(n (log n)^{1-delta}) bound for every n, no explicit delta, and the non-uniform circuit model charges nothing for preparing the scalars (the summary's model counts scalar preparation and indexing).", "definitions_to_check": ["Non-uniform scalar-circuit model: `Gate := add | sub | scale (c : ℂ)` with arbitrary complex constants charged one gate each and no cost for creating `c` (e.g. roots of unity); outputs name existing wires (`outputs : Fin n → Fin (n + 1 + size)`); `fourierMatrix n j k := zeta n ^ (j*k)`"], "external_packages": [], "cone_lines_max": 19103, "machine_check": "comparator-sandboxed-pass", "lab_scope_note": "The formalized result gives exact discrete Fourier transforms with arbitrarily small normalized circuit cost along an unbounded sequence of lengths. For every $c>0$ and every cutoff $N_0\\ge2$, some $n\\ge N_0$ has a circuit computing the unnormalized DFT with fewer than $cn\\log_2 n$ gates. Addition, subtraction, and multiplication by a predetermined complex scalar each cost one gate; diagonal scalings are charged. The result is subsequential, with no all-length, bounded-coefficient, conditioning, or bit-complexity claim.", "lab_docs_url": "https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/docs/130.md", "overview_entry": "family 130 in overview.pdf / CONTENTS.md", "papers": [{"dir": "Finite-tensor-savings-and-exact-Fourier-circuits-September-25-2026", "title": "Finite tensor savings and exact Fourier circuits", "url": "https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Finite-tensor-savings-and-exact-Fourier-circuits-September-25-2026"}], "audit_part": 1, "second_reader": "", "referee_verdict": ""}, {"family": "131", "title": "Polynomial mixing of graph switches with prescribed degrees", "subject": "Theoretical computer science", "headline": "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.", "verdict": "full", "challenges": ["SwitchChain"], "review_note": "Mixing time <= 2 n^8 (TV 1/4, worst-case start), spectral gap, TV bound and connectivity are stated; the exact uniform sampler (expected polynomial bit time) is not.", "definitions_to_check": ["Mixing is defined by hand: `kernelPow` (matrix powers via sums over `allGraphStates`), `tvDistanceFrom := (1/2) ∑ |P^t(G,H) - 1/|Ω||`, `MixedAt`, `mixingTime := Nat.find h` (the proof argument `h` is part of the statement), `hasSpectralGapAtLeast` via Dirichlet form; standard in content, no Mathlib Markov-chain objects"], "external_packages": [], "cone_lines_max": 22691, "machine_check": "comparator-sandboxed-pass", "lab_scope_note": "The Kannan–Tetali–Vempala conjecture asks for polynomial mixing of the switch chain for every graphical degree sequence. The formalization proves the simple undirected case: for $n\\ge4$, the lazy chain that proposes switches on four vertices has total-variation mixing time at distance $1/4$ at most $2n^8$. It also proves switch connectivity, the stated exponential total-variation bound, and a spectral gap of at least $1/(24n^2\\binom n4)$ when more than one graph realizes the degree sequence.\n\nThe paper's exactly uniform sampling algorithm is outside these selected chain estimates.", "lab_docs_url": "https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/docs/131.md", "overview_entry": "family 131 in overview.pdf / CONTENTS.md", "papers": [{"dir": "Polynomial-Mixing-of-the-Switch-Chain-for-Every-Graphical-Degree-Sequence-September-25-2026", "title": "Polynomial mixing of the switch chain for every graphical degree sequence", "url": "https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Polynomial-Mixing-of-the-Switch-Chain-for-Every-Graphical-Degree-Sequence-September-25-2026"}], "audit_part": 1, "second_reader": "", "referee_verdict": ""}, {"family": "132", "title": "A counterexample to the quadratic sensitivity conjecture", "subject": "Theoretical computer science", "headline": "Constructs total Boolean functions with block sensitivity $\\operatorname{bs}(f)\\ge s(f)^\\alpha$ for a fixed $\\alpha>2$, 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.", "verdict": "weaker-statement", "challenges": ["SensitivitySeparation"], "review_note": "Unbounded ratio bs/s^2 only; the fixed exponent alpha > 2 (bs >= s^alpha) of the summary is not stated in the challenge even though the docs claim it.", "definitions_to_check": ["`blockSensitivityAt f x := Finset.univ.sup (fun blocks => if pairwise disjoint nonempty sensitive blocks then blocks.card else 0)` and `sensitivity := sup of sensitivityAt` are standard; the ratio is real division, so `sensitivity f = 0` would make the inequality false rather than vacuous"], "external_packages": [], "cone_lines_max": 3836, "machine_check": "comparator-sandboxed-pass", "lab_scope_note": "The formalized result disproves a universal quadratic bound of block sensitivity by sensitivity for total Boolean functions. For every integer $d\\ge1$, it constructs a nonconstant function with $\\mathrm{bs}(f)/s(f)^2\\ge2^d/(4(d+2)^2)$, making the ratio unbounded. The formalization also gives a fixed exponent $\\alpha>2$ and a sequence with $s(f)^\\alpha\\le\\mathrm{bs}(f,0)$ while the latter tends to infinity; here $\\mathrm{bs}(f,0)$ is block sensitivity at the all-zero input.", "lab_docs_url": "https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/docs/132.md", "overview_entry": "family 132 in overview.pdf / CONTENTS.md", "papers": [{"dir": "A-superquadratic-separation-between-sensitivity-and-block-sensitivity-September-25-2026", "title": "A superquadratic separation between sensitivity and block sensitivity", "url": "https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-superquadratic-separation-between-sensitivity-and-block-sensitivity-September-25-2026"}], "audit_part": 1, "second_reader": "", "referee_verdict": ""}, {"family": "133", "title": "The complexity of Weisfeiler--Leman refinement", "subject": "Theoretical computer science", "headline": "Proves unconditional $n^{\\Omega(k)}$ deterministic time lower bounds for joint and separate $k$-dimensional Weisfeiler--Leman equivalence, for sufficiently large fixed $k$ in the specified sequential adjacency-matrix models. With dimension as input, joint equivalence is EXPTIME-complete even on subcubic graphs; deciding whether refinement identifies a graph is also EXPTIME-complete.", "verdict": "full", "challenges": ["ParityLifts", "WLIdentification", "WeisfeilerLeman", "VariableWL"], "review_note": "None; all three headline claims are stated. The conditional running-time exclusions are not (docs).", "definitions_to_check": ["Hand-rolled computation models: `TM` (multi-tape, fixed finite tables, halting states 1/2), a `RAM` with logarithmic word width (`WordWidth.logarithmic`), memory `Word w → Word w`, `compute (op : WordOperation R)` where a whole word operation (any map implemented by a TM within `constant * (w+1)^exponent`) counts as one instruction and one `step` runs a finite instruction tree; `InEXPTIME` (`2^(c (n+1)^d)` steps), `PolytimeReduces`/`PolytimeManyOne`, `EXPTIMEComplete`; WL colors are canonical nested multisets (`Color c k t`) with `Equivalent := ∀ t, histogram G t = histogram H t`"], "external_packages": [], "cone_lines_max": 24602, "machine_check": "comparator-sandboxed-pass", "lab_scope_note": "The formalization gives the paper's parity-lift graph construction for every Weisfeiler–Leman dimension $k\\ge4$, under both joint and separate conventions. From a finite choice system it constructs two equal-size uncolored graphs that are Weisfeiler–Leman equivalent exactly when the system has no successful compatible choice. The same equivalence holds for the pure variant.\n\nA successful choice is detected by the specified early round: round two in the joint convention and round one in the separate convention. The graph-size bound is linear in the choice-system size with a factor depending on $k$. The paper's conditional running-time exclusions are outside these selected construction statements.\n\nThe formalization proves that deciding whether Weisfeiler–Leman refinement of a supplied dimension identifies a graph is EXPTIME-complete under polynomial-time many-one reductions. The input is a nonempty finite simple uncolored graph encoded by its adjacency matrix together with a positive binary-encoded dimension. Identification quantifies over every comparison graph, and the statement includes both the exponential-time upper bound and hardness.\n\nThe formalization proves an unconditional deterministic time lower bound for deciding $k$-dimensional Weisfeiler–Leman equivalence. For every sufficiently large fixed $k$, every deciding algorithm in the stated multitape Turing-machine or sequential logarithmic-word RAM model requires at least $n^{ck}$ worst-case time for all sufficiently large graph orders $n$, for one absolute $c>0$.\n\nThe bound covers both joint and separate replacement conventions and remains valid for explicit adjacency-matrix inputs restricted to simple connected uncolored graphs of diameter at most two.\n\nThe formalization proves EXPTIME-completeness of deciding joint-update $k$-dimensional Weisfeiler–Leman equivalence when $k\\ge2$ is part of the input in binary and the graphs are encoded by explicit adjacency matrices. It proves both the general result and the restriction to connected simple uncolored graphs of equal positive order and maximum degree at most three. Completeness uses polynomial-time many-one reductions.", "lab_docs_url": "https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/docs/133.md", "overview_entry": "family 133 in overview.pdf / CONTENTS.md", "papers": [{"dir": "Parity-lifts-and-bounded-treewidth-witnesses-for-Weisfeiler-Leman-equivalence-September-25-2026", "title": "Parity lifts and bounded-treewidth witnesses for Weisfeiler–Leman equivalence", "url": "https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Parity-lifts-and-bounded-treewidth-witnesses-for-Weisfeiler-Leman-equivalence-September-25-2026"}, {"dir": "The-complexity-of-identifying-a-graph-by-Weisfeiler-Leman-refinement-September-25-2026", "title": "The complexity of identifying a graph by Weisfeiler–Leman refinement", "url": "https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-complexity-of-identifying-a-graph-by-Weisfeiler-Leman-refinement-September-25-2026"}, {"dir": "Unconditional-time-lower-bounds-for-Weisfeiler-Leman-equivalence-September-25-2026", "title": "Unconditional time lower bounds for Weisfeiler–Leman equivalence", "url": "https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Unconditional-time-lower-bounds-for-Weisfeiler-Leman-equivalence-September-25-2026"}, {"dir": "Variable-dimension-Weisfeiler-Leman-equivalence-on-general-and-subcubic-graphs-September-25-2026", "title": "Variable-dimension Weisfeiler–Leman equivalence on general and subcubic graphs", "url": "https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Variable-dimension-Weisfeiler-Leman-equivalence-on-general-and-subcubic-graphs-September-25-2026"}], "audit_part": 1, "second_reader": "", "referee_verdict": ""}, {"family": "134", "title": "Generalized star height at most three", "subject": "Theoretical computer science", "headline": "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.", "verdict": "full", "challenges": ["GeneralizedStarHeight"], "review_note": "None.", "definitions_to_check": [], "external_packages": [], "cone_lines_max": 11956, "machine_check": "comparator-sandboxed-pass", "lab_scope_note": "Generalized star height measures the nesting of Kleene stars in regular expressions that also allow Boolean operations. The formalization proves that every regular language over a finite alphabet has a generalized expression over the same alphabet of star height at most three. This is stronger than the bound of thirteen in the accompanying finite-monoid paper. The selected statement asserts the uniform expression bound, without separately encoding every construction step of that paper.\n\nThe formalization proves that every regular language over a finite alphabet has a generalized regular expression over that same alphabet with star height at most three. Generalized expressions allow Boolean operations as well as concatenation and Kleene star. The bound of three implies the accompanying paper's bound of four; the selected statement concerns the expression bound itself.\n\nGeneralized star height measures the nesting of Kleene stars in regular expressions that also allow Boolean operations. The formalization proves that every regular language over a finite alphabet has a generalized regular expression over that same alphabet of star height at most three. Complements are taken in the same free monoid. This is the paper's uniform bound.", "lab_docs_url": "https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/docs/134.md", "overview_entry": "family 134 in overview.pdf / CONTENTS.md", "papers": [{"dir": "Finite-Monoid-Computations-and-a-Uniform-Generalized-Star-Height-Bound-September-25-2026", "title": "Finite Monoid Computations and a Uniform Generalized Star-Height Bound", "url": "https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Finite-Monoid-Computations-and-a-Uniform-Generalized-Star-Height-Bound-September-25-2026"}, {"dir": "Generalized-Star-Height-at-Most-Four-September-25-2026", "title": "Generalized Star Height at Most Four", "url": "https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Generalized-Star-Height-at-Most-Four-September-25-2026"}, {"dir": "Generalized-Star-Height-at-Most-Three-September-25-2026", "title": "Generalized Star Height at Most Three", "url": "https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Generalized-Star-Height-at-Most-Three-September-25-2026"}], "audit_part": 1, "second_reader": "", "referee_verdict": ""}, {"family": "135", "title": "Sharp homogeneous depth-five complexity of matrix products", "subject": "Theoretical computer science", "headline": "Over every characteristic-zero field, the $(1,1)$ entry of a product of $n$ independent $n\\times n$ variable matrices requires $n^{\\Theta(\\sqrt n)}$ gates in homogeneous depth-five sum--product circuits. This sharp bound allows shared gates and bottom linear forms involving all variables.", "verdict": "full", "challenges": ["DepthFive"], "review_note": "None.", "definitions_to_check": ["`Depth5Circuit` is a bespoke structure (leaf/bottom/lower/middle/upper/output arrays with `bottomHomogeneous`, `middleHomogeneous`, `outputHomogeneous` degree-equality fields) and `circuitSize` counts all nodes; homogeneity is syntactic and the model charges nodes, not wires"], "external_packages": [], "cone_lines_max": 20323, "machine_check": "comparator-sandboxed-pass", "lab_scope_note": "Let $\\mathrm{IMM}_{n,n}$ be the $(1,1)$ entry of the product of $n$ independent $n\\times n$ variable matrices. The formalization proves that, over every characteristic-zero field, syntactically homogeneous depth-five $\\Sigma\\Pi\\Sigma\\Pi\\Sigma$ circuits computing this polynomial require at least $n^{\\sqrt n/400}$ gates for all sufficiently large $n$, with a threshold independent of the field. Arbitrary bottom support, finite fan-in and fan-out, and sharing are allowed.\n\nIt also gives, over every field and for $n\\ge2$, circuits with at most $n^{\\sqrt n+4}$ gates. These are the lower and upper bounds selected from the paper.", "lab_docs_url": "https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/docs/135.md", "overview_entry": "family 135 in overview.pdf / CONTENTS.md", "papers": [{"dir": "Homogeneous-depth-five-lower-bounds-for-iterated-matrix-multiplication-September-25-2026", "title": "Homogeneous depth-five lower bounds for iterated matrix multiplication", "url": "https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Homogeneous-depth-five-lower-bounds-for-iterated-matrix-multiplication-September-25-2026"}], "audit_part": 1, "second_reader": "", "referee_verdict": ""}, {"family": "139", "title": "Subpolynomial queries for log-concave sampling", "subject": "Theoretical computer science", "headline": "For $C^2$ potentials with a supplied minimizer and $I\\preceq\\nabla^2V\\preceq2I$, proves that sampling within total variation $1/10$ requires only $C_\\varepsilon d^\\varepsilon$ exact value-and-gradient queries for every fixed $\\varepsilon>0$. 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.", "verdict": "full", "challenges": ["LogConcaveQuery"], "review_note": "None; the claim is about the number of exact value-and-gradient queries only (no time/bit-cost statement), as in the summary.", "definitions_to_check": ["Hand-rolled oracle model (`OracleAlgorithm`, `OracleAlgorithm.history`/`run`, `TVAtMost := ∀ s measurable, |μ s - ν s| ≤ ε`, `queryComplexity := sInf` over feasible integer budgets in ℕ∞); non-uniform in d (one algorithm per (d, q))"], "external_packages": [], "cone_lines_max": 63748, "machine_check": "comparator-sandboxed-pass", "lab_scope_note": "The formalized result determines the dimension exponent of exact value-and-gradient query complexity for well-conditioned log-concave sampling. For potentials with $V(0)=0$, $\\nabla V(0)=0$, and $I\\le\\nabla^2V\\le2I$, the least worst-case query budget achieving total-variation error at most $1/10$ is at most $C_\\varepsilon d^\\varepsilon$ for every $\\varepsilon>0$, and at least $c\\log d$ eventually. Hence the infimal polynomial exponent is zero. Algorithms may be measurable, randomized, and adaptive; arithmetic and bit costs are not bounded.", "lab_docs_url": "https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/docs/139.md", "overview_entry": "family 139 in overview.pdf / CONTENTS.md", "papers": [{"dir": "Subpolynomial-query-complexity-for-well-conditioned-log-concave-sampling-September-26-2026", "title": "Subpolynomial query complexity for well-conditioned log-concave sampling", "url": "https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Subpolynomial-query-complexity-for-well-conditioned-log-concave-sampling-September-26-2026"}], "audit_part": 1, "second_reader": "", "referee_verdict": ""}, {"family": "140", "title": "Memory--sample lower bounds for noiseless Gaussian regression", "subject": "Theoretical computer science", "headline": "For fixed $A>0$, a one-pass learner with $Ad^2$ persistent bits needs $\\Omega_A(d\\log(1/\\epsilon))$ noiseless Gaussian samples to recover a unit vector to angular error $0<\\epsilon\\le1/10$ with probability $2/3$, uniformly in accuracy for large $d$. Computation and randomized updates are unrestricted, but output uses only the terminal state, stopping index and fresh randomness.", "verdict": "full", "challenges": ["MemoryPrecision", "NoiselessRegression", "PosteriorReplicas", "GaussianFiniteEntropy", "GaussianInformation", "ProjectionMoments", "GaussianReplacement", "SubsphereCurrent"], "review_note": "None for the headline; supporting results are in separate files. The learner model gives the learner a free shared seed at every step (a stronger model, so the lower bound is stronger).", "definitions_to_check": ["Bespoke finite-memory learner model (`Learner`, `RunStatus`, `Admissible` bundling several `AEMeasurable` hypotheses, `angularError := arccos ⟪ŝ, s⟫`, `uniformSphere` via `Measure.toSphere`); `GaussianInformation.lean` also introduces `local instance ambientSub/ambientInner` re-defining `Sub`/`Inner` on `EuclideanSpace` (supporting file)"], "external_packages": [], "cone_lines_max": 16263, "machine_check": "comparator-sandboxed-pass", "lab_scope_note": "In noiseless Gaussian regression, a learner observes inner products of an unknown unit vector in $\\mathbb R^d$ with independent standard Gaussian vectors. The formalization proves that, for every fixed $A>0$, there is $c_A>0$ such that, in sufficiently large dimension, a learner retaining at most $Ad^2$ bits between observations needs at least $c_A d\\log(1/\\varepsilon)$ observations to attain angular error at most $\\varepsilon$ with probability at least $2/3$ for a uniformly random unit signal, for $0<\\varepsilon\\le1/10$.\n\nFor memory $o(d^2)$, the constant can be universal, and the selected statement also permits success at least $2/3$ separately for every signal. The linked supporting results include inverse Gram-matrix moments, exact-projection densities, regularization bounds, and sphere- and cube-prior block estimates.\n\nThe paper studies how posterior replicas—independent signals drawn conditionally on the same observed data—control information in noiseless Gaussian regression. The formalization establishes the equal-label identity for finite measures, its density and measurable forms, and information bounds for Gaussian, distance, Haar, synthetic, and incidence comparisons. These statements retain exact observation labels, independent side randomness, and randomized transition rules.\n\nFor a learner using $M(d)=o(d^2)$ bits of memory, the resulting bound says that, for sufficiently large $d$ and $0<\\varepsilon\\le1/10$, success probability at least $2/3$ under the uniform unit-sphere prior requires $T\\ge c d\\log(1/\\varepsilon)$ observations, where $c>0$ is universal. The formalized streaming result uses the $2/3$ threshold; the stronger $3/5$ variant is outside this scope.\n\nThe formalization controls information gained from exact Gaussian observations of a spherical image of the uniform cube. Under the stated entropy and size conditions, repeated localization over $t$ observation blocks produces nested disclosures with mutual information at most $C(n+1)t$ and expected final localization level at most $Ct$. The selected estimates include inverse-volume control, kernel coercivity, and drift for the actual observation rows.\n\nFor a law of finite relative entropy with respect to a uniform cell law in sufficiently large dimension, the regularization procedure terminates almost surely in regular cells. Its expected depth and number of attempts are finite, with at most twice one plus the expected depth in expectation; the remaining relative entropy plus $(\\log 2)n/4$ times the expected depth is at most the initial relative entropy plus $2/e$.\n\nThe paper uses projection moments and domination by positive spherical caps to bound the information carried by exact Gaussian observations of an unknown unit vector. The formalization covers integrated frame and sphere moments, cap domination, Riesz-density estimates with fixed offsets, and bounds valid at every radius. It also covers probability bounds for finite-memory learners at a fixed stopping time and after summing over stopping times.\n\nFor $M(d)=o(d^2)$ bits of memory, sufficiently large $d$, and $0<\\varepsilon\\le1/10$, success probability at least $2/3$ under the uniform sphere prior requires $T\\ge c d\\log(1/\\varepsilon)$ observations for a universal $c>0$. A corresponding bound holds when $M\\le A d^2$ for each fixed $A>0$, with the constant allowed to depend on $A$. The randomized learner model allows independent shared seeds; its reduction to a finite-state model is proved for dimensions at least three, with the auxiliary dimension-two case outside this scope.\n\nThe paper compares fresh Gaussian observations with observations coupled to information already held in memory. The formalization covers mixed-moment and actual-row estimates, critical-radius, one-level, and two-label comparisons, and conditional-information bounds across a block of observations. For sufficiently large dimension $d$, the relevant entropy bounds of order $d^2$ bound the information in the fresh experiment by that in the coupled experiment plus a term of order $d$; the associated block information increase also has order $d$.\n\nIt also covers two-point identities on observation fibers and the fiber-block bound, including the boundary case of $d-2$ observed rows. The uniform unit-sphere prior, Gaussian rows, finite memory states, and randomized transition rules remain part of these statements. The separate streaming consequences and several intermediate comparison constructions are outside this selection.\n\nThe paper uses subspheres to bound the number of exact Gaussian observations needed to estimate a unit vector with limited memory. The formalization proves that, for $M(d)=o(d^2)$ bits of memory, sufficiently large $d$, and $0<\\varepsilon\\le1/10$, success probability at least $2/3$ under the uniform sphere prior, or for every unit signal, requires at least $2^{-16}d\\log_2(1/\\varepsilon)$ observations. Under the uniform prior, a lower bound of the same order with a universal positive constant holds at success probability $1/2$.\n\nThe formalization also covers radius-weighted block and terminal estimates, affine and dimension estimates, and a vector-valued beta-mixture comparison with its boundary cases. These statements retain their dimension, width, radius, and suffix hypotheses, as well as the finite-state learner's independent randomness and restrictions on stopping and output.", "lab_docs_url": "https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/docs/140.md", "overview_entry": "family 140 in overview.pdf / CONTENTS.md", "papers": [{"dir": "Memory-and-precision-in-noiseless-Gaussian-regression-September-27-2026", "title": "Memory and precision in noiseless Gaussian regression", "url": "https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Memory-and-precision-in-noiseless-Gaussian-regression-September-27-2026"}, {"dir": "Posterior-replicas-and-conditional-information-in-Gaussian-regression-September-27-2026", "title": "Posterior replicas and conditional information in Gaussian regression", "url": "https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Posterior-replicas-and-conditional-information-in-Gaussian-regression-September-27-2026"}, {"dir": "Localization-costs-and-information-growth-for-exact-Gaussian-observations-September-27-2026", "title": "Localization costs and information growth for exact Gaussian observations", "url": "https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Localization-costs-and-information-growth-for-exact-Gaussian-observations-September-27-2026"}, {"dir": "Projection-moments-positive-cap-domination-and-Riesz-estimates-on-the-sphere-September-27-2026", "title": "Projection moments, positive cap domination, and Riesz estimates on the sphere", "url": "https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Projection-moments-positive-cap-domination-and-Riesz-estimates-on-the-sphere-September-27-2026"}, {"dir": "Replacing-Gaussian-observations-in-memory-constrained-inference-September-27-2026", "title": "Replacing Gaussian observations in memory-constrained inference", "url": "https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Replacing-Gaussian-observations-in-memory-constrained-inference-September-27-2026"}, {"dir": "Subsphere-methods-for-memory-sample-lower-bounds-in-noiseless-Gaussian-regression-September-27-2026", "title": "Subsphere methods for memory-sample lower bounds in noiseless Gaussian regression", "url": "https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Subsphere-methods-for-memory-sample-lower-bounds-in-noiseless-Gaussian-regression-September-27-2026"}], "audit_part": 1, "second_reader": "", "referee_verdict": ""}, {"family": "143", "title": "Uniform limit-cycle bounds in Hilbert's sixteenth problem", "subject": "Dynamical systems and ergodic theory", "headline": "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 Li\\'enard systems, the exact maximum is two limit cycles.", "verdict": "partial", "challenges": ["QuinticLienard"], "review_note": "Only the quintic Lienard system x' = y - F(x), y' = -x (at most 2 limit cycles, exactly 2 attained); the general uniform boundedness theorem for planar polynomial vector fields of degree n is not stated.", "definitions_to_check": ["`IsLimitCycle F C := IsPeriodicOrbit F C ∧ ∃ U open ⊇ C, ∀ periodic orbits C' ⊆ U, C' = C` (isolated nonconstant periodic orbit, standard); counts via `Set.encard`"], "external_packages": [], "cone_lines_max": 16659, "machine_check": "comparator-sandboxed-pass", "lab_scope_note": "For the quintic Liénard system $x'=y-F(x)$, $y'=-x$, the formalized result proves that every real polynomial $F$ of degree at most five yields at most two limit cycles, and that some such $F$ yields exactly two. Limit cycles are isolated images of nonconstant periodic solutions. No sign, parity, hyperbolicity, or amplitude restriction is imposed.", "lab_docs_url": "https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/docs/143.md", "overview_entry": "family 143 in overview.pdf / CONTENTS.md", "papers": [{"dir": "two-limit-cycles-for-quintic-lienard-systems-September-24-2026", "title": "Two limit cycles for quintic Liénard systems", "url": "https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/two-limit-cycles-for-quintic-lienard-systems-September-24-2026"}], "audit_part": 1, "second_reader": "", "referee_verdict": ""}, {"family": "144", "title": "Banach's simple Lebesgue-spectrum problem", "subject": "Dynamical systems and ergodic theory", "headline": "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 $L^2$ space: the bilateral iterates of one real observable form an orthonormal basis of that space.", "verdict": "full", "challenges": ["ThreeTorus"], "review_note": "None.", "definitions_to_check": ["`Smooth` is hand-defined by local smooth lifts through `project : (Fin 3 → ℝ) → Torus` with `ContDiffOn ℝ ((⊤ : ℕ∞) : WithTop ℕ∞)` (C^∞, not analytic); `meanZero` is a hand-built `Submodule` of `Lp ℂ 2 μ`"], "external_packages": [], "cone_lines_max": 33818, "machine_check": "comparator-sandboxed-pass", "lab_scope_note": "Banach's simple Lebesgue-spectrum problem asks for a probability-preserving transformation with simple Lebesgue spectrum on its mean-zero $L^2$ space. The formalization constructs a smooth volume-preserving diffeomorphism of the three-torus with this property. It gives a real-valued cyclic vector whose integer iterates form an orthonormal basis of the entire mean-zero space, and an isometric identification under which the Koopman operator becomes multiplication by the coordinate function on the circle. The transformation is also ergodic.", "lab_docs_url": "https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/docs/144.md", "overview_entry": "family 144 in overview.pdf / CONTENTS.md", "papers": [{"dir": "A-smooth-three-torus-diffeomorphism-with-simple-Lebesgue-spectrum-September-23-2026", "title": "A smooth three-torus diffeomorphism with simple Lebesgue spectrum", "url": "https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-smooth-three-torus-diffeomorphism-with-simple-Lebesgue-spectrum-September-23-2026"}], "audit_part": 1, "second_reader": "", "referee_verdict": ""}, {"family": "145", "title": "Rokhlin's multiple-mixing problem", "subject": "Dynamical systems and ergodic theory", "headline": "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.", "verdict": "full", "challenges": ["Rokhlin"], "review_note": "None.", "definitions_to_check": ["Hand-written `timeMap`, `IsMixing`, `layoutTime`, `MixingOfOrder` (all four are required `definition_names`); all-pairwise-gaps-diverge is encoded as `atTop` on `Fin (k-1) → ℕ+` (product order), standard"], "external_packages": [], "cone_lines_max": 27889, "machine_check": "comparator-sandboxed-pass", "lab_scope_note": "Rokhlin's multiple-mixing problem asks whether ordinary mixing of one invertible probability-preserving transformation implies mixing of every finite order. The formalization proves this implication. For each $k\\ge3$ and every $k$ measurable sets, the measure of their translated intersection tends to the product of their measures as all successive time gaps tend to infinity.", "lab_docs_url": "https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/docs/145.md", "overview_entry": "family 145 in overview.pdf / CONTENTS.md", "papers": [{"dir": "Rokhlins-multiple-mixing-problem-for-one-transformation-September-23-2026", "title": "Rokhlin's multiple-mixing problem for one transformation", "url": "https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Rokhlins-multiple-mixing-problem-for-one-transformation-September-23-2026"}], "audit_part": 1, "second_reader": "", "referee_verdict": ""}, {"family": "146", "title": "Sinai's positive-entropy conjecture for the standard map", "subject": "Dynamical systems and ergodic theory", "headline": "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.", "verdict": "full", "challenges": ["StandardMapComponents", "StandardMapEntropy", "StandardMapLyapunov"], "review_note": "`main_entropy : ∃ k₀ > 0, ∀ k ≥ k₀, 0 < metricEntropy area (standardMap k)` is the headline verbatim (full parameter tail); the Lyapunov and hyperbolic Bernoulli-component claims are also stated (Lyapunov, Components). Parameter is k·sin(2πx) on (ℝ/ℤ)², i.e. Chirikov K = 2πk, immaterial for 'k large'.", "definitions_to_check": ["`metricEntropy μ f := ⨆ r, ⨆ p : FinitePartition r, partitionEntropy μ f p` with `partitionEntropy := ⨅ n, ofReal (blockEntropy μ f p (n+1)/(n+1))`: custom (not in Mathlib); the inf form equals the usual limit only for measure-preserving f, which holds here (checked: the map is a composition of shears) but is not part of the statement", "Two statement files (StandardMapComponents, StandardMapLyapunov) contain the meta command `run_cmd Lean.modifyEnv fun env => Lean.Meta.auxLemmasExt.setState env {}` (clears the aux-lemma cache; not a fidelity issue, but non-standard in a Comparator statement)"], "external_packages": [], "cone_lines_max": 54864, "machine_check": "comparator-sandboxed-pass", "lab_scope_note": "The formalization proves Sinai's positive-entropy conjecture for the standard sine map on the two-dimensional torus in the stronger form of a full positive parameter tail. For every sufficiently large parameter, normalized area has positive metric entropy, and a positive-area set has positive largest Lyapunov exponent with the stated derivative-growth limit.\n\nIt also constructs a positive-area invariant ergodic hyperbolic component. That component splits into finitely many cyclic pieces on each of which the corresponding iterate is Bernoulli. No genericity assumption on the parameter is used.", "lab_docs_url": "https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/docs/146.md", "overview_entry": "family 146 in overview.pdf / CONTENTS.md", "papers": [{"dir": "Positive-Metric-Entropy-for-the-Standard-Map-at-Large-Parameters-September-23-2026", "title": "Positive Metric Entropy for the Standard Map at Large Parameters", "url": "https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Positive-Metric-Entropy-for-the-Standard-Map-at-Large-Parameters-September-23-2026"}], "audit_part": 3, "second_reader": "", "referee_verdict": ""}, {"family": "148", "title": "The dimension formula for self-similar measures on the line", "subject": "Dynamical systems and ergodic theory", "headline": "For every self-similar measure on the line generated by finitely many contracting similarities, proves $\\dim_{\\mathrm H}\\mu=\\min\\{1,h_{\\mathrm{RW}}/\\chi\\}$, where $h_{\\mathrm{RW}}$ is the entropy rate of random composed maps and $\\chi$ the average logarithmic contraction. This resolves the entropy-rate dimension conjecture without a separation assumption, allowing exact overlaps and unequal contraction ratios.", "verdict": "full", "challenges": ["SelfSimilar", "SelfSimilarCorollaries"], "review_note": "`entropy_rate_dimension` gives `lowerHausdorffDimension μ = ofReal (min 1 (entropyRate / lyapunov))` for every `System` (signed ratios 0<|r_i|<1, positive weights, arbitrary offsets) and every probability measure with μ = Σ w_i f_i μ, with no separation hypothesis, so exact overlaps are allowed. The two corollaries (homogeneous ratio, attractor formula under no exact overlaps) are stated in SelfSimilarCorollaries.", "definitions_to_check": ["`lowerHausdorffDimension μ := ⨅ E measurable with 0 < μ E, dimH E` (the inf-dimension, as the docs say; equals dim_H for the exact-dimensional self-similar measure)", "`entropyRate S := sInf (range fun n => walkEntropy (n+1)/(n+1))` with `walkEntropy` the base-2 entropy of the law of the composed affine map (words with equal composition merged), and `lyapunov` also in base 2: custom but matches the standard h_RW/chi"], "external_packages": [], "cone_lines_max": 12093, "machine_check": "comparator-sandboxed-pass", "lab_scope_note": "The formalization proves $\\dim_H\\mu=\\min\\{1,h_{\\mathrm{RW}}/\\chi\\}$ for the lower Hausdorff dimension of every finite real self-similar probability measure with positive weights and nonzero contraction ratios of absolute value below one. Here $h_{\\mathrm{RW}}$ is the entropy rate of random affine compositions and $\\chi$ is the corresponding Lyapunov exponent in the same logarithmic base. Ratios may be signed and unequal, and exact overlaps are allowed.\n\nIt also gives two consequences without exact overlaps: the usual entropy-over-Lyapunov formula for a common positive contraction ratio, and the attractor formula $\\dim_H K=\\min\\{1,s\\}$, where $s\\ge0$ is the unique solution of $\\sum_i|r_i|^s=1$. Absolute continuity is outside these statements.", "lab_docs_url": "https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/docs/148.md", "overview_entry": "family 148 in overview.pdf / CONTENTS.md", "papers": [{"dir": "The-entropy-rate-dimension-formula-for-self-similar-measures-on-the-line-September-24-2026", "title": "The entropy-rate dimension formula for self-similar measures on the line", "url": "https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-entropy-rate-dimension-formula-for-self-similar-measures-on-the-line-September-24-2026"}], "audit_part": 3, "second_reader": "", "referee_verdict": ""}, {"family": "149", "title": "Permanence for weakly reversible reaction networks", "subject": "Dynamical systems and ergodic theory", "headline": "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.", "verdict": "weaker-statement", "challenges": ["MassAction"], "review_note": "Lean gives, for each positive initial state x0, an ε = ε(N, κ, x0) with ε ≤ x_i(t) ≤ 1/ε for all t ≥ 0 (boundedness and persistence of each trajectory). The summary's permanence (a common compact convex forward-invariant absorbing set for every positive stoichiometric class, uniform over its trajectories) is not stated; docs admit 'the bound may depend on ... the initial state'.", "definitions_to_check": ["`IsGlobalForwardSolution N κ x0 x := x 0 = x0 ∧ ∀ t ≥ 0, HasDerivAt x (massAction N κ (x t)) t` (two-sided derivative at t = 0, harmless for forward solutions of a polynomial field)"], "external_packages": [], "cone_lines_max": 7758, "machine_check": "comparator-sandboxed-pass", "lab_scope_note": "The boundedness and persistence conjectures for mass-action systems ask whether positive concentrations remain finite and separated from zero. The formalization proves this for every finite weakly reversible reaction network with positive constant reaction rates and every strictly positive initial state. A global forward solution exists, and one $\\varepsilon\\in(0,1)$ bounds every concentration of every global forward solution between $\\varepsilon$ and $\\varepsilon^{-1}$ for all nonnegative times. The bound may depend on the network, rates, and initial state.", "lab_docs_url": "https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/docs/149.md", "overview_entry": "family 149 in overview.pdf / CONTENTS.md", "papers": [{"dir": "Boundedness-and-persistence-of-weakly-reversible-mass-action-systems-September-25-2026", "title": "Boundedness and persistence of weakly reversible mass-action systems", "url": "https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Boundedness-and-persistence-of-weakly-reversible-mass-action-systems-September-25-2026"}], "audit_part": 3, "second_reader": "", "referee_verdict": ""}, {"family": "150", "title": "Weak mixing of irrational triangular billiards", "subject": "Dynamical systems and ergodic theory", "headline": "Proves that the billiard flow in every nondegenerate Euclidean triangle with at least one angle irrational relative to $\\pi$ is weakly mixing for normalized area times uniform direction. This strengthens ergodicity on the entire irrational-angle class, with no genericity or Diophantine restrictions.", "verdict": "weaker-statement", "challenges": ["IrrationalTriangleBilliard"], "review_note": "`MainConclusion` states ergodicity (a.e.-invariant measurable sets have measure 0 or 1), plus a.e. existence and uniqueness of flight chains, measure preservation and the a.e. flow law; weak mixing, the summary's headline, is strictly stronger and is not stated (docs also only say 'ergodic').", "definitions_to_check": ["`billiardFlow Q t z := if h : Nonempty (FlightChain Q z) then (Classical.choice h).at t else z` (flow defined by choice on chains; sound because the same statement proves chain uniqueness `∀ z c d : FlightChain Q z, c = d`; singular states are frozen)"], "external_packages": [], "cone_lines_max": 12363, "machine_check": "comparator-sandboxed-pass", "lab_scope_note": "The formalization proves that the unit-speed billiard flow in every nondegenerate Euclidean triangle with at least one angle irrational relative to $\\pi$ is ergodic for normalized area times uniform angular measure. It constructs the flow outside the null set of exceptional trajectories, proves uniqueness of the flight chain there, and establishes measure preservation and the flow law almost everywhere. No genericity or Diophantine condition on the irrational angle is assumed.", "lab_docs_url": "https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/docs/150.md", "overview_entry": "family 150 in overview.pdf / CONTENTS.md", "papers": [{"dir": "Ergodicity-of-triangular-billiards-with-an-irrational-angle-September-25-2026", "title": "Ergodicity of triangular billiards with an irrational angle", "url": "https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Ergodicity-of-triangular-billiards-with-an-irrational-angle-September-25-2026"}], "audit_part": 3, "second_reader": "", "referee_verdict": ""}, {"family": "151", "title": "A $C^1$ counterexample to Shub’s entropy conjecture", "subject": "Dynamical systems and ergodic theory", "headline": "Constructs a noninvertible $C^1$ 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 $C^1$ self-maps: homological growth need not force positive orbit complexity.", "verdict": "full", "challenges": ["C1EntropyCounterexample"], "review_note": "Statement: ∃ q > 0 and a non-bijective C¹ self-map f of S¹ × (S²)^(q+1) (C¹ = locally the restriction of an ambient C¹ map) with `Dynamics.coverEntropy f univ = 0`, a nonzero v ∈ H₂(M;ℝ) with f_* v = 2v, and log 2 ≤ log(total real-homology spectral radius): exactly the summary's counterexample. Mathlib's coverEntropy is the topological entropy on compact metric spaces.", "definitions_to_check": ["`totalRealHomologySpectralRadius f := sSup {r | ∃ n a b u v, (u ≠ 0 ∨ v ≠ 0) ∧ f_* u = a•u − b•v ∧ f_* v = b•u + a•v ∧ r = √(a²+b²)}` (custom; moduli of complex eigenvalues over all degrees; if the set were unbounded sSup = 0 and the clause `log 2 ≤ log radius` would fail, so it cannot trivialise)"], "external_packages": [], "cone_lines_max": 22878, "machine_check": "comparator-sandboxed-pass", "lab_scope_note": "Shub's entropy conjecture predicts that a smooth self-map's topological entropy is at least the logarithm of the spectral radius of its action on real homology. The formalization gives a counterexample to the general $C^1$ self-map version: a noninvertible $C^1$ map on a compact smooth manifold without boundary has topological entropy zero, while its action on second real homology has a nonzero eigenvector with eigenvalue $2$. Thus the homological lower bound is strictly positive and fails for this map.", "lab_docs_url": "https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/docs/151.md", "overview_entry": "family 151 in overview.pdf / CONTENTS.md", "papers": [{"dir": "A-C1-Counterexample-to-the-Entropy-Conjecture-September-25-2026", "title": "A C^1 Counterexample to the Entropy Conjecture", "url": "https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-C1-Counterexample-to-the-Entropy-Conjecture-September-25-2026"}], "audit_part": 3, "second_reader": "", "referee_verdict": ""}, {"family": "152", "title": "Zero entropy does not guarantee a smooth positive-volume model", "subject": "Dynamical systems and ergodic theory", "headline": "Constructs a zero-entropy ergodic invertible transformation of a standard nonatomic probability space that is not measurably conjugate to any $C^\\infty$ diffeomorphism preserving a strictly positive smooth probability density on a compact finite-dimensional manifold. One example rules out every finite dimension.", "verdict": "weaker-statement", "challenges": ["SmoothObstruction"], "review_note": "`main` gives an ergodic invertible T on a standard nonatomic probability space with `FiniteKSEntropy` (all finite-partition entropy rates exist and share one real bound) and `¬ HasSmoothPositiveVolumeModel` (no C^∞ positive-density model on any compact manifold, closed of every dimension or with boundary); the summary's zero-entropy requirement is not stated (docs: 'the paper's stronger zero-entropy conclusion is outside this statement'), so the Lean is implied by the headline but does not imply it.", "definitions_to_check": ["`def FiniteKSEntropy μ T := ∃ C : ℝ, ∀ q (P : X → Fin (q+1)), Measurable P → ∃ h ≤ C, Tendsto (fun n => blockEntropy μ T P n / n) atTop (𝓝 h)`: custom; says sup of partition entropies is finite, not zero", "`HasModelWith`/`SmoothPositiveDensity`/`ConullConjugate`: custom manifold-model encoding (density of `Measure.map (extChartAt I x)` against Lebesgue in every chart, conjugacy on invariant conull sets); read as standard"], "external_packages": [], "cone_lines_max": 30090, "machine_check": "comparator-sandboxed-pass", "lab_scope_note": "The paper asks whether a measure-preserving system can be represented by a smooth diffeomorphism preserving positive smooth volume. The formalized supporting result constructs an ergodic invertible transformation of a standard nonatomic probability space with finite Kolmogorov–Sinai entropy that has no such model on any compact finite-dimensional manifold, including manifolds with smooth boundary. The comparison is measurable conjugacy after discarding null sets.\n\nThe linked statement gives finite entropy; the paper's stronger zero-entropy conclusion is outside this statement.", "lab_docs_url": "https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/docs/152.md", "overview_entry": "family 152 in overview.pdf / CONTENTS.md", "papers": [{"dir": "A-finite-entropy-system-without-a-smooth-positive-volume-model-September-25-2026", "title": "A zero-entropy system without a smooth positive-volume model", "url": "https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-finite-entropy-system-without-a-smooth-positive-volume-model-September-25-2026"}], "audit_part": 3, "second_reader": "", "referee_verdict": ""}, {"family": "155", "title": "A counterexample to periodic tiling in dimension three", "subject": "Combinatorics", "headline": "Constructs a finite translational tile in $\\mathbb Z^3$ 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 $\\mathbb R^3$, even with arbitrary real translation vectors.", "verdict": "full", "challenges": ["PeriodicTilingThree"], "review_note": "One conjunction states all three parts: a nonempty finite T ⊆ ℤ³ with `Tiles T A` for some A but no `FullyPeriodic` (finite-index-subgroup invariant) tiling complement; the unit-cube thickening T + [0,1]³ a.e.-tiles ℝ³ by some arbitrary real translation set A but by none invariant under a full-rank lattice; and `IsLeast {d | HasTileWithoutPeriodicComplement d} 3` (minimality, which absorbs the d = 1, 2 periodicity theorems). `Tiles` is exact (bijective F × A → G) and the ℝ³ version uses closed cubes with `AETiles` (unique representation a.e.), matching the summary and docs.", "definitions_to_check": ["`abbrev Lattice (d : ℕ) := Fin d → ℤ` shadows Mathlib's `Lattice` class name inside the namespace (harmless; means ℤ^d)"], "external_packages": [], "cone_lines_max": 16899, "machine_check": "comparator-sandboxed-pass", "lab_scope_note": "The periodic-tiling question asks whether a finite tile that tiles a lattice must admit a periodic tiling. The formalized counterexample is a finite tile in $\\mathbb Z^3$ that tiles but has no complement invariant under a finite-index subgroup. Its unit-cube thickening also tiles $\\mathbb R^3$ almost everywhere but admits no fully periodic tiling, even with arbitrary real translation vectors. The result also establishes that dimension three is the least lattice dimension where this failure occurs.", "lab_docs_url": "https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/docs/155.md", "overview_entry": "family 155 in overview.pdf / CONTENTS.md", "papers": [{"dir": "A-translational-tile-with-no-fully-periodic-tiling-in-dimension-three-September-23-2026", "title": "A translational tile with no fully periodic tiling in dimension three", "url": "https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-translational-tile-with-no-fully-periodic-tiling-in-dimension-three-September-23-2026"}], "audit_part": 3, "second_reader": "", "referee_verdict": ""}, {"family": "156", "title": "Borsuk's conjecture fails in dimension nine", "subject": "Combinatorics", "headline": "Constructs a compact subset of $\\mathbb R^9$ 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 $\\mathbb R^4$, with the Frobenius metric.", "verdict": "full", "challenges": ["BorsukNine"], "review_note": "`main_theorem`: the set {uu^T : ‖u‖ = 1} ⊆ `EuclideanSpace ℝ (Fin 4 × Fin 4)` (Frobenius metric) is compact, lies in the trace-one symmetric matrices, has `Metric.diam = √2`, and admits no cover by ten sets of diameter < √2 (pieces taken inside the set, w.l.o.g.). The routine step not stated in Lean is that the trace-one symmetric 4×4 matrices form a 9-dimensional affine subspace (10 - 1), isometric to ℝ⁹.", "definitions_to_check": [], "external_packages": [], "cone_lines_max": 78964, "machine_check": "comparator-sandboxed-pass", "lab_scope_note": "Borsuk's conjecture predicts that every bounded subset of $\\mathbb R^d$ can be covered by $d+1$ sets of strictly smaller diameter. The formalized counterexample is the compact set of rank-one orthogonal projectors onto lines in $\\mathbb R^4$, with the Frobenius metric. It lies in the nine-dimensional affine space of trace-one symmetric matrices, has diameter $\\sqrt2$, and cannot be covered by ten arbitrary sets of smaller diameter.", "lab_docs_url": "https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/docs/156.md", "overview_entry": "family 156 in overview.pdf / CONTENTS.md", "papers": [{"dir": "A-nine-dimensional-counterexample-to-Borsuks-covering-assertion-September-23-2026", "title": "A nine-dimensional counterexample to Borsuk's covering assertion", "url": "https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-nine-dimensional-counterexample-to-Borsuks-covering-assertion-September-23-2026"}], "audit_part": 3, "second_reader": "", "referee_verdict": ""}, {"family": "157", "title": "Counterexamples to the Hadwiger and Colin de Verdi\\`ere conjectures", "subject": "Combinatorics", "headline": "Disproves Hadwiger's conjecture even for fractional coloring: arbitrarily large finite simple graphs with independence number at most two satisfy $\\chi_f(G)>h(G)$, where $h(G)$ is the largest clique-minor order. Also disproves the fractional Colin de Verdi\\`ere chromatic bound $\\chi_f(G)\\le\\mu(G)+1$. In the positive direction, every finite nonempty graph satisfies $\\chi_{\\mathrm{list}}(G)\\le C h(G)$ for a universal constant $C$.", "verdict": "partial", "challenges": ["ListHadwiger"], "review_note": "Only the third (positive) claim is stated: `∃ C ≥ 1, ∀ finite nonempty G, listChromaticNumber G ≤ C * hadwigerNumber G`. The headline counterexamples (graphs with α ≤ 2 and χ_f(G) > h(G); failure of χ_f ≤ μ(G)+1) have no statement at all, and docs/paper scope covers only the list-Hadwiger bound.", "definitions_to_check": ["`Choosable G k := ∀ (Color : Type) (L : V → Finset Color), (∀ v, k ≤ (L v).card) → ListColorable G L` and `listChromaticNumber := sInf {k | Choosable G k}`: standard list-chromatic number (colors restricted to `Type`, harmless for finite graphs)", "`HasCliqueMinor G t := ∃ B : Fin t → Set V, (∀ i, (G.induce (B i)).Connected) ∧ pairwise disjoint ∧ pairwise adjacent` and `hadwigerNumber := sSup {t | HasCliqueMinor G t}`: standard (Connected forces nonempty branch sets; set bounded by |V|)"], "external_packages": [], "cone_lines_max": 19965, "machine_check": "comparator-sandboxed-pass", "lab_scope_note": "The Linear List Hadwiger conjecture asks for a universal linear bound on list chromatic number in terms of clique-minor size. The formalization proves that there is one integer $C\\ge1$ such that every finite nonempty simple graph $G$ satisfies $\\chi_{\\mathrm{list}}(G)\\le C h(G)$, where $h(G)$ is the largest order of a clique minor. The constant is independent of the graph.", "lab_docs_url": "https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/docs/157.md", "overview_entry": "family 157 in overview.pdf / CONTENTS.md", "papers": [{"dir": "A-linear-list-coloring-bound-in-terms-of-the-Hadwiger-number-September-23-2026", "title": "A linear list-coloring bound in terms of the Hadwiger number", "url": "https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-linear-list-coloring-bound-in-terms-of-the-Hadwiger-number-September-23-2026"}], "audit_part": 3, "second_reader": "", "referee_verdict": ""}, {"family": "158", "title": "The Euclidean plane cannot be colored with five colors", "subject": "Combinatorics", "headline": "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.", "verdict": "full", "challenges": ["EuclideanFiveColor", "PlaneColoring"], "review_note": "`no_proper_five_coloring : ¬ ∃ coloring : ℂ → Fin 5, ∀ p q, ‖p - q‖ = 1 → coloring p ≠ coloring q` is the headline for arbitrary colorings (no measurability or color-class hypothesis), and `properColoring_seven` gives the classical seven-coloring of `EuclideanSpace ℝ (Fin 2)` (so only 6 or 7 remain). The two statements use different plane models (ℂ with its norm vs `EuclideanSpace ℝ (Fin 2)`, namespace `Problem160`), both genuinely the Euclidean plane; no lemma identifies them, which does not matter for either claim.", "definitions_to_check": [], "external_packages": [], "cone_lines_max": 30336, "machine_check": "comparator-sandboxed-pass", "lab_scope_note": "The Hadwiger–Nelson problem asks for the fewest colors needed to color the plane so that points at distance one have different colors. The formalized results prove that five colors do not suffice and that seven colors do suffice. The lower bound applies to arbitrary colorings, with no measurability or continuity assumption; the upper bound includes every boundary point of the coloring regions.", "lab_docs_url": "https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/docs/158.md", "overview_entry": "family 158 in overview.pdf / CONTENTS.md", "papers": [{"dir": "The-Euclidean-plane-is-not-five-colorable-September-23-2026", "title": "The Euclidean plane is not five-colorable", "url": "https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-Euclidean-plane-is-not-five-colorable-September-23-2026"}], "audit_part": 3, "second_reader": "", "referee_verdict": ""}, {"family": "159", "title": "Erd\\H{o}s's reciprocal-sum conjecture and quasipolynomial Szemer\\'edi bounds", "subject": "Combinatorics", "headline": "Proves Erd\\H{o}s's conjecture that every set of positive integers with divergent reciprocal sum contains arithmetic progressions of every finite length. Quantitatively, for each fixed $k\\ge3$, every subset of $\\{1,\\ldots,N\\}$ with no nonconstant $k$-term progression has size at most $C_kN\\exp[-c_k(\\log N)^{\\varepsilon_k}]$, with positive constants depending only on $k$.", "verdict": "partial", "challenges": ["ErdosReciprocal"], "review_note": "`ReciprocalProgressionTheorem : ∀ A : Set ℕ, ¬ Summable (reciprocalTerm A) → ∀ k, HasAP A k` (`HasAP`: a k-term progression a + i·d, d > 0, inside A) is exactly the first claim, stated for all A ⊆ ℕ (0 ∈ A is harmless: it contributes 0 to the sum and the positive-integer case is covered). The paper's quantitative bound |A| ≤ C_k N exp(-c_k (log N)^ε_k) for k-AP-free subsets of {1..N} has no statement (docs: 'outside this statement'); the solution cone is by far the largest in this set (4,774 files, 1,122,983 OAI lines).", "definitions_to_check": [], "external_packages": [], "cone_lines_max": 1122983, "machine_check": "comparator-pass", "lab_scope_note": "Erdős's reciprocal-sum conjecture asks whether every set of positive integers with divergent reciprocal sum contains arithmetic progressions of every finite length. The formalization proves this statement: for every requested length, such a set contains a progression with positive common difference.\n\nThe selected theorem is the reciprocal-sum consequence. The paper's quantitative upper bound for the largest progression-free subset of $\\{1,\\ldots,N\\}$ is outside this statement.", "lab_docs_url": "https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/docs/159.md", "overview_entry": "family 159 in overview.pdf / CONTENTS.md", "papers": [{"dir": "Quasipolynomial-Bounds-for-Arithmetic-Progressions-September-23-2026", "title": "Quasipolynomial Bounds for Arithmetic Progressions", "url": "https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Quasipolynomial-Bounds-for-Arithmetic-Progressions-September-23-2026"}], "audit_part": 3, "second_reader": "", "referee_verdict": ""}, {"family": "160", "title": "Superexponential van der Waerden numbers", "subject": "Combinatorics", "headline": "Resolves Erd\\H{o}s'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 $W_r(k)>k^{ck\\lfloor\\log_2 r\\rfloor}$ for an absolute $c>0$, all $r\\ge2$ and sufficiently large $k$, uniformly in $r$. In particular, $W_r(k)^{1/k}\\to\\infty$ for each fixed $r$.", "verdict": "full", "challenges": ["QuantitativeVanDerWaerden"], "review_note": "`uniform_lower_bound : ∃ K, ∀ k ≥ K, ∀ r ≥ 2, k^((1/100000)·k·Nat.log 2 r) < W r k` is the headline with c = 10⁻⁵ and the uniformity in r; since `W r k = sInf {N > 0 | every r-coloring of ℕ has a mono k-AP inside [0,N)}` and sInf ∅ = 0, the strict inequality also forces finiteness. The 'in particular W_r(k)^{1/k} → ∞' is the routine corollary (⌊log₂ r⌋ ≥ 1 for fixed r ≥ 2), not stated separately.", "definitions_to_check": [], "external_packages": [], "cone_lines_max": 14837, "machine_check": "comparator-sandboxed-pass", "lab_scope_note": "Let $W(r,k)$ be the least interval length forcing a monochromatic $k$-term arithmetic progression in every coloring with at most $r$ colors. The formalized result gives an absolute threshold $K$ such that $W(r,k)>k^{k\\lfloor\\log_2 r\\rfloor/100000}$ for all $k\\ge K$ and $r\\ge2$. It also establishes the associated growth limits, finiteness, and boundary values. The sharper intermediate estimates used in the paper are not part of the described formalization.", "lab_docs_url": "https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/docs/160.md", "overview_entry": "family 160 in overview.pdf / CONTENTS.md", "papers": [{"dir": "Quantitative-Superexponential-Bounds-for-van-der-Waerden-Numbers-September-23-2026", "title": "Quantitative Superexponential Bounds for van der Waerden Numbers", "url": "https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Quantitative-Superexponential-Bounds-for-van-der-Waerden-Numbers-September-23-2026"}], "audit_part": 3, "second_reader": "", "referee_verdict": ""}, {"family": "161", "title": "Counterexamples to Sidorenko's conjecture and the forcing conjecture", "subject": "Combinatorics", "headline": "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.", "verdict": "partial", "challenges": ["SidorenkoCounterexample"], "review_note": "`main : ∃ size > 0, ∃ host : SimpleGraph (Fin size), (host has an edge) ∧ homDensity H host < (edgeDensity host)^66` for a fixed explicit H = incidence graph of 22 triples on 13 points (checked by script: 35 vertices, 66 edges, connected, bipartite by construction), with homomorphism counts including non-injective maps and `edgeDensity = 2e/n²` = t(K₂,G): a faithful Sidorenko counterexample. The second named claim (the same pattern disproves the Skokan-Thoma forcing conjecture) has no statement (docs: 'outside this statement'); connectivity and the counts 35/66 are properties of the defined H, not stated.", "definitions_to_check": ["`homDensity F G := homCount F G / (Nat.card W)^(Nat.card V)` with `homCount := Nat.card {hom : V → W // ∀ l r, F.Adj l r → G.Adj (hom l) (hom r)}` and `edgeDensity G := 2 * Nat.card G.edgeSet / (Nat.card W)^2`: custom but the standard t(H,G) and t(K₂,G) (checked)"], "external_packages": [], "cone_lines_max": 23761, "machine_check": "comparator-sandboxed-pass", "lab_scope_note": "Sidorenko's conjecture predicts that every bipartite graph $H$ has homomorphism density at least the host graph's edge density raised to $|E(H)|$. The formalization disproves this for the paper's fixed bipartite graph with $35$ vertices and $66$ edges: it constructs a nonempty finite simple host graph with $t(H,G) 0, (∃ admissible drawing with Z(m)Z(n) crossings) ∧ (∀ admissible drawings, Z(m)Z(n) ≤ crossings)` state both formulas for all sizes, for drawings by arbitrary continuous simple arcs with crossings counted as points. The minimum ranges only over the custom `AdmissibleDrawing` class (finitely many crossing points, each a topologically transversal crossing, no triple points), which is the usual crossing-number class.", "definitions_to_check": ["`structure AdmissibleDrawing ... extends ContinuousDrawing` with `crossings_finite`, `crossings_proper : ProperCrossing ...` (local chart straightening the two arcs to the axes) and `no_triples`; `ordinaryCrossingNumber := sInf (Set.range crossingCount)` (sInf ∅ = 0, but a nonempty class is forced for n ≥ 5 since hill n > 0); the two files carry separate copies of the definitions (`Zarankiewicz` vs `Paper170` namespaces)"], "external_packages": [], "cone_lines_max": 33427, "machine_check": "comparator-sandboxed-pass", "lab_scope_note": "The formalized result proves Hill's proposed formula for the crossing number of the complete graph: for every $n\\ge3$, $\\mathrm{cr}(K_n)=\\frac14\\lfloor n/2\\rfloor\\lfloor(n-1)/2\\rfloor\\lfloor(n-2)/2\\rfloor\\lfloor(n-3)/2\\rfloor$. It proves the lower bound and constructs a matching two-page drawing. Crossings are spatial points, so repeated crossings of one edge pair count separately.\n\nZarankiewicz's crossing-number conjecture predicts $\\mathrm{cr}(K_{m,n})=\\lfloor m/2\\rfloor\\lfloor(m-1)/2\\rfloor\\lfloor n/2\\rfloor\\lfloor(n-1)/2\\rfloor$. The formalized result proves this equality for every positive $m,n$ and constructs a drawing attaining it. Crossings are counted as spatial points for continuous simple edge paths, including repeated crossings between the same pair of edges.", "lab_docs_url": "https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/docs/165.md", "overview_entry": "family 165 in overview.pdf / CONTENTS.md", "papers": [{"dir": "The-crossing-number-of-complete-graphs-September-23-2026", "title": "The crossing number of complete graphs", "url": "https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-crossing-number-of-complete-graphs-September-23-2026"}, {"dir": "The-crossing-number-of-complete-bipartite-graphs-September-23-2026", "title": "The crossing number of complete bipartite graphs", "url": "https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-crossing-number-of-complete-bipartite-graphs-September-23-2026"}], "audit_part": 3, "second_reader": "", "referee_verdict": ""}, {"family": "167", "title": "Pinned distances and a power saving for planar unit distances", "subject": "Combinatorics", "headline": "Proves the weak pinned Erd\\H{o}s distance conjecture: for every fixed $\\varepsilon>0$, all but $o(n)$ points of any $n$-point planar set determine at least $n^{1-\\varepsilon}$ distinct nonzero distances. A complementary theorem bounds the number of unit-distance pairs by $O(n^{4/3-\\delta})$ for an absolute $\\delta>0$.", "verdict": "full", "challenges": ["PinnedDistances", "PlanarUnitDistances"], "review_note": "`WeakPinned.main : Tendsto (fun n => F n s) atTop (𝓝 0)` with `F n s = sSup {pairFraction P s | P.card = n}` states the rich-pair-fraction form; the summary's 'all but o(n) points determine >= n^{1-eps} distinct distances' follows by a routine pigeonhole step (docs: 'Consequently') that is not itself stated in Lean. The unit-distance theorem IS also stated (`∃ C β, 0 < C ∧ 1 ≤ β ∧ β < 4/3 ∧ ∀ n, (u n:ℝ) ≤ C * n^β`, u = sSup over n-point sets, bounded so a true max); docs paragraph 1 says it 'is not included' but the Lean contradicts that (Lean wins).", "definitions_to_check": [], "external_packages": [], "cone_lines_max": 28113, "machine_check": "comparator-sandboxed-pass", "lab_scope_note": "The formalized result shows that large repeated distance fibers are rare in arbitrary finite planar point sets. For each fixed $s>0$, the largest possible fraction of ordered distinct pairs $(x,y)$ whose distance from the pin $x$ occurs at least $n^s$ times tends to zero as the set size $n$ grows. Consequently, for every $\\varepsilon>0$, the fraction of pins determining fewer than $n^{1-\\varepsilon}$ distances tends uniformly to zero. The separate unit-distance power-saving theorem is not included.\n\nThe planar unit-distance problem asks how many pairs at distance one can occur among $n$ points. The formalization proves that there are absolute constants $C>0$ and $1\\le\\beta<4/3$ such that every finite planar point set of size $n$ determines at most $Cn^\\beta$ unordered unit-distance pairs. The same constants work for every $n$, giving a fixed power improvement over the classical exponent $4/3$.", "lab_docs_url": "https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/docs/167.md", "overview_entry": "family 167 in overview.pdf / CONTENTS.md", "papers": [{"dir": "The-weak-pinned-planar-distance-theorem-September-23-2026", "title": "The weak pinned planar distance theorem", "url": "https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-weak-pinned-planar-distance-theorem-September-23-2026"}, {"dir": "A-power-saving-for-planar-unit-distances-September-23-2026", "title": "A power saving for planar unit distances", "url": "https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-power-saving-for-planar-unit-distances-September-23-2026"}], "audit_part": 2, "second_reader": "", "referee_verdict": ""}, {"family": "168", "title": "Combinatorial invariance of Kazhdan--Lusztig polynomials", "subject": "Combinatorics", "headline": "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.", "verdict": "full", "challenges": ["KLInvariance"], "review_note": "`combinatorial_invariance` takes arbitrary Coxeter systems `cs`, `cs'` (different W, W' allowed) and `ι : Interval cs u b ≃o Interval cs' u' b'` and concludes `klPolynomial cs u b = klPolynomial cs' u' b'`; `BruhatLE` is the genuine strong Bruhat order (ReflTransGen of length-increasing reflection steps) and the R/P normalization is the standard Kazhdan-Lusztig characterization.", "definitions_to_check": ["`noncomputable def klFamilies (cs) := Classical.epsilon (NormalizedKL cs)` and `klPolynomial cs x y := (klFamilies cs).2 x y`: KL polynomials are defined as an arbitrary solution of the normalization axioms (existence/uniqueness not part of the statement); this is the standard characterization and not trivializing, but it is a non-Mathlib, choice-based definition."], "external_packages": [], "cone_lines_max": 42948, "machine_check": "comparator-sandboxed-pass", "lab_scope_note": "The combinatorial invariance conjecture asks whether a Kazhdan–Lusztig polynomial depends only on its Bruhat interval as an ordered set. The formalization proves that every order isomorphism between Bruhat intervals in arbitrary Coxeter systems preserves the corresponding equal-parameter Kazhdan–Lusztig polynomial. The two intervals may come from different Coxeter systems.", "lab_docs_url": "https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/docs/168.md", "overview_entry": "family 168 in overview.pdf / CONTENTS.md", "papers": [{"dir": "Combinatorial-Invariance-of-Kazhdan-Lusztig-Polynomials-September-24-2026", "title": "Combinatorial invariance of Kazhdan–Lusztig polynomials", "url": "https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Combinatorial-Invariance-of-Kazhdan-Lusztig-Polynomials-September-24-2026"}], "audit_part": 2, "second_reader": "", "referee_verdict": ""}, {"family": "169", "title": "The Shareshian--Wachs $e$-positivity conjecture", "subject": "Combinatorics", "headline": "Resolves the elementary-positivity part of the Shareshian--Wachs conjecture: the chromatic quasisymmetric function of every natural unit interval graph has elementary-basis coefficients in $\\mathbb N[q]$. The coefficients count explicitly described permutations, giving a combinatorial explanation of positivity.", "verdict": "full", "challenges": ["ElementaryPositivity"], "review_note": "`elementaryPositivityWitness n G : G.PermutationWitness` (a `def` closed by `sorry`, not a `theorem`) must supply `theta : {σ // G.Nondescent σ} → Nat.Partition n` with `∀ r, G.chromatic r = Σ_σ C(X^(graphInversions σ)) * esymmPart (theta σ)` in `MvPolynomial (Fin r) ℕ[X]`: an explicit ℕ[q]-positive e-expansion for every natural unit interval graph (stronger than bare positivity). Chromatic quasisymmetric function and Hessenberg-function graphs are the standard definitions.", "definitions_to_check": [], "external_packages": [], "cone_lines_max": 74575, "machine_check": "comparator-sandboxed-pass", "lab_scope_note": "The elementary-positivity part of the Shareshian–Wachs conjecture asks whether the chromatic quasisymmetric function of every natural unit interval graph has nonnegative coefficients in the elementary basis. The formalization proves this over $\\mathbb N[q]$. It constructs an explicit elementary-basis expansion indexed by the graph's permitted nondescent permutations, with each term weighted by a nonnegative power of $q$ determined by its graph inversions. The expansion holds for every finite number of color variables.", "lab_docs_url": "https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/docs/169.md", "overview_entry": "family 169 in overview.pdf / CONTENTS.md", "papers": [{"dir": "Elementary-Positivity-of-Chromatic-Quasisymmetric-Functions-September-24-2026", "title": "Elementary positivity of chromatic quasisymmetric functions", "url": "https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Elementary-Positivity-of-Chromatic-Quasisymmetric-Functions-September-24-2026"}], "audit_part": 2, "second_reader": "", "referee_verdict": ""}, {"family": "170", "title": "Sharp logarithmic exponents for off-diagonal Ramsey numbers", "subject": "Combinatorics", "headline": "For every fixed integer $s\\ge5$, proves $r(s,t)=t^{s-1}/(\\log t)^{s-2+o(1)}$ as $t\\to\\infty$, determining the logarithmic exponent and matching the classical upper bound at that scale. Here $r(s,t)$ is the least number of vertices forcing an $s$-clique or a $t$-vertex independent set.", "verdict": "full", "challenges": ["RamseyFive", "SharpLogRamsey"], "review_note": "`RamseyFive.main : SharpBounds ∧ SharpExponent` covers s=5 (lower `t^4/(log t)^(3+ε)`, upper `C t^4/(log t)^3`, and `(4 log t - log r(5,t))/log log t -> 3`); `SharpLogRamsey.main (s) (hs : 6 ≤ s) : MainBounds s ∧ MainLimit s` covers every s >= 6, so together all s >= 5. r(s,t) = `sInf {n | ∀ G : SimpleGraph (Fin n), K_s ∨ independent t-set}` with Mathlib `IsNClique`/`IsNIndepSet`, standard.", "definitions_to_check": [], "external_packages": [], "cone_lines_max": 73678, "machine_check": "comparator-sandboxed-pass", "lab_scope_note": "The formalization determines the sharp logarithmic exponent of the off-diagonal Ramsey number $r(5,t)$. It proves\n$r(5,t)=t^4/(\\log t)^{3+o(1)}$ as $t\\to\\infty$.\nFor every $\\varepsilon>0$, the lower bound $t^4/(\\log t)^{3+\\varepsilon}$ holds eventually, while the upper bound is $Ct^4/(\\log t)^3$ for an absolute $C>0$. The corresponding logarithmic exponent converges to three along all natural values of $t$.\n\nThe formalization determines the sharp logarithmic exponent of the off-diagonal Ramsey number for every fixed integer $s\\ge6$. It proves\n$r(s,t)=t^{s-1}/(\\log t)^{s-2+o(1)}$ as $t\\to\\infty$.\nMore explicitly, for each $\\varepsilon>0$ the lower bound $t^{s-1}/(\\log t)^{s-2+\\varepsilon}$ holds eventually, while the upper bound is $C_s t^{s-1}/(\\log t)^{s-2}$. The logarithmic exponent limit is taken along all natural values of $t$.", "lab_docs_url": "https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/docs/170.md", "overview_entry": "family 170 in overview.pdf / CONTENTS.md", "papers": [{"dir": "The-Sharp-Logarithmic-Exponent-of-r-5-t-September-24-2026", "title": "The sharp logarithmic exponent of r(5,t)", "url": "https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-Sharp-Logarithmic-Exponent-of-r-5-t-September-24-2026"}, {"dir": "Sharp-Logarithmic-Exponents-for-Fixed-Off-Diagonal-Ramsey-Numbers-September-24-2026", "title": "Sharp logarithmic exponents for fixed off-diagonal Ramsey numbers", "url": "https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Sharp-Logarithmic-Exponents-for-Fixed-Off-Diagonal-Ramsey-Numbers-September-24-2026"}], "audit_part": 2, "second_reader": "", "referee_verdict": ""}, {"family": "172", "title": "Classification of finite Euclidean Ramsey configurations", "subject": "Combinatorics", "headline": "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.", "verdict": "partial", "challenges": ["EuclideanRamsey", "EuclideanRamseyCircle", "EuclideanRamseyNine", "EuclideanRamseyQuadratic", "EuclideanRamseySpherical", "EuclideanRamseyTransitive", "GrahamSpherical"], "review_note": "Classification is only for full-affine-span configurations: `classification (hs : 2 ≤ s) (hd : 1 ≤ d) (ha : Injective a) (hspan : affineSpan ℝ (range a) = ⊤) : Ramsey a ↔ FieldCriterion a` (docs: 'for full-affine-span configurations'; the isometric re-embedding reduction to this case is not a challenge statement). The summary's disproof of the Leader-Russell-Walters conjecture has no statement: only the Ramsey half is present (`at_most_five_circle_points_ramsey`, `nine_circle_configuration`, `GrahamSpherical.full_main` = a 12-point non-Ramsey spherical set), nothing says some Ramsey configuration is not contained in a finite transitive set.", "definitions_to_check": [], "external_packages": [], "cone_lines_max": 7309, "machine_check": "comparator-sandboxed-pass", "lab_scope_note": "A finite configuration is Euclidean Ramsey if every finite coloring of some sufficiently high-dimensional Euclidean space contains a monochromatic congruent copy at the original scale. The formalization gives the tensor-field classification for full-affine-span configurations, covers singleton and affine-span reductions, and proves that every Ramsey configuration is cospherical. It also proves that every nonempty subset of a finite transitive configuration and every nonempty set of at most five points on a circle is Ramsey. A further sufficient condition uses linear independence of the quadratic evaluation rows of a spherical configuration.\n\nThe formalization also gives spherical non-Ramsey examples: the specified twelve-point set on the unit circle has a fifty-color obstruction in every positive dimension, and a nine-point circle configuration built from algebraically independent parameters is non-Ramsey.", "lab_docs_url": "https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/docs/172.md", "overview_entry": "family 172 in overview.pdf / CONTENTS.md", "papers": [{"dir": "A-classification-of-finite-Euclidean-Ramsey-configurations-September-23-2026", "title": "A classification of finite Euclidean Ramsey configurations", "url": "https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-classification-of-finite-Euclidean-Ramsey-configurations-September-23-2026"}], "audit_part": 2, "second_reader": "", "referee_verdict": ""}, {"family": "173", "title": "Seymour's second-neighborhood conjecture", "subject": "Combinatorics", "headline": "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.", "verdict": "full", "challenges": ["SeymourSecondNeighborhood"], "review_note": "`exists_goodVertex (r) (hr : IsOriented r) : ∃ v, GoodVertex r v` over a nonempty `Fintype V`, with `GoodVertex r v := (firstNeighbors r v).card ≤ (secondNeighbors r v).card` and second neighbors `{w | w ≠ v ∧ ¬ r v w ∧ ∃ u, r v u ∧ r u w}` (directed distance exactly two); `IsOriented` = loopless + asymmetric. Exactly Seymour's conjecture.", "definitions_to_check": [], "external_packages": [], "cone_lines_max": 4861, "machine_check": "comparator-sandboxed-pass", "lab_scope_note": "Seymour's second-neighborhood conjecture asserts that every nonempty finite oriented graph has a vertex with at least as many second out-neighbors as first out-neighbors. The formalized result proves this assertion, where the second neighborhood consists of vertices at directed distance exactly two. The initial vertex and first neighbors are excluded from that count, and sinks are included.", "lab_docs_url": "https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/docs/173.md", "overview_entry": "family 173 in overview.pdf / CONTENTS.md", "papers": [{"dir": "A-proof-of-Seymours-second-neighborhood-conjecture-September-23-2026", "title": "A proof of Seymour’s second-neighborhood conjecture", "url": "https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-proof-of-Seymours-second-neighborhood-conjecture-September-23-2026"}], "audit_part": 2, "second_reader": "", "referee_verdict": ""}, {"family": "174", "title": "Deterministic construction of strong thin spanning trees", "subject": "Combinatorics", "headline": "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.", "verdict": "full", "challenges": ["AlgorithmicThinTrees", "StrongThinTree"], "review_note": "`StrongThinTree.MainStatement` (`∃ C > 0, ∀ k ≥ 1, ∀ n m, 2 ≤ n → ∀ G : MultiGraph n m, G.EdgeConnected k → ∃ T, G.SpanningTree T ∧ ∀ S, |cut T S| ≤ (C/k) * |cut univ S|`) is the existence theorem; `AlgorithmicStrongThinTrees` adds `∃ M : CurrentKS.Machine, ∃ a degree, ... result.state = none ∧ GoodOutput C x ...` for explicit and binary-multiplicity inputs. Both parts of the summary are stated; binary multiplicities are really binary (`encodeNat n = frame (Computability.encodeNat n)` and Mathlib's `encodeNat` is binary).", "definitions_to_check": ["`CurrentKS.Machine` (hand-rolled multi-stack machine with a lookup table, `Machine.run` with a fuel argument) and the notion of polynomial time `M.run x.encode (a * (x.length + 1) ^ degree)` halting with `state = none`: a custom model, not Mathlib's `TM2ComputableInPolyTime`; multi-stack machines are polynomially equivalent to Turing machines so this looks faithful, but 'polynomial time' is a bespoke definition."], "external_packages": [], "cone_lines_max": 101101, "machine_check": "comparator-sandboxed-pass", "lab_scope_note": "The strong thin-tree conjecture asks for spanning trees that cross every cut sparsely relative to the original graph. The formalized result gives one absolute constant $C>0$ such that every finite loopless $k$-edge-connected multigraph with at least two vertices has a spanning tree crossing each nontrivial cut at most $(C/k)$ times the original cut size. Parallel edges remain distinct. No numerical value of $C$ or construction algorithm is asserted.\n\nThe strong thin-tree problem asks for a spanning tree that crosses every cut sparsely relative to the original graph. The formalization gives one deterministic polynomial-time algorithm and an absolute constant $C>0$ such that a finite $k$-edge-connected loopless multigraph yields a spanning tree using at most a $C/k$ fraction of the edges of every cut. The input may encode parallel-edge multiplicities in binary, and the running time is polynomial in that binary input length.", "lab_docs_url": "https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/docs/174.md", "overview_entry": "family 174 in overview.pdf / CONTENTS.md", "papers": [{"dir": "The-strong-thin-tree-conjecture-September-23-2026", "title": "The strong thin tree conjecture", "url": "https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-strong-thin-tree-conjecture-September-23-2026"}, {"dir": "A-polynomial-time-construction-of-strong-thin-trees-September-23-2026", "title": "A polynomial-time construction of strong thin trees", "url": "https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-polynomial-time-construction-of-strong-thin-trees-September-23-2026"}], "audit_part": 2, "second_reader": "", "referee_verdict": ""}, {"family": "175", "title": "Talagrand's conjectures and graph decompositions at expectation thresholds", "subject": "Combinatorics", "headline": "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.", "verdict": "partial", "challenges": ["TalagrandDiscreteConvexity", "TalagrandExpectationThreshold"], "review_note": "Two of the three claims are stated with explicit constants: `talagrand_expectation_threshold_equivalence : qf F ≤ (25 * 512^4) * q F` (increasing nonempty proper F) and `talagrand_discrete_convexity` with k = 2^75 (`Small p (exceptional (2^75) D)` from `1 - 1/2^75 ≤ familyMeasure p D`). The summary's application, the Ascoli-He-Park-Talagrand graph-decomposition conjecture (also in the family title), has no challenge statement and the docs do not mention it.", "definitions_to_check": [], "external_packages": [], "cone_lines_max": 4834, "machine_check": "comparator-sandboxed-pass", "lab_scope_note": "Talagrand's expectation-threshold conjecture asks whether the fractional and integral expectation thresholds are comparable by an absolute constant. The formalized result proves $q_f(\\mathcal F)\\le25\\cdot512^4 q(\\mathcal F)$ for every nonempty proper increasing family on a finite nonempty ground set. Both thresholds use cover budget $1/2$, and fractional covers may assign weights to every subset, including the empty set.\n\nThe formalized result proves Talagrand's discrete-convexity assertion with $k=2^{75}$. If a family of subsets of a finite nonempty ground set has Bernoulli-$p$ measure at least $1-1/k$, then the sets not contained in a union of $k$ members form a $p$-small family: they have a containment cover of total $p$-cost at most $1/2$. This holds for every $01$ exactly when $n\\in\\{2,3,4,5,7,11,13\\}$.", "verdict": "full", "challenges": ["CirculantHadamard", "EvenBarker"], "review_note": "`exists_iff_order_one_or_four (n) (hn : 0 < n) : ExistsRealCirculantHadamard n ↔ n = 1 ∨ n = 4` with `IsCirculant H := ∃ h, ∀ i j, H i j = h (j - i)` and `IsSignHadamard H := (∀ i j, IsSign (H i j)) ∧ H * Hᵀ = n • 1`: exact statement of the headline. The Barker corollary is only half stated (`even_length_eq_two_or_four : Even n → IsBarker h → n = 2 ∨ n = 4`); odd lengths and the 'exist exactly at {2,3,4,5,7,11,13}' classification are not in Lean (docs: 'outside this selected additional statement').", "definitions_to_check": [], "external_packages": [], "cone_lines_max": 9773, "machine_check": "comparator-sandboxed-pass", "lab_scope_note": "The formalization proves the circulant Hadamard conjecture in exact form: a real circulant Hadamard matrix of positive order $n$ exists exactly when $n=1$ or $n=4$. Explicit witnesses are supplied for both orders, with no restriction on prime factors.\n\nIt also proves the even-length part of the Barker-sequence consequence. A positive even-length sign sequence whose nonzero aperiodic autocorrelations have absolute value at most one must have length $2$ or $4$. The paper's classification of odd Barker lengths is outside this selected additional statement.", "lab_docs_url": "https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/docs/179.md", "overview_entry": "family 179 in overview.pdf / CONTENTS.md", "papers": [{"dir": "The-circulant-Hadamard-conjecture-September-23-2026", "title": "The circulant Hadamard conjecture", "url": "https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-circulant-Hadamard-conjecture-September-23-2026"}], "audit_part": 2, "second_reader": "", "referee_verdict": ""}, {"family": "180", "title": "Barnette's Hamiltonian-cycle conjecture", "subject": "Combinatorics", "headline": "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.", "verdict": "full", "challenges": ["BarnetteHamiltonian"], "review_note": "`Barnette.main : MainStatement` = `∀ V [Fintype V] G, G.IsRegularOfDegree 3 → G.IsBipartite → Planar G → ThreeVertexConnected G → HasHamiltonianCycle G`, with `Planar` an explicit crossing-free topological embedding (`PlaneEmbedding` with `Path` arcs in ℝ×ℝ), `ThreeVertexConnected` = card >= 4 and connected after deleting any <= 2 vertices, and Mathlib `IsHamiltonianCycle`. The summary's 'equivalently ... Pfaffian graph' reformulation is not stated.", "definitions_to_check": [], "external_packages": [], "cone_lines_max": 8936, "machine_check": "comparator-sandboxed-pass", "lab_scope_note": "Barnette's conjecture states that every finite simple cubic bipartite planar graph that is 3-vertex-connected has a Hamiltonian cycle. The formalization proves this statement for every such graph. The selected result is the existence of a cycle visiting every vertex exactly once.", "lab_docs_url": "https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/docs/180.md", "overview_entry": "family 180 in overview.pdf / CONTENTS.md", "papers": [{"dir": "Paired-states-and-Hamiltonian-cycles-in-cubic-bipartite-planar-graphs-September-24-2026", "title": "Paired states and Hamiltonian cycles in cubic bipartite planar graphs", "url": "https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Paired-states-and-Hamiltonian-cycles-in-cubic-bipartite-planar-graphs-September-24-2026"}], "audit_part": 2, "second_reader": "", "referee_verdict": ""}, {"family": "181", "title": "The Erd\\H{o}s--Gallai cycle-decomposition conjecture", "subject": "Combinatorics", "headline": "Proves that the edges of every finite simple undirected graph on $n$ vertices can be partitioned into at most $Cn$ simple cycles and single edges, for an absolute constant $C$. This resolves the Erd\\H{o}s--Gallai cycle-decomposition conjecture, bounding the number of pieces linearly even for dense graphs.", "verdict": "full", "challenges": ["CycleDecomposition"], "review_note": "`MainStatement : ∃ C > 0, ∀ n (G : SimpleGraph (Fin n)), ∃ k, EdgeDecomposition G k ∧ k ≤ C * n` where `EdgeDecomposition` = pairwise-disjoint parts each `CycleOrSingleEdge` (edge set of a Mathlib `IsCycle` walk or `{e}`) covering `G.edgeSet`. Exactly the Erdős–Gallai conjecture with an absolute constant.", "definitions_to_check": [], "external_packages": [], "cone_lines_max": 21451, "machine_check": "comparator-sandboxed-pass", "lab_scope_note": "The Erdős–Gallai cycle-decomposition conjecture asks for a linear bound on the number of cycles and single edges needed to partition a graph's edges. The formalized result gives one absolute constant $C>0$ such that every finite simple graph on $n$ vertices has an edge-disjoint decomposition into at most $Cn$ cycles or singleton edges. Edgeless and small graphs are included. The optimal value of $C$ is not determined.", "lab_docs_url": "https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/docs/181.md", "overview_entry": "family 181 in overview.pdf / CONTENTS.md", "papers": [{"dir": "A-linear-cycle-and-edge-decomposition-of-every-graph-September-24-2026", "title": "A linear cycle-and-edge decomposition of every graph", "url": "https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-linear-cycle-and-edge-decomposition-of-every-graph-September-24-2026"}], "audit_part": 2, "second_reader": "", "referee_verdict": ""}, {"family": "182", "title": "Power savings for polynomial-difference-free sets", "subject": "Combinatorics", "headline": "For every fixed intersective integer polynomial $h$ of degree $k\\ge2$ with positive leading coefficient, proves that a subset of $\\{1,\\ldots,N\\}$ avoiding nonzero values $h(1),h(2),\\ldots$ as differences has size $O_h(N^{1-c_k})$, with $c_k>0$ 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 $h$.", "verdict": "partial", "challenges": ["SquareDifference"], "review_note": "Only the square case h(x) = x^2 is stated: `IsSquareDifferenceFree A := ∀ a ∈ A, ∀ b ∈ A, ∀ m ≥ 1, a - b ≠ m^2` and `∃ c C, 0 < c ∧ ∀ N ≥ 1, ∀ A ⊆ Icc 1 N, ... → A.card ≤ C * N^(1-c)` (docs: 'a fixed power saving for sets with no nonzero square difference'). The summary's general intersective polynomials of degree k >= 2 (with c_k depending only on degree) and the prime-argument result are not formalized.", "definitions_to_check": [], "external_packages": [], "cone_lines_max": 24221, "machine_check": "comparator-sandboxed-pass", "lab_scope_note": "The formalization proves a fixed power saving for sets with no nonzero square difference. There are absolute constants $c>0$ and $C$ such that every $A\\subseteq\\{1,\\ldots,N\\}$ satisfying $a-b\\ne m^2$ for all $a,b\\in A$ and integers $m\\ge1$ has $|A|\\le C N^{1-c}$. The bound is uniform for every integer $N\\ge1$.", "lab_docs_url": "https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/docs/182.md", "overview_entry": "family 182 in overview.pdf / CONTENTS.md", "papers": [{"dir": "A-power-saving-for-square-difference-free-sets-September-24-2026", "title": "A power saving for square-difference-free sets", "url": "https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-power-saving-for-square-difference-free-sets-September-24-2026"}], "audit_part": 2, "second_reader": "", "referee_verdict": ""}, {"family": "183", "title": "Power savings for planar halving lines and $k$-sets", "subject": "Combinatorics", "headline": "Improves the planar halving-line bound to $O(n^{4/3-\\varepsilon})$ for sets with no three collinear and an absolute $\\varepsilon>0$. More generally, an $n$-point set with no three collinear has $O(n(k+1)^{1/3-\\varepsilon_0})$ strictly separable $k$-subsets for $1\\le k\\le n/2$, with an absolute $\\varepsilon_0>0$. The constants and positive exponents are nonquantitative.", "verdict": "partial", "challenges": ["HalvingLines"], "review_note": "First half is stated: `∃ ε>0, C, n₀, ∀ even n ≥ n₀, ∀ P, GeneralPosition P → halvingCount P ≤ C * n^(4/3-ε)`. The k-set claim O(n(k+1)^{1/3-ε0}) for 1 <= k <= n/2 is NOT stated: the second conjunct only bounds `switchCount P k` (level-switch pairs) by `C * n^(4/3-ε)` uniformly in k, for the stronger hypothesis `Generic P` (distinct x-coordinates/slopes plus a segment-crossing condition), i.e. a weaker, k-independent bound on a different count (docs: 'a bound of the same form for all level-switch counts under the additional generic-position assumptions').", "definitions_to_check": [], "external_packages": [], "cone_lines_max": 18237, "machine_check": "comparator-sandboxed-pass", "lab_scope_note": "A halving pair in an even planar point set is a pair whose line leaves equally many remaining points on each side. The formalization proves that some absolute $\\varepsilon>0$ and $C$ bound the number of halving pairs by $Cn^{4/3-\\varepsilon}$ for every sufficiently large even $n$ and every $n$-point set with no three collinear. It also proves a bound of the same form for all level-switch counts under the additional generic-position assumptions in the statement. The constants are existential.", "lab_docs_url": "https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/docs/183.md", "overview_entry": "family 183 in overview.pdf / CONTENTS.md", "papers": [{"dir": "A-power-saving-for-planar-halving-lines-September-25-2026", "title": "A power saving for planar halving lines", "url": "https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-power-saving-for-planar-halving-lines-September-25-2026"}], "audit_part": 2, "second_reader": "", "referee_verdict": ""}, {"family": "184", "title": "Coloring and independence in graphs with forbidden subgraphs", "subject": "Combinatorics", "headline": "Proves the Alon--Krivelevich--Sudakov coloring conjecture in correspondence-coloring form: graphs avoiding any fixed subgraph $F$ need $O_F(\\Delta/\\log\\Delta)$ colors when their maximum degree $\\Delta$ is sufficiently large. Also proves the Ajtai--Erd\\H{o}s--Koml\\'os--Szemer\\'edi independence conjecture: for fixed $r\\ge4$, every $n$-vertex $K_r$-free graph of average degree $d\\ge2$ has an independent set of size $\\Omega_r(n\\log d/d)$.", "verdict": "partial", "challenges": ["CliqueFreeLog"], "review_note": "PRIMARY CLAIM NOT STATED (only a secondary result is formalized). Only the independence half is stated: `logarithmic_independence_bound (r) (hr : 4 ≤ r) : ∃ c > 0, ∀ G, G.CliqueFree r → 2 ≤ averageDegree G → c * n * log d / d ≤ G.indepNum` (averageDegree = 2|E|/|V|). The summary's first and primary claim, the Alon-Krivelevich-Sudakov coloring conjecture in correspondence-coloring form (O_F(Δ/log Δ) colors for any fixed forbidden subgraph F), has no Lean statement, and the docs scope also only mentions the independence bound.", "definitions_to_check": [], "external_packages": [], "cone_lines_max": 5539, "machine_check": "comparator-sandboxed-pass", "lab_scope_note": "The formalized result gives a logarithmic improvement in the independence number of clique-free graphs. For every fixed integer $r\\ge4$, there is $c_r>0$ such that every finite $K_r$-free simple graph on $n$ vertices with average degree $d\\ge2$ has an independent set of size at least $c_r n\\log d/d$. The constant depends only on $r$.", "lab_docs_url": "https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/docs/184.md", "overview_entry": "family 184 in overview.pdf / CONTENTS.md", "papers": [{"dir": "A-Logarithmic-Independence-Bound-for-Clique-Free-Graphs-September-25-2026", "title": "A logarithmic independence bound for clique-free graphs", "url": "https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-Logarithmic-Independence-Bound-for-Clique-Free-Graphs-September-25-2026"}], "audit_part": 2, "second_reader": "", "referee_verdict": ""}, {"family": "185", "title": "Counterexamples to infinite matroid intersection and packing/covering", "subject": "Combinatorics", "headline": "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 Jo\\'o’s partitional-matroid question negatively. They are neither finitary nor cofinitary, so Nash-Williams’ original finitary conjecture remains outside the result.", "verdict": "full", "challenges": ["InfiniteMatroid", "InfiniteMatroidCorollaries"], "review_note": "`InfiniteMatroid.main` gives countable infinite `E`, two self-dual matroids with `∀ I₀ I₁ indep, I₀ ∪ I₁ ≠ univ` and `¬ HasPackingCovering M₀ M₁`; `partitional_intersection_counterexample` adds `IsPartitional M₀ ∧ IsPartitional M₁ ∧ ¬ HasIntersection M₀ M₁` (answering Joó's partitional question), and `separate_covering_packing_counterexamples` refutes the covering and packing conjectures separately. Uses Mathlib's infinite `Matroid`; the conjecture predicates (`HasIntersection`, `HasPackingCovering`, `contractOnto M C := (M.dual ↾ C).dual`) are written out by hand and look standard.", "definitions_to_check": [], "external_packages": [], "cone_lines_max": 4477, "machine_check": "comparator-sandboxed-pass", "lab_scope_note": "The infinite matroid packing/covering conjecture predicts a partition of a common ground set into parts admitting the corresponding packing and covering. The formalization constructs two self-dual partitional matroids on one countably infinite ground set that admit neither an independent covering nor a packing/covering partition. The same pair has no intersection witness, so it also refutes the unrestricted infinite matroid intersection conjecture.\n\nSeparate formalized consequences give counterexamples to the individual covering and packing conjectures. The construction is in ordinary set theory with Choice and assumes no finitary restriction on the matroids.", "lab_docs_url": "https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/docs/185.md", "overview_entry": "family 185 in overview.pdf / CONTENTS.md", "papers": [{"dir": "A-Counterexample-to-the-Infinite-Matroid-Packing-Covering-Conjecture-September-24-2026", "title": "A Counterexample to the Infinite Matroid Packing/Covering Conjecture", "url": "https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-Counterexample-to-the-Infinite-Matroid-Packing-Covering-Conjecture-September-24-2026"}], "audit_part": 2, "second_reader": "", "referee_verdict": ""}, {"family": "186", "title": "The Friedgut--Kalai graph and hypergraph threshold conjectures", "subject": "Combinatorics", "headline": "Proves the Friedgut--Kalai threshold-width conjectures for graphs and fixed-uniformity hypergraphs. For fixed $0<\\varepsilon<1/2$, every nontrivial increasing relabeling-invariant property crosses from probability $\\varepsilon$ to $1-\\varepsilon$ within width $O((\\log n)^{-2})$ for graphs and $O_r((\\log n)^{-r/(r-1)})$ for $r$-uniform hypergraphs, $r\\ge3$. The hypergraph influence bound also applies to nonmonotone properties.", "verdict": "partial", "challenges": ["SharpThreshold"], "review_note": "Only the graph case is stated: `sharp_threshold_width` for vertex-invariant increasing nontrivial `f` on `GraphConfig n`, `graphQuantile (1-ε) f - graphQuantile ε f ≤ (2^19/(log n)^2) * log(1/(2ε))`. The r-uniform hypergraph width O_r((log n)^{-r/(r-1)}) for r >= 3 and the nonmonotone influence bound named in the summary have no statement (docs: graph case only).", "definitions_to_check": [], "external_packages": [], "cone_lines_max": 7699, "machine_check": "comparator-sandboxed-pass", "lab_scope_note": "The Friedgut–Kalai sharp-threshold conjecture concerns how quickly a nontrivial increasing graph property appears in the independent-edge random graph. For every $n\\ge2$, every such property invariant under vertex relabeling, and $0<\\varepsilon<1/2$, the formalization proves that the edge-probability interval between probabilities $\\varepsilon$ and $1-\\varepsilon$ has width at most $2^{19}\\log(1/(2\\varepsilon))/(\\log n)^2$. The constant is uniform over the graph property, $n$, and $\\varepsilon$.", "lab_docs_url": "https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/docs/186.md", "overview_entry": "family 186 in overview.pdf / CONTENTS.md", "papers": [{"dir": "A-Sharp-Threshold-Bound-for-Monotone-Graph-Properties-September-25-2026", "title": "A Sharp Threshold Bound for Monotone Graph Properties", "url": "https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-Sharp-Threshold-Bound-for-Monotone-Graph-Properties-September-25-2026"}], "audit_part": 2, "second_reader": "", "referee_verdict": ""}, {"family": "187", "title": "Snaky in 21 Maker moves", "subject": "Combinatorics", "headline": "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.", "verdict": "full", "challenges": ["SnakyCertificate", "SnakyConditional", "SnakyTwentyOne"], "review_note": "`SnakyPrototype.snaky_winning_strategy_21_with_legal_states : ∃ σ : Policy Cell, (∀ r M B, σ r M B ∉ M ∧ σ r M B ∉ B) ∧ ∀ β, LegalRepliesBeforeFinal σ 21 β ∅ ∅ → HasSnaky (playState σ 21 β ∅ ∅ 21).1 ∧ ...card = 21 ∧ ...` on ℤ×ℤ with all 8 dihedral orientations plus translations, Maker first. The other two challenges are supporting: `SnakyCertificate.certificate_correct` checks a 610-row certificate table (`TableValid ∧ placementCount = 1837 ∧ Output 557 ...`), and `SnakyConditional` is actually the N=35 strategy from the empty board, NOT the 'four conditional winning templates' the docs describe (docs/Lean mismatch).", "definitions_to_check": ["Hand-rolled game model: `Policy α := ℕ → Finset α → Finset α → α`, `playState σ N β M₀ B₀` with Breaker as a fixed sequence `β : ℕ → Cell` (equivalent to an adaptive Breaker against a deterministic σ) and `LegalRepliesBeforeFinal` constraining Breaker only for `k + 1 < N`; looks faithful (final reply unused)."], "external_packages": [], "cone_lines_max": 71326, "machine_check": "comparator-sandboxed-pass", "lab_scope_note": "In the Snaky Maker–Breaker game, the players alternately claim cells of $\\mathbb Z^2$, and Maker seeks a translated, rotated, or reflected copy of the six-cell Snaky shape. The formalization gives a legal strategy that wins within $21$ actual Maker moves against every legal Breaker play from the empty infinite board. The selected theorem checks disjointness and the move counts throughout play; it does not impose the paper's smaller finite-board restriction.\n\nThe linked supplements reconstruct the older $35$-move appendix's finite recursive certificate and prove four conditional winning templates from partial positions, assuming the required Maker cells are present and Breaker avoids the corresponding finite envelope.", "lab_docs_url": "https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/docs/187.md", "overview_entry": "family 187 in overview.pdf / CONTENTS.md", "papers": [{"dir": "Snaky-in-21-Maker-moves-September-25-2026", "title": "Snaky in 21 Maker moves", "url": "https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Snaky-in-21-Maker-moves-September-25-2026"}], "audit_part": 2, "second_reader": "", "referee_verdict": ""}, {"family": "188", "title": "The sharp constant in random triangle removal", "subject": "Combinatorics", "headline": "Starting from the complete graph on $n$ vertices, repeatedly delete a uniformly chosen remaining triangle. The terminal edge count is asymptotic to $n^{3/2}/(2\\sqrt2)$, with mean-square convergence after normalization by $n^{3/2}$. This proves the triangle case of the Joos--K\\\"uhn sharp-constant conjecture.", "verdict": "full", "challenges": ["TriangleRemoval"], "review_note": "`sharp_terminal_leave` states, for the Mathlib-`PMF` Markov chain `evolve (completeGraph n) (n.choose 2)` that deletes a uniformly chosen remaining triangle (`PMF.uniformOfFinset (triangles G)`), (i) `E[(|G|/n^(3/2) - 1/(2√2))^2] → 0`, (ii) convergence in probability, (iii) `E|G|/n^(3/2) → 1/(2√2)`. This is the summary's mean-square convergence; the process is absorbing at triangle-free graphs so running C(n,2) steps gives the terminal law.", "definitions_to_check": [], "external_packages": [], "cone_lines_max": 33800, "machine_check": "comparator-sandboxed-pass", "lab_scope_note": "Start with the complete graph on $n$ vertices and repeatedly delete the three edges of a uniformly chosen remaining triangle. The formalization proves that the number of edges at termination, divided by $n^{3/2}$, converges in $L^2$ to $1/(2\\sqrt2)$. It also states convergence in probability and convergence of the normalized expectation to the same constant. This is the triangle case of the sharp terminal-leave conjecture of Joos and Kühn.", "lab_docs_url": "https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/docs/188.md", "overview_entry": "family 188 in overview.pdf / CONTENTS.md", "papers": [{"dir": "The-Sharp-Terminal-Leave-in-Random-Triangle-Removal-September-25-2026", "title": "The sharp terminal leave in random triangle removal", "url": "https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-Sharp-Terminal-Leave-in-Random-Triangle-Removal-September-25-2026"}], "audit_part": 2, "second_reader": "", "referee_verdict": ""}, {"family": "189", "title": "Exact cycle--clique Ramsey numbers", "subject": "Combinatorics", "headline": "Proves the Erd\\H{o}s--Faudree--Rousseau--Schelp conjecture: $R(C_m,K_n)=(m-1)(n-1)+1$ for every $m\\ge n\\ge3$, except $R(C_3,K_3)=6$. This is the exact threshold forcing a red $m$-cycle or a blue $n$-clique in every red--blue coloring of a complete graph.", "verdict": "full", "challenges": ["CycleCliqueRamsey"], "review_note": "`thm_main : (∀ m n : ℤ, n ≤ m → 3 ≤ n → (m, n) ≠ (3, 3) → (cycleCliqueRamsey m.toNat n.toNat : ℤ) = (m - 1) * (n - 1) + 1) ∧ cycleCliqueRamsey 3 3 = 6`, with `RamseyProperty m n N := ∀ G : SimpleGraph (Fin N), cycleGraph m ⊑ G ∨ (⊤ : SimpleGraph (Fin n)) ⊑ Gᶜ` (Mathlib `IsContained`) and `sInf` over ℕ: the exact EFRS statement. Note the import cone is 2.8M lines of `lean/OAI`, the largest in this range.", "definitions_to_check": [], "external_packages": [], "cone_lines_max": 2795707, "machine_check": "none", "lab_scope_note": "The cycle–clique Ramsey conjecture predicts the exact number of vertices forcing either a cycle $C_m$ or a clique $K_n$ in complementary colors. The formalization proves $R(C_m,K_n)=(m-1)(n-1)+1$ for all integers $m\\ge n\\ge3$ except $(m,n)=(3,3)$, where it proves $R(C_3,K_3)=6$. This is the complete parameter range of the paper's main theorem.", "lab_docs_url": "https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/docs/189.md", "overview_entry": "family 189 in overview.pdf / CONTENTS.md", "papers": [{"dir": "Cycle-clique-Ramsey-numbers-September-25-2026", "title": "Cycle--clique Ramsey numbers", "url": "https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Cycle-clique-Ramsey-numbers-September-25-2026"}], "audit_part": 2, "second_reader": "", "referee_verdict": ""}, {"family": "190", "title": "Polynomial removal fails for ordered binary matrices", "subject": "Combinatorics", "headline": "Disproves polynomial ordered binary matrix removal with one fixed $66\\times66$ zero--one pattern. Matrices can require many binary-entry changes to become pattern-free while their copy density is smaller than every proposed polynomial bound in that distance. Copies preserve row and column orders and match both zeros and ones.", "verdict": "full", "challenges": ["MatrixRemoval"], "review_note": "`no_polynomial_removal_bound : ∀ c C > 0, ∃ n ≥ 1, ∃ ε ∈ (0,1), ∃ A : BinaryMatrix n, fixedDistance A ≥ ε ∧ copyCount fixedH A < c * ε^C * n^132` for the explicit 66x66 `fixedH`; `copyCount` counts pairs of strictly increasing row/column maps with `A (r i) (c j) = H i j` (ordered, zeros and ones matched) and `fixedMinEdits` is a true min Hamming distance to pattern-free matrices (junk branch n*n+1 exceeds any distance). Matches the summary and docs.", "definitions_to_check": [], "external_packages": [], "cone_lines_max": 7609, "machine_check": "comparator-sandboxed-pass", "lab_scope_note": "The formalized result disproves a polynomial removal bound for one explicit $66\\times66$ binary pattern. For every $c,C>0$, there is an $n\\times n$ binary matrix at normalized edit distance at least $\\varepsilon>0$ from being pattern-free but with fewer than $c\\varepsilon^C n^{132}$ induced ordered copies. Row and column indices are independently increasing, and every zero and one entry must match. The construction gives an explicit sequence of such matrices, allowing edits in both directions.", "lab_docs_url": "https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/docs/190.md", "overview_entry": "family 190 in overview.pdf / CONTENTS.md", "papers": [{"dir": "Polynomial-removal-fails-for-ordered-binary-matrices-September-25-2026", "title": "Polynomial removal fails for ordered binary matrices", "url": "https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Polynomial-removal-fails-for-ordered-binary-matrices-September-25-2026"}], "audit_part": 2, "second_reader": "", "referee_verdict": ""}, {"family": "191", "title": "A power improvement in the Heilbronn triangle problem", "subject": "Combinatorics", "headline": "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.", "verdict": "weaker-statement", "challenges": ["HeilbronnTriangle"], "review_note": "`heilbronn_power_lower_bound` only yields an unbounded sequence of sizes (`∃ n P, Tendsto n atTop atTop ∧ ∀ j, ... triangleAreasAtLeast (P j) ((n j)^(-2 + heilbronnExponent))`), not 'for every sufficiently large n' (docs: 'The linked construction is for an unbounded sequence of sizes; the paper's statement for every sufficiently large n is broader'). The disproof of the almost-n^-2 upper bound is stated (`almost_n_minus_two_refuted : ¬ eventualAlmostUpperBound (heilbronnExponent/2)`) and follows from the subsequence, so only the all-large-n uniformity is missing; the exponent is an astronomically small explicit constant `1/(100000 * K)`.", "definitions_to_check": [], "external_packages": [], "cone_lines_max": 24275, "machine_check": "comparator-sandboxed-pass", "lab_scope_note": "Heilbronn's triangle problem asks how large the smallest triangle determined by $n$ points in the unit square can be. The formalization constructs an unbounded sequence of sizes $n$ and point sets for which every triangle has area at least $n^{-2+\\eta}$, for one fixed $\\eta>0$. It consequently refutes the proposed upper bound of order $n^{-2+\\varepsilon}$ for every $\\varepsilon>0$. The linked construction is for an unbounded sequence of sizes; the paper's statement for every sufficiently large $n$ is broader.", "lab_docs_url": "https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/docs/191.md", "overview_entry": "family 191 in overview.pdf / CONTENTS.md", "papers": [{"dir": "A-power-improvement-in-the-Heilbronn-triangle-lower-bound-September-25-2026", "title": "A power improvement in the Heilbronn triangle lower bound", "url": "https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-power-improvement-in-the-Heilbronn-triangle-lower-bound-September-25-2026"}], "audit_part": 2, "second_reader": "", "referee_verdict": ""}, {"family": "192", "title": "A counterexample to the Gopalan--Servedio conjecture", "subject": "Combinatorics", "headline": "Disproves the proposed square-root bound relating a Boolean function's linear Fourier coefficients to its polynomial degree. For every $C>0$, there is a sign-valued Boolean function $f$ with $\\sum_i\\widehat f(\\{i\\})>C\\sqrt{\\deg(f)}$. Thus its total signed correlation with individual input bits can exceed the proposed bound by an arbitrary factor.", "verdict": "full", "challenges": ["SquareRootDegree"], "review_note": "`main : SignedViolations ∧ ¬ BddAbove AbsoluteRatios` with `SignedViolations := ∀ C > 0, ∃ n > 0, ∃ f : Cube (Fin n) → ℝ, IsBoolean f ∧ Nonconstant f ∧ C * √(fourierDegree f) < singletonSum f`; Fourier coefficients are the uniform-average `E[f * χ_s]` over the ±1 cube and `fourierDegree` is the max |s| with nonzero coefficient (real multilinear degree, as the file comment says). Standard definitions, matches the summary exactly.", "definitions_to_check": [], "external_packages": [], "cone_lines_max": 3641, "machine_check": "comparator-sandboxed-pass", "lab_scope_note": "The formalization disproves the Gopalan–Servedio square-root degree conjecture by an unbounded factor. For every $C>0$, it gives a nonconstant Boolean function on a finite sign cube for which the sum of its linear Fourier coefficients exceeds $C\\sqrt{\\deg f}$, where $\\deg f$ is its real multilinear degree.\n\nIt also proves that the ratio of the sum of the absolute values of those coefficients to $\\sqrt{\\deg f}$ is unbounded among positive-degree Boolean functions.", "lab_docs_url": "https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/docs/192.md", "overview_entry": "family 192 in overview.pdf / CONTENTS.md", "papers": [{"dir": "Unbounded-Violations-of-the-Square-Root-Degree-Bound-September-26-2026", "title": "Unbounded Violations of the Square-Root Degree Bound", "url": "https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Unbounded-Violations-of-the-Square-Root-Degree-Bound-September-26-2026"}], "audit_part": 2, "second_reader": "", "referee_verdict": ""}, {"family": "194", "title": "Lech's multiplicity conjecture", "subject": "Algebra", "headline": "Proves $e(R)\\le e(S)$ for every flat local homomorphism of nonzero Noetherian local rings, where $e$ is Hilbert--Samuel multiplicity. This resolves Lech's conjecture in every dimension and characteristic.", "verdict": "supporting-only", "challenges": ["DuttaDomain"], "review_note": "`dutta_domain : DuttaDomainClaim` is only a characteristic-p statement over a complete Noetherian local DOMAIN D (`IsShortComplex D F → ... Tendsto (duttaSequence D p F) atTop (𝓝 (duttaMultiplicity D p F)) ∧ multiplicity D ≤ duttaMultiplicity D p F`); docs: 'The paper's full flat-local result in arbitrary characteristic is outside it'. The headline `e(R) ≤ e(S)` for flat local homomorphisms is not stated. `multiplicity` (Hilbert-Samuel via `limUnder` of `d! * colength / N^d`) is standard.", "definitions_to_check": [], "external_packages": [], "cone_lines_max": 53953, "machine_check": "comparator-sandboxed-pass", "lab_scope_note": "Lech's multiplicity conjecture compares Hilbert–Samuel multiplicities across flat local maps. The linked formalization covers a supporting characteristic-$p$ comparison over a complete Noetherian local domain $D$: for a complex satisfying the stated short-complex and finite-length homology conditions, its Frobenius multiplicity sequence converges to its Dutta multiplicity, and the Hilbert–Samuel multiplicity of $D$ is at most that Dutta multiplicity.\n\nThis selected statement is the complete-domain Dutta comparison. The paper's full flat-local result in arbitrary characteristic is outside it.", "lab_docs_url": "https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/docs/194.md", "overview_entry": "family 194 in overview.pdf / CONTENTS.md", "papers": [{"dir": "Lechs-multiplicity-conjecture-September-23-2026", "title": "Lech's multiplicity conjecture", "url": "https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Lechs-multiplicity-conjecture-September-23-2026"}], "audit_part": 2, "second_reader": "", "referee_verdict": ""}, {"family": "196", "title": "A counterexample to Kaplansky's zero-divisor conjecture", "subject": "Algebra", "headline": "Constructs a finitely presented torsion-free group $G$ whose group algebra $\\mathbb F_2[G]$ has nonzero zero divisors, disproving Kaplansky's zero-divisor conjecture. The group has a finite two-dimensional classifying space.", "verdict": "full", "challenges": ["TorsionFreeZeroDivisors"], "review_note": "`MainTheorem : ∃ G, Group.IsFinitelyPresented G ∧ TorsionFree G ∧ HasFiniteTwoDimensionalClassifyingSpace G ∧ ∃ α β : MonoidAlgebra (ZMod 2) G, α ≠ 0 ∧ β ≠ 0 ∧ α * β = 0`. `TorsionFree` is the usual `g^n = 1 → g = 1`; the K(G,1) clause asks for a finite 2-dimensional Hausdorff path-connected Mathlib `CWComplex` X with `G ≃* FundamentalGroup X x` and a contractible covering space, which is the standard meaning.", "definitions_to_check": [], "external_packages": [], "cone_lines_max": 44340, "machine_check": "comparator-sandboxed-pass", "lab_scope_note": "Kaplansky's zero-divisor conjecture asserts that the group algebra of a torsion-free group over a field has no zero divisors. The formalized result constructs a finitely presented torsion-free group $G$ and nonzero elements $\\alpha,\\beta\\in\\mathbb F_2[G]$ with $\\alpha\\beta=0$, giving a counterexample.\n\nThe same group admits a finite two-dimensional classifying space $K(G,1)$.", "lab_docs_url": "https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/docs/196.md", "overview_entry": "family 196 in overview.pdf / CONTENTS.md", "papers": [{"dir": "A-Torsion-Free-Group-Algebra-with-Zero-Divisors-September-23-2026", "title": "A Torsion-Free Group Algebra with Zero Divisors", "url": "https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-Torsion-Free-Group-Algebra-with-Zero-Divisors-September-23-2026"}], "audit_part": 2, "second_reader": "", "referee_verdict": ""}, {"family": "197", "title": "Nonsofic groups and group-ring counterexamples", "subject": "Algebra", "headline": "Constructs a finitely presented torsion-free nonsofic group whose group algebra over $\\mathbb F_2$ 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.", "verdict": "partial", "challenges": ["GroupRingDeterminant", "KaplanskyDirectFiniteness", "KaplanskyFinitelyPresented", "OddKaplansky"], "review_note": "Determinant counterexample is stated (`GroupRingDeterminant.main`: f.g. G, n >= 1, integral matrix `A` invertible over ℚ[G], bounded left-regular `T` invertible with `0 < fkDet T < 1`, `fkDet T = exp(Re tr(CFC.log(T*T))/2)`) and a Gottschalk counterexample is stated only for odd characteristic (`OddKaplansky`: K of order p^4, f.g. G with torsion, `ab = 1 ∧ ba ≠ 1`, `cellular b` injective and not surjective). The summary's headline for Kaplansky is a finitely presented TORSION-FREE NONSOFIC group over F_2: the char-2 statements (`KaplanskyDirectFiniteness`, `KaplanskyFinitelyPresented`) give a f.g./f.p. group that has an element of odd prime order, and nonsofic/torsion-free are not stated (docs: 'The further conclusion that the group is nonsofic is outside these statements').", "definitions_to_check": ["`@[irreducible] def sourcePrime : ℕ := Nat.minFac ((Nat.factorial sourceM)^2 + 1)` with `sourceM := Nat.choose 1200 600` in `OddKaplansky`: the characteristic is a specific astronomically large, non-computable prime; existence of a counterexample is claimed only for this prime (not suspicious in logic, but the statement is for one prescribed p)."], "external_packages": [], "cone_lines_max": 18517, "machine_check": "comparator-sandboxed-pass", "lab_scope_note": "Kaplansky's direct-finiteness conjecture asserts that $ab=1$ implies $ba=1$ for $a,b\\in K[G]$, for every field $K$ and group $G$. The formalized results construct a finite field of characteristic two and a group algebra violating this implication. One statement records the counterexample for a finitely generated group; the more detailed construction gives a finitely presented group with an element of odd prime order.\n\nThe detailed construction also gives a precise recipe for choosing the data and proves that the required search terminates. The further conclusion that the group is nonsofic is outside these statements.\n\nA nonsingular integer matrix has absolute determinant at least $1$. The group-ring Determinant Conjecture extends this bound to matrices over $\\mathbb Z[G]$ for every discrete group $G$. If $T_A$ is the operator induced by such a matrix $A$ on finite direct sums of $\\ell^2(G)$, and $\\mu_A$ is the spectral measure of $T_A^*T_A$ with respect to the group trace, the conjecture asserts\n\n$\\displaystyle \\int_{(0,\\infty)}\\log t\\,d\\mu_A(t)\\ge0,$\n\nwith the zero spectral atom omitted. For an invertible square matrix, this is equivalent to its Fuglede–Kadison determinant being at least $1$.\n\nThe formalized result contradicts that bound. It gives a finitely generated group $G$ and an $n\\times n$ matrix over $\\mathbb Z[G]$, with $n\\ge1$, that is invertible over $\\mathbb Q[G]$. Its bounded left-regular operator is invertible and has determinant strictly between $0$ and $1$.\n\nThe formalization uses the trace of $\\log(T^*T)$ to define the determinant for the invertible operator $T$. The paper's additional spectral-measure integral conclusion is not included.\n\nKaplansky's direct-finiteness conjecture asserts that $ab=1$ implies $ba=1$ in every group algebra over a field. Let $p$ be the smallest prime divisor of $\\bigl(\\binom{1200}{600}!\\bigr)^2+1$; this prime is odd. The formalized result constructs a field $K$ of order $p^4$, a finitely generated group $G$ with torsion, and elements $a,b$ violating this implication. The associated cellular automaton on all configurations $K^G$ is injective but not surjective.\n\nThe formalization also contains fixed-field transfer results, separate from the statement linked below.", "lab_docs_url": "https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/docs/197.md", "overview_entry": "family 197 in overview.pdf / CONTENTS.md", "papers": [{"dir": "A-Counterexample-to-Kaplanskys-Direct-Finiteness-Conjecture-in-Characteristic-Two-September-23-2026", "title": "A Counterexample to Kaplansky's Direct-Finiteness Conjecture in Characteristic Two", "url": "https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-Counterexample-to-Kaplanskys-Direct-Finiteness-Conjecture-in-Characteristic-Two-September-23-2026"}, {"dir": "A-Counterexample-to-the-Group-Ring-Determinant-Conjecture-September-23-2026", "title": "A Counterexample to the Group-Ring Determinant Conjecture", "url": "https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-Counterexample-to-the-Group-Ring-Determinant-Conjecture-September-23-2026"}, {"dir": "A-Counterexample-to-Kaplanskys-Direct-Finiteness-Conjecture-in-Odd-Characteristic-September-26-2026", "title": "A Counterexample to Kaplansky's Direct-Finiteness Conjecture in Odd Characteristic", "url": "https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-Counterexample-to-Kaplanskys-Direct-Finiteness-Conjecture-in-Odd-Characteristic-September-26-2026"}], "audit_part": 2, "second_reader": "", "referee_verdict": ""}, {"family": "198", "title": "A counterexample to the little finitistic-dimension conjecture", "subject": "Algebra", "headline": "Constructs a finite-dimensional complex algebra whose finite-dimensional modules have unbounded finite projective dimensions. This disproves the little finitistic-dimension conjecture.", "verdict": "full", "challenges": ["FinitisticAsymmetry", "LittleFinitistic"], "review_note": "`exists_counterexample : ∃ A (_ : Ring A) (_ : Algebra ℂ A), FiniteDimensional ℂ A ∧ littleFinitisticDimension A = ⊤ ∧ ∀ m > 0, ∃ N, Module.Finite A N ∧ (2m-2 : WithBot ℕ∞) ≤ projectiveDimension N ∧ projectiveDimension N < ⊤`, where `littleFinitisticDimension A := ⨆ (M) (_ : Module.Finite A M) (_ : projectiveDimension M < ⊤), projectiveDimension M` (Mathlib `projectiveDimension`): unbounded finite projective dimensions over a finite-dimensional complex algebra, as in the summary. `FinitisticAsymmetry` is an extra secondary result.", "definitions_to_check": [], "external_packages": [], "cone_lines_max": 50191, "machine_check": "comparator-sandboxed-pass", "lab_scope_note": "The little finitistic-dimension conjecture predicts that finitely generated modules of finite projective dimension over a fixed finite-dimensional algebra have bounded projective dimensions. The formalized counterexample is a finite-dimensional complex algebra with a finitely generated module of projective dimension at least $2m-2$ and still finite for every $m\\ge1$.\n\nA separate formalized result gives a finite-dimensional complex algebra whose little and big finitistic dimensions are infinite on the left and zero on the right. Injectives fail to generate its unbounded derived category on the left and generate it on the right.", "lab_docs_url": "https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/docs/198.md", "overview_entry": "family 198 in overview.pdf / CONTENTS.md", "papers": [{"dir": "An-algebra-of-infinite-little-finitistic-dimension-September-23-2026", "title": "An algebra of infinite little finitistic dimension", "url": "https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/An-algebra-of-infinite-little-finitistic-dimension-September-23-2026"}], "audit_part": 2, "second_reader": "", "referee_verdict": ""}, {"family": "199", "title": "Counterexamples to conjectures of Auslander--Reiten, Tachikawa, and Nakayama", "subject": "Algebra", "headline": "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.", "verdict": "partial", "challenges": ["AuslanderReiten", "Tachikawa"], "review_note": "Stated: `ArExplicit.Statement.main` (`FullStatement`: over K = Frac(F₂[X₁,X₂,X₃]) a finite-dimensional algebra and a non-projective Gorenstein-projective module Z (`TotallyAcyclicWitness`) with `Ext^i(Z,Z)` and `Ext^i(Z,A)` subsingleton for i > 0, A/rad ≅ K^8, rad^4 ≠ 0, all conclusions persisting under every field extension E ⊗_K −) and `Tachikawa.main_theorem` (symmetric finite-dimensional algebra over k with a non-projective f.d. module with vanishing self-Ext). Not stated: the 'associated endomorphism algebra' consequences (classical/generalized/strong Nakayama, Auslander-Gorenstein, Wakamatsu tilting) named in the summary's second sentence (docs: 'The later field-extension, endomorphism-algebra, and related homological consequences are not included'; the docs also contradict themselves on field-extension persistence, which the Lean does state).", "definitions_to_check": [], "external_packages": [], "cone_lines_max": 55531, "machine_check": "comparator-sandboxed-pass", "lab_scope_note": "The Auslander–Reiten conjecture predicts that a finitely generated module $M$ over an Artin algebra $A$ is projective if $\\mathrm{Ext}^i_A(M,M\\oplus A)=0$ for every $i>0$. The formalized counterexample is a finite-dimensional algebra over $k=\\mathbb F_2(q,H_1,H_2)$ and a finite-dimensional nonprojective Gorenstein-projective module with this vanishing.\n\nIt also establishes $A/\\mathrm{rad}\\,A\\cong k^8$, $(\\mathrm{rad}\\,A)^4\\ne0$, and preservation of the listed properties after every field extension. The Tachikawa companion is separate.\n\nTachikawa's second conjecture predicts that a finite-dimensional module over a finite-dimensional self-injective algebra is projective if all its positive-degree self-Ext groups vanish. The formalized counterexample gives a finite-dimensional symmetric algebra over $\\mathbb F_2(q,H_1,H_2)$ and a finite-dimensional nonprojective module $M$ with $\\mathrm{Ext}^i(M,M)=0$ for every $i>0$. Symmetric algebras are self-injective, so this contradicts the conjecture.\n\nThe later field-extension, endomorphism-algebra, and related homological consequences are not included.", "lab_docs_url": "https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/docs/199.md", "overview_entry": "family 199 in overview.pdf / CONTENTS.md", "papers": [{"dir": "An-explicit-counterexample-to-the-Auslander-Reiten-conjecture-September-23-2026", "title": "An explicit counterexample to the Auslander-Reiten conjecture", "url": "https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/An-explicit-counterexample-to-the-Auslander-Reiten-conjecture-September-23-2026"}, {"dir": "A-counterexample-to-Tachikawas-second-conjecture-September-23-2026", "title": "A counterexample to Tachikawa's second conjecture", "url": "https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-counterexample-to-Tachikawas-second-conjecture-September-23-2026"}], "audit_part": 2, "second_reader": "", "referee_verdict": ""}, {"family": "205", "title": "Saxl's conjecture and universal tensor squares", "subject": "Algebra", "headline": "Proves Saxl's conjecture: the tensor square of every staircase representation contains every irreducible complex representation of the corresponding symmetric group. More generally, every $S_n$ with $n\\notin\\{2,4,9\\}$ has an irreducible representation whose tensor square contains all irreducibles.", "verdict": "full", "challenges": ["Saxl", "UniversalTensorSquares"], "review_note": "`Saxl.saxl_conjecture : ∀ m ≥ 1, ∀ μ with μ.card = (staircase m).card, 0 < kronecker (staircase m) (staircase m) μ` and `universal_tensor_square (n) (hn : 0 < n) (h2 : n ≠ 2) (h4 : n ≠ 4) (h9 : n ≠ 9) : ∃ λ, (∀ ν, 0 < kronecker λ λ ν) ∧ IsIrreducible (spechtRep λ) ∧ ∀ irreducible ρ, ∃ injective intertwiner ρ → S^λ ⊗ S^λ`. Both summary claims are stated.", "definitions_to_check": ["Specht modules are hand-built: `spechtSub t := cyclic (wordRep n d) (polytabloid t)` as the orbit span of the column-antisymmetrized row-word inside `(Fin n → Fin d) → ℂ`, and `kronecker := finrank ℂ (IntertwiningMap (spechtRep t) ((spechtRep a).tprod (spechtRep b)))`. This is the standard polytabloid construction (and `IsIrreducible` of the chosen Specht module is part of the universal statement), but it is not Mathlib's notion, so Saxl's statement is only as faithful as this construction."], "external_packages": [], "cone_lines_max": 217185, "machine_check": "comparator-sandboxed-pass", "lab_scope_note": "The formalization proves the universal tensor-square conjecture for symmetric groups in the stated range. For every positive integer $n\\notin\\{2,4,9\\}$, it constructs an irreducible complex representation of $S_n$ whose tensor square contains every irreducible complex representation of $S_n$. Equivalently, all corresponding Kronecker coefficients are positive. The statement also supplies injective intertwining maps for arbitrary finite-dimensional irreducible representations.\n\nSaxl's conjecture asserts that the tensor square of each staircase Specht module contains every irreducible representation of the corresponding symmetric group. The formalization establishes this for every $m\\ge1$: if $\\rho_m=(m,m-1,\\ldots,1)$, then $g(\\rho_m,\\rho_m,\\mu)>0$ for every partition $\\mu$ of $m(m+1)/2$.\n\nThe stronger claim that every constituent appears in the orbit span of one prescribed tensor is not included.", "lab_docs_url": "https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/docs/205.md", "overview_entry": "family 205 in overview.pdf / CONTENTS.md", "papers": [{"dir": "Universal-Tensor-Squares-for-Symmetric-Groups-September-24-2026", "title": "Universal Tensor Squares for Symmetric Groups", "url": "https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Universal-Tensor-Squares-for-Symmetric-Groups-September-24-2026"}, {"dir": "A-Cyclic-Polytabloid-Proof-of-Saxls-Conjecture-September-24-2026", "title": "A Cyclic Polytabloid Proof of Saxl's Conjecture", "url": "https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-Cyclic-Polytabloid-Proof-of-Saxls-Conjecture-September-24-2026"}], "audit_part": 2, "second_reader": "", "referee_verdict": ""}, {"family": "206", "title": "Finite lattice representation: counterexamples and undecidability", "subject": "Algebra", "headline": "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.", "verdict": "supporting-only", "challenges": ["FiniteCongruenceGraph"], "review_note": "`graph_criterion (Lat) [Fintype Lat] [Lattice Lat] [OrderBot Lat] : Representable Lat ↔ HasGraphWitness Lat` is only the paper's colored-graph characterization (docs: 'This selected theorem is the equivalence with the graph criterion; the paper's undecidability and subgroup-interval conclusions are outside it'). Nothing states that some finite lattice is NOT representable (the headline negative answer to the finite lattice representation problem) or that representability/subgroup-interval membership is undecidable.", "definitions_to_check": [], "external_packages": [], "cone_lines_max": 469, "machine_check": "comparator-sandboxed-pass", "lab_scope_note": "The finite lattice representation problem asks which finite lattices occur as the full congruence lattice of a finite algebra. The formalization proves the paper's colored-graph characterization: a finite nonempty lattice is representable exactly when it admits the specified finite nonempty graph witness, whose edge colors encode the required congruence relations. This selected theorem is the equivalence with the graph criterion; the paper's undecidability and subgroup-interval conclusions are outside it.", "lab_docs_url": "https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/docs/206.md", "overview_entry": "family 206 in overview.pdf / CONTENTS.md", "papers": [{"dir": "Finite-Congruence-Lattices-Characterization-and-Undecidability-September-24-2026", "title": "Finite congruence lattices: characterization and undecidability", "url": "https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Finite-Congruence-Lattices-Characterization-and-Undecidability-September-24-2026"}], "audit_part": 2, "second_reader": "", "referee_verdict": ""}, {"family": "207", "title": "The $\\ell^1$-Bass and complex Bass trace conjectures", "subject": "Algebra", "headline": "Proves the $\\ell^1$-Bass conjecture for every discrete group: Hattori--Stallings traces of idempotent matrices over $\\ell^1(G)$ are supported on finitely many finite-order conjugacy classes. The algebraic companion proves the integral Bass trace conjecture and Kaplansky's idempotent conjecture for torsion-free groups over every commutative unital characteristic-zero domain.", "verdict": "weaker-statement", "challenges": ["BassTorsionFree", "BassTrace"], "review_note": "Headline is the ℓ¹-Bass conjecture over `ℓ¹(G)`, but `bassTraceModules_vanishing_and_support` is only about the group algebra `MonoidAlgebra ℂ G` (Hattori-Stallings trace of every K0 class vanishes off finite-order classes and is supported on them); no ℓ¹(G) appears anywhere (docs: 'proves the complex Bass trace conjecture for every group'; docs title says ℓ¹ but scope does not). The algebraic companion IS stated: `torsion_free_corollary` gives trace = augmentation rank on K0(ℂ[G]) and `∀ R comm. domain with CharZero, e^2 = e → e = 0 ∨ e = 1` for torsion-free G.", "definitions_to_check": ["Hand-rolled K0: `BassTrace.K0.Group R := FreeAbelianGroup (Idempotent R) ⧸ relations R` (relations: e ~ f when `a*b = e ∧ b*a = f`, plus direct sums) and `ModuleK0` of f.g. projective right modules, `HattoriStallings` into `ConjClasses G →₀ ℂ`; standard presentations, not Mathlib objects."], "external_packages": [], "cone_lines_max": 24157, "machine_check": "comparator-sandboxed-pass", "lab_scope_note": "The formalization proves the complex Bass trace conjecture for every group: the Hattori–Stallings trace of each virtual class of finitely generated projective right $\\mathbb C[G]$-modules vanishes on conjugacy classes of infinite-order elements. No finiteness, countability, or geometric hypothesis on $G$ is imposed.\n\nFor torsion-free $G$, it identifies the trace on $K_0(\\mathbb C[G])$ with the integral augmentation rank and proves the characteristic-zero Kaplansky idempotent conjecture: for every commutative unital domain $R$ of characteristic zero, an idempotent in $R[G]$ is $0$ or $1$.", "lab_docs_url": "https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/docs/207.md", "overview_entry": "family 207 in overview.pdf / CONTENTS.md", "papers": [{"dir": "The-Bass-trace-conjecture-for-complex-group-rings-September-24-2026", "title": "The Bass trace conjecture and the characteristic-zero Kaplansky idempotent conjecture", "url": "https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-Bass-trace-conjecture-for-complex-group-rings-September-24-2026"}], "audit_part": 2, "second_reader": "", "referee_verdict": ""}, {"family": "210", "title": "Foulkes’ conjecture for sixth powers and quadratic stabilization", "subject": "Algebra", "headline": "Proves the sixth case of Foulkes’ conjecture: $\\operatorname{Sym}^6(\\operatorname{Sym}^bV)$ embeds equivariantly in $\\operatorname{Sym}^b(\\operatorname{Sym}^6V)$ for every $b\\ge6$ and finite-dimensional complex $V$. More generally, the canonical multiplication map $\\operatorname{Sym}^b(\\operatorname{Sym}^aV)\\to\\operatorname{Sym}^a(\\operatorname{Sym}^bV)$ is surjective for $a\\ge2$ and $b\\ge a(a-1)$, giving dimension-independent quadratic stabilization.", "verdict": "partial", "challenges": ["FoulkesHowe"], "review_note": "Quadratic stabilization is stated: `canonical_foulkes_howe_surjective : ∀ V f.d., ∀ a b, 2 ≤ a → a * (a - 1) ≤ b → ∃ μ, IsFoulkesMap a b V μ ∧ Function.Surjective μ ∧ (unique)`. The headline sixth case of Foulkes' conjecture for all b >= 6 is only reached for b >= 30 = 6*5 (docs: 'The sixth-power specialization is covered for b ≥ 30; the companion's full range b ≥ 6 is not included'); the equivariant embedding is only implicit (dual of surjectivity).", "definitions_to_check": ["`foulkesFormula a b V v := (a!)^(-b) • Σ_{σ : Fin b → Perm (Fin a)} sym_a(fun i => sym_b(fun j => v j (σ j i)))` and `IsFoulkesMap` define the 'canonical' map by hand on `SymPow n V` := span of degree-n products in `SymmetricAlgebra ℂ V`; the theorem also asserts uniqueness, which pins the definition."], "external_packages": [], "cone_lines_max": 5029, "machine_check": "comparator-sandboxed-pass", "lab_scope_note": "The formalized result proves surjectivity of the canonical averaged Foulkes–Howe map $\\mathrm{Sym}^b(\\mathrm{Sym}^a\\,V)\\to\\mathrm{Sym}^a(\\mathrm{Sym}^b\\,V)$ for every finite-dimensional complex vector space, $a\\ge2$, and $b\\ge a(a-1)$. It also covers the bijective $a=1$ case, the vanishing-on-products consequence, and the corresponding equivariant embedding in the reverse direction. The sixth-power specialization is covered for $b\\ge30$; the companion's full range $b\\ge6$ is not included.", "lab_docs_url": "https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/docs/210.md", "overview_entry": "family 210 in overview.pdf / CONTENTS.md", "papers": [{"dir": "Quadratic-Stabilization-of-the-Canonical-Foulkes-Howe-Map-September-25-2026", "title": "Quadratic stabilization of the canonical Foulkes--Howe map", "url": "https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Quadratic-Stabilization-of-the-Canonical-Foulkes-Howe-Map-September-25-2026"}], "audit_part": 2, "second_reader": "", "referee_verdict": ""}, {"family": "211", "title": "Geometry, diffusion, and spectra of random planar maps", "subject": "Probability and statistical mechanics", "headline": "Critical Fortuin--Kasteleyn planar maps converge to Liouville quantum gravity spheres for $04$, 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.", "verdict": "partial", "challenges": ["FKCRT"], "review_note": "Only the q > 4 Brownian-CRT limit is stated: `MainStatement : ∀ q > 4, ∃ c > 0, ... Tendsto (fun n => finiteExpectation q (c / √(n+1)) n F) atTop (𝓝 (brownianExpectation P B F))` for every GHP-test F, with the standard excursion realized as the norm of three independent Mathlib `IsBrownianReal` bridges. Not stated: the 0 < q <= 4 convergence to Liouville quantum gravity spheres, the Liouville Brownian motion limits of random walks, and the FK-Ising spectral/heat-trace convergence (docs: 'For every fixed q>4 ...').", "definitions_to_check": ["Entire topology is hand-rolled: `MetricMeasureData`, `ghpEDist := ⨅ amalgamating pseudometrics on X ⊕ Y` (Hausdorff/Lévy-Prokhorov), `IsGHPTest F := (∃ C, ∀ X, |F X| ≤ C) ∧ ∀ X valid, ∀ ε, ∃ δ, ∀ Y valid, ghpEDist X Y < δ → |F Y - F X| < ε` (convergence in distribution defined by bounded tests continuous at valid points); maps are pairs of dart permutations with `fkWeight q M A := q^(k(A) + (|A| - |V|)/2)`; `treeSpace` is a quotient of unitInterval by `excursionDist`. Plausible and documented in the file header, but none of it is Mathlib's notion."], "external_packages": [], "cone_lines_max": 40208, "machine_check": "comparator-sandboxed-pass", "lab_scope_note": "For every fixed $q>4$, the formalization proves the Brownian continuum-random-tree limit for critical finite Fortuin–Kasteleyn planar maps. After rescaling graph distances in an $n$-edge map by a constant depending on $q$ times $n^{-1/2}$ and using normalized degree measure, the metric-measure space converges in distribution to the Brownian continuum random tree in the Gromov–Hausdorff–Prokhorov topology. The convergence holds through all positive integer sizes.", "lab_docs_url": "https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/docs/211.md", "overview_entry": "family 211 in overview.pdf / CONTENTS.md", "papers": [{"dir": "Brownian-continuum-random-tree-limits-of-finite-Fortuin-Kasteleyn-maps-above-four-September-24-2026", "title": "Brownian continuum random tree limits of finite Fortuin–Kasteleyn maps above four", "url": "https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Brownian-continuum-random-tree-limits-of-finite-Fortuin-Kasteleyn-maps-above-four-September-24-2026"}], "audit_part": 2, "second_reader": "", "referee_verdict": ""}, {"family": "212", "title": "No bigeodesics and smooth limit shapes in planar first-passage percolation", "subject": "Probability and statistical mechanics", "headline": "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 $C^1$ boundary. Differentiability also holds for every Gamma law with positive shape and rate.", "verdict": "partial", "challenges": ["GammaPassage", "PlanarFirstPassage"], "review_note": "PRIMARY CLAIM NOT STATED (only a secondary result is formalized). Stated: exponential case `manuscriptMain` (norm `μ` with `IsTimeConstant`, `DifferentiableAwayFromOrigin`, `UniqueNormalizedSupports`, `C1UnitSphere`, unique supporting line, `HasC1CurveChart` on the frontier) and `GammaFPP.gamma_differentiability` for every shape/rate > 0. Not stated: the first headline claim (no doubly infinite geodesic for iid nonatomic weights with finite second moment of the min of four) and strict convexity of the exponential limit shape (docs: 'Strict convexity is outside these selected differentiability statements'; bigeodesics are not mentioned in the docs scope).", "definitions_to_check": [], "external_packages": [], "cone_lines_max": 20106, "machine_check": "comparator-sandboxed-pass", "lab_scope_note": "The formalization proves differentiability of the planar first-passage time-constant norm for independent Gamma edge weights of every positive shape and rate. The norm is differentiable away from the origin, and its unit sphere has a $C^1$ boundary. This includes the exponential model, for which the linked statement also gives a unique supporting line at every boundary point.\n\nThe paper also claims strict convexity for the exponential limit shape. Strict convexity is outside these selected differentiability statements.", "lab_docs_url": "https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/docs/212.md", "overview_entry": "family 212 in overview.pdf / CONTENTS.md", "papers": [{"dir": "Strict-convexity-and-differentiability-of-the-planar-exponential-first-passage-limit-shape-September-24-2026", "title": "Strict convexity and differentiability of the planar exponential first-passage limit shape", "url": "https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Strict-convexity-and-differentiability-of-the-planar-exponential-first-passage-limit-shape-September-24-2026"}], "audit_part": 2, "second_reader": "", "referee_verdict": ""}, {"family": "213", "title": "No infinite critical clusters on quasi-transitive graphs", "subject": "Probability and statistical mechanics", "headline": "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 $\\mathbb Z^3$.", "verdict": "full", "challenges": ["CriticalPercolation", "CriticalZ3"], "review_note": "`no_percolation_at_criticality [Infinite V] (hconnected : G.fullGraph.Connected) (hlocal : G.LocallyFinite) (hquasi : G.QuasiTransitive) (hpc : G.criticalProbability < 1) : G.law G.criticalProbability G.percolates = 0` on `BondGraph V E` (multigraphs, `law p = setBernoulli univ p`, `criticalProbability = sInf {p | 0 < P_p(∃ infinite cluster)}`), plus `CriticalZ3.critical_no_infinite`: `∀ᵐ ω ∂bondLaw bondCritical, ∀ x, (bondCluster ω x).Finite` and the same for site percolation, with `bondCritical = sInf {p | 0 < P_p(origin cluster infinite)}`. Both summary claims are stated.", "definitions_to_check": [], "external_packages": [], "cone_lines_max": 68224, "machine_check": "comparator-sandboxed-pass", "lab_scope_note": "The critical-percolation question asks whether an infinite cluster can remain at the threshold. The formalized result proves that, at their respective critical probabilities, nearest-neighbor bond and site percolation on $\\mathbb Z^3$ almost surely have no infinite cluster. Equivalently, almost surely every vertex belongs to a finite cluster in each model.\n\nThe formalization proves absence of an infinite cluster at criticality for Bernoulli bond percolation on every infinite connected locally finite quasi-transitive graph with critical probability $p_c<1$. At $p=p_c$, the probability that any infinite cluster exists is zero. The graph model permits bond multiplicities.", "lab_docs_url": "https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/docs/213.md", "overview_entry": "family 213 in overview.pdf / CONTENTS.md", "papers": [{"dir": "Critical-bond-and-site-percolation-on-the-cubic-lattice-September-24-2026", "title": "Critical bond and site percolation on the cubic lattice", "url": "https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Critical-bond-and-site-percolation-on-the-cubic-lattice-September-24-2026"}, {"dir": "No-percolation-at-criticality-on-quasi-transitive-graphs-September-24-2026", "title": "No percolation at criticality on quasi-transitive graphs", "url": "https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/No-percolation-at-criticality-on-quasi-transitive-graphs-September-24-2026"}], "audit_part": 2, "second_reader": "", "referee_verdict": ""}, {"family": "214", "title": "The Benjamini--Schramm nonuniqueness conjecture", "subject": "Probability and statistical mechanics", "headline": "Proves $p_c 0, ∃ A : Finset Λ, A.Nonempty ∧ ∀ k ∈ K, ((A.image (· * k)) \\ A).card < ε * A.card` (and the identical `AmenableGroup` in CayleyPercolation) is the standard right-Folner criterion, not Mathlib's; for general graphs nonamenability is `0 < hV G` with `hV G := sInf {|∂A| / |A| : A finite nonempty}` (outer vertex boundary). Standard but hand-written."], "external_packages": [], "cone_lines_max": 151059, "machine_check": "comparator-sandboxed-pass", "lab_scope_note": "The formalization proves the nonuniqueness phase conjecture for Bernoulli bond percolation on infinite connected locally finite nonamenable quasi-transitive graphs. It establishes $p_c 0, ∃ A m, 0 < m ∧ ∀ finite lattice graph G, b ∈ [0,β] → 0 ≤ correlation ≤ A * exp(-m * siteDistance x y)` (free boundary, finite volume, uniform in the graph; docs: 'The infinite-volume limit and the separate O(4) spectral-gap claim are not included'). The summary's headline results, the canonical O(3) continuum limit with positive mass gap and the exact O(4) asymptotic m_lat(β) ~ 32 e^{π/4-1/2} √β e^{-πβ}, are not stated.", "definitions_to_check": [], "external_packages": [], "cone_lines_max": 12907, "machine_check": "comparator-sandboxed-pass", "lab_scope_note": "The formalized result proves exponential decay of spin correlations for the two-dimensional classical $O(n)$ model at every temperature when $n\\ge3$. For each interaction bound $\\beta>0$, constants $A$ and $m>0$ work for every finite square-lattice subgraph with free boundary and nonnegative edge strengths at most $\\beta$: the correlation between sites $x,y$ is at most $Ae^{-m|x-y|}$. The infinite-volume limit and the separate $O(4)$ spectral-gap claim are not included.", "lab_docs_url": "https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/docs/215.md", "overview_entry": "family 215 in overview.pdf / CONTENTS.md", "papers": [{"dir": "Exponential-decay-in-two-dimensional-classical-On-models-September-23-2026", "title": "Exponential decay in two-dimensional classical O(n) models", "url": "https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Exponential-decay-in-two-dimensional-classical-On-models-September-23-2026"}], "audit_part": 2, "second_reader": "", "referee_verdict": ""}, {"family": "218", "title": "Conformal universality for weakly interacting and disordered Ising models", "subject": "Probability and statistical mechanics", "headline": "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 $\\mathrm{SLE}_3$ 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.", "verdict": "supporting-only", "challenges": ["BufferedIsing"], "review_note": "The only statement is a technical finite-graph inequality: `finite_graph_comparison : exp(-4 * artanh (qZero ..)) ≤ mixtureProb .. mix / mixtureProb .. mix' ∧ ... ≤ exp(4 * artanh (qZero ..))` for Ising with arbitrary fields and boundary mixtures (docs: 'The paper's finite stopping-band approximation theorem is outside this selected statement'). None of the headline results (universality of critical bulk spin/energy limits under weak multispin perturbations, SLE_3 interface limits, weak iid bond disorder) is stated.", "definitions_to_check": [], "external_packages": [], "cone_lines_max": 3114, "machine_check": "comparator-sandboxed-pass", "lab_scope_note": "The formalized result compares conditional Ising probabilities on finite graphs when two boundary mixtures differ but a common set of spins is pinned. If $q_0$ is the associated zero-field Fortuin–Kasteleyn connection probability after deleting the pinned vertices, the likelihood ratio lies between $\\exp(-4\\mathrm{artanh}\\,q_0)$ and $\\exp(4\\mathrm{artanh}\\,q_0)$. The statement allows arbitrary external fields and arbitrary probability mixtures satisfying the stated disjointness conditions. The paper's finite stopping-band approximation theorem is outside this selected statement.", "lab_docs_url": "https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/docs/218.md", "overview_entry": "family 218 in overview.pdf / CONTENTS.md", "papers": [{"dir": "Buffered-comparison-and-stopping-band-resolution-in-critical-Ising-September-23-2026", "title": "Buffered comparison and stopping-band resolution in critical Ising", "url": "https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Buffered-comparison-and-stopping-band-resolution-in-critical-Ising-September-23-2026"}], "audit_part": 2, "second_reader": "", "referee_verdict": ""}, {"family": "220", "title": "Directional zero--one laws and ballisticity in random environments", "subject": "Probability and statistical mechanics", "headline": "On $\\mathbb Z^d$, $d\\ge3$, 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 $d\\ge2$, almost-sure directional transience implies a deterministic limiting velocity with positive projection in that direction, resolving the ballisticity conjecture.", "verdict": "partial", "challenges": ["DirectionalBallisticity", "DirectionalWalk", "VelocityHemisphere"], "review_note": "Stated: `directional_zero_one (d) (hd : 3 ≤ d) (hell : StrictEllipticity μ) (ℓ ≠ 0) : annealed μ 0 (escape ℓ) = 0 ∨ ... = 1` for IID environments (`environmentLaw μ = infinitePi (fun _ => μ)`), and `directional_transience_implies_ballisticity` (d >= 2, iid uniformly elliptic, `DirectionallyTransient ν ℓ → ∃ v, 0 < dot v ℓ ∧ HasVelocity ν v`), plus the velocity-hemisphere refinement. Not stated: the zero-one law for stationary ergodic finite-range-dependent environments under uniform ellipticity (the docs title says 'beyond iid environments' but the scope and Lean are iid only).", "definitions_to_check": [], "external_packages": [], "cone_lines_max": 116839, "machine_check": "comparator-sandboxed-pass", "lab_scope_note": "The directional zero–one conjecture asks whether a random walk's probability of escape in a fixed direction must be zero or one. The formalization proves this for nearest-neighbor walks in independent identically distributed strictly elliptic environments on $\\mathbb Z^d$, for every $d\\ge3$ and every nonzero real direction. Strict ellipticity means that every allowed transition probability is positive almost surely; no uniform positive lower bound or moment condition is assumed. The probability is the annealed law from the origin.\n\nFor an i.i.d. uniformly elliptic nearest-neighbor random environment on $\\mathbb Z^d$, the formalization proves that almost-sure directional transience implies convergence of $X_n/n$ to a deterministic velocity with positive projection in that direction for every $d\\ge2$.\n\nFor $d\\ge3$, positive probability of transience in any nonzero direction already suffices. The limiting velocity is unique, and the unit directions with positive transience probability are exactly the open hemisphere having positive inner product with that velocity; transience has probability one on that hemisphere and zero on its complement. The environment is sampled once and retained along the walk, and the probabilities use the annealed law from the origin.", "lab_docs_url": "https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/docs/220.md", "overview_entry": "family 220 in overview.pdf / CONTENTS.md", "papers": [{"dir": "A-directional-zero-one-law-under-strict-ellipticity-September-23-2026", "title": "A directional zero–one law under strict ellipticity", "url": "https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-directional-zero-one-law-under-strict-ellipticity-September-23-2026"}, {"dir": "Directional-transience-implies-ballisticity-September-23-2026", "title": "Directional transience implies ballisticity", "url": "https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Directional-transience-implies-ballisticity-September-23-2026"}], "audit_part": 2, "second_reader": "", "referee_verdict": ""}, {"family": "221", "title": "The M\\'ezard--Parisi formula for diluted spin glasses", "subject": "Probability and statistical mechanics", "headline": "Proves the M\\'ezard--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.", "verdict": "full", "challenges": ["DilutedSpin"], "review_note": "`mezard_parisi (M : Model p) (hM : Admissible M) : Tendsto (pressure M) atTop (𝓝 (variationalValue M))` for Poisson-diluted even-arity (p >= 2 even) Ising models with the factorization `exp(θ s) = a (1 + b ∏ f_l(s_l))`, independence/identical-distribution of the f_l, b-moment positivity `0 ≤ E[(-b)^n]` and only first-moment integrability of θ and the field; `variationalValue M := ⨅ r, phi M r` with `phi M r := sInf {functional M r ζ m | ζ ∈ Hierarchy (r+1), 0 < m_1 < ... < m_r < 1}`. Matches the summary (Viana-Bray etc. are instances of the class, not separately stated). Caveat: everything is Bochner integrals, so a non-integrable integrand silently evaluates to 0.", "definitions_to_check": ["Custom 'free energy': `pressure M N := (∫ k ∂Poisson(αN), ∫ θ, ∫ h, avg_indices log Σ_σ exp(logWeight) ...) / N` and the hierarchical functional `functional M r ζ m` over `Hierarchy : ℕ → TopCat` (H_0 = ℝ, H_{r+1} = ProbabilityMeasure (H_r)) with `logMean`/`trialLog`; all hand-written, and `phi` is an `sInf` over reals (junk value 0 if unbounded below) with Bochner integrals defaulting to 0 off the integrable class. Not trivializing (convergence to a nontrivial inf is asserted) but the junk-value behaviour is a fidelity risk."], "external_packages": [], "cone_lines_max": 56285, "machine_check": "comparator-sandboxed-pass", "lab_scope_note": "The formalization proves the Mézard–Parisi hierarchical cavity formula for diluted even-arity Ising models in the Panchenko–Talagrand class. Under the class's factorization, independence, integrability, and positivity assumptions, the finite-system pressure converges to the infimum of the trial functional over all finite hierarchy depths and trial laws. The arity is any even integer at least two and the interaction density is positive.", "lab_docs_url": "https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/docs/221.md", "overview_entry": "family 221 in overview.pdf / CONTENTS.md", "papers": [{"dir": "The-Mezard-Parisi-formula-for-diluted-spin-glasses-September-23-2026", "title": "The Mézard–Parisi formula for diluted spin glasses", "url": "https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-Mezard-Parisi-formula-for-diluted-spin-glasses-September-23-2026"}], "audit_part": 2, "second_reader": "", "referee_verdict": ""}, {"family": "222", "title": "Perceptron free energies and microscopic jamming", "subject": "Probability and statistical mechanics", "headline": "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.", "verdict": "partial", "challenges": ["IsingFiniteness", "PerceptronFreeEnergy", "SphericalField"], "review_note": "Stated: the spherical Gaussian perceptron formula `SphericalPerceptronFreeEnergy.main : ∀ α β > 0, ∀ φ : ℝ →ᵇ ℝ, ∃ p, variationalValue P α β φ = p ∧ expectedPressure → p ∧ pressure → p in probability`, plus a supporting `spherical_linear_field_formula`. For the Ising perceptron only well-definedness is stated: `variationalValue_real_of_continuous : ∃ p : ℝ, variationalValue α f = (p : EReal)` (docs: 'supporting well-definedness result. It does not assert convergence of the finite-system pressure or ... all bounded Borel log-potentials'). Not stated: Ising free-energy convergence, the bi-orthogonally invariant spherical extension, and the margin -1 jamming gap/force laws.", "definitions_to_check": ["Hand-written variational 'free energies': Ising `variationalValue α f := ⨅ q : OverlapPath, (α * patternFunctional f q : ℝ) + isingEntropy q` with `patternFunctional f q := Filter.limUnder atTop (uniformPattern f q)` (junk value if the limit does not exist) and `isingEntropy q := ⨆ h : FieldStep, ...` in EReal; spherical `variationalValue P α β φ := ⨅ m : Trial, α * controlValue P (β • φ) m + entropy m` via Brownian stochastic control (`Progressive`, `usualBrownianSigma`). Finiteness of a `limUnder`-defined inf is a weak guarantee."], "external_packages": [], "cone_lines_max": 71237, "machine_check": "comparator-sandboxed-pass", "lab_scope_note": "The linked formalization proves finiteness of the variational value used for the Ising random perceptron. For every nonnegative pattern density and every bounded continuous log-potential, the infimum over admissible overlap paths is a finite real number.\n\nThis is a supporting well-definedness result. It does not assert convergence of the finite-system pressure or the paper's extension to all bounded Borel log-potentials.\n\nThe formalization proves the variational formula for the spherical random perceptron's limiting pressure for every positive pattern density and inverse temperature and every bounded continuous single-pattern potential. It shows that the variational value is finite and that the pressure converges to it both in expectation and in probability.\n\nThe linked spherical-field result supplies a supporting dual formula: finite hierarchy field values converge uniformly on compact parameter sets to an attained dual minimum, and the corresponding bounded stationary-field approximation holds for monotone overlap quantiles bounded away from one.", "lab_docs_url": "https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/docs/222.md", "overview_entry": "family 222 in overview.pdf / CONTENTS.md", "papers": [{"dir": "The-free-energy-of-the-Ising-random-perceptron-September-24-2026", "title": "The free energy of the Ising random perceptron", "url": "https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-free-energy-of-the-Ising-random-perceptron-September-24-2026"}, {"dir": "The-free-energy-of-the-spherical-random-perceptron-September-24-2026", "title": "The free energy of the spherical random perceptron", "url": "https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-free-energy-of-the-spherical-random-perceptron-September-24-2026"}], "audit_part": 2, "second_reader": "", "referee_verdict": ""}, {"family": "227", "title": "The dynamical phase transition in the Sherrington--Kirkpatrick model", "subject": "Probability and statistical mechanics", "headline": "For zero-field Gaussian SK heat-bath dynamics with rate-one updates per spin, proves worst-start cutoff on the $\\log n$ scale for fixed $0\\le\\beta<1$, mixing time $n^{2/3+o(1)}$ at $\\beta=1$, and stretched-exponential mixing from a Gibbs-sampled fixed starting configuration for $\\beta>1$, 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.", "verdict": "partial", "challenges": ["CriticalSK", "CriticalSKMixing", "SKBarriers", "SKHighTemperature", "SKRatio"], "review_note": "Stated: stretched-exponential barrier for β > 1 (`SK.main : Tendsto continuousBadMass atTop (𝓝 1) ∧ Tendsto discreteBadMass atTop (𝓝 1)` at time exp(n^(1/10000)), expected bad Gibbs mass over disorder); high-temperature spectral gap `sk_main` (0<β<1: Poincaré constant C and gap ≥ 1/(Cn)) as a supporting result; ratio cutoff `ratio_cutoff` only for 0 ≤ β < 1/2. Critical case is one-sided: `critical_mixing_bounds` gives `n^(2/3-ε) ≤ continuousMixingTime ≤ exp(εn)` (discrete `n^(5/3-ε) ≤ ... ≤ exp(εn)`), so the headline 'mixing time n^{2/3+o(1)} at β=1' has no n^{2/3+ε} upper bound in Lean. Not stated: cutoff for 1/2 <= β < 1 and its log n location, and the critical autocorrelation-process limits (Gaussian/Rademacher, quench relaxing to stationary).", "definitions_to_check": ["Custom 'mixing time': `continuousMixingTime W := sInf {t | 0 ≤ t ∧ ∀ x, continuousDistance W t x ≤ 1/4}` (matrix exponential kernel `NormedSpace.exp (t • generator W)`), `discreteMixingTime W := sInf {k : ℕ | ∀ x, discreteDistance W k x ≤ 1/4}`, `SKRatio.mixingTime g ε := sInf {k | discreteDistance g k ≤ ε}`: standard worst-start total-variation mixing (sets are nonempty so sInf is a true min), but hand-written; `SKBarriers` uses its own `heatBath` kernel and a Poissonized `continuousKernel` (series in k)."], "external_packages": [], "cone_lines_max": 119320, "machine_check": "comparator-sandboxed-pass", "lab_scope_note": "For every fixed inverse temperature $0<\\beta<1$, the formalization proves a dimension-independent Poincaré inequality for the zero-field Gaussian Sherrington–Kirkpatrick model with probability tending to one over the disorder. Equivalently, the unscaled single-site heat-bath spectral gap stays bounded away from zero. For dynamics that choose one site uniformly at each discrete step, the spectral gap is at least $1/(Cn)$ with probability tending to one, for a constant $C$ depending on $\\beta$.\n\nThe formalized supporting result proves ratio cutoff for discrete single-site heat-bath dynamics in the zero-field Gaussian Sherrington–Kirkpatrick model for $0\\le\\beta<1/2$. For every $0<\\varepsilon<1/2$ and $\\eta>0$, the probability over the disorder that $t_{\\mathrm{mix}}(\\varepsilon)/t_{\\mathrm{mix}}(1-\\varepsilon)>1+\\eta$ tends to zero as $n\\to\\infty$. The denominator is positive for all sufficiently large sizes.\n\nThe paper's full range $\\beta<1$, explicit cutoff location, and continuous-time conclusion are outside this selected statement.\n\nAt criticality in the zero-field Sherrington–Kirkpatrick model, the formalization proves that for every fixed $\\varepsilon>0$, the continuous-time mixing time lies between $n^{2/3-\\varepsilon}$ and $e^{\\varepsilon n}$, and the discrete-time mixing count lies between $n^{5/3-\\varepsilon}$ and $e^{\\varepsilon n}$, with probability tending to one over the Gaussian disorder. The disorder has independent off-diagonal entries of variance $1/n$. Continuous updates have rate one per site; discrete time counts uniformly chosen single-site update attempts. Mixing is worst-start total variation at threshold $1/4$, with holding allowed.\n\nFor deterministic times $t_n=o(n^{2/3})$ or update counts $k_n=o(n^{5/3})$, the Gibbs mass of realized starting states still farther than $1/4$ from equilibrium tends to one in probability. For every deterministic positive sequence $M_n\\to\\infty$, the covariance operator norm and the linear Rayleigh supremum, using the unscaled heat-bath Dirichlet form, both lie between $n^{2/3}/M_n$ and $M_n n^{2/3}$ with probability tending to one. The covariance result does not give an upper bound for the full inverse spectral gap.\n\nThe formalization proves a stretched-exponential mixing obstruction for the zero-field Sherrington–Kirkpatrick model at every fixed inverse temperature $\\beta>1$. At time $\\exp(n^{1/10000})$, the expected Gibbs mass of initial configurations whose total-variation distance from equilibrium exceeds $1/4$ tends to one. Since this mass lies in $[0,1]$, it also tends to one in probability over the disorder.\n\nThe result covers both rate-one-per-site continuous-time heat-bath dynamics and the same stated number of discrete update attempts.", "lab_docs_url": "https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/docs/227.md", "overview_entry": "family 227 in overview.pdf / CONTENTS.md", "papers": [{"dir": "A-spectral-gap-throughout-the-high-temperature-Sherrington-Kirkpatrick-phase-September-24-2026", "title": "A spectral gap throughout the high-temperature Sherrington–Kirkpatrick phase", "url": "https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-spectral-gap-throughout-the-high-temperature-Sherrington-Kirkpatrick-phase-September-24-2026"}, {"dir": "Cutoff-throughout-the-high-temperature-Sherrington-Kirkpatrick-phase-September-24-2026", "title": "Cutoff throughout the high-temperature Sherrington–Kirkpatrick phase", "url": "https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Cutoff-throughout-the-high-temperature-Sherrington-Kirkpatrick-phase-September-24-2026"}, {"dir": "Critical-slowing-down-in-the-Sherrington-Kirkpatrick-model-September-24-2026", "title": "Critical slowing down in the Sherrington–Kirkpatrick model", "url": "https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Critical-slowing-down-in-the-Sherrington-Kirkpatrick-model-September-24-2026"}, {"dir": "Stretched-exponential-barriers-for-typical-SK-initial-states-September-24-2026", "title": "Stretched-exponential barriers for typical SK initial states", "url": "https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Stretched-exponential-barriers-for-typical-SK-initial-states-September-24-2026"}], "audit_part": 2, "second_reader": "", "referee_verdict": ""}, {"family": "228", "title": "Continuum phase transitions for radial pair potentials", "subject": "Probability and statistical mechanics", "headline": "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.", "verdict": "full", "challenges": ["ContinuumTransition", "RadialDensityInterval", "RadialTransition"], "review_note": "`ContinuumTemperature.main_theorem : ∃ φ f I βc, Admissible φ ∧ φ = o(r^{-3}) ∧ HasCanonicalFreeEnergy φ f ∧ I open interval ⊆ (0,∞) ∧ βc ∈ (1/2, 3/2) ∧ ∀ ρ ∈ I, dplus < dminus` (Admissible = stable, `φ → +∞` at 0 (divergent core), integrable tail) and `RadialTransition.densityInterval` (bounded continuous φ, `|φ r| ≤ C r^(-3-1/32)`, stable, a nonempty open density interval around 5p/3, common βc ∈ [7/8, 9/8], `StrictTemperatureCorner`); both potentials of the summary and the density-interval statement are covered.", "definitions_to_check": ["Custom 'free energy': `HasCanonicalFreeEnergy φ f := ∀ β ρ > 0, Tendsto (L^{-3} log Z(β, L, ⌊ρL^3⌋)) atTop (𝓝 (-β f β ρ))` / `IsCanonicalFreeEnergy` with explicit `partition φ β L N := (N!)^{-1} ∫_{cube^N} exp(-β energy)` and `energy` in EReal (⊤ at coincident points) for the first potential; standard thermodynamic-limit definitions, free boundary conditions."], "external_packages": [], "cone_lines_max": 14988, "machine_check": "comparator-sandboxed-pass", "lab_scope_note": "The formalization constructs one stable radial pair potential in three-dimensional continuum space with a divergent repulsive core and an integrable tail that is $o(r^{-3})$. Its canonical free energy exists for every positive inverse temperature and density. At one inverse temperature $\\beta_c\\in(1/2,3/2)$, the free energy has a strict finite jump between its one-sided derivatives with respect to inverse temperature, for every density in one nonempty open interval.\n\nThe formalization constructs a bounded continuous stable radial pair potential in $\\mathbb R^3$ satisfying $|\\phi(r)|\\le Cr^{-3-1/32}$ for $r\\ge1$. There is a nonempty open interval of positive densities around $5p/3$, where $p$ is the unit-separated packing-density limit, on which the canonical free energy exists at every positive inverse temperature and has a strict downward derivative jump at one common $\\beta_c\\in[7/8,9/8]$.\n\nThe earlier fixed-density statement at $5p/3$ is also retained. Both statements concern the derivative with respect to inverse temperature.", "lab_docs_url": "https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/docs/228.md", "overview_entry": "family 228 in overview.pdf / CONTENTS.md", "papers": [{"dir": "A-continuum-temperature-singularity-for-a-radial-pair-potential-September-24-2026", "title": "A continuum temperature singularity for a radial pair potential", "url": "https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-continuum-temperature-singularity-for-a-radial-pair-potential-September-24-2026"}, {"dir": "A-radial-continuum-phase-transition-with-algebraic-decay-September-24-2026", "title": "A radial continuum phase transition with algebraic decay", "url": "https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-radial-continuum-phase-transition-with-algebraic-decay-September-24-2026"}], "audit_part": 2, "second_reader": "", "referee_verdict": ""}, {"family": "229", "title": "Sharp three- and four-state reconstruction thresholds", "subject": "Probability and statistical mechanics", "headline": "Proves the exact reconstruction threshold $d\\lambda^2>1$, with nonreconstruction at equality, for three-state symmetric and four-state ferromagnetic broadcasting on regular trees ($d\\ge2$) 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 $\\lambda$ and gives the exact weak-recovery threshold for the symmetric three-community stochastic block model.", "verdict": "weaker-statement", "challenges": ["ThreeStateSupercritical", "ThreeStateTreeClauses"], "review_note": "Only the supercritical (Kesten-Stigum) direction for the three-state symmetric channel is stated: `regular_supercritical (hcrit : 1 < b * lam^2) : Reconstructs ..` and `poisson_supercritical (hd : 1 < d) (hcrit : 1 < d * lam^2) : Reconstructs ..` (advantage `∑' y, marginal y * (∑ i |posterior y i - 1/3|)/2` converges to L > 0), docs: 'Non-reconstruction at or below the threshold and the stochastic-block-model consequences in the paper are outside them'. The summary's headline is the EXACT threshold dλ²>1 with nonreconstruction at equality, plus the four-state ferromagnetic case and the SBM weak-recovery threshold: none of those is stated (the formalized direction is the classical one, the sharpness is missing).", "definitions_to_check": [], "external_packages": [], "cone_lines_max": 7792, "machine_check": "comparator-sandboxed-pass", "lab_scope_note": "The formalization proves the supercritical direction of reconstruction for the symmetric three-state broadcast channel, whose parameter satisfies $-1/2\\le\\lambda\\le1$. On a regular $b$-ary tree, reconstruction holds when $b\\lambda^2>1$; on an observed Poisson Galton–Watson tree of mean $d$, it holds when $d\\lambda^2>1$. In each case the root-estimation advantage converges to a positive limit, with the Poisson advantage averaged over trees and spins.\n\nThe selected statements cover reconstruction above the Kesten–Stigum threshold. Non-reconstruction at or below the threshold and the stochastic-block-model consequences in the paper are outside them.", "lab_docs_url": "https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/docs/229.md", "overview_entry": "family 229 in overview.pdf / CONTENTS.md", "papers": [{"dir": "The-exact-reconstruction-threshold-for-the-three-state-symmetric-channel-September-25-2026", "title": "The exact reconstruction threshold for the three-state symmetric channel", "url": "https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-exact-reconstruction-threshold-for-the-three-state-symmetric-channel-September-25-2026"}], "audit_part": 2, "second_reader": "", "referee_verdict": ""}, {"family": "230", "title": "Exact Hausdorff measure for SLE", "subject": "Probability and statistical mechanics", "headline": "Resolves Schramm’s Hausdorff-measure question for chordal $\\mathrm{SLE}_\\kappa$, $0<\\kappa<8$. The explicit gauge $r^d(\\log\\log(1/r))^{(2-d)/2}$, $d=1+\\kappa/8$, gives almost surely positive finite measure to every trace segment $\\gamma([s,t])$ with $0 0 via `scaledSpectralLaw`), `ground_state_limit_unconditional` (`Tendsto (thermalVariationalValue ν b) atTop (𝓝 M)` with M the ground-state limit), plus magnetic-field, random-spectrum and Gaussian-pattern variants. All summary claims are stated.", "definitions_to_check": ["Custom 'free energy': `logPartition H := log ((card X)⁻¹ * Σ_x exp(H x))` (normalized by 2^{-N}, i.e. pressure minus log 2), `rotatedPressure`, the Parisi-type `variationalFunctional R := ⨅ p : OverlapPath, entropyFunctional p + spectralFunctional R p` (EReal) and `measureR μ e x` (R-transform via a `h.choose` inverse of the resolvent on (e, ∞)); hand-written variational formulas, with the final value taken via `.toReal` of an EReal (junk if infinite)."], "external_packages": [], "cone_lines_max": 210755, "machine_check": "none", "lab_scope_note": "The formalization gives variational limits for orthogonally invariant Ising spin glasses whose eigenvectors have Haar law and whose empirical eigenvalue distributions converge to a compactly supported law with the stated control of extreme eigenvalues. The pressure converges both in expectation and almost surely to an explicit functional of the limiting spectral law. A version with external fields assumes convergence of their empirical laws in Wasserstein distance and gives the corresponding magnetic-field functional.\n\nThe formalization also covers every positive temperature, a ground-state limit, random spectral data with the stated conditional Haar law, and Gaussian-pattern specializations. These hypotheses identify the invariant models to which the limits apply.", "lab_docs_url": "https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/docs/234.md", "overview_entry": "family 234 in overview.pdf / CONTENTS.md", "papers": [{"dir": "All-temperature-pressure-for-orthogonally-invariant-Ising-spin-glasses-September-25-2026", "title": "All-temperature pressure for orthogonally invariant Ising spin glasses", "url": "https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/All-temperature-pressure-for-orthogonally-invariant-Ising-spin-glasses-September-25-2026"}], "audit_part": 2, "second_reader": "", "referee_verdict": ""}, {"family": "235", "title": "Random-SAT thresholds, sharp variance and computability", "subject": "Probability and statistical mechanics", "headline": "For random $k$-SAT with independent uniformly signed proper clauses sampled with replacement, proves finite positive limiting thresholds and hitting-time variance $\\Theta_k(n)$ for every fixed $k\\ge3$, and computability of the 3-SAT threshold. We credit Gaia Carenini with priority for resolving the threshold-existence conjecture in her concurrent \\href{https://eccc.weizmann.ac.il/report/2026/229/}{ECCC TR26-229}, made public October 5, 2026; this family supplies another proof and the sharper variance and computability results.", "verdict": "partial", "challenges": ["FixedClauseThreshold", "SATComputability", "SATSharpness", "SATVariance"], "review_note": "Stated: `FixedClauseThreshold.main (k) (hk : 3 ≤ k) : HasLimitingThreshold k` (∃ α > 0, P(SAT) → 1 for c < α and → 0 for c > α with m = ⌊c n⌋ i.i.d. uniform proper clauses), `SATComputability.main` (3-SAT threshold α with a `Nat.Partrec.Code` producing rationals q r, |q r - α| ≤ 2^-r) and `variance_main`. But the variance claim is only Θ(n) for k >= 4: for k = 3 the Lean gives `Var(H) ≤ C n ell` with `ell = log(e n)` (an n log n upper bound) and a linear lower bound, so the summary's 'hitting-time variance Θ_k(n) for every fixed k ≥ 3' is not stated for k = 3 (docs: 'They do not show that the logarithmic factor in the k=3 variance bound is necessary').", "definitions_to_check": [], "external_packages": [], "cone_lines_max": 87536, "machine_check": "comparator-sandboxed-pass", "lab_scope_note": "The random $k$-SAT threshold problem asks whether satisfiability changes at one limiting clause density. The formalization proves that every fixed integer $k\\ge3$ has a finite positive threshold $\\alpha_k$: for every fixed density $c<\\alpha_k$, the satisfiability probability tends to one as the number of variables grows, while for every $c>\\alpha_k$ it tends to zero. Clauses use distinct variables with independent uniform signs and are sampled independently with replacement. The statement makes no assertion at the threshold itself.\n\nLet $H_n$ be the first unsatisfiable prefix length in random $k$-SAT with independent uniformly signed clauses on $k$ distinct variables, sampled with replacement. The formalization proves $\\mathrm{Var}(H_n)=\\Theta_k(n)$ for every fixed $k\\ge4$. For $k=3$, it proves a positive linear lower bound and an $O(n\\log n)$ upper bound. The same upper bounds hold after clipping at any fixed positive multiple of $n$, and linear lower bounds hold for sufficiently high clipping levels.\n\nThe linked supplements prove sharpness of the separate survival-lifetime and clause-replacement estimates used in the argument. They do not show that the logarithmic factor in the $k=3$ variance bound is necessary.\n\nThe formalization proves that the limiting threshold $\\alpha$ for uniform random $3$-SAT is a computable positive real. Satisfiability tends to one at every fixed density below $\\alpha$ and to zero at every fixed density above it. One deterministic machine produces a rational number $q_r$ for every requested precision $r$ with $|q_r-\\alpha|\\le2^{-r}$. No computable rate of finite-size convergence is assumed in the statement.", "lab_docs_url": "https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/docs/235.md", "overview_entry": "family 235 in overview.pdf / CONTENTS.md", "papers": [{"dir": "A-Limiting-Satisfiability-Threshold-for-Every-Fixed-Clause-Size-September-25-2026", "title": "A Limiting Satisfiability Threshold for Every Fixed Clause Size", "url": "https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-Limiting-Satisfiability-Threshold-for-Every-Fixed-Clause-Size-September-25-2026"}, {"dir": "Variance-of-the-Random-k-SAT-Hitting-Time-September-27-2026", "title": "Variance of the Random k-SAT Hitting Time", "url": "https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Variance-of-the-Random-k-SAT-Hitting-Time-September-27-2026"}, {"dir": "Computing-the-Random-3-SAT-Threshold-September-27-2026", "title": "Computing the Random 3-SAT Threshold", "url": "https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Computing-the-Random-3-SAT-Threshold-September-27-2026"}], "audit_part": 2, "second_reader": "", "referee_verdict": ""}, {"family": "236", "title": "The factor-of-IID threshold for free Ising states on trees", "subject": "Probability and statistical mechanics", "headline": "Determines when the free zero-field ferromagnetic Ising state on the infinite $d$-regular tree is a factor of independent vertex labels: exactly when $\\tanh\\beta\\le(d-1)^{-1/2}$, including equality, for $d\\ge3$ and $\\beta\\ge0$. The construction uses no root and is almost surely equivariant for each fixed tree automorphism, resolving the ferromagnetic case of Lyons's question.", "verdict": "full", "challenges": ["FreeIsing"], "review_note": "`free_ising_factor_iff_threshold : ∀ d β, 3 ≤ d → 0 ≤ β → (IsFactorOfIID d (tanh β) ↔ tanh β ≤ 1 / √(d-1))`, with `TreeVertex d` the reduced words in Fin d (the d-regular tree), `HasFreeIsingLaw θ` the free zero-field Ising cylinder weights `1/2 ∏_edges (1 ± θ)/2`, and `IsFactorOfIID` = ∃ measurable `Phi` pushing iid Uniform[0,1] labels to that law and a.s. equivariant for each fixed tree automorphism `g`. Exactly the summary's iff, including equality.", "definitions_to_check": [], "external_packages": [], "cone_lines_max": 117435, "machine_check": "comparator-sandboxed-pass", "lab_scope_note": "The formalized result determines the factor-of-IID threshold for the free zero-field Ising law on the infinite $d$-regular tree. For every $d\\ge3$ and $\\beta\\ge0$, the law is a factor of IID exactly when $\\tanh\\beta\\le1/\\sqrt{d-1}$, including equality. The factor uses independent uniform vertex labels and is almost surely equivariant for each fixed tree automorphism. Finitary coding and coding-radius bounds are not included.", "lab_docs_url": "https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/docs/236.md", "overview_entry": "family 236 in overview.pdf / CONTENTS.md", "papers": [{"dir": "The-sharp-factor-of-IID-threshold-for-the-free-Ising-model-on-regular-trees-September-26-2026", "title": "The sharp factor-of-IID threshold for the free Ising model on regular trees", "url": "https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-sharp-factor-of-IID-threshold-for-the-free-Ising-model-on-regular-trees-September-26-2026"}], "audit_part": 2, "second_reader": "", "referee_verdict": ""}, {"family": "237", "title": "The three-quarter diameter exponent for honeycomb walks", "subject": "Probability and statistical mechanics", "headline": "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.", "verdict": "supporting-only", "challenges": ["CriticalStripMass", "HoneycombBridgeFiniteness", "HoneycombFreeEnergy"], "review_note": "Only technical inputs are stated: `HoneycombBridgeFiniteness.finite_bridge_sums` (summability of bridge weights and first length moments, also in the corridor |x| ≤ h (log h)^2), `CriticalStripMass.critical_strip_mass` (`c * archMass N + bridgeMass N = 1`, `Comparable bridgeMass N^(-1/4)`, `Comparable moment N^(3/4)`, bridge mass monotone) and `HoneycombFreeEnergy.logPartition_tendsto` (existence of the free-energy limit). The summary's headline, that a uniform n-step honeycomb SAW has diameter n^{3/4+o(1)} (and local mass/covering numbers of exponent 4/3, with high polynomial probability), has no Lean statement (docs: 'supporting summability statements ... outside them'; 'does not state ... the 3/4 spatial and moment laws').", "definitions_to_check": ["`freeEnergy o e s := Filter.limUnder atTop (logPartition o e s)` in `HoneycombFreeEnergy` (the theorem `logPartition_tendsto` asserts the limit exists, which makes the junk-value risk moot); SAW partition functions are written with explicit `walkLists`/`saws` and critical activity `rho = 1/√(2+√2)`."], "external_packages": [], "cone_lines_max": 62844, "machine_check": "comparator-sandboxed-pass", "lab_scope_note": "The linked formalization establishes finiteness of the critical bridge measures used in the paper. For every strip height $h\\ge1$, the total weight and first length moment of self-avoiding honeycomb bridges from a fixed initial port to a free terminal port are finite. The corresponding sums are also finite when the bridge is confined to the corridor of horizontal width $h(\\log h)^2$, including the normalized first-moment sums.\n\nThese are supporting summability statements. The paper's $h^{4/3+o(1)}$ mean-length law and its central-visit and hexagonal-chord exponents are outside them.\n\nThe linked formalization proves existence of the infinite-length free energy for the critical honeycomb self-avoiding-walk partition function. For every starting vertex, force direction, and real force parameter, the logarithm of the length-$n$ partition function divided by $n$ converges to its stated free-energy limit.\n\nThis selected result supplies the limiting free energy. It does not state the paper's small-force exponent, near-critical correlation scale, or the $3/4$ spatial and moment laws.\n\nFor critical self-avoiding walks on the honeycomb lattice, the formalization proves that the total weight of paths crossing a strip of height $N$ is comparable to $N^{-1/4}$, while the first horizontal-displacement moment of return paths is comparable to $N^{3/4}$. It also proves finiteness of the strip sums, the exact arch–bridge balance identity, comparison of successive moment increments with bridge mass, and monotonicity of bridge mass. The comparison constants are uniform in the strip height.", "lab_docs_url": "https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/docs/237.md", "overview_entry": "family 237 in overview.pdf / CONTENTS.md", "papers": [{"dir": "Polynomial-vacuum-representations-and-bridge-mass-for-honeycomb-walks-September-26-2026", "title": "Polynomial vacuum representations and bridge mass for honeycomb walks", "url": "https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Polynomial-vacuum-representations-and-bridge-mass-for-honeycomb-walks-September-26-2026"}, {"dir": "Renewal-and-changes-of-law-for-critical-honeycomb-walks-September-26-2026", "title": "Renewal and changes of law for critical honeycomb walks", "url": "https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Renewal-and-changes-of-law-for-critical-honeycomb-walks-September-26-2026"}, {"dir": "Critical-strip-crossing-mass-on-the-honeycomb-lattice-September-26-2026", "title": "Critical strip-crossing mass on the honeycomb lattice", "url": "https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Critical-strip-crossing-mass-on-the-honeycomb-lattice-September-26-2026"}], "audit_part": 2, "second_reader": "", "referee_verdict": ""}, {"family": "238", "title": "Optimal logarithmic mixing of the Thorp shuffle", "subject": "Probability and statistical mechanics", "headline": "Proves that the Thorp shuffle randomizes $N=2^d$ labeled cards in $\\Theta(\\log N)$ physical shuffles, settling its optimal mixing order for power-of-two deck sizes. Convergence is in total variation from the worst initial ordering and concerns the entire permutation, not just individual card positions.", "verdict": "full", "challenges": ["BinarySweep", "CoordinateSweeps", "CoordinateTrace", "FourRowPermanent", "OccupiedOverlap", "PartialPermutation", "SignedSweepMoment", "SpinAngle", "ThorpFirstReciprocal", "ThorpRemaining", "ThorpRouting", "ThorpWeightedCompatibility", "WeightedSweepMoments"], "review_note": "`ThorpResults.remaining_main` (ThorpRemaining) is a conjunction whose last conjunct `OptimalOrderMain := IsTheta atTop (mixingTime d) (log (2^d))` states exactly the Θ(log N) mixing for N = 2^d cards, with `mixingTime d := sInf {t | distance d t ≤ 1/4}`, `distance d t := tv (law d t) (uniform d)` the total-variation distance of the full permutation law (and `InformationMain` records that `lawFrom` from any starting deck has the same distance, plus `distance d (2048 d) → 0`); the 12 other challenge files are supporting Fourier/representation estimates (`regular_trace_smoothing`, `binary_sweep_contraction_and_mixing`, `robust_permanent`, ...). All of them state estimates for the same hand-built Thorp shuffle.", "definitions_to_check": ["Custom Thorp shuffle and 'mixing time': `step (d+1) c := (pairSwitch d c on the head bit, then rotate the coordinate order)` on `State d := Equiv.Perm (Fin d → Bool)` with `Coins (d+1) = (Fin d → Bool) → Bool` iid fair coins, `fairMass`, `tv`, `mixingTime d := sInf {t | distance d t ≤ 1/4}`; this is the standard pair-swap-and-interleave Thorp shuffle and TV-from-uniform with ε = 1/4, but it is a hand-written model and the Θ-statement hides explicit constants (1600d, 2048d, ...). `OccupiedOverlap` shows an `axiom_files` hit (HookModel.lean) that is a doc-comment false positive."], "external_packages": [], "cone_lines_max": 142145, "machine_check": "comparator-sandboxed-pass", "lab_scope_note": "The formalization proves optimal-order mixing of the Thorp shuffle on $2^d$ cards. After $1600d$ complete shuffles, the full permutation law converges in total variation to uniform as $d\\to\\infty$, uniformly over initial decks. Together with the support lower bound of $2d-O(1)$, this gives mixing time $\\Theta(d)=\\Theta(\\log(2^d))$.\n\nThe linked supporting results include frame and conditional-list estimates, regular trace and spectral bounds, signed moments, and full-density $L^2$ control. They also bound reciprocal Specht-dimension sums and give the eight-block Fourier estimate used in the mixing argument.\n\nThe formalized supporting result turns bounds on partial-permutation cosets into Fourier bounds. Partition a finite set into $b\\ge1$ nonempty blocks, and let $f\\ge0$ be a subprobability weight on its symmetric group whose left-coset masses for each block subgroup are at most $B$. For every irreducible unitary representation of dimension $D$ and every $u>0$, both the squared Hilbert–Schmidt norm and squared operator norm of its Fourier transform are at most $bB\\,C(u)D^{-1+(u+2)/b}$, where $C(u)$ is the stated symmetric-group degree constant.\n\nThe paper's asymptotic conclusion about the product of two random permutations is outside this selected finite estimate.\n\nThe formalized results give averaged conditional mixing for half-permutations in the revealed-path model and full-deck mixing for the Thorp shuffle. In particular, the full-deck total-variation distance tends to zero after $16040400d$ steps on $2^d$ cards, uniformly over deterministic initial decks. The conditional estimate is averaged over the actual outside-path law, rather than asserted for each individual environment. The formalization also includes the associated overlay, reset minorization, and two-color conditional constructions.\n\nThe formalization covers nine routing and representation estimates for Thorp sweeps: adaptive operator and fourth-trace bounds, Casimir moments, dense truncation, sparse contact, harmonic-cycle contraction, smoothing, and three tail or sparse-saving estimates. They bound routing-density deviations and Fourier operators in dense and sparse regimes, including exponential savings in the level scale and representation dimension.\n\nThe conditional high-height estimate requires a sufficiently large cutoff height $J$ that exceeds twice the number $s$ of coordinates in the fixed low block, so $J>2s$. These are the earlier nine statements associated with the paper; its changed later statements and six other result blocks are outside this scope.\n\nThe formalization proves uniform representation bounds for one coordinate sweep of the Thorp shuffle. For each irreducible representation of dimension $D$, it chooses a weight between $D^{3/4}$ and $D$ and a Schatten exponent bounded by one absolute constant so that the weighted Schatten moment is at most one. The sweep's operator norm is consequently at most $D^{-c}$ for one absolute $c>0$.\n\nThe supporting row–column estimate bounds the total squared overlap over all multiplicity copies on an occupied board, with explicit dependence on the row and column representation dimensions, the global dimension, the number of missing cells, and the number of rows or columns containing the global Young diagram. The paper's full physical-mixing conclusion is outside these selected representation estimates.\n\nThe formalization proves a uniform trace-smoothing estimate for coordinate sweeps of the Thorp shuffle on $2^d$ cards. One absolute sweep parameter works for every $d\\ge1$: the regular trace is at most $1+(2^d)^{-10}$, and the full permutation law after the corresponding fixed number of sweeps is within $\\tfrac12(2^d)^{-5}$ of uniform in total variation, for every initial deck. Thus the selected upper bound uses $O(d)$ physical shuffles.\n\nThe formalized result is the weighted compatibility theorem for balanced $A\\times D$ rectangles, with $n=AD$ and $\\sqrt n/2\\le A,D\\le2\\sqrt n$. For arbitrary nonnegative weights on the row and column permutation groups, the normalized compatibility average is bounded by $\\exp(C_0n^{54/100})$ times the product of the marginal $L^{1/\\theta}$ factors, where $\\theta=1-L/\\log\\sqrt n>0$ and the constants are absolute. Compatibility means injectivity in every original column. The bounded regular-moment and other moment/rank conclusions are not included.\n\nThe formalization gives signed representation estimates for coordinate sweeps of the Thorp shuffle. For every signed occurrence of a Young diagram of size $2^d$, it bounds the logarithm of a fixed-order weighted sweep moment by a small multiple of the signed partition entropy plus an explicit remainder-size budget. The moment order is uniform over the diagrams and decompositions after the two small coefficients are fixed.\n\nThe linked supporting angle bound controls the overlap of row, column, and global signed-type projections on an occupied rectangle, with explicit entropy, dimension, and missing-cell factors. These selected estimates support the paper's full-density argument; the full $L^2$ and total-variation mixing conclusion is outside them.\n\nThe formalized binary-sweep theorem gives a universal power contraction in every irreducible representation: for sufficiently large $d$, the averaged sweep on $2^d$ slots has operator norm at most $D^{-g}$ in representation dimension $D$, for some $g>0$. The sign expectation is zero, and a fixed number of independent sweeps approaches uniform in total variation from every initial deck. A separate conditional moment estimate covers allowed power-of-two grids and disjoint coordinate-respecting trajectories, with its stated feasibility premise. Physical-shuffle law identification is not included.\n\nThe formalization proves the paper's strict four-row permanent inequality. There are constants $4/30$ such that every probability law on $S_4$ within total-variation distance $\\varepsilon$ of uniform, and with exactly uniform coordinate marginals, satisfies the permanent bound by the product of the four $L^p$ row norms for every nonnegative matrix. The exponent and neighborhood are uniform over those laws. The Thorp mixing consequence is outside this selected statement.", "lab_docs_url": "https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/docs/238.md", "overview_entry": "family 238 in overview.pdf / CONTENTS.md", "papers": [{"dir": "Optimal-order-mixing-of-the-Thorp-shuffle-September-26-2026", "title": "Optimal-order mixing of the Thorp shuffle", "url": "https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Optimal-order-mixing-of-the-Thorp-shuffle-September-26-2026"}, {"dir": "From-partial-permutation-information-to-Fourier-bounds-September-26-2026", "title": "From partial permutation information to Fourier bounds", "url": "https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/From-partial-permutation-information-to-Fourier-bounds-September-26-2026"}, {"dir": "Conditional-permutations-in-a-revealed-switching-environment-September-26-2026", "title": "Conditional permutations in a revealed switching environment", "url": "https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Conditional-permutations-in-a-revealed-switching-environment-September-26-2026"}, {"dir": "Routing-densities-and-representation-contraction-for-Thorp-sweeps-September-26-2026", "title": "Routing densities and representation contraction for Thorp sweeps", "url": "https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Routing-densities-and-representation-contraction-for-Thorp-sweeps-September-26-2026"}, {"dir": "Row-column-symmetry-and-contraction-of-coordinate-sweeps-September-26-2026", "title": "Row–column symmetry and contraction of coordinate sweeps", "url": "https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Row-column-symmetry-and-contraction-of-coordinate-sweeps-September-26-2026"}, {"dir": "Random-subspace-tests-and-trace-smoothing-for-coordinate-sweeps-September-26-2026", "title": "Random-subspace tests and trace smoothing for coordinate sweeps", "url": "https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Random-subspace-tests-and-trace-smoothing-for-coordinate-sweeps-September-26-2026"}, {"dir": "Compatibility-entropy-and-the-spectrum-of-a-Thorp-sweep-September-26-2026", "title": "Compatibility entropy and the spectrum of a Thorp sweep", "url": "https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Compatibility-entropy-and-the-spectrum-of-a-Thorp-sweep-September-26-2026"}, {"dir": "Signed-tensor-densities-and-diagram-budgets-for-coordinate-sweeps-September-26-2026", "title": "Signed tensor densities and diagram budgets for the Thorp shuffle", "url": "https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Signed-tensor-densities-and-diagram-budgets-for-coordinate-sweeps-September-26-2026"}, {"dir": "Conditional-coordinate-sweeps-and-analytic-transfer-September-26-2026", "title": "Conditional coordinate sweeps and analytic transfer", "url": "https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Conditional-coordinate-sweeps-and-analytic-transfer-September-26-2026"}, {"dir": "A-strict-four-row-permanent-inequality-and-permutation-moments-September-26-2026", "title": "A strict four-row permanent inequality and permutation moments", "url": "https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-strict-four-row-permanent-inequality-and-permutation-moments-September-26-2026"}], "audit_part": 2, "second_reader": "", "referee_verdict": ""}, {"family": "240", "title": "Shelah's eventual categoricity conjecture", "subject": "Mathematical logic", "headline": "Proves Shelah's eventual categoricity conjecture in ZFC: for each bound on the L\\\"owenheim--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.", "verdict": "partial", "challenges": ["CHObstruction"], "review_note": "PRIMARY CLAIM NOT STATED (only a secondary result is formalized). `CHObstruction.main (hCH : CH) : ∃ L countable relational, ∃ K, K.IsAEC ∧ K.HasLSNumber ℵ₀ ∧ endpoint = hanf ℵ₀ ∧ K.TwoModels endpoint ∧ ∀ μ, tailThreshold ≤ μ → K.Categorical μ` is the summary's last sentence (under CH a prescribed Hanf threshold need not suffice), formalized with hand-written AEC axioms on `ZFSet` models. The headline, Shelah's eventual categoricity conjecture proved in ZFC, is not stated (docs: 'The later canonical-point obstruction and the separate eventual-categoricity theorem are not included').", "definitions_to_check": ["Hand-rolled abstract elementary classes: `structure ClassData` (`objects`, `strong`), `IsAEC` (iso-closure, partial order, coherence, chain unions over arbitrary `γ.ToType`, LS bound), `LSBound/HasLSNumber`, `Categorical μ := (∃ M, objects M ∧ card = μ) ∧ ∀ M N of size μ, Nonempty (M.Iso N)`, `hanf κ := beth (succ (2^κ)).ord`; standard Shelah axioms but entirely custom, and `antisymm` is literal equality `M = N` of `RelModel`s."], "external_packages": [], "cone_lines_max": 5089, "machine_check": "comparator-sandboxed-pass", "lab_scope_note": "Under the continuum hypothesis, the formalized result refutes the proposed transfer of categoricity down to the Hanf threshold. It gives an abstract elementary class in a countable relational language, with Löwenheim–Skolem number $\\aleph_0$, that has two nonisomorphic models at $\\beth_{\\omega_2}=h(\\aleph_0)$ but is categorical in every cardinal at least $\\beth_{(2^{\\aleph_1})^+}$.\n\nNo amalgamation, joint embedding, tameness, or absence-of-maximal-models assumption is imposed. The later canonical-point obstruction and the separate eventual-categoricity theorem are not included.", "lab_docs_url": "https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/docs/240.md", "overview_entry": "family 240 in overview.pdf / CONTENTS.md", "papers": [{"dir": "A-CH-Obstruction-to-a-Prescribed-Categoricity-Threshold-September-24-2026", "title": "A CH obstruction to a prescribed categoricity threshold", "url": "https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-CH-Obstruction-to-a-Prescribed-Categoricity-Threshold-September-24-2026"}], "audit_part": 2, "second_reader": "", "referee_verdict": ""}, {"family": "241", "title": "Rigidity of the Turing degrees", "subject": "Mathematical logic", "headline": "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.", "verdict": "full", "challenges": ["DegreeRigidity"], "review_note": "`MainTheorem : ∀ π : Degree ≃o Degree, ∀ a : Degree, π a = a` with `Oracle := ℕ → Bool` (all subsets of ℕ), `Reduces A B := TuringReducible (oracleFunction A) (oracleFunction B)` (Mathlib) and `Degree := Antisymmetrization Oracle Reduces`: every order automorphism of the Turing degrees is the identity. No assumptions; the representation corollaries are not stated.", "definitions_to_check": [], "external_packages": [], "cone_lines_max": 84081, "machine_check": "comparator-sandboxed-pass", "lab_scope_note": "The rigidity problem asks whether the ordering of Turing degrees by relative computability has any nontrivial automorphism. The formalized result gives a negative answer: every order automorphism of the full Turing degrees of subsets of $\\mathbb N$ fixes every degree, with no definability or genericity assumption. The additional representation corollaries are not included.", "lab_docs_url": "https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/docs/241.md", "overview_entry": "family 241 in overview.pdf / CONTENTS.md", "papers": [{"dir": "Rigidity-of-the-Turing-degrees-September-24-2026", "title": "Rigidity of the Turing degrees", "url": "https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Rigidity-of-the-Turing-degrees-September-24-2026"}], "audit_part": 2, "second_reader": "", "referee_verdict": ""}, {"family": "242", "title": "Single-fold Diophantine representations", "subject": "Mathematical logic", "headline": "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.", "verdict": "full", "challenges": ["SingleFold"], "review_note": "`MainStatement : ∀ n ≥ 1, ∀ S : Set (Fin n → ℕ), REPred (· ∈ S) → ∃ m ≥ 1, ∃ P : MvPolynomial (Fin n ⊕ Fin m) ℤ, Represents P S` with `Represents P S := ∀ a, (a ∈ S ↔ ∃ w, P(a,w) = 0) ∧ ∀ w v, P(a,w) = 0 → P(a,v) = 0 → w = v` (witnesses in ℕ): exactly the single-fold representation of every r.e. set. The summary's corollary, undecidability under an at-most-one-solution promise, is not separately stated (docs say so).", "definitions_to_check": [], "external_packages": [], "cone_lines_max": 7598, "machine_check": "comparator-sandboxed-pass", "lab_scope_note": "The single-fold Diophantine conjecture asks whether every recursively enumerable set has a polynomial representation with a unique auxiliary witness for each member. The formalization establishes this for every recursively enumerable subset of $\\mathbb N^n$, $n\\ge1$: an integer polynomial has exactly one complete natural-number witness tuple for members and none for nonmembers.\n\nThe consequence that solvability remains undecidable under an at-most-one-solution promise is not separately formalized.", "lab_docs_url": "https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/docs/242.md", "overview_entry": "family 242 in overview.pdf / CONTENTS.md", "papers": [{"dir": "Single-fold-Diophantine-representations-September-24-2026", "title": "Single-fold Diophantine representations", "url": "https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Single-fold-Diophantine-representations-September-24-2026"}], "audit_part": 2, "second_reader": "", "referee_verdict": ""}, {"family": "243", "title": "Choiceless counting does not capture polynomial time", "subject": "Mathematical logic", "headline": "Confirms the Blass--Gurevich--Shelah noncapture conjecture: consistency of a linear system over $\\mathbb F_3$ 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.", "verdict": "full", "challenges": ["ChoicelessPolynomialTime", "WitnessedChoice"], "review_note": "`CPTSeparation.main : (∀ S T, S.Iso T → (S.query ↔ T.query)) ∧ OrdinaryPolynomialTime ∧ ¬ FullCPT.EvaluationDefinable (fun I => I.query)`, where `query` = consistency of an F₃ linear system coded by 8 relations, `OrdinaryPolynomialTime` uses Mathlib `Turing.TM2ComputableInPolyTime` with finite stack alphabets on ordered encodings, and `EvaluationDefinable` is polynomially bounded choiceless counting over hereditarily finite sets of arbitrary rank; and `WitnessedChoice.main : ∃ φ, φ.formula.wscCount = 1 ∧ φ.BooleanOnAllInputs ∧ ∀ ψ, ψ.isCPT → φ.TrueModels ≠ ψ.TrueModels`. Both separations of the summary are stated.", "definitions_to_check": ["Custom 'choiceless polynomial time with counting', twice and in two different hand-written formalisms: `FullCPT` (ASM-style `Program` with `Rule`/`Term` over `HF A` = `Lists A` quotient, `evaluationAccepts time space := ∃ h ≤ time(|A|), ... (P.occurring rel h).card ≤ space(|A|)`) in ChoicelessPolynomialTime, and the term/formula logic `WitnessedChoice.BGS` (`Term.iterate`, `resource p n := ⌊p(n)⌋₊`, `Formula.wsc`, `isCPT φ := wscCount = 0`) in WitnessedChoice; the nonexpressibility statements are only as good as these encodings of CPTC. 'Polynomial time' for the query itself is Mathlib's standard `TM2ComputableInPolyTime`."], "external_packages": [], "cone_lines_max": 26381, "machine_check": "comparator-sandboxed-pass", "lab_scope_note": "The formalized result separates polynomial time from choiceless polynomial time with counting. It gives an explicit query on finite structures with eight relations that is invariant under isomorphism and decidable in polynomial time, but is not definable in the stated hereditarily finite-set language with cardinality and polynomial bounds on stages and intermediate objects. The query applies to all finite inputs without a promise. The separate witnessed-symmetric-choice result is not included.\n\nThe formalization proves that witnessed symmetric choice strictly increases the expressive power of choiceless polynomial time with counting. It constructs one fixed sentence with exactly one witnessed-choice occurrence that gives a Boolean answer on every finite input, while no sentence of the original counting formalism defines the same class of inputs.", "lab_docs_url": "https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/docs/243.md", "overview_entry": "family 243 in overview.pdf / CONTENTS.md", "papers": [{"dir": "Choiceless-polynomial-time-with-counting-does-not-capture-polynomial-time-September-23-2026", "title": "Choiceless polynomial time with counting does not capture polynomial time", "url": "https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Choiceless-polynomial-time-with-counting-does-not-capture-polynomial-time-September-23-2026"}, {"dir": "Witnessed-symmetric-choice-is-strictly-stronger-than-choiceless-polynomial-time-with-counting-September-24-2026", "title": "Witnessed symmetric choice is strictly stronger than choiceless polynomial time with counting", "url": "https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Witnessed-symmetric-choice-is-strictly-stronger-than-choiceless-polynomial-time-with-counting-September-24-2026"}], "audit_part": 2, "second_reader": "", "referee_verdict": ""}, {"family": "244", "title": "The partition principle does not imply choice", "subject": "Mathematical logic", "headline": "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.", "verdict": "partial", "challenges": ["PartitionConsistency", "PartitionPrinciple"], "review_note": "Stated: `PartitionConsistency.main_consistency : Consistent ZF → Consistent (ZF ∪ {PP, ACwo, ¬AC})` (a hand-coded first-order `Derives` calculus, full ZF with arbitrary Separation/Replacement instances, `PP` = every internal surjection X ↠ Y yields an internal injection Y ↪ X, `ACwo` = choice for ordinal-indexed families), which is the summary's first two claims. The third claim, a transitive symmetric extension of any countable transitive model of ZFC with no new countable sequences, appears only as `exists_model_partitionPrinciple_without_choice` and needs an extra hypothesis (an internal strongly inaccessible `K` in the ground model, docs: 'containing an internal strongly inaccessible cardinal') and its conclusion lacks ordinal-indexed choice and the no-new-sequences property (docs: 'stronger transitive-model preservation assertions are outside').", "definitions_to_check": ["Hand-written proof theory and set theory: `Formula`, `Derives` (natural deduction), `Consistent T := ∀ n, ¬ Derives (T.at n) .bot`, `ZF := {p | IsZFAxiom p}` with `separation`/`replacement` coded by variable-index arithmetic, `PP`, `AC`, `ACwo`. The theorem is an implication from `Consistent ZF`, so a coding error that made the encoded ZF inconsistent would make it vacuous; `PartitionPrinciple` uses its own `SetFormula`/`Realize` semantics on `ZFSet`. (`axiom_files` hit for LocalRecords.lean is a doc-comment false positive.)"], "external_packages": [], "cone_lines_max": 233891, "machine_check": "comparator-sandboxed-pass", "lab_scope_note": "The Partition Principle says that every surjection admits an injection in the reverse direction. The formalization proves the relative-consistency implication: if ZF is consistent, then so is ZF with the Partition Principle, Choice for ordinal-indexed families, and failure of the Axiom of Choice.\n\nThe linked model-construction result also constructs a transitive model of the Partition Principle without Choice from an externally countable transitive ground model satisfying the stated axioms, including Choice, and containing an internal strongly inaccessible cardinal. The paper's stronger transitive-model preservation assertions are outside that selected construction.", "lab_docs_url": "https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/docs/244.md", "overview_entry": "family 244 in overview.pdf / CONTENTS.md", "papers": [{"dir": "The-Partition-Principle-does-not-imply-Choice-September-24-2026", "title": "The Partition Principle does not imply Choice", "url": "https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-Partition-Principle-does-not-imply-Choice-September-24-2026"}], "audit_part": 2, "second_reader": "", "referee_verdict": ""}, {"family": "245", "title": "The $\\beta$-Barendregt--Geuvers--Klop conjecture", "subject": "Mathematical logic", "headline": "Proves that weak normalization implies strong normalization for every pure type system: if every legal expression in every valid context has a $\\beta$-normal form, every $\\beta$-reduction sequence terminates. This resolves the $\\beta$-Barendregt--Geuvers--Klop conjecture, including nonfunctional rules and open contexts.", "verdict": "full", "challenges": ["TypeSystemNormalization"], "review_note": "`weak_implies_strong (P : Specification S) (h : SystemWeaklyNormalizing P) : SystemStronglyNormalizing P` for every sort type `S`, arbitrary `axioms`/`rule` relations (nonfunctional allowed), annotated `lam`/`pi` syntax with β-reduction inside annotations (`lam_domain`, `pi_domain`, `pi_body`), typing rules ax/var/weaken/product/abstraction/application/conversion, valid contexts as lists, legal = `∃ A, M : A ∨ A : M`, WN = some β-normal form, SN = `Acc`. Exactly the β-BGK conjecture including open contexts.", "definitions_to_check": [], "external_packages": [], "cone_lines_max": 29547, "machine_check": "comparator-sandboxed-pass", "lab_scope_note": "The formalization proves the $\\beta$-Barendregt–Geuvers–Klop conjecture for pure type systems: if every legal expression in every valid context has some terminating $\\beta$-reduction sequence, then every $\\beta$-reduction sequence from every such expression terminates. Reduction is allowed inside type annotations, and the specification may have arbitrary sorts and nonfunctional axioms or rules.", "lab_docs_url": "https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/docs/245.md", "overview_entry": "family 245 in overview.pdf / CONTENTS.md", "papers": [{"dir": "Weak-and-strong-normalization-in-pure-type-systems-September-25-2026", "title": "Weak and strong normalization in pure type systems", "url": "https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Weak-and-strong-normalization-in-pure-type-systems-September-25-2026"}], "audit_part": 2, "second_reader": "", "referee_verdict": ""}, {"family": "246", "title": "Cannon's conjecture", "subject": "Group theory", "headline": "Every word-hyperbolic group with boundary homeomorphic to $S^2$ 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.", "verdict": "full", "challenges": ["CannonGeometricAction"], "review_note": "`cannon (G) [Group G] [DiscreteTopology G] (D : CayleyData G) (hthin : D.ThinTriangles) (hsphere : Nonempty (Boundary D ≃ₜ SphereTwo)) : ∃ ρ : G →* H3Isom, ProperAction ρ ∧ CocompactAction ρ ∧ (ρ.ker : Set G).Finite`, with hyperbolicity as uniformly thin geodesic triangles in the Cayley graph, `Boundary D` the quotient of based geodesic rays by bounded synchronous distance (quotient of the pointwise topology), H³ the upper half-space with `arcosh(1 + |x-y|²/(2 x₃ y₃))` and its full isometry group. The summary's corollary (torsion-free ⇒ closed hyperbolic 3-manifold group) is not stated.", "definitions_to_check": ["Custom hyperbolic-group boundary: `BasedRay D := {r : ℕ → G // r 0 = 1 ∧ ∀ i j, dist (r i) (r j) = Nat.dist i j}`, `Boundary D := Quotient (raySetoid D)` with `raySetoid r s := ∃ C, ∀ n, dist (r n) (s n) ≤ C`, topology inherited from `ℕ → G` (discrete G): the standard ray model of the Gromov boundary but hand-built rather than a Mathlib notion."], "external_packages": [], "cone_lines_max": 69536, "machine_check": "comparator-sandboxed-pass", "lab_scope_note": "Cannon's conjecture asks whether a hyperbolic group with boundary homeomorphic to the two-sphere acts geometrically on hyperbolic three-space. The formalization proves that such a group admits an isometric action on $\\mathbb H^3$ that is proper and cocompact and has finite kernel. Hyperbolicity is expressed through uniformly thin geodesic triangles in a Cayley graph, and the boundary hypothesis is a homeomorphism with $S^2$.", "lab_docs_url": "https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/docs/246.md", "overview_entry": "family 246 in overview.pdf / CONTENTS.md", "papers": [{"dir": "A-Modulus-Proof-of-Cannons-Conjecture-September-23-2026", "title": "A Modulus Proof of Cannon’s Conjecture", "url": "https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-Modulus-Proof-of-Cannons-Conjecture-September-23-2026"}], "audit_part": 2, "second_reader": "", "referee_verdict": ""}, {"family": "247", "title": "An infinite finitely presented residually finite $2$-group", "subject": "Group theory", "headline": "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 $\\mathbb F_2$-algebra and a finitely presented infinite-dimensional algebraic unitization, giving negative answers to the corresponding nilpotence and Kurosh finiteness questions.", "verdict": "weaker-statement", "challenges": ["PeriodicGroup"], "review_note": "`thm_main : MainStatement ∧ BurnsideStatement` only gives an infinite finitely presented PERIODIC group (`Periodic G := ∀ g, ∃ m > 0, g^m = 1`, namely `Steinberg 12 R` for some F₂-algebra R, and a general `∃ G` version). The summary's group is a residually finite 2-group (every element of 2-power order) with accompanying nil-algebra consequences; neither 'residually finite' nor '2-power order' nor the algebra statements is stated (docs: 'no common exponent is asserted ... separate nil-algebra and radical-algebra conclusions are outside these selected statements'). The negative answer to the finitely presented Burnside problem itself is covered.", "definitions_to_check": ["`Steinberg n R := PresentedGroup {r | SteinbergRel n R r}` with relations x_{ij}(a+b) = x_{ij}(a) x_{ij}(b), [x_{ij}(a), x_{kl}(b)] = 1 for p.col ≠ q.row ∧ p.row ≠ q.col, and [x_{ij}(a), x_{jk}(b)] = x_{ik}(ab): the standard Steinberg presentation, hand-written (the carrier ring R is existentially chosen)."], "external_packages": [], "cone_lines_max": 38076, "machine_check": "comparator-sandboxed-pass", "lab_scope_note": "The finitely presented Burnside question asks whether a finitely presented group in which every element has finite order must be finite. The formalization gives a negative answer by constructing an infinite finitely presented periodic group, including a witness realized as a Steinberg group over an algebra of characteristic two. Periodicity means that each element has some finite order; no common exponent is asserted. The paper's separate nil-algebra and radical-algebra conclusions are outside these selected statements.", "lab_docs_url": "https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/docs/247.md", "overview_entry": "family 247 in overview.pdf / CONTENTS.md", "papers": [{"dir": "An-infinite-finitely-presented-periodic-group-September-23-2026", "title": "An infinite finitely presented periodic group", "url": "https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/An-infinite-finitely-presented-periodic-group-September-23-2026"}], "audit_part": 2, "second_reader": "", "referee_verdict": ""}, {"family": "248", "title": "Thompson's group $F$ is nonamenable", "subject": "Group theory", "headline": "Proves that Thompson's group $F$, the group of dyadic piecewise linear homeomorphisms of the interval, is nonamenable, resolving its longstanding amenability problem.", "verdict": "full", "challenges": ["ThompsonNonamenability"], "review_note": "`thompson_F_nonamenable_composition : ∃ group : Group F, (∀ h g x, (h * g) x = h (g x)) ∧ ¬ Nonempty (InvariantMean F)` where `F := {f : IntervalHomeomorph // StrictMono f ∧ HasDyadicPLPieces f}` (dyadic breakpoints, slopes 2^k via `DyadicPLWitness`) with group law forced to be composition, and `InvariantMean G` = positive normalized left-invariant linear functional on `lp (fun _ : G => ℝ) ∞`. Exactly the nonamenability of Thompson's F.", "definitions_to_check": ["Custom 'amenable': `InvariantMean G` (positive, `toLinearMap 1 = 1`, `toLinearMap (leftPull h f) = toLinearMap f` on ℓ^∞(G, ℝ)) is the standard invariant-mean definition but is not a Mathlib notion; F itself is hand-defined (Group structure is existentially provided inside the theorem)."], "external_packages": [], "cone_lines_max": 9457, "machine_check": "comparator-sandboxed-pass", "lab_scope_note": "The amenability problem for Thompson's group $F$ asks whether it admits a positive normalized left-invariant mean on bounded real functions. The formalized result rules out such a mean for the standard group of dyadic piecewise-linear homeomorphisms of the interval, proving nonamenability. No explicit boundary constant or prescribed generating set is given.", "lab_docs_url": "https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/docs/248.md", "overview_entry": "family 248 in overview.pdf / CONTENTS.md", "papers": [{"dir": "Thompsons-group-F-is-nonamenable-September-23-2026", "title": "Thompson's group F is nonamenable", "url": "https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Thompsons-group-F-is-nonamenable-September-23-2026"}], "audit_part": 2, "second_reader": "", "referee_verdict": ""}, {"family": "249", "title": "A finitely generated counterexample to Eilenberg--Ganea", "subject": "Group theory", "headline": "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.", "verdict": "full", "challenges": ["EilenbergGanea"], "review_note": "`source_main : Group.FG SourceGroup ∧ Group.ResiduallyFinite SourceGroup ∧ cohomologicalDimension SourceGroup = 2 ∧ HasClassifyingSpace.{0} SourceGroup 3 ∧ ¬ HasClassifyingSpace.{u} SourceGroup 2`, with `SourceGroup` the kernel of the height map of an explicit Artin group, `cohomologicalDimension := projectiveDimension (trivial ℤ[G]-module)` and `HasClassifyingSpace Γ d` = aspherical T2 path-connected CW complex with cells only up to dimension d (arbitrary cardinality of cells, any universe). Matches the summary including 'no two-dimensional classifying space, even with infinitely many cells'.", "definitions_to_check": [], "external_packages": [], "cone_lines_max": 34123, "machine_check": "comparator-sandboxed-pass", "lab_scope_note": "The Eilenberg–Ganea conjecture predicts that a group of integral cohomological dimension two has a two-dimensional classifying space. The formalization constructs a finitely generated residually finite group of cohomological dimension two that has a three-dimensional classifying space but no two-dimensional one. Thus its geometric dimension is three, contradicting the conjecture.", "lab_docs_url": "https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/docs/249.md", "overview_entry": "family 249 in overview.pdf / CONTENTS.md", "papers": [{"dir": "A-finitely-generated-counterexample-to-the-Eilenberg-Ganea-conjecture-September-23-2026", "title": "A finitely generated counterexample to the Eilenberg–Ganea conjecture", "url": "https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-finitely-generated-counterexample-to-the-Eilenberg-Ganea-conjecture-September-23-2026"}], "audit_part": 2, "second_reader": "", "referee_verdict": ""}, {"family": "250", "title": "The Boone--Higman conjecture and higher finiteness", "subject": "Group theory", "headline": "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 $F_\\infty$: a classifying space with finitely many cells in each dimension. A single group of type $F_\\infty$ can also contain every finitely presented group.", "verdict": "full", "challenges": ["BooneHigman", "SimpleOvergroups", "UniversalFInfinity"], "review_note": "`BooneHigman.main (G) [Group.FG G] : HasDecidableWordProblem G ↔ EmbedsInFinitelyPresentedSimpleGroup G` (`ComputablePred` of `evalWord g w = 1` for some finite generating tuple; Mathlib `Group.IsFinitelyPresented`, `IsSimpleGroup`), `SimpleOvergroups.main` (G f.g. with decidable word problem ⇒ ∃ nontrivial simple H of type F∞ and injective `G →* H`) and `UniversalFInfinity.universal_group_of_type_FInfinity` (∃ H of type F∞ into which every `Group.IsFinitelyPresented` group embeds). All three claims are stated; type F∞ = aspherical Hausdorff connected CW complex with `FiniteType` (finitely many cells per dimension).", "definitions_to_check": [], "external_packages": [], "cone_lines_max": 90766, "machine_check": "comparator-sandboxed-pass", "lab_scope_note": "The Boone–Higman conjecture characterizes finitely generated groups with decidable word problem by embeddings into finitely presented simple groups. The formalization proves the equivalence in full: a finitely generated group has a computable word problem for a finite generating set exactly when it embeds by an injective homomorphism into a finitely presented simple group.\n\nThe formalization proves the higher-finiteness strengthening of the Boone–Higman conjecture: every finitely generated group with decidable word problem embeds by an injective homomorphism into a nontrivial simple group of type $F_\\infty$. Here type $F_\\infty$ means that the group has a classifying CW complex with finitely many cells in each dimension. No additional finiteness property of the original group is assumed.\n\nThe formalized result constructs one group $H$ of type $F_\\infty$ containing every finitely presented group. Here type $F_\\infty$ means that $H$ has a classifying space with finitely many cells in each dimension.\n\nThe group $H$ is fixed before the groups embedded into it are chosen. No word-problem assumption is imposed, and the classifying space need not be finite-dimensional or have finitely many cells in total. The paper's converse about recursively presented subgroups is not included.", "lab_docs_url": "https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/docs/250.md", "overview_entry": "family 250 in overview.pdf / CONTENTS.md", "papers": [{"dir": "Finite-algebraic-envelopes-and-the-Boone-Higman-conjecture-September-23-2026", "title": "Finite algebraic envelopes and the Boone–Higman conjecture", "url": "https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Finite-algebraic-envelopes-and-the-Boone-Higman-conjecture-September-23-2026"}, {"dir": "Simple-F-infinity-overgroups-of-groups-with-decidable-word-problem-September-23-2026", "title": "Simple F∞ overgroups of groups with decidable word problem", "url": "https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Simple-F-infinity-overgroups-of-groups-with-decidable-word-problem-September-23-2026"}, {"dir": "A-universal-group-of-type-F-infinity-September-23-2026", "title": "A universal group of type F∞", "url": "https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-universal-group-of-type-F-infinity-September-23-2026"}], "audit_part": 2, "second_reader": "", "referee_verdict": ""}, {"family": "251", "title": "Amenability, unitarizability, and Ulam stability", "subject": "Group theory", "headline": "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.", "verdict": "partial", "challenges": ["Dixmier", "DixmierAllDiscrete"], "review_note": "Stated: Dixmier's equivalence for all discrete groups (`CurrentMainTheorem : ∀ G discrete, (Amenable G ↔ Unitarizable G) ∧ ∀ ε > 0, ¬ Amenable G → ∃ nonunitarizable π with ‖π g‖ ≤ 1 + ε` on a complete Hilbert space, separable if G is countable) and the countable-group separable witness with bound 101 (`Dixmier.main_theorem_all_universes`). Not stated: the summary's second claim, equivalence of amenability with strong Ulam stability for countable discrete groups (it is not mentioned in the docs scope either, although the family title names Ulam stability).", "definitions_to_check": ["Custom 'amenable': `def Amenable (G) [DiscreteTopology G] : Prop := ∃ m : (G →ᵇ ℂ) →ₗ[ℂ] ℂ, m 1 = 1 ∧ (∀ f, (∀ x, 0 ≤ (f x).re ∧ (f x).im = 0) → 0 ≤ (m f).re ∧ (m f).im = 0) ∧ ∀ g f, m (leftTranslate g f) = m f`, the standard positive left-invariant mean on ℓ^∞(G, ℂ); `SimilarToUnitary π := ∃ S : H ≃L[ℂ] H, ∀ g x, ‖S (π g (S.symm x))‖ = ‖x‖` and `Unitarizable` quantify over Hilbert spaces in one universe (the theorem is universe polymorphic)."], "external_packages": [], "cone_lines_max": 4092, "machine_check": "comparator-sandboxed-pass", "lab_scope_note": "Dixmier's unitarizability problem asks whether a discrete group is amenable exactly when every uniformly bounded Hilbert-space representation is similar to a unitary one. The formalization establishes this equivalence for every discrete group.\n\nFor every nonamenable group and every $\\varepsilon>0$, it also gives a nonunitarizable representation with uniform operator bound at most $1+\\varepsilon$. The Hilbert space can be chosen separable for countable groups; a separate linked statement records a separable witness with uniform bound $101$ in that case.", "lab_docs_url": "https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/docs/251.md", "overview_entry": "family 251 in overview.pdf / CONTENTS.md", "papers": [{"dir": "Unitarizability-Implies-Amenability-for-Countable-Groups-September-23-2026", "title": "Unitarizability implies amenability for discrete groups", "url": "https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Unitarizability-Implies-Amenability-for-Countable-Groups-September-23-2026"}], "audit_part": 2, "second_reader": "", "referee_verdict": ""}, {"family": "252", "title": "A non-residually-finite torsion-free hyperbolic group", "subject": "Group theory", "headline": "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.", "verdict": "full", "challenges": ["TorsionFreeHyperbolic"], "review_note": "`main : ∃ (G : Type) (_ : Group G), TorsionFree G ∧ WordHyperbolic G ∧ ¬ Group.ResiduallyFinite G` with `WordHyperbolic` = finite generating set whose Cayley graph has uniformly thin geodesic triangles (all three sides, `SideThin`). The summary's nonlinearity claim (one fixed element killed by every finite-dimensional representation over every commutative field) is not separately stated; it follows from the stated non-residual-finiteness via Mal'cev's theorem (docs: 'Nonlinearity over every field is not a separate conclusion').", "definitions_to_check": ["Custom hyperbolicity: `WordHyperbolic G := ∃ S finite, (mulCayley S).Connected ∧ ∃ δ, ∀ geodesic triangles, SideThin ...` (standard thin-triangles definition on the Cayley graph, hand-written, not a Mathlib notion)."], "external_packages": [], "cone_lines_max": 10737, "machine_check": "comparator-sandboxed-pass", "lab_scope_note": "The residual-finiteness question for hyperbolic groups asks whether every nonidentity element survives in some finite quotient. The formalized result gives a negative answer by constructing a torsion-free word-hyperbolic group that is not residually finite.\n\nNonlinearity over every field is not a separate conclusion of the statement linked below.", "lab_docs_url": "https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/docs/252.md", "overview_entry": "family 252 in overview.pdf / CONTENTS.md", "papers": [{"dir": "a-torsion-free-hyperbolic-group-that-is-not-residually-finite-September-23-2026", "title": "A torsion-free hyperbolic group that is not residually finite", "url": "https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/a-torsion-free-hyperbolic-group-that-is-not-residually-finite-September-23-2026"}], "audit_part": 2, "second_reader": "", "referee_verdict": ""}, {"family": "253", "title": "An infinite finitely presented simple amenable group", "subject": "Group theory", "headline": "Constructs an infinite finitely presented simple amenable group, answering the longstanding question of whether these properties can occur simultaneously.", "verdict": "full", "challenges": ["SimpleAmenable"], "review_note": "`main : ∃ (G : Type) (_ : Group G), Infinite G ∧ Group.IsFinitelyPresented G ∧ IsSimpleGroup G ∧ FolnerAmenable G`, with `FolnerAmenable G := ∀ K : Finset G, ∀ ε > 0, ∃ D, D.Nonempty ∧ ∀ g ∈ K, |(g • D) ∆ D| < ε |D|` (the left Folner criterion). Exactly the summary's existence statement.", "definitions_to_check": ["Custom 'amenable': `FolnerAmenable G := ∀ K : Finset G, ∀ ε : ℝ, 0 < ε → ∃ D : Finset G, D.Nonempty ∧ ∀ g ∈ K, (((D.image (g * ·)) ∆ D).card : ℝ) < ε * D.card`; this is Folner's criterion (equivalent to amenability for discrete groups) but not Mathlib's notion, and the group is existentially chosen."], "external_packages": [], "cone_lines_max": 84848, "machine_check": "comparator-sandboxed-pass", "lab_scope_note": "The formalized result answers the existence question affirmatively: there is one group that is infinite, finitely presented, simple, and amenable. Amenability is expressed by the Følner condition.\n\nThe later general claims about central kernels and enlargements across families are not included.", "lab_docs_url": "https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/docs/253.md", "overview_entry": "family 253 in overview.pdf / CONTENTS.md", "papers": [{"dir": "An-Infinite-Finitely-Presented-Simple-Amenable-Group-September-23-2026", "title": "An infinite finitely presented simple amenable group", "url": "https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/An-Infinite-Finitely-Presented-Simple-Amenable-Group-September-23-2026"}], "audit_part": 2, "second_reader": "", "referee_verdict": ""}, {"family": "254", "title": "Classifying spaces and geometric obstructions for Artin groups", "subject": "Group theory", "headline": "The Salvetti complex of every finite-rank Artin group is aspherical, proving the Artin $K(\\pi,1)$ conjecture. Arbitrary intersections of its parabolic subgroups are parabolic, proving the Parabolic Intersection Conjecture. An explicit Artin group admits no proper cocompact isometric action on any nonempty proper CAT$(0)$ space.", "verdict": "full", "challenges": ["ArtinCAT0", "ArtinParabolicIntersections", "HarmonicArtin"], "review_note": "All three claims are stated: `HarmonicArtin.salvetti_cover_contractible [Finite S] (M : CoxeterMatrix S) : ContractibleSpace (SalvettiCover M)` (K(π,1): the cover is the realization of the nerve of the lifted spherical-cell poset), `ArtinCAT0.main` (the explicit 116-generator matrix has entries in {1,2,3,∞} and `¬ GeometricAction (ArtinGroup explicitMatrix) X` for every nonempty proper metric space X with a MulAction, `CAT0` via the CN inequality), and `ArtinParabolicIntersections` (`unconditional_arbitrary_intersections`: `sInf F` of any family of parabolics is parabolic and equals `sInf` of at most `Nat.card S` members; unique parabolic closure; acylindrical hyperbolicity and weak malnormality for irreducible infinite-label groups). Config notes: `ArtinParabolicIntersections.json` has `enable_nanoda: true` (an extra independent-kernel check, not a weakening); tmp/import_cones.json flags external `OAI` for 254 only because it mis-parses the module `MulBlockDiagonal'Entry` (apostrophe), so the true cone is Mathlib-only and ~32 lines larger.", "definitions_to_check": ["`SalvettiCover M := SSet.toTop.obj (nerve (LiftedCell M))` with `LiftedCell M := Artin M × SphericalType M` and the face order generated by `LiftedFace` (minimal-length Coxeter coset representatives lifted to the Artin group): a hand-built model of the universal cover of the Salvetti complex (nerve of its face poset). Nothing in the challenge statement ties this space to a Mathlib notion of Salvetti complex or to the action of the Artin group, so 'asphericity of the Salvetti complex' is only as faithful as this model (the docs call it the equivalent statement that the universal cover is contractible)."], "external_packages": [], "cone_lines_max": 73097, "machine_check": "comparator-sandboxed-pass", "lab_scope_note": "The Artin $K(\\pi,1)$ conjecture asserts that the standard Salvetti complex of every finite-rank Artin group is aspherical. The formalized result proves the equivalent statement that its universal cover is contractible.\n\nIt applies to every Coxeter matrix on a finite generating set, including empty and disconnected diagrams and infinite edge labels. The associated Coxeter and Artin groups may be infinite. Later corollaries in the manuscript are not included.\n\nThe geometric-action question asks whether every Artin group acts properly and cocompactly by isometries on a proper CAT(0) space. The formalized result gives a counterexample: an Artin group defined by a symmetric matrix on $116$ generators, with diagonal entries $1$ and off-diagonal entries in $\\{2,3,\\infty\\}$.\n\nIt admits no such action on any nonempty proper CAT(0) space, in any dimension.\n\nThe formalization proves that the intersection of any family of parabolic subgroups of a finite-rank Artin group is parabolic. Every such intersection is already the intersection of at most the rank many members, and every subset has a unique parabolic closure. Both finite and infinite Coxeter labels are allowed.\n\nFor irreducible Artin groups with an infinite Coxeter label, it also proves acylindrical hyperbolicity and weak malnormality of every proper parabolic subgroup. These are the intersection and structural consequences selected from the paper.", "lab_docs_url": "https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/docs/254.md", "overview_entry": "family 254 in overview.pdf / CONTENTS.md", "papers": [{"dir": "Harmonic-heights-and-the-Artin-K-pi-1-conjecture-September-23-2026", "title": "Harmonic heights and the Artin K(pi,1) conjecture", "url": "https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Harmonic-heights-and-the-Artin-K-pi-1-conjecture-September-23-2026"}, {"dir": "An-Artin-group-with-no-geometric-CAT-0-action-September-23-2026", "title": "An Artin group with no geometric CAT(0) action", "url": "https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/An-Artin-group-with-no-geometric-CAT-0-action-September-23-2026"}, {"dir": "Parabolic-intersections-in-Artin-groups-September-23-2026", "title": "Parabolic intersections in Artin groups", "url": "https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Parabolic-intersections-in-Artin-groups-September-23-2026"}], "audit_part": 2, "second_reader": "", "referee_verdict": ""}, {"family": "255", "title": "Quasi-isometric rigidity of virtually polycyclic groups", "subject": "Group theory", "headline": "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.", "verdict": "full", "challenges": ["PolycyclicRecognition"], "review_note": "`group_recognition (e : GroupCoarseEquivalence P J) (hP : IsVirtuallyPolycyclic P) [P, J f.g.] : IsVirtuallyPolycyclic J ∧ ∃ Γ finite-index, ∃ S Lie model, Λ, IsUniformLattice Λ ∧ Nonempty (Γ ≃* Λ)` and `lattice_recognition` (a finite-covolume lattice in a simply connected solvable Lie group, quasi-isometric to a f.g. J, makes J virtually a uniform lattice in possibly another such group). 'Quasi-isometric' is encoded as `GroupCoarseEquivalence` (bornologous maps with finite-set errors), which for f.g. groups is equivalent to a quasi-isometry; polycyclic = explicit cyclic subnormal series. Both summary claims are stated.", "definitions_to_check": ["Hand-built coarse geometry: `GroupBornologous f := ∀ S finite, ∃ T finite, ∀ x y, x⁻¹ y ∈ S → (f x)⁻¹ (f y) ∈ T`, `GroupCoarseEquivalence` (maps in both directions plus finite error sets) used instead of word-metric quasi-isometry; `IsUniformLattice Λ := DiscreteTopology Λ ∧ ∃ K compact, ∀ g, ∃ γ k, k ∈ K ∧ γ k = g`; `SimplyConnectedSolvableLieModel` packages Mathlib `LieGroup`/`SimplyConnectedSpace`/`Group.IsSolvable`. Plausible and standard but non-Mathlib. External package `Gromov` is in the import cone."], "external_packages": ["Gromov"], "cone_lines_max": 232617, "machine_check": "none", "lab_scope_note": "The formalization proves quasi-isometric recognition of virtually polycyclic groups: every finitely generated group quasi-isometric to a finitely generated virtually polycyclic group is itself virtually polycyclic. It also realizes a finite-index subgroup of the recognized group as a uniform lattice in a simply connected solvable Lie group, which may differ from the original ambient group.\n\nThe linked structural estimate controls the height-coordinate behavior of self quasi-isometries of the stated unimodular solvable Lie models, up to bounded error and one of finitely many linear height symmetries.", "lab_docs_url": "https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/docs/255.md", "overview_entry": "family 255 in overview.pdf / CONTENTS.md", "papers": [{"dir": "quasi-isometric-recognition-of-virtually-polycyclic-groups-September-24-2026", "title": "Quasi-isometric recognition of virtually polycyclic groups", "url": "https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/quasi-isometric-recognition-of-virtually-polycyclic-groups-September-24-2026"}], "audit_part": 2, "second_reader": "", "referee_verdict": ""}, {"family": "256", "title": "Howie's conjecture on equations over groups", "subject": "Group theory", "headline": "Proves that every finite system of equations over an arbitrary group whose exponent-sum matrix has full row rank over $\\mathbb Q$ 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.", "verdict": "partial", "challenges": ["Kervaire"], "review_note": "Only the Kervaire special case is stated: `coefficient_injective (A) (w : WordGroup A) (hw : Unimodular w) : Function.Injective (coefficientMap w)` with `WordGroup A := Monoid.Coprod A (Multiplicative ℤ)` and `Unimodular w := exponentSum w = 1 ∨ exponentSum w = -1` (one equation in one unknown with exponent sum ±1; docs: 'the selected unimodular-relator result underlying the conjecture'). The summary's headline, Howie's conjecture for arbitrary finite systems whose exponent-sum matrix has full row rank over ℚ (simultaneous solution in an overgroup), is not stated; the Kervaire companion claim is covered (coefficient injectivity implies nontriviality).", "definitions_to_check": [], "external_packages": [], "cone_lines_max": 11273, "machine_check": "comparator-sandboxed-pass", "lab_scope_note": "Kervaire's conjecture states that the free product of a nontrivial group with an infinite cyclic group cannot be normally generated by one element. The linked formalization proves the stronger coefficient-injectivity statement used in the paper: for any group $A$ and any relator in $A*\\mathbb Z$ whose exponent sum in the cyclic generator is $1$ or $-1$, the natural map from $A$ into the quotient by that relator is injective. This is the selected unimodular-relator result underlying the conjecture.", "lab_docs_url": "https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/docs/256.md", "overview_entry": "family 256 in overview.pdf / CONTENTS.md", "papers": [{"dir": "The-Kervaire-Theorem-for-Groups-September-24-2026", "title": "The Kervaire theorem for groups", "url": "https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-Kervaire-Theorem-for-Groups-September-24-2026"}], "audit_part": 2, "second_reader": "", "referee_verdict": ""}, {"family": "257", "title": "A hyperbolic group with no geometric CAT(0) action", "subject": "Group theory", "headline": "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 $\\operatorname{CAT}(0)$ space, in any dimension.", "verdict": "full", "challenges": ["HyperbolicObstruction"], "review_note": "`MainStatement : ∃ K : FiniteComplex, ∃ x C, ConnectedSpace K.Carrier ∧ K.Aspherical x ∧ 0 ≤ C ∧ K.LinearDiskFilling C ∧ NoGeometricCATZeroAction (FundamentalGroup K.Carrier x) ∧ WordHyperbolic (FundamentalGroup K.Carrier x) ∧ ∀ L, K ≃ₕ L → ¬ CompatibleLocallyCATNegOne L`: a finite aspherical complex (hence a finite classifying space) with word-hyperbolic π₁ (four-point condition) admitting no geometric action on any nonempty proper complete CAT(0) space (`NoGeometricCATZeroAction`, CN-inequality CAT(0)). Matches the summary.", "definitions_to_check": ["Hand-written geometry: `CATZero X := ∀ x y, ∃ γ, IsSegment γ x y ∧ ∀ z, ∀ t ∈ [0,1], dist z (γ t)^2 ≤ (1-t) dist z x^2 + t dist z y^2 - t(1-t) dist x y^2` (CN inequality), `WordHyperbolic` as the four-point condition with `WordDistance` from `sInf`-defined word length, `FiniteComplex` as barycentric realizations of `AbstractSimplicialComplex (Fin n)`; standard formulations but not Mathlib notions."], "external_packages": [], "cone_lines_max": 56644, "machine_check": "comparator-sandboxed-pass", "lab_scope_note": "The formalized result constructs a finite connected aspherical complex with a linear disk-filling bound whose fundamental group is word-hyperbolic but admits no geometric action on a nonempty proper complete $\\mathrm{CAT}(0)$ space. It also excludes locally $\\mathrm{CAT}(-1)$ geodesic metrics on every finite complex homotopy equivalent to the witness. The stronger claims that the witness is two-dimensional and that those models admit no locally $\\mathrm{CAT}(0)$ metric are not included.", "lab_docs_url": "https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/docs/257.md", "overview_entry": "family 257 in overview.pdf / CONTENTS.md", "papers": [{"dir": "A-hyperbolic-group-with-no-geometric-CAT0-action-September-25-2026", "title": "A hyperbolic group with no geometric CAT(0) action", "url": "https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-hyperbolic-group-with-no-geometric-CAT0-action-September-25-2026"}], "audit_part": 2, "second_reader": "", "referee_verdict": ""}, {"family": "260", "title": "Spacetime Penrose inequalities and rigidity", "subject": "Mathematical physics", "headline": "Proves the sharp enclosing-area spacetime Penrose inequality for smooth one-ended asymptotically flat initial data in every spatial dimension $n\\ge3$, 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.", "verdict": "supporting-only", "challenges": ["CKSBondiPenrose"], "review_note": "The two challenge theorems are `CKSSourceExterior.area_controlled_end_replacement` (a 3-dimensional Cha-Khuri-Sakovich hyperboloidal end can be replaced by asymptotically flat ends keeping the compact interior, completeness and DEC, with `Tendsto (spatialADMEnergy G) ... (√(m² - |p|²) + η)` and `ε R → 0` area loss) and `schwarzschild_equality_examples (m) (hm : 0 < m)` (the Schwarzschild data satisfy `MainHypotheses`, `bondiMass d = m`, `minimumEnclosingArea g = 16π m²`). Neither states the Penrose inequality: the summary's headline (invariant ADM mass ≥ √(min enclosing area/16π) for asymptotically flat data in every dimension n >= 3 under DEC/weak future trapping, with equality rigidity and charged bounds) is absent; docs: 'The paper's general Bondi-Penrose inequality ... remains unformalized' and the conditional inequality 'is not selected'. The challenge deals with the dimension-3 Bondi/CKS class only.", "definitions_to_check": ["Entire geometric-analysis stack is hand-built inside one 3.6k-line challenge file (`MetricJet`, `TensorJet`, `PhysicalDEC`, `CKSData`, `minimumEnclosingArea`, `bondiCharge`, `spatialADMEnergy`, ...) on top of Mathlib manifolds; none is a Mathlib notion and the file has dozens of auxiliary definitions."], "external_packages": [], "cone_lines_max": 58903, "machine_check": "comparator-sandboxed-pass", "lab_scope_note": "The formalization replaces a three-dimensional Cha–Khuri–Sakovich hyperboloidal end by asymptotically flat ends while preserving each fixed compact interior, completeness, and the dominant energy condition. For strictly future-timelike initial-data charge, the ADM masses approach the invariant Bondi mass and the loss of enclosing area tends to zero.\n\nThe Comparator also includes canonical Schwarzschild equality examples at every positive mass, with $m_{\\mathrm{Bondi}}=m$ and enclosing area $16\\pi m^2$. The paper's general Bondi–Penrose inequality for possibly disconnected weakly trapped boundaries remains unformalized.\n\nThe implementation retains a conditional inequality for the connected marginal-boundary case, assuming the asymptotically flat exterior Penrose inequality. This supporting result is not selected as a substitute for the paper's main inequality.", "lab_docs_url": "https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/docs/260.md", "overview_entry": "family 260 in overview.pdf / CONTENTS.md", "papers": [{"dir": "Area-controlled-end-replacement-and-the-Bondi-Penrose-inequality-in-the-CKS-class-September-27-2026", "title": "Area-controlled end replacement and the Bondi Penrose inequality in the CKS class", "url": "https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Area-controlled-end-replacement-and-the-Bondi-Penrose-inequality-in-the-CKS-class-September-27-2026"}], "audit_part": 2, "second_reader": "", "referee_verdict": ""}, {"family": "261", "title": "Localization and delocalization in the Anderson model", "subject": "Mathematical physics", "headline": "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 $d\\ge3$, sufficiently weak positive disorder gives purely absolutely continuous spectrum on a fixed open interval with nonzero spectral weight.", "verdict": "weaker-statement", "challenges": ["PlanarAndersonSpectrum"], "review_note": "`anderson_spectrum_ae (hh : 0 < h) : ∀ᵐ v ∂disorderLaw h, ∃ H, IsAndersonOperator v H ∧ IsSelfAdjoint H ∧ spectrum ℂ H = spectralInterval h` identifies only the spectrum `[-4-h, 4+h]` of the 2D nearest-neighbor Anderson operator (iid Uniform[-h,h] potentials); docs: 'It does not assert the pure-point spectral type claimed in the accompanying paper'. The summary's headline claims (almost-sure pure-point spectrum in d = 2 at every positive disorder, and purely absolutely continuous spectrum on an open interval for weak disorder in every d >= 3) are both unstated.", "definitions_to_check": [], "external_packages": [], "cone_lines_max": 767, "machine_check": "comparator-sandboxed-pass", "lab_scope_note": "The linked formalization proves a supporting spectral statement for the nearest-neighbor Anderson operator on $\\mathbb Z^2$. For every disorder strength $h>0$ with independent site potentials uniform on $[-h,h]$, it constructs the bounded self-adjoint operator almost surely and identifies its spectrum as the real interval $[-4-h,4+h]$.\n\nThis selected statement identifies the spectral set. It does not assert the pure-point spectral type claimed in the accompanying paper.", "lab_docs_url": "https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/docs/261.md", "overview_entry": "family 261 in overview.pdf / CONTENTS.md", "papers": [{"dir": "Pure-Point-Spectrum-for-the-Two-Dimensional-Anderson-Model-at-Every-Positive-Disorder-September-23-2026", "title": "Pure-Point Spectrum for the Two-Dimensional Anderson Model at Every Positive Disorder", "url": "https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Pure-Point-Spectrum-for-the-Two-Dimensional-Anderson-Model-at-Every-Positive-Disorder-September-23-2026"}], "audit_part": 2, "second_reader": "", "referee_verdict": ""}, {"family": "262", "title": "Sharp one-dimensional Lieb--Thirring inequalities", "subject": "Mathematical physics", "headline": "Proves the sharp one-dimensional Lieb--Thirring inequality for $1/2<\\gamma<3/2$ and arbitrary finite-matrix potentials $W\\ge0$ with $\\int\\operatorname{tr}(W^{\\gamma+1/2})<\\infty$: 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 $\\operatorname{sech}^2$ solitons with independent scales and centers, and zero channels.", "verdict": "weaker-statement", "challenges": ["LiebThirring"], "review_note": "Only scalar potentials are treated: `MainClaim : ∀ γ ∈ (1/2, 3/2), (∀ W : ℝ → ℝ, Admissible γ W → negativeMoment γ W ≤ sharpConstant γ * potentialMass γ W) ∧ optimalConstant γ = sharpConstant γ ∧ oneStateConstant γ = sharpConstant γ ∧ equality at (r+1) sech²(r x)` with `negativeMoment` the sup over finite orthonormal families of H¹ weak negative eigenfunctions. The summary's finite-MATRIX-valued potentials with constant independent of matrix size, and the complete classification of equality cases (direct sums of sech² solitons with independent scales/centers and zero channels), are not stated: this is the scalar bound with attainment at one potential but without the equality-case classification.", "definitions_to_check": ["Custom spectral definitions: `H1` as an L² function with an L² weak derivative, `schrodingerForm`, `IsNegativeEigenfunction W k u` (weak eigenfunction at energy -k²) and `negativeMoment γ W := ⨆ N (u k) orthonormal eigenfunctions, Σ (ofReal k_i)^(2γ)` in ℝ≥0∞ instead of Mathlib spectral theory; constants `semiclassicalConstant`, `sharpConstant` are written explicitly (Γ-function formulas)."], "external_packages": [], "cone_lines_max": 13717, "machine_check": "comparator-sandboxed-pass", "lab_scope_note": "The formalized result determines the sharp one-dimensional Lieb–Thirring constant for $1/2<\\gamma<3/2$. For every nonnegative $W\\in L^{\\gamma+1/2}(\\mathbb R)$, it bounds the full negative-eigenvalue moment of $-d^2/dx^2-W$ by the one-bound-state constant times the potential integral. This constant is optimal and is attained by $(r+1)\\mathrm{sech}^2(rx)$, where $r=(\\gamma-1/2)^{-1}$. No finite spectral cutoff is imposed.", "lab_docs_url": "https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/docs/262.md", "overview_entry": "family 262 in overview.pdf / CONTENTS.md", "papers": [{"dir": "Sharp-One-Dimensional-Lieb-Thirring-Constants-September-23-2026", "title": "Sharp one-dimensional Lieb–Thirring constants", "url": "https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Sharp-One-Dimensional-Lieb-Thirring-Constants-September-23-2026"}], "audit_part": 2, "second_reader": "", "referee_verdict": ""}, {"family": "263", "title": "The ionization and generalized ionization conjectures", "subject": "Mathematical physics", "headline": "For the full nonrelativistic Coulomb model with two electron spin states, proves that a molecule with $M$ fixed nuclei of charges at least one and total charge $Z$ strictly binds at most $Z+CM$ electrons. Neutral-atom first ionization energies and radii containing all but an expected half-electron have universal positive upper and lower bounds. The energy cost of removing $m$ electrons has Thomas--Fermi asymptotics as $m\\to\\infty$ and $Z/m\\to\\infty$; neutral-atom outer radii have the corresponding iterated-limit asymptotics, taking $Z\\to\\infty$ first.", "verdict": "partial", "challenges": ["CoulombIonization", "CoulombRadii"], "review_note": "PRIMARY CLAIM NOT STATED (only a secondary result is formalized). Stated: `CoulombIonization.generalized_ionization : ∃ a > 0, TFCharacterization a ∧ JointLimit a ∧ IteratedLimits a` (I_m(Z)/m^{7/3} → a_TF as m → ∞ and Z/m → ∞, and the Z → ∞ first iterated limsup/liminf; energies are infima of the full antisymmetric two-spin form with `FormAdmissible`) and `CoulombRadii.generalized_outer_radii` (`m^{1/3} R_m → (81π²/2)^{1/3}` for both limsup and liminf over any sequence of normalized ground states). Not stated: the summary's first claim, a molecule with M nuclei and total charge Z strictly binds at most Z + CM electrons, and the universal positive upper and lower bounds on first ionization energies/half-electron radii (docs: only 'large-ionization asymptotics').", "definitions_to_check": ["Custom 'energy': `energy Z N := if N = 0 then 0 else sInf {e | ∃ ψ : FormVector N, FormAdmissible ψ ∧ formEnergy Z ψ = e}` (a real `sInf`, junk 0 if unbounded below, which atoms are not) and `IteratedLimits` uses real-valued `limsup`/`liminf` of `ionization m` (junk 0 if unbounded); `CoulombRadii` instead uses EReal-valued limsup/liminf and takes the ground states `Ψ N` as hypotheses (`IsNormalizedGroundState`), so it is vacuous if such minimizers did not exist (they do for neutral atoms). TF functional constant `tfKinetic = (3/10)(3π²)^(2/3)` written by hand."], "external_packages": [], "cone_lines_max": 95362, "machine_check": "comparator-sandboxed-pass", "lab_scope_note": "The formalization gives the large-ionization asymptotics for the full nonrelativistic two-spin Coulomb atom. Let $I_m(Z)$ be the energy needed to remove $m$ electrons from a neutral atom of nuclear charge $Z$. There is a positive coefficient $a_{\\mathrm{TF}}$ with the Thomas–Fermi variational characterization such that $I_m(Z)/m^{7/3}\\to a_{\\mathrm{TF}}$ whenever $m\\to\\infty$ and $Z/m\\to\\infty$, with integers $Z>m\\ge1$. The corresponding iterated limsup and liminf limits hold with $Z\\to\\infty$ first. The energy uses the full antisymmetric Sobolev form domain; fixed-$m$ convergence and ground-state attainment are outside this statement.\n\nFor neutral Coulomb atoms with two electron spin states, the formalization proves the asymptotic law for generalized outer-electron radii defined by an expected exterior electron mass $m$. For every choice of normalized ground states, both the upper and lower large-nuclear-charge radius limits satisfy\n$m^{1/3}R_m\\longrightarrow (81\\pi^2/2)^{1/3}$ as $m\\to\\infty$. The large-nuclear-charge limit is taken first. Ground states are inputs to the statement; their existence is not asserted separately.", "lab_docs_url": "https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/docs/263.md", "overview_entry": "family 263 in overview.pdf / CONTENTS.md", "papers": [{"dir": "Generalized-ionization-energies-for-full-Coulomb-atoms-September-24-2026", "title": "Generalized ionization energies for full Coulomb atoms", "url": "https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Generalized-ionization-energies-for-full-Coulomb-atoms-September-24-2026"}, {"dir": "Generalized-outer-electron-radii-of-neutral-Coulomb-atoms-September-24-2026", "title": "Generalized outer-electron radii of neutral Coulomb atoms", "url": "https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Generalized-outer-electron-radii-of-neutral-Coulomb-atoms-September-24-2026"}], "audit_part": 2, "second_reader": "", "referee_verdict": ""}, {"family": "266", "title": "Exactly three mutually unbiased bases in dimension six", "subject": "Mathematical physics", "headline": "Proves $N(6)=3$, resolving Zauner\\textquotesingle s dimension-six mutually unbiased bases conjecture: three such bases exist in $\\mathbb C^6$, 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\\textquotesingle s cubic equivalence class.", "verdict": "weaker-statement", "challenges": ["HadamardCubeFiber", "MUBSix"], "review_note": "`MUB6.fourier_and_family_bound` states `(∀ H, IsHadamard H → ¬ Equivalent H tao → ∀ π, g H (permuteCharge π alpha) = 0) ∧ (∀ n, Attainable n → n ≤ 5)`: the Fourier-vanishing statement (the Matolcsi-Ruzsa-Weiner companion) and only the weak family bound `n ≤ 5` for mutually unbiased orthonormal bases of ℂ⁶ (`IsMUBFamily B := ∀ r ≠ s, ∀ i j, |⟨B r i, B s j⟩|² = 1/6`). Not stated: N(6) = 3, i.e. neither `Attainable 3` nor the exclusion of four bases (docs: 'does not establish the paper's upper bound of three or its computer-assisted exclusion of four'). `n ≤ 5` is the non-existence of 6 MUBs (with Weiner's gap theorem, not in Lean, this would give N(6) <= 4); the binary64 certificate for excluding four is outside the Lean statement. Cone is 2.83M lines of OAI, the largest in this range.", "definitions_to_check": [], "external_packages": [], "cone_lines_max": 2834627, "machine_check": "comparator-sandboxed-pass", "lab_scope_note": "The paper claims that at most three mutually unbiased orthonormal bases exist in $\\mathbb C^6$. The linked formalization proves a weaker family bound: every family in its mutually unbiased bases model has at most five members. It also proves a Fourier character-sum vanishing statement for order-six complex Hadamard matrices not equivalent to the Tao matrix, uniformly over coordinate permutations.\n\nThe selected statement does not establish the paper's upper bound of three or its computer-assisted exclusion of four arbitrary bases.\n\nThe linked formalization proves a cancellation lemma used in the order-six complex Hadamard analysis. Let $H$ and its entrywise square both be complex Hadamard matrices, and fix two distinct rows. If the cubes of their six entrywise ratios take only two distinct values, then the sum of the row-ratio terms over either specified cube fiber is zero.\n\nThis is a supporting Fourier cancellation statement. The paper's full character-sum vanishing theorem and its mutually unbiased bases bound are outside this selected statement.", "lab_docs_url": "https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/docs/266.md", "overview_entry": "family 266 in overview.pdf / CONTENTS.md", "papers": [{"dir": "The-maximum-number-of-mutually-unbiased-bases-in-dimension-six-September-24-2026", "title": "The maximum number of mutually unbiased bases in dimension six", "url": "https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-maximum-number-of-mutually-unbiased-bases-in-dimension-six-September-24-2026"}, {"dir": "Exact-Fourier-certificates-for-complex-Hadamard-matrices-of-order-six-September-24-2026", "title": "Exact Fourier certificates for complex Hadamard matrices of order six", "url": "https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Exact-Fourier-certificates-for-complex-Hadamard-matrices-of-order-six-September-24-2026"}], "audit_part": 2, "second_reader": "", "referee_verdict": ""}, {"family": "267", "title": "Positive-temperature Bose--Einstein condensation and quantum depletion", "subject": "Mathematical physics", "headline": "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.", "verdict": "weaker-statement", "challenges": ["HardSphere"], "review_note": "`condensation_with_mixed : ∃ ε₀ c₀ > 0, ∀ a ρ, ρ a^3 < ε₀ → ∀ thermodynamic sequences (N_k, L_k), (∀ pure ground vectors Ψ_k, c₀ ≤ liminf occupation Ψ_k) ∧ (∀ density operators supported on the ground space, c₀ ≤ liminf mixedOccupation)` is ZERO-temperature ground-state condensation of the hard-sphere gas with a non-sharp lower bound c₀ (docs: 'concerns ground states, without a positive-temperature assertion'). The summary's headline claims are not stated: BEC for the exact canonical Gibbs state at a strictly positive volume-independent temperature, and the sharp Bogoliubov leading quantum-depletion law (thermodynamic limit before dilute limit) for hard spheres and bounded finite-range potentials.", "definitions_to_check": ["Hand-built hard-sphere gas: Dirichlet form domain `dirichletDomain N L a := closure (range SmoothTest.jet)` (smooth compactly supported functions vanishing on `d(x_i,x_j) ≤ a` on the torus `AddCircle L ^3`), `energy u := Σ ‖∇_p u‖²` (no 1/2), `bosonic`, `IsGroundVector`, `occupation ψ := L^{-3} ∫_Y |∫_x ψ(x,Y)|²` and a custom trace-class `Density.Operator`; the zero-mode occupation fraction matches the standard condensate fraction, but all objects are hand-written."], "external_packages": [], "cone_lines_max": 71853, "machine_check": "comparator-sandboxed-pass", "lab_scope_note": "The formalization proves ground-state Bose–Einstein condensation in the dilute hard-sphere gas. There are absolute constants $\\varepsilon_0,c_0>0$ such that, whenever the density $\\rho$ and hard-sphere radius $a$ satisfy $\\rho a^3<\\varepsilon_0$, the condensate occupation fraction has limit inferior at least $c_0$ along every thermodynamic sequence with $N/L^3\\to\\rho$.\n\nThe bound covers every pure ground state, every mixed state supported on the ground space, and the normalized ground-space projection. It is uniform in the gas parameter within this range and concerns ground states, without a positive-temperature assertion.", "lab_docs_url": "https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/docs/267.md", "overview_entry": "family 267 in overview.pdf / CONTENTS.md", "papers": [{"dir": "Ground-state-condensation-in-the-dilute-hard-sphere-gas-September-24-2026", "title": "Ground-state condensation in the dilute hard-sphere gas", "url": "https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Ground-state-condensation-in-the-dilute-hard-sphere-gas-September-24-2026"}], "audit_part": 2, "second_reader": "", "referee_verdict": ""}, {"family": "269", "title": "The Laughlin gap and stability under scalar disorder", "subject": "Mathematical physics", "headline": "Proves the fermionic Laughlin spectral-gap conjecture for the full $V_1$ 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.", "verdict": "partial", "challenges": ["Laughlin", "LaughlinFock", "LaughlinGap", "LaughlinPlanar"], "review_note": "Stated: the unperturbed spherical Laughlin gap at flux Q = 3(N-1): `Laughlin.MainTarget` (1/100) and `LaughlinGap.MainTarget` (`(1/25) * distanceToLaughlinSq ψ ≤ energy ψ` for every antisymmetric ψ and N >= N₀), the Fock-space inequality `thm_fock` (H² ≥ γ H for all γ < γ* ≈ 0.4617 and large flux, independent of particle number) and a planar version. Not stated: the summary's second claim, stability of the uniform gap and uniqueness of the ground state under weak bounded real scalar one-body potentials projected to the lowest Landau level (docs: 'Stability under projected one-body potentials and uniqueness of the perturbed ground state are outside it'); nor that `laughlinVector` is a zero-energy state (the inequality is stated against the line through it).", "definitions_to_check": ["Hand-coded spherical V₁ Hamiltonian: `energy ψ := Σ_{i1/25$.\n\nFor the untruncated planar model, the formalization proves the corresponding inequality at the endpoint $\\gamma_*$ on every homogeneous particle sector. These statements use coefficient one for each pair projector.", "lab_docs_url": "https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/docs/269.md", "overview_entry": "family 269 in overview.pdf / CONTENTS.md", "papers": [{"dir": "Uniform-Stability-of-the-Spherical-Laughlin-Gap-October-5-2026", "title": "Uniform Stability of the Spherical Laughlin Gap", "url": "https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Uniform-Stability-of-the-Spherical-Laughlin-Gap-October-5-2026"}, {"dir": "A-Fock-space-inequality-and-the-Laughlin-spectral-gap-September-24-2026", "title": "A Fock-space inequality and the Laughlin spectral gap", "url": "https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-Fock-space-inequality-and-the-Laughlin-spectral-gap-September-24-2026"}], "audit_part": 2, "second_reader": "", "referee_verdict": ""}, {"family": "271", "title": "Bloch's law and spontaneous ferromagnetic order", "subject": "Mathematical physics", "headline": "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 $\\mathbb Z^3$. 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 $d\\ge3$ and determines the first lattice correction for three-dimensional nearest-neighbor couplings.", "verdict": "partial", "challenges": ["Heisenberg"], "review_note": "PRIMARY CLAIM NOT STATED (only a secondary result is formalized). Only `spontaneous_magnetization (d ℓ) (hd : 3 ≤ d) (hℓ : 1 ≤ ℓ) : DynamicsConverges d ℓ ∧ ∃ β₀ > 0, ∀ β ≥ β₀, ∃ ω, TranslationInvariant ω ∧ IsKMS β ω ∧ (ω.functional (spinZ d ℓ 0)).im = 0 ∧ ℓ/8 ≤ (ω.functional (spinZ d ℓ 0)).re` (nearest-neighbor Heisenberg ferromagnet, every spin S = ℓ/2) is stated, i.e. the summary's third claim. The headline, Bloch's T^{3/2} law with its exact coefficient for general finite-range couplings (and the first lattice correction for 3D nearest-neighbor couplings), has no Lean statement (docs: 'The formalization proves spontaneous magnetization ...').", "definitions_to_check": ["Entire quantum-spin setup is hand-built: `Configuration d ℓ := Site d →₀ Level ℓ` (incomplete tensor product over the all-zero reference configuration), `QuasiLocal d ℓ := closure of the *-algebra of matrix units`, `hamiltonian` with edges counted via `WellOrderingRel`, `dynamics := Filter.limUnder atTop (finiteDynamics .. (box d n) ..)` (with convergence asserted separately by `DynamicsConverges`) and a hand-written `IsKMS` (analytic strip function with F(t) = ω(A α_t B), F(t+iβ) = ω(α_t B A)). Plausible, but none is Mathlib's notion of a KMS state or quantum spin system."], "external_packages": [], "cone_lines_max": 25400, "machine_check": "comparator-sandboxed-pass", "lab_scope_note": "The formalization proves spontaneous magnetization for the nearest-neighbor isotropic quantum Heisenberg ferromagnet on $\\mathbb Z^d$ for every $d\\ge3$ and every spin $S\\in\\{\\tfrac12,1,\\tfrac32,\\ldots\\}$. At every sufficiently low positive temperature, it constructs a translation-invariant equilibrium state satisfying the KMS condition for the zero-field dynamics and having magnetization at least $S/4$.\n\nIt also proves convergence of the finite-volume dynamics to the infinite-volume dynamics used in the KMS statement.", "lab_docs_url": "https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/docs/271.md", "overview_entry": "family 271 in overview.pdf / CONTENTS.md", "papers": [{"dir": "Spontaneous-magnetization-in-the-quantum-Heisenberg-ferromagnet-September-24-2026", "title": "Spontaneous magnetization in the quantum Heisenberg ferromagnet", "url": "https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Spontaneous-magnetization-in-the-quantum-Heisenberg-ferromagnet-September-24-2026"}], "audit_part": 2, "second_reader": "", "referee_verdict": ""}, {"family": "272", "title": "Entanglement without secret key and the PPT-square conjecture", "subject": "Mathematical physics", "headline": "Constructs an entangled state on $\\mathbb C^{10}\\otimes\\mathbb C^{10}$ with zero distillable secret key for the specified local-instrument protocols that complete almost surely. These allow joint local processing and authenticated two-way public communication, with no other shared private resource and an eavesdropper holding the input purification and public record. A trace-preserving PPT channel on $M_{21}(\\mathbb C)$ whose square is not entanglement breaking disproves Christandl\\textquotesingle s PPT-square conjecture.", "verdict": "partial", "challenges": ["DimensionTenChannel", "DimensionTenPair"], "review_note": "PRIMARY CLAIM NOT STATED (only a secondary result is formalized). Stated: `DimensionTenChannel.exists_channel_fin21 : ∃ Θ : Map (Fin 21) (Fin 21), PPT Θ ∧ TracePreserving Θ ∧ ¬ EntanglementBreaking (Θ.comp Θ)` (the PPT-square counterexample with `PPT F := CP F ∧ CP (T ∘ F)` and `EntanglementBreaking` via separability of every amplification) and `DimensionTenPair.main_pair` (two explicit PPT maps on 10x10 matrices whose composition is not entanglement breaking and whose Choi matrix has no nonzero product vector in its range). Not stated: the summary's first claim, an entangled state on ℂ¹⁰⊗ℂ¹⁰ with zero distillable secret key under the specified local-instrument/authenticated-communication protocols (docs: 'outside these two selected Comparator statements').", "definitions_to_check": [], "external_packages": [], "cone_lines_max": 44673, "machine_check": "comparator-sandboxed-pass", "lab_scope_note": "The paper's entanglement construction yields counterexamples to PPT-composition claims. The linked formalization covers these consequences: two explicitly specified PPT maps on $10\\times10$ complex matrices have a composition that is not entanglement breaking, and the nonzero Choi matrix of that composition has no nonzero product vector in its range. A separate trace-preserving PPT channel on $21\\times21$ matrices has a square that is not entanglement breaking.\n\nThe paper's zero-distillable-secret-key statement is outside these two selected Comparator statements.", "lab_docs_url": "https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/docs/272.md", "overview_entry": "family 272 in overview.pdf / CONTENTS.md", "papers": [{"dir": "Entanglement-with-zero-distillable-secret-key-in-local-dimension-ten-September-27-2026", "title": "Entanglement with zero distillable secret key in local dimension ten", "url": "https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Entanglement-with-zero-distillable-secret-key-in-local-dimension-ten-September-27-2026"}], "audit_part": 2, "second_reader": "", "referee_verdict": ""}, {"family": "273", "title": "The entropy photon-number inequality", "subject": "Mathematical physics", "headline": "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\\textquotesingle s entropy photon number is at least the transmissivity-weighted average of the inputs\\textquotesingle. Arbitrary entanglement within each input is allowed, and product thermal inputs attain equality even when their entropies differ.", "verdict": "full", "challenges": ["EntropyPhotonNumber"], "review_note": "`entropy_photon_number_inequality (n) (hn : 1 ≤ n) (ρA ρB ρC : State n) (hA : FiniteEnergy ρA) (hB : FiniteEnergy ρB) (η ∈ [0,1]) (hC : IsBeamSplitterOutput η ρA ρB ρC) : η * gInv (S ρA / n) + (1 - η) * gInv (S ρB / n) ≤ gInv (S ρC / n)` on the untruncated Fock space ℓ²(ℕⁿ) with arbitrary (entangled) inputs and `gInv` the inverse of the thermal entropy `g t = (t+1) log(t+1) - t log t`. The summary's remark that product thermal inputs attain equality is not separately stated. Caveat: the output state is specified only by the hypothesis `IsBeamSplitterOutput` (explicit matrix-element formula), so the theorem is vacuous if that formula were wrong or unsatisfiable.", "definitions_to_check": ["Hand-written quantum optics: `State n` = positive operator with `HasSum` diagonal 1 on `lp ℂ 2` over `ℕⁿ`, `entropy ρ := ∑' k, (entry (cfc (-t log t) ρ.op) k k).re` (real `tsum`, junk 0 if not summable; finite energy ensures it is), `gInv s := sInf {t ≥ 0 | s ≤ g t}`, and the beam-splitter output defined by explicit coefficients `oneModeBeamCoefficient η k e a b` (binomial/factorial formula) and `IsBeamSplitterOutput`: ∀ k l, HasSum (outputBlock ..) (entry ρC k l)."], "external_packages": [], "cone_lines_max": 27838, "machine_check": "comparator-sandboxed-pass", "lab_scope_note": "The formalization proves the entropy photon-number inequality for two independent bosonic inputs with finite mean energy and any finite positive number $n$ of modes. Write $N(\\rho)=g^{-1}(S(\\rho)/n)$, where $S$ is von Neumann entropy and $g(x)=(x+1)\\log(x+1)-x\\log x$ is the thermal entropy per mode. For a beam splitter of transmissivity $0\\le\\eta\\le1$, its output satisfies $N(\\rho_C)\\ge\\eta N(\\rho_A)+(1-\\eta)N(\\rho_B)$.\n\nEntanglement among modes within either input is allowed. The thermal-attenuator minimum-output-entropy and broadcast-capacity consequences in the paper are outside this selected inequality.", "lab_docs_url": "https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/docs/273.md", "overview_entry": "family 273 in overview.pdf / CONTENTS.md", "papers": [{"dir": "The-entropy-photon-number-inequality-September-24-2026", "title": "The entropy photon-number inequality", "url": "https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-entropy-photon-number-inequality-September-24-2026"}], "audit_part": 2, "second_reader": "", "referee_verdict": ""}, {"family": "274", "title": "Moore's parity conjecture for $\\mathrm{QAC}^0$", "subject": "Mathematical physics", "headline": "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.", "verdict": "full", "challenges": ["QACParity", "RegularParity"], "review_note": "`QAC.parity_lower_bound : ParityStatement` = `∀ d, c ≥ 1, 0 < ε ≤ 1/2, ∃ n₀, ∀ n ≥ n₀, ∀ N, n ≤ N ≤ n^c, ∀ layers (≤ d layers of disjoint one-qubit unitaries / unbounded-arity Toffolis), ∀ out, ¬ (∀ x : Word n, 1/2 + ε ≤ successProbability (physicalCircuitMatrix layers) out x)` with ancillas initialized to zero (`inputWord`) and `successProbability` the Born probability of the measured output qubit equalling the parity, summing over all garbage. This is exactly Moore's parity conjecture in the measured-output model; `RegularParity` adds the explicit '< 2/3' polynomial-size specialization. Xu-Li's majority reduction is only cited in the summary.", "definitions_to_check": [], "external_packages": [], "cone_lines_max": 3658, "machine_check": "comparator-sandboxed-pass", "lab_scope_note": "The formalization rules out bounded-error parity computation by constant-depth quantum circuits with polynomially many zero-initialized ancillary qubits. For every fixed depth, polynomial bound on the total number of qubits, and $0<\\varepsilon\\le1/2$, every sufficiently large input length has an input on which the measured-output parity success probability is less than $1/2+\\varepsilon$. A companion specialization gives success strictly below $2/3$. The formalization also includes the product-projection localization estimate supporting this bound.\n\nThe formalized result supplies the polynomial-size parity consequence of the paper. For every fixed circuit depth and polynomial bound on the number of qubits, all sufficiently large input lengths have an input on which any such circuit computes measured-output parity with probability strictly below $2/3$. Ancillary qubits start at zero. The formalization also contains the general positive-advantage parity bound and its product-projection localization estimate; regular-trajectory propagation and pruning are not separate formalized statements here.", "lab_docs_url": "https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/docs/274.md", "overview_entry": "family 274 in overview.pdf / CONTENTS.md", "papers": [{"dir": "Product-projection-localization-and-the-QAC0-parity-lower-bound-September-24-2026", "title": "Product-projection localization and the QAC0 parity lower bound", "url": "https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Product-projection-localization-and-the-QAC0-parity-lower-bound-September-24-2026"}, {"dir": "Regular-trajectories-pruning-and-quantum-parity-September-24-2026", "title": "Regular trajectories, pruning and quantum parity", "url": "https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Regular-trajectories-pruning-and-quantum-parity-September-24-2026"}], "audit_part": 2, "second_reader": "", "referee_verdict": ""}, {"family": "275", "title": "QMA-hardness of continuum Coulomb energy", "subject": "Mathematical physics", "headline": "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.", "verdict": "full", "challenges": ["ContinuumCoulombHardness"], "review_note": "`unit_coulomb_qmaHard : QMAHard unitCoulombCodec.encode unitCoulombPromise` and `binary_coulomb_qmaHard`: every promise problem in QMA (verifier circuits over {H, T, CNOT} generated by a Mathlib `Turing.TM2ComputableInPolyTime`, completeness 2/3, soundness 1/3) has a deterministic polynomial-time many-one reduction (`PolynomialManyOne`, also Mathlib TM2 poly-time) to deciding `groundEnergy ≤ lower` versus `upper ≤ groundEnergy` with `1 ≤ upper - lower`, distinct rational nuclear positions, unary electron count and (for the unit version) unit charges; `groundEnergy` is the `sInf` over normalized antisymmetric H¹ states of the two-spin Coulomb form `kinetic - nuclearEnergy + pairEnergy`. External package `RellichKondrachov` is in the cone.", "definitions_to_check": ["Custom complexity setup: hand-written `QMAGate`/`QMACircuit`/`QuantumVerifier`/`InQMA` (QMA over gate set {H, T, CNOT}, acceptance = Born probability of the last qubit being 1), `PromiseProblem`, `QMAHard`, and a `Codec` library for binary/unary encodings; polynomial time itself is Mathlib's `Turing.TM2ComputableInPolyTime` (not custom). `groundEnergy` is an EReal-valued `sInf` of Bochner-integral forms on H¹ vectors, which is fine because Hardy's inequality makes the Coulomb terms integrable."], "external_packages": ["RellichKondrachov"], "cone_lines_max": 222484, "machine_check": "build-in-progress", "lab_scope_note": "The formalization proves QMA-hardness of approximating the electronic Coulomb spectral infimum in three-dimensional continuum space when positive integer nuclear charges are encoded in binary. The instances have distinct rational nuclear positions and a unary electron count, and the energy ranges over antisymmetric continuum states and all spin sectors. A deterministic polynomial-time many-one reduction produces the promise gap with threshold separation at least one, while keeping the complete output length polynomial. The same Comparator file also contains the companion unit-charge result.\n\nThe formalization proves QMA-hardness of approximating the electronic ground-energy infimum for clamped unit-charge nuclei in the full spinful fermionic continuum. The reduction is deterministic and polynomial time, with rational nuclear positions and separated rational energy thresholds. No orbital basis, magnetic field, additional external potential, or binding premise is part of the input.\n\nThe same Comparator file also includes the companion hardness result when positive integer nuclear charges are encoded in binary.", "lab_docs_url": "https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/docs/275.md", "overview_entry": "family 275 in overview.pdf / CONTENTS.md", "papers": [{"dir": "Continuum-Coulomb-hardness-with-binary-nuclear-charges-September-24-2026", "title": "Continuum Coulomb hardness with binary nuclear charges", "url": "https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Continuum-Coulomb-hardness-with-binary-nuclear-charges-September-24-2026"}, {"dir": "QMA-hardness-of-continuum-Coulomb-energy-with-unit-nuclear-charges-September-24-2026", "title": "QMA-hardness of continuum Coulomb energy with unit nuclear charges", "url": "https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/QMA-hardness-of-continuum-Coulomb-energy-with-unit-nuclear-charges-September-24-2026"}], "audit_part": 2, "second_reader": "", "referee_verdict": ""}, {"family": "276", "title": "The classical capacity of generalized amplitude damping", "subject": "Mathematical physics", "headline": "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.", "verdict": "partial", "challenges": ["AmplitudeDamping"], "review_note": "Stated: `GAD.main`: for all γ, ν ∈ [0,1], `holevo γ ν n = n * holevo γ ν 1` (additivity across n uses of the same channel), `∃ p, Maximizes γ ν p ∧ holevo n = n / log 2 * objective γ ν p` (explicit one-variable optimization), `capacity γ ν = holevo γ ν 1` (operational capacity as sup of achievable rates with unrestricted collective decoding) and that every maximizing phase-pair product ensemble attains the block value. Not stated: the summary's additivity of Holevo capacity, minimum output entropy and regularized capacity when tensored with ANY finite-dimensional channel (docs: 'Additivity with an arbitrary different partner channel ... not included').", "definitions_to_check": ["Hand-written information theory: `entropy P := (trace (cfc Real.negMulLog P)).re`, `holevo γ ν n := sSup (range ensembleValue)` over arbitrary finite ensembles, GAD Kraus operators `kraus γ ν r`, `Achievable`/`capacity := sSup {R | Achievable γ ν R}` with a `Tendsto` error and liminf-rate condition; standard definitions but not Mathlib's."], "external_packages": [], "cone_lines_max": 6206, "machine_check": "comparator-sandboxed-pass", "lab_scope_note": "The formalized result determines the unassisted classical capacity of every generalized amplitude-damping channel with parameters in $[0,1]^2$. Its unrestricted Holevo information is additive across every number of repeated uses, equals an attained scalar maximum, and agrees with the operational capacity per use. Every maximizing equiprobable phase-pair product ensemble attains the corresponding block value. Additivity with an arbitrary different partner channel and decoding by separate measurements are not included.", "lab_docs_url": "https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/docs/276.md", "overview_entry": "family 276 in overview.pdf / CONTENTS.md", "papers": [{"dir": "Classical-capacity-and-entropy-inequalities-for-generalized-amplitude-damping-September-24-2026", "title": "Classical capacity and entropy inequalities for generalized amplitude damping", "url": "https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Classical-capacity-and-entropy-inequalities-for-generalized-amplitude-damping-September-24-2026"}], "audit_part": 2, "second_reader": "", "referee_verdict": ""}, {"family": "277", "title": "Threshold repetition for entangled games", "subject": "Mathematical physics", "headline": "Proves exponential threshold repetition for every finite two-player one-round game: if its entangled value is $v<1$, the probability of winning at least a fraction $v+\\delta$ of $k$ independent repetitions decays exponentially in $k$, for $0<\\delta<1-v$. Arbitrary joint finite-dimensional entangled strategies and correlated question distributions are allowed.", "verdict": "full", "challenges": ["EntangledGames"], "review_note": "`threshold_parallel_repetition : ∃ κ₀ > 0, ∀ games G with entangledValue G < 1, ∀ 0 < δ < 1 - v, ∀ k ≥ 1, (∀ joint finite-dimensional k-fold strategies S, thresholdProbability G δ S ≤ exp(-κ₀ δ^13 k / (1 + log((a+1)(b+1))))) ∧ thresholdValue G δ ≤ the same bound`, with `Game` an arbitrary correlated question distribution plus predicate, `Strategy` a finite-dimensional bipartite state with POVMs and `entangledValue := sSup` of single-copy success, and `thresholdProbability` = probability of at least ⌈(v+δ)k⌉ wins. Exactly the summary's exponential threshold repetition (explicit δ^13 exponent).", "definitions_to_check": [], "external_packages": [], "cone_lines_max": 7234, "machine_check": "comparator-sandboxed-pass", "lab_scope_note": "The formalization gives exponential threshold parallel repetition for every finite two-player game with arbitrary correlated questions. If its finite-dimensional entangled value is $v<1$, its answer sets are $A,B$, and $0<\\delta<1-v$, then the probability of winning at least the fraction $v+\\delta$ of $k$ repetitions is at most $\\exp(-\\kappa\\delta^{13}k/(1+\\log(|A||B|)))$ for one universal $\\kappa>0$. The bound applies to every joint finite-dimensional strategy and to their supremum, without assuming attainment. The separate distribution-dependent cubic bound is outside this scope.", "lab_docs_url": "https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/docs/277.md", "overview_entry": "family 277 in overview.pdf / CONTENTS.md", "papers": [{"dir": "Threshold-parallel-repetition-for-finite-dimensional-entangled-games-September-25-2026", "title": "Threshold parallel repetition for finite-dimensional entangled games", "url": "https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Threshold-parallel-repetition-for-finite-dimensional-entangled-games-September-25-2026"}], "audit_part": 2, "second_reader": "", "referee_verdict": ""}, {"family": "279", "title": "Exact quantum factoring over a fixed finite gate set", "subject": "Mathematical physics", "headline": "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.", "verdict": "full", "challenges": ["ExactQuantumFactoring"], "review_note": "`exact_quantum_factoring : MainTheorem := ∃ family : ℕ → Circuit, Uniform family ∧ PolynomialResources family ∧ ∀ ℓ N, 2 ≤ N → N.size = ℓ → ℓ + (paddedLength ℓ)^2 ≤ (family ℓ).qubits ∧ (family ℓ).correctProbability ℓ N = 1` over a fixed 20-gate set (NOT/CNOT/Toffoli/Hadamard/phase with optional inverse and one control), where `correctProbability` is the exact Born probability of outputting the sorted prime factorization with multiplicities (`CorrectEncoding`) and `Uniform` uses a Mathlib `TM2ComputableInPolyTime` generator with finite alphabets. Matches the summary (exact, polynomial gates and qubits, fixed finite gate set).", "definitions_to_check": ["Hand-built circuit model: `Gate := (primitive, inverse, controlled)`, `Primitive.matrix` entries on `List Bool`, `Instruction.matrix`, `Circuit.apply := foldl` over instructions; the polynomial-time notion is Mathlib's `TM2ComputableInPolyTime` applied to unary input `replicate ℓ true` with `Finite (Γ k)`, which is standard; the quantum gate semantics are written by hand."], "external_packages": [], "cone_lines_max": 54007, "machine_check": "comparator-sandboxed-pass", "lab_scope_note": "The formalization constructs a polynomial-time uniform quantum circuit family that outputs the complete prime factorization of every integer $N\\ge2$ with probability exactly one. The circuits use one fixed finite set of bounded-arity gates, and both the gate count and number of qubits have polynomial worst-case bounds in the bit length of $N$. The probability-one conclusion is exact, rather than an asymptotic or bounded-error guarantee.", "lab_docs_url": "https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/docs/279.md", "overview_entry": "family 279 in overview.pdf / CONTENTS.md", "papers": [{"dir": "Exact-quantum-factoring-over-a-fixed-finite-gate-set-September-25-2026", "title": "Exact quantum factoring over a fixed finite gate set", "url": "https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Exact-quantum-factoring-over-a-fixed-finite-gate-set-September-25-2026"}], "audit_part": 2, "second_reader": "", "referee_verdict": ""}, {"family": "280", "title": "Strong locality for strongly rational unitary vertex operator algebras", "subject": "Mathematical physics", "headline": "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.", "verdict": "partial", "challenges": ["VertexAlgebraNet"], "review_note": "`MinimalVertex.main (hSimple : A.toVertexAlgebra.IsSimple) (hSR : A.IsStronglyRational) : A.PolynomialEnergyBounds ∧ CKLWStrongLocal U _ ∧ Nonempty (IrreducibleConformalNetStructure (intervalAlgebra U _))`: polynomial energy bounds, strong locality (`intervalAlgebra I ≤ (intervalAlgebra I.complement).commutant`) and an irreducible conformal net (isotony, locality, Möbius and Diff(S¹) covariance, unique vacuum, positive energy) for a simple unitary strongly rational VOA, i.e. the strong-locality core of the summary. Not stated: complete rationality of the net (the summary says it generates a COMPLETELY RATIONAL net), unitarizability of all simple modules, and the braided unitary tensor equivalence of Rep(V) with the net's finite-index sectors (docs: 'does not include complete rationality ... unitarizability ... the braided tensor equivalence').", "definitions_to_check": ["Entire VOA/conformal-net stack is hand-built (`VertexAlgebra`, `CFTTypeVOA`, `IsStronglyRational := IsSelfContragredient ∧ IsRational ∧ C2Cofinite`, `smearedMap`, `closedSmoothField`, `localAlgebra`, `IrreducibleConformalNetStructure`), with many definitions using junk defaults: `smearedMap := if h : ∀ v, Summable (..) then tsumMap else 0`, `smoothSmearedField := if h : .. then .. else 0`, `resolvent`/`flow := if h : ∃ L, .. then h.choose else 0`, `projectiveStandardAction := if h : ∃ L, .. then h.choose else 1`. Not Mathlib notions; fidelity rests on these encodings."], "external_packages": [], "cone_lines_max": 28634, "machine_check": "comparator-sandboxed-pass", "lab_scope_note": "The linked formalization proves the first construction step relating strongly rational unitary vertex operator algebras to conformal nets. For a simple unitary strongly rational vertex operator algebra, it establishes polynomial energy bounds and strong locality and constructs an irreducible conformal net with the stated covariance, vacuum, and positive-energy properties.\n\nThe selected statement does not include complete rationality of the net, unitarizability of all simple modules, the braided tensor equivalence, or the classification of local extensions described in the paper.", "lab_docs_url": "https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/docs/280.md", "overview_entry": "family 280 in overview.pdf / CONTENTS.md", "papers": [{"dir": "Strongly-rational-unitary-vertex-operator-algebras-and-conformal-nets-September-25-2026", "title": "Strongly rational unitary vertex operator algebras and conformal nets", "url": "https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Strongly-rational-unitary-vertex-operator-algebras-and-conformal-nets-September-25-2026"}], "audit_part": 2, "second_reader": "", "referee_verdict": ""}, {"family": "281", "title": "QAOA optimality for the SK model", "subject": "Mathematical physics", "headline": "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.", "verdict": "supporting-only", "challenges": ["SKFullSupport", "SKValue"], "review_note": "No QAOA statement exists. The challenges are `SKValue.value_consequences` (conditional on `IsMinimizer W γ` and a diffusion `X`: the finite-size SK ground-state energy per spin `groundStateSequence n` converges to `groundStateValue = parisi W γ`, with martingale/curvature-integral formulas and convergence of finite Gaussian coefficient sums) and `SKFullSupport.full_support` (full support of zero-temperature Parisi minimizers). Docs: 'it does not itself assert convergence of QAOA circuit energies', and 'does not separately assert existence of a minimizer'. The summary's headline (QAOA with finite depth and size/disorder-independent angles reaches the SK ground-state energy per spin, size before depth, plus MaxCut on random regular graphs) is absent.", "definitions_to_check": ["Custom Parisi/stochastic-control objects: `phi W γ t x := sSup {controlPayoff ..}` over `Admissible` progressive controls (real `sSup`), `gradient := deriv (phi W γ t)`, `curvature := deriv (gradient ..)`, `parisi`, `IsMinimizer`, `groundStateValue := limUnder atTop groundStateSequence` (convergence asserted in `ScalarValueConclusion`); derivatives are Mathlib `deriv` (junk 0 where not differentiable)."], "external_packages": [], "cone_lines_max": 29844, "machine_check": "comparator-sandboxed-pass", "lab_scope_note": "The linked formalization supplies variational identities for the Gaussian zero-field Sherrington–Kirkpatrick ground-state energy used in the paper's QAOA argument. Conditional on an admissible Parisi minimizer and its associated diffusion, it proves convergence of the finite-system ground-state energy, identifies its limit with the Parisi value, and gives equivalent terminal-martingale and integrated-curvature formulas.\n\nIt also proves convergence of finite Gaussian coefficient sums to the curvature integral, which approaches the ground-state value as the terminal time tends to one. The selected statement covers these value and approximation results; it does not itself assert convergence of QAOA circuit energies.\n\nThe formalization proves that every admissible integrable minimizer of the zero-temperature Parisi functional for the pure zero-field Sherrington–Kirkpatrick model has full relative Stieltjes support on $[0,1)$. Equivalently, the order parameter increases strictly between every two overlap values below one, so its support has no gap. The covariance normalization is $\\xi(t)=t^2/2$.\n\nThe statement also constructs the associated diffusion and proves its selected self-consistency moment identities. It is conditional on the order parameter being a minimizer and does not separately assert existence of a minimizer.", "lab_docs_url": "https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/docs/281.md", "overview_entry": "family 281 in overview.pdf / CONTENTS.md", "papers": [{"dir": "QAOA-attains-the-SK-ground-state-energy-in-the-thermodynamic-first-limit-September-25-2026", "title": "QAOA attains the SK ground-state energy in the thermodynamic-first limit", "url": "https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/QAOA-attains-the-SK-ground-state-energy-in-the-thermodynamic-first-limit-September-25-2026"}, {"dir": "Full-support-of-the-zero-temperature-Sherrington-Kirkpatrick-order-parameter-September-27-2026", "title": "Full support of the zero-temperature Sherrington-Kirkpatrick order parameter", "url": "https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Full-support-of-the-zero-temperature-Sherrington-Kirkpatrick-order-parameter-September-27-2026"}], "audit_part": 2, "second_reader": "", "referee_verdict": ""}, {"family": "287", "title": "All nonabelian free group factors are isomorphic", "subject": "Operator algebras", "headline": "Resolves the free group factor isomorphism problem: $L(\\mathbb F_2)\\cong L(\\mathbb F_3)$, and hence all interpolated free group factors, including $L(\\mathbb F_\\infty)$, are isomorphic. Their common factor has fundamental group $\\mathbb R_{>0}$.", "verdict": "full", "challenges": ["InterpolatedFactors"], "review_note": "`allInterpolatedIsomorphic (r s : ℝ≥0∞) (hr : 1 < r) (hs : 1 < s) : Nonempty (NormalTracialEquiv (interpolatedTrace r) (interpolatedTrace s) (interpolatedTopology r) (interpolatedTopology s))`: every pair of interpolated free group factors L(F_r), L(F_s) with r,s ∈ (1,∞] (integer ranks via `groupVonNeumann (FreeGroup (Fin n))`, `FreeGroup ℕ` for ∞, and non-integer r as the corner of `L(F_2) ⊗ B(ℓ²)` by a projection of trace `1/√(r-1)`) is related by a *-isomorphism that preserves the canonical trace and is ultraweakly bicontinuous. The summary's statement L(F_2) ≅ L(F_3) and all interpolated factors; the fundamental-group conclusion is a consequence, not stated.", "definitions_to_check": ["Hand-built II_1 factor models: `groupVonNeumann G := centralizer (centralizer (leftTranslations G))` (double commutant of left translations), `canonicalTrace`, `ultraweakOperatorTopology := ⨅ ξ η, induced (T ↦ Σ' k ⟨η k, T (ξ k)⟩)`, corners `Corner p hp` and `selectedProjection a := Classical.epsilon (stabilizedProjectionTrace p = ofReal a)` (junk default projection 0 if none existed, which does not happen in L(F_2)⊗B(ℓ²)); standard but not Mathlib's `VonNeumannAlgebra`-with-trace."], "external_packages": [], "cone_lines_max": 26804, "machine_check": "comparator-sandboxed-pass", "lab_scope_note": "The free group factor problem asks whether the von Neumann algebras of free groups of different ranks are isomorphic. The formalization proves that interpolated free group factors with any parameters $r,s>1$, including the infinite parameter, are normally trace-preservingly isomorphic. The paper's fundamental-group conclusion is a further consequence rather than a separate selected statement here.", "lab_docs_url": "https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/docs/287.md", "overview_entry": "family 287 in overview.pdf / CONTENTS.md", "papers": [{"dir": "An-isomorphism-of-the-free-group-factors-September-23-2026", "title": "An isomorphism of the free group factors", "url": "https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/An-isomorphism-of-the-free-group-factors-September-23-2026"}], "audit_part": 2, "second_reader": "", "referee_verdict": ""}, {"family": "288", "title": "Kadison's similarity conjecture", "subject": "Operator algebras", "headline": "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.", "verdict": "full", "challenges": ["KadisonSimilarity", "UniformCommutator"], "review_note": "`similarityTheorem : SimilarityTheorem.{u,v} := ∀ (A : Type u) [CStarAlgebra A] (K : Type v) [Hilbert space K] (π : BoundedUnitalHom A K), SimilarToStar A K π` with `BoundedUnitalHom A H := {π : A →ₐ[ℂ] (H →L[ℂ] H) // Continuous π}` and `SimilarToStar π := ∃ S : (H →L[ℂ] H)ˣ, ∀ a, S π(a⋆) S⁻¹ = (S π(a) S⁻¹)⋆`: Kadison's similarity conjecture for arbitrary Hilbert spaces, plus `universalHyperreflexivity` (one constant for all von Neumann algebras) and the uniform amplified commutator estimate.", "definitions_to_check": [], "external_packages": [], "cone_lines_max": 127680, "machine_check": "comparator-sandboxed-pass", "lab_scope_note": "Kadison's similarity problem asks whether every bounded unital homomorphism from a unital complex $C^*$-algebra into operators on a Hilbert space is similar to a $*$-homomorphism. The formalization proves this for arbitrary Hilbert spaces. It also gives one universal hyperreflexivity constant for all unital von Neumann algebras.\n\nThe linked commutator estimate is uniform over all finite matrix amplifications: its constant is independent of the algebra, Hilbert space, and amplification size. This is the quantitative derivation estimate supporting the similarity result.", "lab_docs_url": "https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/docs/288.md", "overview_entry": "family 288 in overview.pdf / CONTENTS.md", "papers": [{"dir": "Kadisons-similarity-theorem-through-uniform-derivation-estimates-September-23-2026", "title": "Kadison's similarity theorem through uniform derivation estimates", "url": "https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Kadisons-similarity-theorem-through-uniform-derivation-estimates-September-23-2026"}], "audit_part": 2, "second_reader": "", "referee_verdict": ""}, {"family": "289", "title": "The strong Kadison--Kastler conjecture", "subject": "Operator algebras", "headline": "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.", "verdict": "partial", "challenges": ["StrongKadisonKastler"], "review_note": "Stated: `universal_strong_stability : ∀ ε > 0, ∃ δ > 0, ∀ H, ∀ von Neumann algebras M N on H (Mathlib `VonNeumannAlgebra`), kkDistance M N < δ → ∃ unitary v, v M v* = N ∧ ‖v - 1‖ < ε` with `kkDistance` the Hausdorff distance of the unit balls. Not stated: the summary's two counterexamples (near-identity conjugacy fails for one-sided near inclusions; arbitrarily close norm-separable C*-algebras need not be ambiently unitarily conjugate); the docs scope also only mentions the stability theorem although the docs title says 'spatial boundaries'.", "definitions_to_check": [], "external_packages": [], "cone_lines_max": 326060, "machine_check": "none", "lab_scope_note": "The formalization proves the strong Kadison–Kastler conjecture uniformly. For every $\\varepsilon>0$, there is a $\\delta>0$ such that any two unital von Neumann algebras on the same complex Hilbert space at Kadison–Kastler distance below $\\delta$ are conjugate by a unitary $u$ with $\\lVert u-1\\rVert<\\varepsilon$. The tolerance depends only on $\\varepsilon$, uniformly over the algebras, their representations, and the Hilbert space.", "lab_docs_url": "https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/docs/289.md", "overview_entry": "family 289 in overview.pdf / CONTENTS.md", "papers": [{"dir": "Universal-strong-Kadison-Kastler-stability-September-23-2026", "title": "Universal strong Kadison–Kastler stability", "url": "https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Universal-strong-Kadison-Kastler-stability-September-23-2026"}], "audit_part": 2, "second_reader": "", "referee_verdict": ""}, {"family": "290", "title": "Connes' bicentralizer conjecture and relative bicentralizers", "subject": "Operator algebras", "headline": "Proves Connes' bicentralizer conjecture for every type $\\mathrm{III}_1$ 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.", "verdict": "supporting-only", "challenges": ["BoundedRecovery"], "review_note": "Only a technical lemma is stated: `bounded_recovery (S : StandardModularData H) (hscalar : S.ScalarCentralizer) (omega free ultrafilter) (T) (s) (delta) ... (hpositive : 0 < limsup (fixedMean omega (fun t => ‖T (D.unitary t (h n))‖²))) : RecoveryConclusion S T s delta` (a subsequence of uniformly bounded algebra elements v_j with `eta ≤ ‖T (v_j ξ)‖` and spectral support in bands of 4x the width). The summary's headline, Connes' bicentralizer conjecture for every type III₁ factor with separable predual and every faithful normal state (and the relative bicentralizer statement producing an amenable P with P' ∩ c(M) = N' ∩ c(M)), is not stated (docs: 'The absolute bicentralizer conjecture is not included').", "definitions_to_check": ["Hand-built Tomita-Takesaki setting: `StandardModularData` (M, ξ, a hand-written `RealSpectralCalculus` D, J, `tomita_graph` equal to the closure of the graph of S₀ composed with half-exponential cutoffs), `fixedMean omega f := Filter.limUnder omega (symmetricAverage f)` (junk if the ultralimit does not exist, but ultrafilter limits on bounded sequences do exist)."], "external_packages": [], "cone_lines_max": 8104, "machine_check": "comparator-sandboxed-pass", "lab_scope_note": "For a faithful normal state with scalar centralizer in the stated standard-space representation, the formalized result converts positive modular spectral averages into uniformly bounded algebra elements. Given unit vectors in shrinking spectral bands around a real number $s$ and a positive limiting symmetric average for a bounded operator $T$, it finds a subsequence of bounded algebra elements whose images under $T$ stay uniformly nonzero and whose spectral bands have four times the original widths. The absolute bicentralizer conjecture is not included.", "lab_docs_url": "https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/docs/290.md", "overview_entry": "family 290 in overview.pdf / CONTENTS.md", "papers": [{"dir": "Bounded-recovery-for-modular-spectral-averages-September-23-2026", "title": "Bounded recovery for modular spectral averages", "url": "https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Bounded-recovery-for-modular-spectral-averages-September-23-2026"}], "audit_part": 2, "second_reader": "", "referee_verdict": ""}, {"family": "291", "title": "Toms--Winter and equivariant Jiang--Su stability", "subject": "Operator algebras", "headline": "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 Szab\\'o'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.", "verdict": "supporting-only", "challenges": ["UniformGamma"], "review_note": "The single challenge is `CurrentMain.main : MainClaim`, i.e. `Statement` (for a simple separable nuclear stably finite infinite-dimensional C*-algebra with tracial states and a free ultrafilter U, if the uniform tracial ultrapower of the uniform tracial completion has real rank zero then `UniformPropertyGammaAt A U`), preceded by existential witnesses for the hand-built `NormConstruction`/`NullConstruction`/`CauchyConstruction`/`LimitConstruction`/`TraceConstruction` classes. This is a supporting technical implication (docs: 'real rank zero ... implies uniform property Γ'). The summary's headline results, equivariant Jiang-Su stability for every countable discrete amenable group action and the unital Toms-Winter conjecture (strict comparison ⇔ finite nuclear dimension ⇔ Jiang-Su stability), are not stated.", "definitions_to_check": ["The uniform tracial completion and ultrapowers are hand-built quotients: `UniformTracialCompletion := (familyCauchyNullIdeal τ).Quotient`, `FamilyUltrapower := (familyNullIdeal τ U).Quotient`, with `class NormConstruction : Prop`/`NullConstruction`/`CauchyConstruction`/`LimitConstruction`/`TraceConstruction` and `MainClaim := ∃ h0 : NormConstruction, ... ∃ h4 : TraceConstruction, Statement`, i.e. the C*-algebra structure is posited as existentially provable classes inside the claim; `IsNuclear` and `RealRankZero` are also local definitions."], "external_packages": [], "cone_lines_max": 17302, "machine_check": "comparator-sandboxed-pass", "lab_scope_note": "The formalization proves that real rank zero of the uniform tracial ultrapower of the uniform tracial completion implies uniform property $\\Gamma$ for a simple separable unital infinite-dimensional nuclear stably finite $C^*$-algebra with traces. The conclusion holds at every specified free ultrafilter under the corresponding real-rank-zero hypothesis.\n\nThe linked comparison results also prove Jiang–Su absorption and uniform property $\\Gamma$ for simple separable unital infinite-dimensional nuclear algebras with strict comparison, and for the stated stably projectionless nuclear algebras with bounded densely finite traces, a nonempty compact normalized trace base, and the prescribed finite-target-rank comparison condition.", "lab_docs_url": "https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/docs/291.md", "overview_entry": "family 291 in overview.pdf / CONTENTS.md", "papers": [{"dir": "Tracial-projection-methods-and-uniform-property-Gamma-September-23-2026", "title": "Tracial projection methods and uniform property Gamma", "url": "https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Tracial-projection-methods-and-uniform-property-Gamma-September-23-2026"}], "audit_part": 2, "second_reader": "", "referee_verdict": ""}, {"family": "292", "title": "A counterexample to Kirchberg's norm-ultrapower embedding problem", "subject": "Operator algebras", "headline": "Constructs an explicit separable unital full group $C^*$-algebra that cannot embed unitally into the norm ultrapower of any fixed nonzero unital nuclear $C^*$-algebra, for any free ultrafilter on the natural numbers. Taking the target to be $\\mathcal O_2$ answers Kirchberg's norm-ultrapower embedding problem negatively.", "verdict": "full", "challenges": ["NuclearUltrapower"], "review_note": "`main_no_embedding : MainStatementWithCuntzCorollary` states: there is a separable unital C*-algebra `A` with `ι : G →* unitary A` satisfying the universal property of the FULL group C*-algebra of `G = (ℤ[1/2])³ ⋊ (SL₃(ℤ) × ℤ)` (`IsFullGroupAlgebra`), and for every nonzero unital nuclear `B` (nuclear = min and max C*-tensor norms agree on `B ⊗ C` for all C) and every free ultrafilter ω on ℕ there is no injective unital *-homomorphism `A →⋆ₐ[ℂ] NormUltrapower B ω`; the same for the universal Cuntz algebra O₂. Exactly the summary, including the O₂ corollary.", "definitions_to_check": ["Hand-built objects: `NormUltrapower B ω := (ultraCon B ω).Quotient` (bounded sequences modulo ω-null), nuclearity via `minTensorNorm`/`maxTensorNorm` defined as `sSup` over representations on Hilbert spaces of one universe, `IsFullGroupAlgebra` (dense span + universal property), `IsCuntzTwoAlgebra` (two Cuntz isometries). Standard formulations, but none is a Mathlib notion, and the tensor-norm `sSup`s are real-valued."], "external_packages": [], "cone_lines_max": 18473, "machine_check": "none", "lab_scope_note": "Kirchberg's norm-ultrapower embedding problem asks whether separable $C^*$-algebras embed into norm ultrapowers of nuclear algebras. The formalization constructs an explicit separable unital full group $C^*$-algebra that has no unital embedding into $B^\\omega$ for any nonzero unital nuclear $C^*$-algebra $B$ and any free ultrafilter $\\omega$ on $\\mathbb N$. In particular, it does not embed unitally into an ultrapower of the Cuntz algebra $\\mathcal O_2$.", "lab_docs_url": "https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/docs/292.md", "overview_entry": "family 292 in overview.pdf / CONTENTS.md", "papers": [{"dir": "An-explicit-obstruction-to-nuclear-norm-ultrapower-embeddings-September-23-2026", "title": "An explicit obstruction to nuclear norm-ultrapower embeddings", "url": "https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/An-explicit-obstruction-to-nuclear-norm-ultrapower-embeddings-September-23-2026"}], "audit_part": 2, "second_reader": "", "referee_verdict": ""}, {"family": "293", "title": "A counterexample to the hyperinvariant-subspace problem", "subject": "Operator algebras", "headline": "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 $\\mathrm{II}_1$ factor.", "verdict": "full", "challenges": ["BackwardIntertwiners", "ContinuousCircleWeight", "FiniteFactor", "HyperinvariantSubspaces", "IrrationalRotation", "ProductBrown"], "review_note": "Both summary claims are stated: `Hyperinvariant.main_theorem (hInf : ¬FiniteDimensional ℂ H)` (also `BackwardIntertwiners.direct_algebra_corollary`) for every separable infinite-dimensional complex Hilbert space: `∃ T ≠ 0, ‖T^n‖^(1/n) → 0 ∧ TransitiveCommutant T ∧ commutant T ≠ ⊤ ∧ SOT-closed commutant` (no nonzero proper closed hyperinvariant subspace), and for the invariant-projection counterexamples `FiniteFactor.main_theorem` (a II₁ factor with separable predual and a nonzero quasinilpotent T ∈ M with `(1 - p) T p = 0 → p = 0 ∨ p = 1`), `IrrationalRotation.prescribed_irrational_counterexample` (every irrational θ: continuous weight f with `Rotation.IsContinuousCounterexample θ f`, T in the irrational-rotation algebra, nonzero, norm-quasinilpotent, trivial invariant projections) and `ProductBrown` (Brown measure δ₀). 'Hyperfinite' is only implicit (FiniteFactor asserts a II₁ factor with separable predual; the rotation algebra is the hyperfinite one by construction).", "definitions_to_check": ["Hand-written operator-algebra notions: `TransitiveCommutant T := ∀ K closed submodule, (∀ A commuting with T, A K ⊆ K) → K = ⊥ ∨ K = ⊤`, `HasOnlyTrivialInvariantProjections`, `NormQuasinilpotent`, `vectorFugledeKadisonDeterminant`, `IsVectorBrownMeasure`, `strongOperatorTopology`, `HasSeparablePredual M := ∃ separable Banach E, Nonempty (StrongDual ℂ E ≃ₗᵢ⋆[ℂ] M)`; quasinilpotence is via `‖T^n‖^(1/n) → 0` and `ContinuousCircleWeight.MainStatement` does not itself state quasinilpotence (only trivial invariant projections for a weight with log-integral -∞)."], "external_packages": [], "cone_lines_max": 81127, "machine_check": "comparator-sandboxed-pass", "lab_scope_note": "For every irrational rotation angle $\\theta\\in(0,1)$, the formalization constructs a continuous nonnegative circle weight with a single zero and logarithmic integral $-\\infty$ whose weighted rotation is nonzero and quasinilpotent. In its associated hyperfinite type $\\mathrm{II}_1$ factor, the operator has no nontrivial invariant projection: $(1-p)Tp=0$ forces $p=0$ or $p=1$. The linked statements also include an earlier existence construction, a product realization, and the selected product model's Brown measure $\\delta_0$.\n\nThe formalization for the companion paper [Backward intertwiners and a transitive commutant](../../preprints/Backward-intertwiners-and-a-transitive-commutant-September-27-2026/paper.pdf) gives, on every separable infinite-dimensional complex Hilbert space, a nonzero quasinilpotent operator whose commutant is transitive, proper, and closed in the strong operator topology. Thus no nontrivial closed subspace is invariant under every operator in the commutant.\n\nThe hyperinvariant-subspace problem asks whether every bounded operator on a complex Hilbert space has a nontrivial closed subspace invariant under every operator that commutes with it. The formalization gives a negative answer on every separable infinite-dimensional complex Hilbert space: there is a nonzero quasinilpotent operator with a transitive commutant. Its commutant is a proper unital operator algebra closed in the strong operator topology. The construction therefore has no nonzero proper closed hyperinvariant subspace.", "lab_docs_url": "https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/docs/293.md", "overview_entry": "family 293 in overview.pdf / CONTENTS.md", "papers": [{"dir": "Invariant-projection-counterexamples-for-every-irrational-rotation-September-27-2026", "title": "Invariant-projection counterexamples for every irrational rotation", "url": "https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Invariant-projection-counterexamples-for-every-irrational-rotation-September-27-2026"}, {"dir": "Backward-intertwiners-and-a-transitive-commutant-September-27-2026", "title": "Backward intertwiners and a transitive commutant", "url": "https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Backward-intertwiners-and-a-transitive-commutant-September-27-2026"}], "audit_part": 2, "second_reader": "", "referee_verdict": ""}, {"family": "294", "title": "A counterexample to Kaplansky's quasitrace conjecture", "subject": "Operator algebras", "headline": "Disproves Kaplansky's quasitrace conjecture by constructing a separable unital complex $C^*$-algebra admitting normalized $2$-quasitraces, all of which are nonadditive. As a consequence, two unital simple stably finite $C^*$-algebras can have a properly infinite minimal tensor product, with one factor $C_r^*(\\mathbb F_2)$.", "verdict": "full", "challenges": ["KaplanskyQuasitrace", "KaplanskyStableFiniteness"], "review_note": "`kaplansky_quasitrace_counterexample : MainTarget := ∃ separable C*-algebra A, (∃ τ : OneQuasitrace A, τ.Normalized ∧ τ.IsTwo) ∧ ∃ positive contractions a b, ∀ normalized 2-quasitraces τ, 1/144 ≤ Re(τ(a+b) - τ a - τ b)` (all normalized 2-quasitraces are non-additive), and `stable_finiteness_bundle := SimpleStableTensorCounterexample ∧ ReducedQuasitraceLoss ∧ TraceFreeStablyFinite` (C*_r(F₂) and a simple stably finite M whose spatial tensor product `tensorAlgebra ρ` is `ProperlyInfinite`, plus the quasitrace-loss and trace-free examples). Both parts of the summary are stated.", "definitions_to_check": ["Hand-written quasitraces: `OneQuasitrace` (positivity, τ(x*x) = τ(xx*), real/imaginary additivity on self-adjoints, linearity on closed abelian subalgebras) and `IsTwo τ := ∃ σ : OneQuasitrace (M₂(A)), ∀ a, σ (upperLeft a) = τ a` (a 2-quasitrace as one that extends to M₂(A), rather than requiring τ ⊗ Tr₂ to be a quasitrace); `C*_r(F₂)` is built as `closure (adjoin (left translations))` on ℓ²(F₂), `ProperlyInfinite A := ∃ s t, s*s = 1 ∧ t*t = 1 ∧ s*t = 0`."], "external_packages": [], "cone_lines_max": 37896, "machine_check": "comparator-sandboxed-pass", "lab_scope_note": "Kaplansky's quasitrace conjecture predicts that every $2$-quasitrace on a $C^*$-algebra is a trace. The formalization constructs a separable $C^*$-algebra with normalized $2$-quasitraces and fixed positive contractions $a,b$ for which every such quasitrace satisfies $\\mathrm{Re}(\\tau(a+b)-\\tau(a)-\\tau(b))\\ge1/144$. Thus none is additive on this pair.\n\nThe formalization also gives simple stably finite $C^*$-algebras whose spatial tensor product is properly infinite, a separable stably finite algebra with no tracial state, and a tensor-product example in which normalized $2$-quasitraces are lost. These are the stable-finiteness consequences named in the paper's title.", "lab_docs_url": "https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/docs/294.md", "overview_entry": "family 294 in overview.pdf / CONTENTS.md", "papers": [{"dir": "A-counterexample-to-Kaplanskys-quasitrace-conjecture-September-23-2026", "title": "A counterexample to Kaplansky's quasitrace conjecture and failure of tensor-product stable finiteness", "url": "https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-counterexample-to-Kaplanskys-quasitrace-conjecture-September-23-2026"}], "audit_part": 2, "second_reader": "", "referee_verdict": ""}, {"family": "295", "title": "The Kadison--Ringrose cohomology conjecture", "subject": "Operator algebras", "headline": "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.", "verdict": "full", "challenges": ["KadisonRingrose"], "review_note": "`main_result {M : Type u} [CStarAlgebra M] [PartialOrder M] [StarOrderedRing M] [WStarAlgebra M] (n) (f : Cochain M (n+2)) (hf : ∀ x, differentialValue f x = 0) : ∃ g : Cochain M (n+1), ∀ x, differentialValue g x = f x` with `Cochain M n := ContinuousMultilinearMap ℂ (fun _ : Fin n => M) M` (bounded multilinear) and the standard Hochschild differential `v₀ f(v₁..vₙ) + Σ_j (-1)^{j+1} f(.., v_j v_{j+1}, ..) + (-1)^{n+1} f(v₀..v_{n-1}) vₙ` with coefficients in M itself: vanishing of bounded Hochschild cohomology in every degree >= 2 for every von Neumann algebra (Mathlib `WStarAlgebra`), no separability or type assumption. Config note: `KadisonRingrose.json` has no `enable_nanoda` key (index.json shows null).", "definitions_to_check": [], "external_packages": [], "cone_lines_max": 62822, "machine_check": "comparator-sandboxed-pass", "lab_scope_note": "The formalization proves vanishing of bounded Hochschild cohomology in every degree at least two for a complex von Neumann algebra with coefficients in itself. Every bounded multilinear cocycle of such a degree is the Hochschild differential of a bounded multilinear cochain one degree lower. No separability or type restriction is imposed. The degree-one inner-derivation theorem is outside this selected statement.", "lab_docs_url": "https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/docs/295.md", "overview_entry": "family 295 in overview.pdf / CONTENTS.md", "papers": [{"dir": "Vanishing-of-higher-bounded-Hochschild-cohomology-September-23-2026", "title": "Vanishing of higher bounded Hochschild cohomology", "url": "https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Vanishing-of-higher-bounded-Hochschild-cohomology-September-23-2026"}], "audit_part": 2, "second_reader": "", "referee_verdict": ""}, {"family": "296", "title": "The generator problem for finite factors", "subject": "Operator algebras", "headline": "Proves that every type $\\mathrm{II}_1$ 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 $G_\\delta$ subset in the trace $2$-norm topology.", "verdict": "full", "challenges": ["FactorGeneration", "RelativeGeneration"], "review_note": "`single_generation_of_II1_separable_predual (S) (hS : IsII1Factor S) (hsep : HasSeparablePredual S) : SinglyGenerated S` (`∃ x ∈ S, wstar {x} = S`, `wstar` the smallest WOT-closed unital *-subalgebra containing x) and `RelativeGeneration.main_theorem : MainTarget`: for every irreducible inclusion `small ≤ large` of concrete II₁ factors with `HasSeparablePredual large`, there is a faithful normal normalized trace on `large` such that `relativeGeneratorLocus small large = {u | wstar (small ∪ {u}) = large}` is a dense Gδ in `unitary large` for the trace 2-norm topology (`traceTopology`). Both summary claims are stated; 'II₁ factor' is encoded as nonzero + WOT-closed + scalar center + finite (isometries unitary) + diffuse.", "definitions_to_check": ["Hand-written von Neumann-algebra notions: `IsII1Factor` (not Mathlib's `VonNeumannAlgebra` with trace), `HasSeparablePredual := ∃ separable Banach X, Nonempty (StrongDual ℂ X ≃ₗᵢ⋆[ℂ] S)`, `wstar := sInf {S | WOTClosed S ∧ s ⊆ S}`, `ultraweakTopology` via summable vector functionals and `traceTopology` generated by `traceDistance` balls on the unitary group; standard formulations."], "external_packages": [], "cone_lines_max": 36021, "machine_check": "comparator-sandboxed-pass", "lab_scope_note": "The formalization answers the generator problem affirmatively for type $\\mathrm{II}_1$ factors with separable predual: one bounded operator generates the factor under adjoints and weak-operator closure. It imposes no separability assumption on the representing Hilbert space or on the operator-norm topology.\n\nIt also proves the paper's relative-generation result. For an irreducible inclusion of type $\\mathrm{II}_1$ factors with separable predual for the larger factor, the unitaries that generate the larger factor together with the smaller one form a dense $G_\\delta$ set in the normalized trace two-norm topology. The normalized faithful normal trace is included in the assertion.", "lab_docs_url": "https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/docs/296.md", "overview_entry": "family 296 in overview.pdf / CONTENTS.md", "papers": [{"dir": "Relative-generation-and-the-generator-problem-for-finite-factors-September-23-2026", "title": "Relative generation and the generator problem for finite factors", "url": "https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Relative-generation-and-the-generator-problem-for-finite-factors-September-23-2026"}], "audit_part": 2, "second_reader": "", "referee_verdict": ""}, {"family": "297", "title": "A ZFC counterexample to Naimark's problem", "subject": "Operator algebras", "headline": "Gives an alternative to \\href{https://arxiv.org/abs/2609.26930v1}{Tanaka's ZFC construction} 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.", "verdict": "full", "challenges": ["Naimark"], "review_note": "`main : ∃ (A : Type u) (_ : CStarAlgebra A), IsSimpleCStar A ∧ ¬ FiniteDimensional ℂ A ∧ (∃ τ, IsFaithfulTracialState τ) ∧ HasUniqueIrreducibleRepresentation A ∧ ∀ (H : Type v) [Hilbert], ¬ IsIsomorphicToCompacts A H` (universe-polymorphic): a unital infinite-dimensional simple C*-algebra with a faithful tracial state all of whose irreducible representations (nonzero, no nontrivial closed invariant subspace) are unitarily equivalent, yet not isomorphic to the compact operators on any Hilbert space. Matches the summary (nonseparability is a consequence, not stated; the summary's 'alternative to Tanaka' framing is irrelevant to the statement).", "definitions_to_check": ["Hand-written C*-algebra representation theory: `IsSimpleCStar`, `IsFaithfulTracialState`, `IsIrreducible π := π ≠ 0 ∧ no nontrivial closed π-invariant subspace`, `UnitarilyEquivalent`, `IsIsomorphicToCompacts A H := ∃ injective φ : A →⋆ₙₐ[ℂ] (H →L H), range = compact operators`; standard, quantified over Hilbert spaces in explicit universes."], "external_packages": [], "cone_lines_max": 25931, "machine_check": "comparator-sandboxed-pass", "lab_scope_note": "Naimark's problem asks whether a $C^*$-algebra whose nonzero irreducible representations are all unitarily equivalent must be an algebra of compact operators. The formalization gives a counterexample in ZFC: a unital infinite-dimensional simple complex $C^*$-algebra with a faithful tracial state has that uniqueness property for irreducible representations but is not isomorphic to the compact operators on any Hilbert space. No additional set-theoretic assumption is used.", "lab_docs_url": "https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/docs/297.md", "overview_entry": "family 297 in overview.pdf / CONTENTS.md", "papers": [{"dir": "A-counterexample-to-Naimarks-problem-in-ZFC-September-24-2026", "title": "A counterexample to Naimark's problem in ZFC", "url": "https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-counterexample-to-Naimarks-problem-in-ZFC-September-24-2026"}], "audit_part": 2, "second_reader": "", "referee_verdict": ""}, {"family": "298", "title": "A counterexample to Voiculescu’s free-entropy equality conjecture", "subject": "Operator algebras", "headline": "Constructs a bounded self-adjoint tuple in a tracial von Neumann algebra whose microstates and nonmicrostates free entropies satisfy $-\\infty<\\chi<\\chi^*<\\infty$. This answers Voiculescu’s finite-entropy equality question negatively: the matrix-approximation and free-Fisher-information definitions differ even when both are finite.", "verdict": "full", "challenges": ["FiniteEntropySeparation"], "review_note": "`main : ∃ H M τ n X S, 2 ≤ n ∧ X_i, S_i ∈ M self-adjoint ∧ FreeStandardNoise τ X S ∧ ⊥ < chi τ X ∧ chi τ X ≤ chiStar τ X S - 1/2 ∧ chiStar τ X S - 1/2 < ⊤`: a bounded self-adjoint tuple in a von Neumann algebra with faithful normal trace (`TracialVector`), with S a freely independent semicircular family free from X, and `chi` (microstates with operator-norm cutoff R, limsup over matrix sizes, inf over moment orders m and tolerances ε, sup over R) strictly below the nonmicrostates `chiStar` (free Fisher information integral along X + √s S) by at least 1/2, both finite. Matches the summary (n >= 2 variables, finite and unequal).", "definitions_to_check": ["Custom 'free entropy': `chiCutoff τ X R := ⨅ m ε, limsup_d ((d+1)^{-2} (volume (microstates τ X R m (d+1) ε)).log + (n/2) log(d+1))` in EReal, `chi := ⨆ R, chiCutoff`, `freeFisher := ⨅ conjugate systems ξ, ‖ξ‖²` (∞ if none), `chiStar := (n/2) log(2πe) + 1/2 ∫⁻ (n/(1+s) - Φ*(X + √s S))⁺ - 1/2 ∫⁻ (Φ* - n/(1+s))⁺` via `ConjugateSystem` (polynomial moments) and `FreeSubalgebras`; standard Voiculescu definitions written out by hand (EReal ⊤ - ⊤ conventions are not an issue because the claim bounds chiStar - 1/2 below ⊤)."], "external_packages": [], "cone_lines_max": 15557, "machine_check": "comparator-sandboxed-pass", "lab_scope_note": "The formalization disproves equality of microstates and nonmicrostates free entropy even when both quantities are finite. It constructs a bounded self-adjoint tuple $X$ in a von Neumann algebra with a faithful normal tracial state such that $-\\infty<\\chi(X)\\le\\chi^*(X)-1/2<\\infty$. Here $\\chi$ is the microstates entropy with an operator-norm cutoff and a limsup over matrix sizes, and $\\chi^*$ is the nonmicrostates entropy defined using free semicircular noise. The example has a fixed finite number of variables.", "lab_docs_url": "https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/docs/298.md", "overview_entry": "family 298 in overview.pdf / CONTENTS.md", "papers": [{"dir": "A-finite-entropy-separation-of-microstates-and-nonmicrostates-free-entropy-September-25-2026", "title": "A finite-entropy separation of microstates and nonmicrostates free entropy", "url": "https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-finite-entropy-separation-of-microstates-and-nonmicrostates-free-entropy-September-25-2026"}], "audit_part": 2, "second_reader": "", "referee_verdict": ""}, {"family": "299", "title": "The Kirchberg--R\\o rdam character criterion", "subject": "Operator algebras", "headline": "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--R\\o rdam character question. Also, the infinite minimal tensor power of every such algebra without characters is Jiang--Su stable, answering the Dadarlat--Toms question.", "verdict": "partial", "challenges": ["CharacterCriterion"], "review_note": "Stated: `character_criterion (A) [CStarAlgebra A] [Nontrivial A] [SeparableSpace A] (ω) (hω : ω ≤ cofinite) : HasNoCharacters (NormUltrapower.CentralAlgebra A ω) ↔ Nonempty (A ≃⋆ₐ[ℂ] MinTensor.Algebra A JiangSu.Algebra)` (for unital separable A: the relative commutant A_ω ∩ A' has no nonzero *-character iff A ≅ A ⊗_min 𝒵; for unital A the annihilator quotient is trivial). Not stated: the summary's second claim, that the infinite minimal tensor power of every such algebra without characters is Jiang-Su stable (the Dadarlat-Toms question; docs only mention the criterion).", "definitions_to_check": ["The Jiang-Su algebra is not Mathlib's: `JiangSu.Algebra := CStarInductiveLimit.Algebra StandardPrimeModel.presentation.system`, a hand-built inductive limit of prime dimension-drop algebras with explicit multiplicities (`copies p := (p-1)(p^4+p^2+1)`, `size n = 2^(7^n)`), so 'A ≅ A ⊗_min 𝒵' is only as faithful as this model of 𝒵; likewise `NormUltrapower.Algebra`, `MinTensor.Algebra` (spatial/min tensor norm via representation pairs) and the central subalgebra are hand-built (2.1k-line challenge file)."], "external_packages": [], "cone_lines_max": 25294, "machine_check": "comparator-sandboxed-pass", "lab_scope_note": "The Kirchberg–Rørdam criterion relates Jiang–Su absorption to characters of the central-sequence algebra. The formalized result proves that, for every nonzero unital separable complex $C^*$-algebra $A$ and every free ultrafilter on $\\mathbb N$, the norm central-sequence algebra has no nonzero character exactly when $A\\cong A\\otimes_{\\min}\\mathcal Z$. No nuclearity, simplicity, trace, or comparison hypothesis is imposed.", "lab_docs_url": "https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/docs/299.md", "overview_entry": "family 299 in overview.pdf / CONTENTS.md", "papers": [{"dir": "The-Kirchberg-Rordam-character-criterion-September-25-2026", "title": "The Kirchberg–Rørdam character criterion", "url": "https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-Kirchberg-Rordam-character-criterion-September-25-2026"}], "audit_part": 2, "second_reader": "", "referee_verdict": ""}, {"family": "303", "title": "From ordinary to strong pure infiniteness", "subject": "Operator algebras", "headline": "Resolves the ordinary-to-strong pure-infiniteness question of Kirchberg and R{\\o}rdam for complex $C^*$-algebras. For exact algebras, proper infiniteness of one fixed finite amplification of every positive element also suffices. Consequently, every separable nuclear algebra with this property absorbs $\\mathcal O_\\infty$, without unitality or simplicity assumptions.", "verdict": "full", "challenges": ["ExactInfiniteness", "IndividualStrongInfiniteness"], "review_note": "`individual_to_strong_main (hP : PropertyP A) : unrestricted diagonalization ∧ StronglyPurelyInfinite A` holds for every non-unital C*-algebra A, where `PropertyP` = every positive element properly infinite (Kirchberg-Rørdam's characterization of ordinary pure infiniteness is the one routine identification step), and `ExactInfiniteness` gives the fixed-amplification version for exact A (WeaklyPurelyInfinite with some n ≥ 1 ⇒ `ProperlyInfinite (diagonalCopies 1 a)` and strongly purely infinite). The summary's final consequence (separable nuclear algebras with this property absorb O_∞) is not stated.", "definitions_to_check": ["`def Exact (A : Type u) ... := ∀ (C D : Type u) (q : C →⋆ₙₐ[ℂ] D), Surjective q → ∀ z : MinimalTensor A C, (∀ π : SpatialRepresentation A D, z (π.precompRight q) = 0) → z.val ∈ tensorKernelIdeal q`: hand-built exactness via the spatial (min) tensor product as an ℓ^∞ sum over all pairs of Hilbert-space representations; matches the standard exactness of A ⊗_min - but tests only C, D in the universe of A", "Two different hand-built notions of proper infiniteness: matrix-level `CuntzSubequiv (blockSum a a) a` (ε-approximation by v : CStarMatrix k m A) in ExactInfiniteness, and `CuntzLE` in the completed stabilization (`MainGap.Stable.Stabilization`, direct limit of CStarMatrix algebras) in IndividualStrongInfiniteness; both read as standard, the Lean theorem uses the weaker (stable) form as hypothesis in the ordinary-to-strong direction"], "external_packages": [], "cone_lines_max": 150750, "machine_check": "comparator-sandboxed-pass", "lab_scope_note": "The formalization proves that a complex $C^*$-algebra in which every positive element is properly infinite is strongly purely infinite, with the full positive-element diagonalization property. No exactness, nuclearity, unitality, or simplicity assumption is needed for this implication.\n\nFor exact algebras, proper infiniteness of one fixed finite amplification of every positive element already implies individual proper infiniteness and strong pure infiniteness.", "lab_docs_url": "https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/docs/303.md", "overview_entry": "family 303 in overview.pdf / CONTENTS.md", "papers": [{"dir": "Weak-pure-infiniteness-and-O-infinity-absorption-September-25-2026", "title": "Weak pure infiniteness and O-infinity absorption", "url": "https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Weak-pure-infiniteness-and-O-infinity-absorption-September-25-2026"}], "audit_part": 3, "second_reader": "", "referee_verdict": ""}, {"family": "307", "title": "Failure of rational injectivity for maximal coarse assembly", "subject": "Topology", "headline": "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.", "verdict": "partial", "challenges": ["CoarseAssembly"], "review_note": "`bounded_geometry_graph_counterexample_unconditional` gives finite connected bounded-degree graphs `fiberedGraph ... j` whose coarse union `VertexUnion` is proper, has uniformly finite balls and separation ≥ 1, and α ∈ `KX1 VertexUnion` of infinite order with `constructedCoarseAssembly α = 0`, `1 ⊗ α ≠ 0` and non-injectivity also after `⊗ ℚ`, but the target is K₁ of the *reduced* Roe algebra (`DiscreteRoe.algebra`; the file header says 'ordinary reduced coarse assembly map'). That is the summary's companion claim (the rational coarse Novikov counterexample); the headline failure for the *maximal* assembly map is strictly stronger (reduced = quotient ∘ maximal, so ker(max) ⊆ ker(reduced)) and has no statement. Title and summary say 'maximal' while docs, lean_scope and the Lean say 'reduced', and Lean wins.", "definitions_to_check": ["The assembly map is built with totalizing fallbacks: `separatedIndexFamily` is `if h : (∀ r s h a, f s (map r s h a) = f r a) then {atScale := f, compatible := h} else {atScale := fun _ => 0, ..}` and `analyticFamilyIndex` is `if h : relations A ≤ (analyticFamilyIndexFree ..).ker then QuotientAddGroup.lift .. else 0`, so from the statement alone the map could be the zero map; the solution proves the genuine branches are taken (`separatedIndexFamily_atScale`, `analyticFamilyIndex_eq_lift` in OAI/Topology/CoarseAssembly)", "6,648-line hand-built stack with no Mathlib counterpart: coarse K-homology `KX1 X := (coarsening X).Homology` (direct limit over Rips complexes of `OddK C₀(P_r X, ℂ)`, odd Fredholm modules mod `CycleRelation` degenerate/unitary/directSum/homotopy), `K1 A := ker (K₁(A⁺) → K₁ ℂ)` via `TopologicalK1.Relation` (mul/stabilize/homotopy), and `vertexUnionMetric` = graph metrics on the fibres glued by `separatedDist` (d(x,o_i) + i + j + 2 + d(o_j,y)); read as standard, not independently re-derived. The statement does not itself say that the fibre metrics are the graph metrics."], "external_packages": [], "cone_lines_max": 44622, "machine_check": "none", "lab_scope_note": "The coarse Novikov conjecture predicts rational injectivity of the ordinary coarse assembly map for uniformly discrete spaces of bounded geometry. The formalization constructs a counterexample from a coarse disjoint union of finite connected graphs with uniformly bounded degree. Its degree-one coarse $K$-homology contains an infinite-order class whose image under ordinary coarse assembly into the reduced locally compact Roe algebra vanishes. The class remains nonzero after tensoring with $\\mathbb Q$, so rational injectivity fails.", "lab_docs_url": "https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/docs/307.md", "overview_entry": "family 307 in overview.pdf / CONTENTS.md", "papers": [{"dir": "A-counterexample-to-the-coarse-Novikov-conjecture-September-23-2026", "title": "A counterexample to the coarse Novikov conjecture", "url": "https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-counterexample-to-the-coarse-Novikov-conjecture-September-23-2026"}], "audit_part": 3, "second_reader": "", "referee_verdict": ""}, {"family": "312", "title": "The Grothendieck homotopy hypothesis", "subject": "Topology", "headline": "Proves the Grothendieck homotopy hypothesis for $\\infty$-groupoids associated with every Grothendieck coherator in the Ara--Henry convention: these algebraic objects recover the homotopy theory of spaces.", "verdict": "supporting-only", "challenges": ["GrothendieckElementaryExpansion"], "review_note": "Only `elementary_expansion`: for a coherator C (Ara-Henry convention, encoded by hand), a cellular model X and a pushout of `J n : disk n → disk (n+1)` along any `disk n → X`, the inclusion X → Y is a `WeakEquivalence` (bijection on π₀ and on every π_n at every basepoint). The homotopy hypothesis itself (comparison of these ∞-groupoid models with spaces) is not stated; docs: 'the later semi-model structure and full comparison with the homotopy theory of spaces are not included'.", "definitions_to_check": ["`GlobularTheory`, `CellularPresentation`, `IsCoherator C := Nonempty C.CellularPresentation ∧ ∀ p : AdmissiblePair, ∃ z, Fills (Morphism.id C) p z` and `Model` are hand-built encodings of Grothendieck coherators and their models", "`noncomputable def unit C hC n := Classical.choose (hC.2 (C.unitPair n))`: the degenerate-cell operation used to define `baseCell`, `Loops`, `HomotopyGroup` and hence `WeakEquivalence` is chosen by `Classical.choose` (any choice is valid, but the notion is not canonical)"], "external_packages": [], "cone_lines_max": 12753, "machine_check": "comparator-sandboxed-pass", "lab_scope_note": "The Grothendieck homotopy hypothesis asks whether algebraic $\\infty$-groupoids recover the homotopy theory of spaces. The formalized result is the elementary-expansion theorem used in this approach: for every Grothendieck coherator in the Ara–Henry convention and every boundary-cellular model, attaching an $(n+1)$-disk along its source $n$-face induces a weak equivalence.\n\nThe later semi-model structure and full comparison with the homotopy theory of spaces are not included.", "lab_docs_url": "https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/docs/312.md", "overview_entry": "family 312 in overview.pdf / CONTENTS.md", "papers": [{"dir": "The-Grothendieck-homotopy-hypothesis-via-elementary-expansions-September-24-2026", "title": "The Grothendieck homotopy hypothesis via elementary expansions", "url": "https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-Grothendieck-homotopy-hypothesis-via-elementary-expansions-September-24-2026"}], "audit_part": 3, "second_reader": "", "referee_verdict": ""}, {"family": "317", "title": "Thomason model structures in every strict higher dimension", "subject": "Topology", "headline": "Resolves the Ara--Maltsiniotis conjecture: for every $n\\ge1$ and $n=\\omega$, 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.", "verdict": "full", "challenges": ["ThomasonModelStructures"], "review_note": "`Thomason.main : FullMain` asserts, for strict ω-categories and for every n ≥ 1 for n-categories (`NCategory n`), an `ExactModelEndpoint` for the adjunction (Sd² ⋙ categorification) ⊣ (Street nerve ⋙ Ex ⋙ Ex): a model structure with weak equivalences and fibrations created by the right adjoint, cofibrantly generated by boundary/horn images, complete, cocomplete, locally presentable, left and right proper, with the Quillen-equivalence criterion against simplicial sets (plus the Kan-Quillen model on SSet). `nerveDetection_allDimensions` identifies the weak equivalences with Street-nerve detection for ω and every n, matching the summary and docs.", "definitions_to_check": ["`structure OmegaCategory` (cells with `source/target : ℕ → Cell → Cell`, partial `comp k`, `interchange`, `finite_dimension : ∃ k, source k x = x`) and `NCategory n := {C // ∀ x, C.source n x = x}` are hand-built strict globular ω-/n-categories (no Mathlib counterpart); `Classical.choose` only picks cell dimensions in the presentation colimit", "`ExactModelEndpoint` bundles the model structure, its classes and the Quillen-equivalence criterion by hand (`quillenEquivalence : ∀ K Y fibrant, ∀ f : L K ⟶ Y, weq f ↔ sweq (adj.homEquiv f)`); `simplicialWeakEquivalences` = weak homotopy equivalences of geometric realizations (π₀ bijection and π_n for all n ≥ 1 at all basepoints)"], "external_packages": [], "cone_lines_max": 65284, "machine_check": "comparator-sandboxed-pass", "lab_scope_note": "The formalization proves the higher-dimensional Thomason model-structure theorem for small strict globular $n$-categories in every positive finite dimension and in dimension $\\omega$. Each category has the stated proper combinatorial model structure, Quillen equivalent to simplicial sets through the twice-subdivided categorification and twice-extended Street nerve adjunction.\n\nThe selected statements also identify weak equivalences by the Street nerve in all these dimensions. Thus the constructions model the homotopy theory of spaces throughout the stated range.", "lab_docs_url": "https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/docs/317.md", "overview_entry": "family 317 in overview.pdf / CONTENTS.md", "papers": [{"dir": "Thomason-Model-Structures-in-Every-Strict-Higher-Dimension-September-25-2026", "title": "Thomason Model Structures in Every Strict Higher Dimension", "url": "https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Thomason-Model-Structures-in-Every-Strict-Higher-Dimension-September-25-2026"}], "audit_part": 3, "second_reader": "", "referee_verdict": ""}, {"family": "322", "title": "Tingley's sphere-isometry problem", "subject": "Functional analysis", "headline": "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.", "verdict": "full", "challenges": ["TingleySphereIsometry"], "review_note": "`tingley_sphere_isometry_full`: for nontrivial complete real normed spaces X, Y and any surjective `Isometry f : {x // ‖x‖ = 1} → {y // ‖y‖ = 1}` there is `T : X ≃ₗᵢ[ℝ] Y` (bijective real-linear isometry) extending f (radial formula, T 0 = 0) and any ℝ-linear L (not assumed continuous) extending f equals T, so existence, surjectivity and uniqueness are all in the conclusion with no dimension, separability or reflexivity hypothesis.", "definitions_to_check": [], "external_packages": [], "cone_lines_max": 9273, "machine_check": "comparator-sandboxed-pass", "lab_scope_note": "Tingley's problem asks whether a surjective isometry between the unit spheres of Banach spaces extends to a linear isometry of the spaces. The formalized result gives a unique surjective real-linear isometric extension for arbitrary nonzero real Banach spaces, with no finite-dimensionality, separability, reflexivity, or convexity assumptions. For complex spaces viewed as real spaces, the conclusion is real linearity rather than complex linearity.", "lab_docs_url": "https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/docs/322.md", "overview_entry": "family 322 in overview.pdf / CONTENTS.md", "papers": [{"dir": "A-positive-solution-to-Tingleys-problem-September-23-2026", "title": "A positive solution to Tingley’s problem", "url": "https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-positive-solution-to-Tingleys-problem-September-23-2026"}], "audit_part": 3, "second_reader": "", "referee_verdict": ""}, {"family": "323", "title": "Relative independence of the separable quotient problem", "subject": "Functional analysis", "headline": "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.", "verdict": "partial", "challenges": ["SeparableQuotientNegative"], "review_note": "`negative_main (hCH : CH) : ¬ SQ.{u} ℝ ∧ ¬ SQ.{u} ℂ` (CH := `Cardinal.mk ℝ = aleph 1`; `SQ` = every complete infinite-dimensional normed space has a bounded linear surjection onto an infinite-dimensional separable Banach space) is the continuum-hypothesis counterexample half over both fields. The other half of the independence claim, consistency of the positive assertion relative to a measurable cardinal, has no statement (docs: 'the positive consistency direction ... outside the selected statement').", "definitions_to_check": [], "external_packages": [], "cone_lines_max": 14178, "machine_check": "comparator-sandboxed-pass", "lab_scope_note": "The separable quotient problem asks whether every infinite-dimensional Banach space has an infinite-dimensional separable quotient. The linked formalization proves the negative direction under the continuum hypothesis: over both the real and complex fields, there is an infinite-dimensional Banach space admitting no bounded linear surjection onto an infinite-dimensional separable Banach space.\n\nThis is the conditional counterexample direction. The positive consistency direction and the paper's full relative-independence conclusions are outside the selected statement.", "lab_docs_url": "https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/docs/323.md", "overview_entry": "family 323 in overview.pdf / CONTENTS.md", "papers": [{"dir": "Relative-independence-of-the-separable-quotient-problem-September-23-2026", "title": "Relative independence of the separable quotient problem", "url": "https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Relative-independence-of-the-separable-quotient-problem-September-23-2026"}], "audit_part": 3, "second_reader": "", "referee_verdict": ""}, {"family": "324", "title": "Lipschitz equivalence without linear isomorphism", "subject": "Functional analysis", "headline": "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.", "verdict": "full", "challenges": ["C0Absorption", "LipschitzEquivalence"], "review_note": "`LipschitzEquivalence.main`: ∃ separable real Banach spaces X, Y and a bijection Ψ with (4/21)‖s-t‖ ≤ ‖Ψ s - Ψ t‖ ≤ (76/25)‖s-t‖, `IsEmpty (X ≃L[ℝ] Y)`, X ⊇ isometric c₀(ℓ₂), Y has no bounded-below linear image of c₀(ℓ₂): exactly the summary's bi-Lipschitz-equivalent but not linearly isomorphic pair. C0Absorption adds the second paper's single space Z (no linear c₀, `Z × c₀` bi-Lipschitz onto Z, metrically universal, `¬ Nonempty ((Z × c₀) ≃L Z)`).", "definitions_to_check": [], "external_packages": [], "cone_lines_max": 10807, "machine_check": "comparator-sandboxed-pass", "lab_scope_note": "The formalized counterexample gives separable real Banach spaces $X,Y$ that are bi-Lipschitz equivalent but not linearly isomorphic. The bijection has lower Lipschitz bound $4/21$ and upper bound $76/25$. The linear obstruction is explicit: $X$ contains a linear isometric copy of $c_0(\\ell_2)$, whereas $Y$ contains no bounded linear copy of that space.\n\nThe formalized result constructs one separable real Banach space $X$ that is bi-Lipschitz equivalent to $X\\times c_0$ but contains no closed linear subspace isomorphic to $c_0$. The same space contains a bi-Lipschitz copy of $c_0$, is bi-Lipschitz universal for separable metric spaces, and is not linearly isomorphic to $X\\times c_0$. Thus nonlinear absorption of $c_0$ does not force a linear copy of it.", "lab_docs_url": "https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/docs/324.md", "overview_entry": "family 324 in overview.pdf / CONTENTS.md", "papers": [{"dir": "Lipschitz-Equivalent-Separable-Banach-Spaces-Need-Not-Be-Linearly-Isomorphic-September-24-2026", "title": "Lipschitz Equivalent Separable Banach Spaces Need Not Be Linearly Isomorphic", "url": "https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Lipschitz-Equivalent-Separable-Banach-Spaces-Need-Not-Be-Linearly-Isomorphic-September-24-2026"}, {"dir": "Bi-Lipschitz-Absorption-of-c0-Without-a-Linear-Copy-of-c0-September-26-2026", "title": "Bi-Lipschitz Absorption of c0 Without a Linear Copy of c0", "url": "https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Bi-Lipschitz-Absorption-of-c0-Without-a-Linear-Copy-of-c0-September-26-2026"}], "audit_part": 3, "second_reader": "", "referee_verdict": ""}, {"family": "325", "title": "The complete Crouzeix conjecture", "subject": "Functional analysis", "headline": "Resolves the complete Crouzeix conjecture: for every bounded operator $A$ on a complex Hilbert space and every finite matrix-valued polynomial $P$, one has $\\lVert P[A]\\rVert\\le2\\sup_{z\\in W(A)}\\lVert P(z)\\rVert$, where $W(A)$ is the numerical range. The constant $2$ is sharp, independent of the matrix size, and valid in infinite dimensions.", "verdict": "full", "challenges": ["CompleteCrouzeix", "CrouzeixHilbert", "DirectCrouzeix", "HilbertCrouzeix", "StructuralCrouzeix"], "review_note": "`CrouzeixHilbert.hilbert` states, for every complete complex inner-product space H (any dimension, no separability) and bounded A, ‖Σ_k A^k ⊗ B_k‖ ≤ 2·sup_{W(A)} ‖Σ z^k B_k‖ for all m > 0, degrees d and complex m×m coefficients (numerical range {⟨x,Ax⟩ : ‖x‖ = 1}, amplification = completed H ⊗ ℂ^m), plus an explicit `SharpConstant` witness (A on ℂ², scalar degree-1 polynomial, sup = 1 and ‖P[A]‖ = 2); CompleteCrouzeix/DirectCrouzeix give the matrix case with `UniversalBound 2 ∧ ∀ C, UniversalBound C → 2 ≤ C`. HilbertCrouzeix (compactness/convexity of the closed numerical range, finite compressions, trivial space) is supporting-only and StructuralCrouzeix (optimal similarity and boundary density for regular analytic convex Jordan domains) is an extra finite-dimensional result.", "definitions_to_check": ["HilbertCrouzeix: `def chosenContour (K U : Set ℂ) : ℝ → ℂ := Classical.epsilon (IsCauchyContour K U)` feeds `holomorphicCalculus`, but the only theorem using it (`zeroConclusion`) is for a subsingleton H, where every operator is 0, so the choice cannot matter", "CrouzeixHilbert: `holomorphicEval A U f := if h : Nonempty (CalculusContour (numericalClosure A) U) then contourEval A (Classical.choice h).toSmoothContour f else 0`; harmless because `NonzeroConclusion` itself asserts the contour exists and that the value equals `contourEval A Γ` for every contour Γ", "`supNorm S F := sSup (insert 0 ((fun z => ‖F z‖) '' S))` (the `insert 0` only handles the empty numerical range of the zero space)"], "external_packages": [], "cone_lines_max": 15655, "machine_check": "comparator-sandboxed-pass", "lab_scope_note": "The complete Crouzeix inequality bounds a matrix-valued polynomial evaluated at a matrix by its maximum norm on the numerical range. The formalized result proves the bound with constant $2$ for every positive matrix and coefficient size, every polynomial degree, and arbitrary complex coefficients. It also proves that no smaller universal constant works. Normality of the matrix and nonempty interior of its numerical range are not assumed.\n\nThe formalization proves the complete Crouzeix inequality with sharp constant two for every bounded operator on an arbitrary complex Hilbert space. For every matrix-valued polynomial, its operator evaluation has norm at most twice its supremum norm on the numerical range. The same bound holds for finite matrix-valued functions holomorphic near the closure of the numerical range and for rational functions with poles outside that closure. No separability assumption is required.\n\nFor finite matrices in a bounded convex domain with regular real-analytic Jordan boundary, the structural result gives an attained optimal similarity with condition number at most two and one continuous positive boundary density of mass the identity representing every matrix-valued analytic test. The linked supporting statements include numerical-range geometry and finite-compression identities.", "lab_docs_url": "https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/docs/325.md", "overview_entry": "family 325 in overview.pdf / CONTENTS.md", "papers": [{"dir": "A-direct-proof-of-the-complete-Crouzeix-inequality-September-26-2026", "title": "A direct proof of the complete Crouzeix inequality", "url": "https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-direct-proof-of-the-complete-Crouzeix-inequality-September-26-2026"}, {"dir": "The-complete-Crouzeix-theorem-September-23-2026", "title": "The complete Crouzeix theorem: optimal similarity and a common positive boundary representation", "url": "https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-complete-Crouzeix-theorem-September-23-2026"}], "audit_part": 3, "second_reader": "", "referee_verdict": ""}, {"family": "326", "title": "The cotype--cotype conjecture under the approximation property", "subject": "Functional analysis", "headline": "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.", "verdict": "full", "challenges": ["Cotype"], "review_note": "`mainTarget_proved : [CompleteSpace X] [Nontrivial X], ApproximationProperty X → (KConvex X ↔ FiniteCotype X ∧ FiniteCotype (X →L[ℝ] ℝ))` is the headline for every nonzero real Banach space with AP; the summary's 'equivalently, these cotype assumptions force nontrivial type' is the classical K-convex ⇔ nontrivial-type reformulation (Pisier), which is not stated in Lean.", "definitions_to_check": ["`KConvex E := ∃ K ≥ 0, ∀ n (f : Cube n → E), cubeL2 (radProjection f) ≤ K * cubeL2 f` (uniform L²-boundedness of the Rademacher projection, the standard characterization) and `HasCotype q := 2 ≤ q ∧ ∃ C ≥ 0, ∀ n (x : Fin n → E), (∑ ‖x i‖^q)^(1/q) ≤ C * cubeL2 (rademacherSum x)`: hand-built, standard (L² average in place of the L^q average, equivalent by Kahane)"], "external_packages": [], "cone_lines_max": 11759, "machine_check": "comparator-sandboxed-pass", "lab_scope_note": "The cotype–cotype problem asks whether finite cotype of a Banach space and its dual characterizes $K$-convexity. The formalized result establishes this equivalence for every nonzero real Banach space with the approximation property: $X$ is $K$-convex exactly when both $X$ and $X^*$ have finite cotype.", "lab_docs_url": "https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/docs/326.md", "overview_entry": "family 326 in overview.pdf / CONTENTS.md", "papers": [{"dir": "The-cotype-cotype-conjecture-under-the-approximation-property-September-23-2026", "title": "The cotype–cotype conjecture under the approximation property", "url": "https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-cotype-cotype-conjecture-under-the-approximation-property-September-23-2026"}], "audit_part": 3, "second_reader": "", "referee_verdict": ""}, {"family": "327", "title": "Markov type characterizes superreflexivity", "subject": "Functional analysis", "headline": "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.", "verdict": "full", "challenges": ["MarkovType"], "review_note": "`hasNontrivialMarkovType_iff_hasEquivalentUCNorm [CompleteSpace X] : HasNontrivialMarkovType X ↔ HasEquivalentUCNorm X` states both the new direction (Markov type p > 1 ⇒ equivalent uniformly convex norm) and the known converse. The final identification 'admits an equivalent UC norm' = superreflexive (Enflo/Pisier) is not stated in Lean.", "definitions_to_check": ["`MarkovTypeWith p K X := 0 < p ∧ 0 < K ∧ ∀ m (C : FiniteReversibleChain (Fin m)) (f : Fin m → X) (n ≥ 1), E_π‖f(Z_n) - f(Z_0)‖^p ≤ K^p · n · E_π‖f(Z_1) - f(Z_0)‖^p` (stateCost via explicit path weights π(ω₀)∏P): hand-built but matches Ball's stationary reversible Markov type; `UniformConvex` and `EquivalentNorm` (seminorm with two-sided norm bounds) are the usual modulus-of-convexity definitions"], "external_packages": [], "cone_lines_max": 27054, "machine_check": "comparator-sandboxed-pass", "lab_scope_note": "The formalized result proves that a real Banach space has nontrivial Markov type exactly when it admits an equivalent uniformly convex norm, hence exactly when it is superreflexive. Nontrivial Markov type means a uniform Markov-type bound for some exponent $p>1$ over all finite stationary reversible chains and all positive times. The exponent and equivalent norm may depend on the space, and the zero space is included. Additional formalized consequences concern uniformly smooth renormings and reflexivity of finitely representable spaces.", "lab_docs_url": "https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/docs/327.md", "overview_entry": "family 327 in overview.pdf / CONTENTS.md", "papers": [{"dir": "Nontrivial-Markov-Type-Forces-Superreflexivity-September-23-2026", "title": "Nontrivial Markov Type Forces Superreflexivity", "url": "https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Nontrivial-Markov-Type-Forces-Superreflexivity-September-23-2026"}], "audit_part": 3, "second_reader": "", "referee_verdict": ""}, {"family": "328", "title": "Fixed points of nonexpansive maps in reflexive Banach spaces", "subject": "Functional analysis", "headline": "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.", "verdict": "full", "challenges": ["ReflexiveFixedPoints"], "review_note": "`exists_fixedPoint_of_canonicallyReflexive`: for a complete real normed space X whose canonical map into the continuous bidual is surjective (`CanonicallyReflexive`), every self-map F of a nonempty closed bounded convex C ⊆ X with ‖F a - F b‖ ≤ ‖a - b‖ in the original norm has a fixed point, with no uniform convexity or normal-structure hypothesis, which is exactly the headline.", "definitions_to_check": [], "external_packages": [], "cone_lines_max": 11540, "machine_check": "comparator-sandboxed-pass", "lab_scope_note": "The fixed-point question asks whether reflexivity suffices for nonexpansive maps on bounded convex sets. The formalized result gives an affirmative answer: every nonexpansive self-map of a nonempty closed bounded convex subset of a real reflexive Banach space has a fixed point in the original norm. Zero and nonseparable spaces are included, with no uniform convexity, normal structure, or weak-continuity assumption.", "lab_docs_url": "https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/docs/328.md", "overview_entry": "family 328 in overview.pdf / CONTENTS.md", "papers": [{"dir": "Fixed-Points-of-Nonexpansive-Maps-in-Reflexive-Banach-Spaces-September-24-2026", "title": "Fixed Points of Nonexpansive Maps in Reflexive Banach Spaces", "url": "https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Fixed-Points-of-Nonexpansive-Maps-in-Reflexive-Banach-Spaces-September-24-2026"}], "audit_part": 3, "second_reader": "", "referee_verdict": ""}, {"family": "329", "title": "A counterexample to Pietsch's metric-entropy duality conjecture", "subject": "Functional analysis", "headline": "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.", "verdict": "full", "challenges": ["MetricEntropyDuality"], "review_note": "For all a, b ≥ 1 there are n > 0 and an origin-symmetric convex body K ⊆ ℝⁿ (compact, convex, symmetric, nonempty interior) with b·log N(B_∞°, a⁻¹K°) < log N(K, B_∞), where N is the least number of translates covering (`coveringNumber`, ℕ-valued sInf), and `Coverable` hypotheses make both numbers finite and positive; this is the violation of every universal-constant comparison with the cube as covering body.", "definitions_to_check": ["`coveringNumber A B := sInf {M | ∃ centers : Fin M → ℝⁿ, Covers A B centers}` and `polar K := {y | ∀ x ∈ K, ⟨x,y⟩ ≤ 1}` are hand-built (ℝⁿ as `ι → ℝ` with the sum pairing), standard; sInf ∅ = 0 is excluded by the stated `Coverable` clauses"], "external_packages": [], "cone_lines_max": 6520, "machine_check": "comparator-sandboxed-pass", "lab_scope_note": "Metric-entropy duality predicts a universal comparison between covering numbers of convex bodies and their polars. The formalized result disproves such a comparison: for every $a,b\\ge1$, there is an origin-symmetric convex body $K$ in a positive finite dimension with $\\log N(K,B_\\infty)>b\\log N(B_\\infty^\\circ,a^{-1}K^\\circ)$, where $N$ is the least number of translates in a finite cover. A further construction makes the dual-to-primal logarithmic entropy ratio tend to zero as the dimension grows.", "lab_docs_url": "https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/docs/329.md", "overview_entry": "family 329 in overview.pdf / CONTENTS.md", "papers": [{"dir": "Counterexamples-to-the-duality-conjecture-for-metric-entropy-September-24-2026", "title": "Counterexamples to the duality conjecture for metric entropy", "url": "https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Counterexamples-to-the-duality-conjecture-for-metric-entropy-September-24-2026"}], "audit_part": 3, "second_reader": "", "referee_verdict": ""}, {"family": "330", "title": "A negative answer to Kalton's Lipschitz-free approximation question", "subject": "Functional analysis", "headline": "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.", "verdict": "full", "challenges": ["DiscreteLipschitzFree", "RealL1Renorming"], "review_note": "`MainStatement` (inside `source_endpoints`) gives a countable metric space M with all distinct distances ≥ 1, unbounded and not proper, such that the real Lipschitz-free space `FreeSpace o` has `HasAP` but `¬ HasBAP` for every Λ ≥ 1, exactly the summary; `BlockStatement` (finite quantitative blocks) and the ℓ¹ renorming with 2B_p-BAP but not p-BAP are the retained earlier results.", "definitions_to_check": ["`FreeSpace o := closure of span {evaluation o x} ⊆ (Lip0 o →L[ℝ] ℝ)` with `Lip0 o` normed by the slope map into ℓ^∞(M × M): hand-built Lipschitz-free space, standard; `HasAP`/`HasBAP` state approximation of the identity on every compact set by finite-rank operators (with ‖T‖ ≤ Λ for BAP)"], "external_packages": [], "cone_lines_max": 4516, "machine_check": "comparator-sandboxed-pass", "lab_scope_note": "Kalton's question asks whether uniform discreteness of a metric space forces its Lipschitz-free space to have the bounded approximation property. The formalization constructs a countable metric space with all distinct points at distance at least one whose real Lipschitz-free space has the approximation property but fails every bounded approximation bound. The metric space is unbounded and not proper.\n\nThe earlier quantitative renorming result for real $\\ell_1$ is retained: for each integer $p\\ge1$, an equivalent complete norm has the $2B_p$-bounded approximation property but fails the $p$-bounded approximation property, where $B_p=300p^2+150p+5$.", "lab_docs_url": "https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/docs/330.md", "overview_entry": "family 330 in overview.pdf / CONTENTS.md", "papers": [{"dir": "Failure-of-Bounded-Approximation-in-a-Lipschitz-Free-Space-over-a-Uniformly-Discrete-Metric-Space-September-26-2026", "title": "A uniformly discrete counterexample to bounded approximation in Lipschitz-free spaces", "url": "https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Failure-of-Bounded-Approximation-in-a-Lipschitz-Free-Space-over-a-Uniformly-Discrete-Metric-Space-September-26-2026"}], "audit_part": 3, "second_reader": "", "referee_verdict": ""}, {"family": "331", "title": "Asymptotic midpoint uniform convexity without asymptotically uniformly convex renorming", "subject": "Functional analysis", "headline": "Constructs a real reflexive Banach space with an asymptotically midpoint uniformly convex norm but no equivalent asymptotically uniformly convex norm, extending Baudier's separation to reflexive spaces. In the same space, depth-$k$ countably branching diamonds require distortion at least $\\sqrt{1+k/12}$, so midpoint uniform convexity does not force uniformly bounded diamond distortion even under reflexivity.", "verdict": "full", "challenges": ["BoundedTreePotentials", "DaugavetModuli", "DiamondDistortion", "ForestSpace", "IndependentProducts", "MidpointLenses", "RecursivePotentials"], "review_note": "`ForestSpace.main_theorem` (unconditional) states for `FullDual Vertex` (continuous dual of the completed disjoint-segment-norm space over a forest): infinite-dimensional, separable, canonically reflexive, no equivalent norm is AUC, averaged midpoint modulus ≥ √(1+t²/12) - 1 for all t > 0 (this implies positivity of the usual max-version midpoint modulus, since max ≥ average: the one routine step), and every embedding of the depth-k countably branching diamond has distortion ≥ √(1+k/12). The other six challenges are companion results of the seven papers (lens tail bound, all-pairs diamond bound 1+k/4 ≤ D², exact Daugavet-subspace moduli, tree-potential, independent-product and recursive-potential spaces), not needed for the headline.", "definitions_to_check": ["`aucModulus`, `averageModulus` and `Cofinite` are hand-built (`inf_{N x=1} sup_{F cofinite closed} inf_{y∈F, N y=1} (N(x+ty)-1)`, resp. with the average of N(x±ty) minus 1), matching the standard asymptotic moduli; `IsAUC N := ∀ t > 0, 0 < aucModulus N t` and `EquivalentNorm` as a seminorm with α‖x‖ ≤ N x ≤ β‖x‖", "`Diamond.diamond k := base.iterate k` (each edge replaced by ℕ-many 2-edge paths, graph distance via `SimpleGraph.dist`) and `EmbeddingBound` (∃ scale s > 0, s·d ≤ ‖f u - f v‖ ≤ C·s·d): standard distortion; docs note 22 companion inputs are assumed in some graph deductions, not in the headline theorem"], "external_packages": [], "cone_lines_max": 13085, "machine_check": "comparator-sandboxed-pass", "lab_scope_note": "The paper constructs a Banach space whose midpoint geometry is asymptotically uniformly convex even though no equivalent norm is asymptotically uniformly convex in the usual one-sided sense. The formalization proves that the full dual of the specified segment-norm completion is infinite-dimensional, separable, and reflexive; its averaged midpoint modulus is at least $\\sqrt{1+t^2/12}-1$ for every $t>0$, while every equivalent norm fails asymptotic uniform convexity. It also proves that an embedding of a depth-$k$ countably branching diamond has distortion at least $\\sqrt{1+k/12}$.\n\nThe general-forest and word-forest results use the closed span of the coordinate functionals. Their conclusions retain the finite-ancestor and equivalent-norm hypotheses, including the scale $\\alpha/(2\\beta)$ when $0<\\alpha\\le\\beta$ and the new norm lies between $\\alpha$ and $\\beta$ times the original norm; they do not identify that span with the full dual or assert reflexivity for every forest.\n\nThe paper controls midpoint lenses in Banach spaces defined by tree-segment norms. The formalization proves the following estimate in both the full dual of the finite-height forest space and the coordinate predual of the infinite-tree space. If $x$ is supported on a finite ancestral set $H$, $R\\ge0$, and $\\|x+y\\|,\\|x-y\\|\\le R$, then the part of $y$ outside $H$ has norm at most $2\\sqrt{R^2-\\|x\\|^2}$.\n\nThe formalization also covers selected consequences for averaged midpoint moduli and separated families of vectors, together with reflexivity and renorming obstructions. This scope concerns the segment-space results; the companion results on diamond energies remain separate.\n\nThe paper studies embeddings of countably branching diamond graphs into Banach spaces built from tree segments. The formalization proves that every embedding of the depth-$k$ diamond with distortion $D$, at any positive scale, satisfies $1+k/4\\le D^2$ in either of two spaces: the full dual of the completed finite-height forest space and the closed coordinate span in the dual of the infinite-tree space. The bound applies to all pairs of vertices and is unconditional.\n\nThe formalization also contains selected graph deductions under explicit companion assumptions. Twenty-two of those companion inputs are assumed in these deductions; the diamond distortion bound above does not require them.\n\nThe paper exhibits a real $L_1$ subspace with positive averaged midpoint convexity but no equivalent asymptotically uniformly convex norm. For every closed infinite-dimensional subspace whose unit ball is precompact in measure and which has the Daugavet property, the formalization computes the moduli at every unit vector: the averaged midpoint modulus is $\\max(t/2,t-1)$ and the usual one-sided asymptotic modulus is $\\max(0,t-2)$ for every $t>0$. The same formulas hold after taking the infimum over unit vectors, and every equivalent norm fails asymptotic uniform convexity.\n\nThe formalization also constructs such a subspace on a countable product of unit intervals and proves that it has the stated measure-precompactness and Daugavet properties, so the modulus formulas apply to an actual example.\n\nThe paper constructs Banach spaces from bounded tree potentials and path costs to separate averaged midpoint convexity from asymptotic uniform convexity. For the specified tree-potential completions, the formalization proves completeness, separability, infinite dimension, and an averaged midpoint modulus of at least $\\sqrt{1+t^2/4}-1$ for $00$, the averaged midpoint modulus nevertheless has explicit positive lower bounds. If $G$ is a standard real Gaussian, the exponential-multiplier bound is $\\frac{7t}{480}\\mathbb E[(|G|-20/t)_+]$, where $u_+=\\max(u,0)$. For squared-Gaussian multipliers, the bounds are $\\frac18\\Pr(|G|\\ge4\\sqrt2(2+\\sqrt2)/t)$ and $\\frac14\\Pr(|G|\\ge8\\sqrt6/t)$.\n\nThe paper constructs three Banach spaces from recursive tree potentials: a root-sum space, a zero-root space, and a joined space. The formalization proves that all three are infinite-dimensional and have no equivalent asymptotically uniformly convex norm. The root-sum and joined spaces have averaged midpoint modulus at least $t^3/128$ for $00$. The zero-root estimate retains its finite ancestral support and tail hypotheses, and the zero-root and joined spaces are reflexive.\n\nThe formalization also covers a separated-midpoint result for the joined space and a variable-exponent model with its recursive norm, duality, projections, renorming obstruction, and sixth- and third-order stability estimates. The separation, sequence, support, and head assumptions in these results are essential parts of their statements.", "lab_docs_url": "https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/docs/331.md", "overview_entry": "family 331 in overview.pdf / CONTENTS.md", "papers": [{"dir": "Asymptotic-midpoint-uniform-convexity-and-unbounded-diamond-distortion-in-a-reflexive-tree-space-September-27-2026", "title": "Asymptotic midpoint uniform convexity and unbounded diamond distortion in a reflexive tree space", "url": "https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Asymptotic-midpoint-uniform-convexity-and-unbounded-diamond-distortion-in-a-reflexive-tree-space-September-27-2026"}, {"dir": "Midpoint-lenses-in-segment-spaces-September-27-2026", "title": "Midpoint lenses in segment spaces", "url": "https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Midpoint-lenses-in-segment-spaces-September-27-2026"}, {"dir": "Diamond-distortion-from-midpoint-and-tree-energies-September-27-2026", "title": "Distortion of countably branching diamonds from midpoint and tree energies", "url": "https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Diamond-distortion-from-midpoint-and-tree-energies-September-27-2026"}, {"dir": "Exact-asymptotic-moduli-in-a-Daugavet-subspace-of-L1-September-27-2026", "title": "Exact asymptotic moduli in a Daugavet subspace of L1", "url": "https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Exact-asymptotic-moduli-in-a-Daugavet-subspace-of-L1-September-27-2026"}, {"dir": "Midpoint-convexity-from-bounded-tree-potentials-and-path-costs-September-27-2026", "title": "Midpoint convexity from bounded tree potentials and path costs", "url": "https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Midpoint-convexity-from-bounded-tree-potentials-and-path-costs-September-27-2026"}, {"dir": "Independent-products-in-real-L1-asymptotic-midpoint-convexity-without-AUC-renormings-September-27-2026", "title": "Independent products in real L1: asymptotic midpoint convexity without AUC renormings", "url": "https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Independent-products-in-real-L1-asymptotic-midpoint-convexity-without-AUC-renormings-September-27-2026"}, {"dir": "Midpoint-convexity-from-two-recursive-potentials-September-27-2026", "title": "Midpoint convexity from two recursive potentials", "url": "https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Midpoint-convexity-from-two-recursive-potentials-September-27-2026"}], "audit_part": 3, "second_reader": "", "referee_verdict": ""}, {"family": "333", "title": "Smooth isometric immersions of surfaces into $\\mathbb R^4$", "subject": "Differential geometry", "headline": "Every closed smooth Riemannian surface admits a smooth isometric immersion into $\\mathbb R^4$, resolving the closed-surface form of the four-dimensional isometric-immersion problem. This includes nonorientable surfaces and metrics of arbitrary Gaussian curvature.", "verdict": "full", "challenges": ["SurfaceImmersion"], "review_note": "`smooth_isometric_immersion (g : SmoothMetric M) : ∃ F : M → ℝ⁴, ContMDiff ∞ F ∧ ∀ p v w, ⟪dF v, dF w⟫ = g.inner p v w` for every compact Hausdorff second-countable C^∞ 2-manifold without boundary and every C^∞ Riemannian metric (Mathlib's `ContMDiffRiemannianMetric`), with no orientability or curvature hypothesis; 'immersion' follows from positive-definiteness (injective differential), and the case is global and closed only, as in the summary.", "definitions_to_check": [], "external_packages": ["SphereEversion"], "cone_lines_max": 167865, "machine_check": "none", "lab_scope_note": "The formalization proves that every closed smooth Riemannian surface admits a smooth isometric immersion into $\\mathbb R^4$. The differential preserves the Riemannian inner product on every tangent space. No orientability assumption is imposed, and the statement asks for an immersion rather than an embedding.", "lab_docs_url": "https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/docs/333.md", "overview_entry": "family 333 in overview.pdf / CONTENTS.md", "papers": [{"dir": "Smooth-isometric-immersions-of-closed-surfaces-into-Euclidean-four-space-September-23-2026", "title": "Smooth isometric immersions of closed surfaces into Euclidean four-space", "url": "https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Smooth-isometric-immersions-of-closed-surfaces-into-Euclidean-four-space-September-23-2026"}], "audit_part": 3, "second_reader": "", "referee_verdict": ""}, {"family": "334", "title": "A smooth surface metric with no local immersion in $\\mathbb R^3$", "subject": "Differential geometry", "headline": "Constructs a smooth positive-definite metric on $(-1,1)^2$ for which no neighborhood of the origin admits a smooth isometric immersion into $\\mathbb R^3$. This answers the unrestricted smooth local isometric realization problem for surfaces negatively, even after shrinking the neighborhood.", "verdict": "full", "challenges": ["IsometricImmersion"], "review_note": "There is a smooth (C^∞, positive-definite) metric g on (-1,1)² such that for every open U ⊆ the square containing 0 and every F : U → ℝ³ it is false that F is C^∞ with ⟨dF v, dF w⟩ = vᵀ g w at every point of U, so no neighbourhood of the origin admits a smooth isometric immersion, as in the summary.", "definitions_to_check": [], "external_packages": [], "cone_lines_max": 56586, "machine_check": "comparator-sandboxed-pass", "lab_scope_note": "The local isometric-immersion problem asks whether every smooth surface metric can be realized locally in Euclidean three-space. The formalized counterexample is a smooth positive-definite metric on $(-1,1)^2$ for which no neighborhood of the origin admits a smooth isometric immersion into $\\mathbb R^3$. The obstruction applies to every smooth candidate map on every such neighborhood.", "lab_docs_url": "https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/docs/334.md", "overview_entry": "family 334 in overview.pdf / CONTENTS.md", "papers": [{"dir": "A-Smooth-Metric-with-No-Local-Isometric-Immersion-into-Three-Space-September-24-2026", "title": "A Smooth Metric with No Local Isometric Immersion into Three-Space", "url": "https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-Smooth-Metric-with-No-Local-Isometric-Immersion-into-Three-Space-September-24-2026"}], "audit_part": 3, "second_reader": "", "referee_verdict": ""}, {"family": "337", "title": "Cartan--Hadamard isoperimetry and CAT$(0)$ fillings", "subject": "Differential geometry", "headline": "Proves generalized Cartan--Hadamard isoperimetry in every dimension: in a complete simply connected manifold with sectional curvature at most $\\kappa\\le0$, every finite-volume finite-perimeter set satisfies the sharp comparison with the equal-volume model ball. Bounded positive-volume equality regions for $\\kappa=0$ are Euclidean balls. Also proves sharp Euclidean filling bounds for compactly supported integral $n$-cycles, $n\\ge2$, in arbitrary proper CAT$(0)$ spaces.", "verdict": "partial", "challenges": ["FillingCoefficient", "SharpCAT0Filling"], "review_note": "`sharp_integral_filling` (every compactly supported integral n-cycle, n ≥ 2, in a proper CAT(0) space has a compactly supported integral filling of mass ≤ C_n·mass^((n+1)/n)) and `coefficient_optimal` (Euclidean sharpness) cover the summary's second claim only. The headline, the sharp Cartan-Hadamard isoperimetric inequality for finite-perimeter sets in complete simply connected manifolds with curvature ≤ κ ≤ 0 (and the Euclidean-ball equality case for κ = 0), has no statement; the docs concede only 'the selected integral-current form' is proved, and the Lean form does not imply the set-based inequality.", "definitions_to_check": ["Hand-built metric currents: `Functional X k := (X → ℝ) → (Fin k → X → ℝ) → ℝ`, `IsMetricCurrent` (off-domain zero, multilinearity, sequential continuity, locality, finite mass), `mass T := sInf {μ.real univ | Controls T μ}`, `IntegerRectifiable` via countably many bi-Lipschitz `IntegerChart`s with integer multiplicity, `boundarySucc T b π := if Admissible b π then T 1 (b, π) else 0`; a Lipschitz-functional model of Ambrosio-Kirchheim integral currents, not in Mathlib", "`IsCAT0 X := ∃ segment, geodesic-parametrization ∧ CN-comparison inequality` (hand-built)"], "external_packages": [], "cone_lines_max": 50219, "machine_check": "comparator-sandboxed-pass", "lab_scope_note": "The formalization proves the sharp Euclidean filling inequality for every compactly supported integral $n$-cycle in a proper $\\mathrm{CAT}(0)$ space, for $n\\ge2$. It constructs a compactly supported integral filling whose mass is at most $C_n$ times the boundary mass to the power $(n+1)/n$, where $C_n$ is the Euclidean isoperimetric coefficient. Integer multiplicities and the ambient dimension are unrestricted.\n\nA second statement proves that this coefficient is optimal by giving, in Euclidean space, a cycle for which every compactly supported integral filling has at least that mass. This yields the selected integral-current form of the Cartan–Hadamard isoperimetric conjecture.", "lab_docs_url": "https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/docs/337.md", "overview_entry": "family 337 in overview.pdf / CONTENTS.md", "papers": [{"dir": "Sharp-integral-fillings-in-CAT(0)-spaces-September-23-2026", "title": "Sharp integral fillings in CAT(0) spaces", "url": "https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Sharp-integral-fillings-in-CAT(0)-spaces-September-23-2026"}], "audit_part": 3, "second_reader": "", "referee_verdict": ""}, {"family": "342", "title": "Donaldson's tamed-to-compatible conjecture", "subject": "Differential geometry", "headline": "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.", "verdict": "full", "challenges": ["TamingCompatibility"], "review_note": "`taming_implies_compatibility`: for a closed connected smooth 4-manifold X and a smooth almost complex structure J, if some symplectic form tames J then some symplectic form η satisfies `Compatible η J` (tames and J-invariant), with J fixed and no integrability or cohomology-class condition.", "definitions_to_check": ["Differential forms are hand-built: `TwoForm X := ∀ x, (T_x X)[⋀²]→L[ℝ] ℝ` with `IsSmooth` (pullbacks along every smooth chart-like map f : ℝ⁴ → X are C^∞) and `IsClosed` (`extDerivWithin` of the pullback vanishes); `AlmostComplexStructure` with smoothness as a map of tangent bundles. Reads as the standard notions."], "external_packages": [], "cone_lines_max": 83060, "machine_check": "comparator-sandboxed-pass", "lab_scope_note": "Donaldson's tamed-to-compatible conjecture asks whether an almost-complex structure tamed by a symplectic form also admits a compatible symplectic form. The formalized result establishes this for every closed connected smooth four-manifold, keeping the almost-complex structure fixed. No integrability assumption or prescribed cohomology class for the compatible form is required.", "lab_docs_url": "https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/docs/342.md", "overview_entry": "family 342 in overview.pdf / CONTENTS.md", "papers": [{"dir": "Taming-implies-compatibility-on-four-manifolds-September-23-2026", "title": "Taming implies compatibility on four-manifolds", "url": "https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Taming-implies-compatibility-on-four-manifolds-September-23-2026"}], "audit_part": 3, "second_reader": "", "referee_verdict": ""}, {"family": "343", "title": "Sharp symplectic ball-packing criteria in higher dimensions", "subject": "Differential geometry", "headline": "Resolves the Siegel--Yao conjecture for arbitrary capacities in every dimension $2n\\ge6$. Finitely many closed symplectic balls of capacities $R_1,\\ldots,R_k$ embed disjointly into an open ball of capacity $R$ exactly when $\\sum_iR_i^n 0): `HasPacking n k R r ↔ (∑ r_i^n < R^n ∧ ∀ i≠j, r_i + r_j < R)`, where a packing is k disjointly-imaged smooth symplectic embeddings (defined on open neighbourhoods of the closed capacity-r_i balls in ℂⁿ with the standard form, capacity π|z|²) into the open ball of capacity R; BallPackingNecessity repeats the forward direction.", "definitions_to_check": [], "external_packages": [], "cone_lines_max": 89175, "machine_check": "comparator-sandboxed-pass", "lab_scope_note": "The formalization proves Siegel and Yao's ball-packing criterion in every symplectic dimension $2n$ with $n\\ge3$. For $k\\ge1$ closed standard balls of positive capacities $R_1,\\ldots,R_k$, disjoint symplectic embeddings into the interior of a ball of capacity $R$ exist exactly when\n$\\sum_i R_i^n0$, ruling out even arbitrarily small power losses.", "verdict": "partial", "challenges": ["NodalLength", "SmoothYau", "YauCounterexample"], "review_note": "`NodalLength.Main` is the headline: on every compact connected smooth Riemannian surface there is C with H¹(nodal set) ≤ C√λ for every nonzero smooth eigenfunction with -Δu = λu, λ > 0; `sphere_three` (every C^∞ neighbourhood of the round S³ metric contains a metric with unbounded nodal ratio) and `sphere_two_torus_two` give dimensions 3 and 4. For dimension five, YauCounterexample (S⁴×S¹) only gives H⁴(nodal)/√λ → ∞, not the summary's power-law growth λ^(1/2+ε₀), so that claim is only stated in a weaker form.", "definitions_to_check": ["Laplace-Beltrami and nodal measures are hand-written: `laplaceBeltrami` (det(g)^(-1/2) ∂_i(det(g)^(1/2) g^{ij} ∂_j u) in the centre chart of each point), `nodalMeasure` = d-dimensional Hausdorff measure for `EMetricSpace.ofRiemannianMetric`; NodalLength uses a `MetricSpace M` instance tied to the Riemannian structure only through `[IsRiemannianManifold]`", "`IsSmoothNeighborhood g₀ N` (finitely many compact-chart C^k seminorm tests with one ε) as the C^∞ topology on metrics; `IsRound` as the pullback of the Euclidean inner product"], "external_packages": [], "cone_lines_max": 99645, "machine_check": "comparator-sandboxed-pass", "lab_scope_note": "Yau's nodal-set conjecture predicts nodal size of order $\\sqrt\\lambda$ for Laplace eigenfunctions of eigenvalue $\\lambda$. The formalization proves the upper bound on every fixed smooth closed connected Riemannian surface: there is a surface-dependent constant $C$ such that the one-dimensional Hausdorff measure of the zero set of every nonzero real eigenfunction with $\\lambda>0$ is at most $C\\sqrt\\lambda$. The known lower bound is not part of this selected statement.\n\nThe formalized results contradict Yau's proposed $O(\\sqrt\\lambda)$ upper bound for nodal measure in dimensions three and four. On $S^3$, every prescribed smooth neighborhood of the round metric contains one fixed smooth metric with an eigenfunction sequence of unbounded nodal-measure ratio. A second fixed metric on $S^2\\times T^2$ has the same property. In both cases the eigenvalues tend to infinity and the intrinsic nodal measures are finite.\n\nYau's nodal upper-bound conjecture predicts that the nodal measure of an eigenfunction is bounded by a constant times the square root of its eigenvalue. The formalized counterexample fixes one smooth metric on $S^4\\times S^1$ and a sequence of nonzero smooth eigenfunctions whose positive eigenvalues tend to infinity, while the intrinsic four-dimensional nodal measures divided by the square roots of the eigenvalues tend to infinity. The metric is fixed throughout the sequence.", "lab_docs_url": "https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/docs/350.md", "overview_entry": "family 350 in overview.pdf / CONTENTS.md", "papers": [{"dir": "Sharp-nodal-length-on-smooth-surfaces-September-23-2026", "title": "Sharp nodal length on smooth surfaces", "url": "https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Sharp-nodal-length-on-smooth-surfaces-September-23-2026"}, {"dir": "Smooth-counterexamples-to-Yaus-nodal-upper-bound-in-dimensions-three-and-four-September-23-2026", "title": "Smooth counterexamples to Yau's nodal upper bound in dimensions three and four", "url": "https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Smooth-counterexamples-to-Yaus-nodal-upper-bound-in-dimensions-three-and-four-September-23-2026"}, {"dir": "Power-law-violations-of-Yaus-nodal-upper-bound-September-23-2026", "title": "Power-law violations of Yau's nodal upper bound", "url": "https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Power-law-violations-of-Yaus-nodal-upper-bound-September-23-2026"}], "audit_part": 3, "second_reader": "", "referee_verdict": ""}, {"family": "353", "title": "The sharp dimension threshold for affine Bernstein rigidity", "subject": "Differential geometry", "headline": "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.", "verdict": "partial", "challenges": ["AffineBernstein"], "review_note": "`affine_bernstein` (3 ≤ n ≤ 9): a smooth function with positive-definite Hessian on a nonempty open convex Ω solving the affine-maximal equation with graph complete for the induced Euclidean path metric forces Ω = ℝⁿ, u a positive-definite quadratic plus affine term, and the graph an affine image of the standard paraboloid. The sharpness claim (smooth entire nonquadratic example in dimension ten) and the extension to hypersurfaces complete for the affine Berwald-Blaschke metric have no statement (docs: immersed extension outside the statement).", "definitions_to_check": ["`AffineMaximalOn` := Σ U^{ij} ∂_ij ((det D²u)^(-(n+1)/(n+2))) = 0 with U the cofactor matrix (the classical affine maximal equation); `EuclideanGraphComplete` is sequential completeness for the hand-built path metric `graphEDist` = inf of lengths ∫√(|γ'|²+(Du·γ')²) of C¹ paths in Ω"], "external_packages": [], "cone_lines_max": 40454, "machine_check": "comparator-sandboxed-pass", "lab_scope_note": "The formalization proves the Euclidean-complete affine Bernstein theorem for graph dimensions $3\\le n\\le9$. If a smooth function on a nonempty open convex domain has positive-definite Hessian, satisfies the affine maximal equation, and its graph is complete in the induced Euclidean metric, then the domain is all of $\\mathbb R^n$ and the function is a positive-definite quadratic polynomial plus an affine term. Its graph is therefore an elliptic paraboloid.\n\nThe paper's extension to immersed hypersurfaces without an initial graph assumption is outside this selected statement.", "lab_docs_url": "https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/docs/353.md", "overview_entry": "family 353 in overview.pdf / CONTENTS.md", "papers": [{"dir": "The-affine-Bernstein-theorem-in-dimensions-three-through-nine-September-24-2026", "title": "The affine Bernstein theorem in dimensions three through nine", "url": "https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-affine-Bernstein-theorem-in-dimensions-three-through-nine-September-24-2026"}], "audit_part": 3, "second_reader": "", "referee_verdict": ""}, {"family": "354", "title": "Isoperimetric regions in the cubic three-torus", "subject": "Differential geometry", "headline": "Determines the isoperimetric profile of the unit cubic flat three-torus and classifies every finite-perimeter minimizer: balls, circular tubes around shortest closed geodesics, coordinate slabs, and their complements. The transition volumes are $4\\pi/81$ and $1/\\pi$, with exactly the adjacent two types minimizing at each transition.", "verdict": "full", "challenges": ["CubicTorus"], "review_note": "`unit_cubic_isoperimetric` for every V ∈ (0,1): a minimizer exists; every minimizer has perimeter `candidateProfile V` = min((36π)^(1/3)v^(2/3), 2√(πv), 2) with v = min(V,1-V); and minimizers are exactly (up to isometry and null sets, with complements for V > 1/2) balls, tubes about a shortest geodesic, or slabs on the ranges V ≤ 4π/81, 4π/81 ≤ V ≤ 1/π, V ≥ 1/π, with both adjacent types allowed at the transition volumes.", "definitions_to_check": ["`perimeter E := ⨆ F admissible periodic C^∞ field with ‖F‖ ≤ 1, ofReal ∫_{cube ∩ E} div F` (De Giorgi perimeter by hand on the cover) and `torusVolume := map quotientMap (volume.restrict fundamentalCube)`: standard"], "external_packages": [], "cone_lines_max": 140491, "machine_check": "none", "lab_scope_note": "The formalization determines the isoperimetric profile and all minimizers in the unit cubic flat three-torus. At every volume $0 1, full-support reference measure finite on bounded sets), a bounded Lipschitz F whose weak Hessian is ≤ G·metric (G bounded continuous) satisfies the distributional bound (F∘σ)'' ≤ G(σ)·d(σ(0),σ(1))² along every constant-speed geodesic. Gigli's conjecture (Alexandrov curvature ≥ κ ⇔ RCD((n-1)κ,n) with H^n and distributional sectional curvature ≥ κ) is not stated.", "definitions_to_check": ["Entire RCD/Sobolev calculus is hand-built: `CurvatureDimension` (unreduced CD(K,N) via optimal dynamical plans, `tauCoeff`/`sigmaCoeff`), `QuadraticCheegerEnergy`, `IsTestPlan`, `IsLocalWeakUpperGradient`, `weakGradient`/`laplacian` defined by `Classical.choose` with fallback 0, `weakHessian`; the cone is 65,542 OAI lines here (recomputed; tmp/import_cones.json says 64,010)"], "external_packages": [], "cone_lines_max": 65542, "machine_check": "comparator-sandboxed-pass", "lab_scope_note": "The formalization transfers a weak Hessian upper bound to every prescribed minimizing geodesic in an $\\mathrm{RCD}(K,N)$ space with finite $N>1$. If a bounded globally Lipschitz function $F$ has weak Hessian bounded above by $G$ times the metric, where $G$ is bounded and continuous, then every constant-speed geodesic $\\gamma:[0,1]\\to X$ satisfies $(F\\circ\\gamma)''\\le G(\\gamma)\\,d(\\gamma(0),\\gamma(1))^2$ in the distributional sense. Constant geodesics are included. The space is complete and separable with full support and measure finite on bounded sets; compactness and metric nonbranching are not assumed.", "lab_docs_url": "https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/docs/356.md", "overview_entry": "family 356 in overview.pdf / CONTENTS.md", "papers": [{"dir": "Weak-Hessian-bounds-along-every-geodesic-in-RCD-spaces-September-24-2026", "title": "Weak Hessian bounds along every geodesic in RCD spaces", "url": "https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Weak-Hessian-bounds-along-every-geodesic-in-RCD-spaces-September-24-2026"}], "audit_part": 3, "second_reader": "", "referee_verdict": ""}, {"family": "358", "title": "A three-manifold with no conjugate points and no nonpositively curved metric", "subject": "Differential geometry", "headline": "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.", "verdict": "full", "challenges": ["ConjugatePoints"], "review_note": "`MainStatement`: there is a compact connected Hausdorff second-countable C^∞ 3-manifold with oriented atlas carrying a smooth metric with `NoConjugatePoints` (for every geodesic and Jacobi field on any open interval, J vanishing at two distinct parameters vanishes identically, constant geodesics included) while no smooth metric satisfies `NonpositiveSectionalCurvature`.", "definitions_to_check": ["Hand-written chart-wise Christoffel symbols, curvature `curvatureTerm`, geodesics and Jacobi fields (`IsGeodesicOn`, `IsJacobiFieldOn`); sign conventions checked: ⟨R(u,v)v,u⟩ ≤ 0 is sectional curvature ≤ 0 and the Jacobi equation J'' + R(J,γ')γ' = 0 is the standard one"], "external_packages": [], "cone_lines_max": 24974, "machine_check": "comparator-sandboxed-pass", "lab_scope_note": "The formalized result separates absence of conjugate points from nonpositive sectional curvature. It constructs a closed connected orientable smooth three-manifold with a smooth Riemannian metric having no conjugate points, while the same manifold admits no smooth metric of everywhere nonpositive sectional curvature. The no-conjugate-points assertion ranges over all geodesics and tangent-bundle Jacobi fields, including constant geodesics and unbounded time intervals. The separate no-focal-points and CAT(0) consequences are not included.", "lab_docs_url": "https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/docs/358.md", "overview_entry": "family 358 in overview.pdf / CONTENTS.md", "papers": [{"dir": "A-Three-Manifold-Without-Conjugate-Points-and-Without-a-Nonpositively-Curved-Metric-September-24-2026", "title": "A Three-Manifold Without Conjugate Points and Without a Nonpositively Curved Metric", "url": "https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-Three-Manifold-Without-Conjugate-Points-and-Without-a-Nonpositively-Curved-Metric-September-24-2026"}], "audit_part": 3, "second_reader": "", "referee_verdict": ""}, {"family": "359", "title": "Negative K\\\"ahler curvature without bounded holomorphic coordinates", "subject": "Differential geometry", "headline": "Constructs a contractible domain in $\\mathbb C^3$ with a complete negatively pinched K\\\"ahler 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.", "verdict": "partial", "challenges": ["PinchedKahler"], "review_note": "`main_theorem` gives a nonempty open contractible M ⊆ ℂ³ with a smooth Kähler metric that is geodesically complete (all geodesics extend for all time) and negatively pinched (-B ≤ sectional ≤ -A < 0), with no bounded holomorphic map with everywhere-nonvanishing Jacobian and no biholomorphism onto a bounded domain; `controlled_potential` and `metric_transfer` support it. The summary's second claim (a higher-dimensional example with sectional curvature ≤ -1, only constant bounded holomorphic functions, curvature not bounded below) is not stated.", "definitions_to_check": ["Kähler geometry is hand-coded on ℂ²×ℂ: `dz`/`dbar` from real derivatives, `curvature g` (-∂̄_b∂_a g_{c d̄} + g^{-1}∂g ∂̄g), `sectional := (R(u,ū,v,v̄) - R(u,v̄,u... ))/(2·realArea)`, `IsKahlerMetric` (symmetry, positivity, ∂_i g_{kj̄} = ∂_k g_{ij̄}), `GeodesicallyComplete` via smooth solutions of the geodesic ODE; `NoBoundedCoordinates` only excludes bounded holomorphic F with nonvanishing Jacobian (stronger than excluding injective coordinate charts)"], "external_packages": [], "cone_lines_max": 18688, "machine_check": "comparator-sandboxed-pass", "lab_scope_note": "The formalized result constructs a nonempty contractible complex threefold with a complete Kähler metric whose real sectional curvatures lie between two fixed negative constants. Nevertheless, it has no system of bounded holomorphic coordinates and is not biholomorphic to a bounded domain. The metric is obtained from an explicitly controlled potential. This is the negatively pinched construction; the different companion with only one-sided curvature control is separate.", "lab_docs_url": "https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/docs/359.md", "overview_entry": "family 359 in overview.pdf / CONTENTS.md", "papers": [{"dir": "A-negatively-pinched-Kahler-threefold-without-bounded-holomorphic-coordinates-September-25-2026", "title": "A negatively pinched Kähler threefold without bounded holomorphic coordinates", "url": "https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-negatively-pinched-Kahler-threefold-without-bounded-holomorphic-coordinates-September-25-2026"}], "audit_part": 3, "second_reader": "", "referee_verdict": ""}, {"family": "360", "title": "Villani’s convexity conjecture and regular optimal transport", "subject": "Differential geometry", "headline": "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 H\\\"older continuous.", "verdict": "full", "challenges": ["BiholderTransport", "WeakMTWGlobalSupport"], "review_note": "`current_main_convexity`: on a compact connected smooth Riemannian manifold of dimension n ≥ 2, weak MTW (-3/2·∂⁴c ≥ 0 for orthogonal ξ, η at interior velocities) implies every open tangent injectivity domain is convex. `uniform_biHolder_transport`: for squared-distance cost and densities between fixed bounds there is one (α, C) such that for all pairs of densities the optimal map exists, is a homeomorphism, is a.e. unique, and T and T⁻¹ are α-Hölder with constant C.", "definitions_to_check": ["Riemannian exponential is defined by `if h : ∃ γ, complete geodesic with initial data then (Classical.choose h) 1 else x` (geodesics characterized metrically as locally constant-speed distance realizers), `injectivityDomain x := {v | ∃ a > 1, dist x (exp x (a•v)) = a‖v‖}`, and `mtw` is an iterated ordinary derivative of the cost along exp-curves; calibrated Hausdorff measure `metricVolume` as volume"], "external_packages": [], "cone_lines_max": 74032, "machine_check": "comparator-sandboxed-pass", "lab_scope_note": "On a compact connected smooth Riemannian manifold of dimension at least two, the formalized result proves that weak MTW curvature implies convexity of every open tangent injectivity domain. Prior convexity or nonfocality is not assumed. Additional formalized results give global support at ordinary subgradients, compact convex lifted gap sections with the stated diameter bound, and scale and local-injectivity estimates for finite target families. The latter finite-family result does not cover the paper's arbitrary-potential local-injectivity assertion.\n\nThe formalized result gives uniform bi-Hölder optimal transport for squared-distance cost on each fixed compact connected smooth Riemannian manifold of dimension at least two satisfying weak MTW. For probability densities between fixed positive lower and finite upper bounds, one exponent and constant work for every pair of densities. The optimal map is a homeomorphism, is unique almost everywhere, and both it and its inverse satisfy the global Hölder bound. The constants are uniform over the density class on the fixed manifold, not over varying metrics.", "lab_docs_url": "https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/docs/360.md", "overview_entry": "family 360 in overview.pdf / CONTENTS.md", "papers": [{"dir": "Global-Support-and-Convex-Injectivity-Domains-under-Weak-MTW-September-25-2026", "title": "Global Support and Convex Injectivity Domains under Weak MTW", "url": "https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Global-Support-and-Convex-Injectivity-Domains-under-Weak-MTW-September-25-2026"}, {"dir": "Uniform-Bi-Holder-Transport-from-Weak-MTW-September-25-2026", "title": "Uniform Bi-Holder Transport from Weak MTW", "url": "https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Uniform-Bi-Holder-Transport-from-Weak-MTW-September-25-2026"}], "audit_part": 3, "second_reader": "", "referee_verdict": ""}, {"family": "361", "title": "Counterexamples to Yau's harmonic dimension bound", "subject": "Differential geometry", "headline": "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 $\\mathbb R^3$ 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 $k$.", "verdict": "weaker-statement", "challenges": ["HarmonicGrowth"], "review_note": "`MainClaim` is ∃ n (even, ≥ 8), k ≥ 2 and one complete smooth metric on ℝⁿ with Ric ≥ 0, Euclidean near 0, asymptotic volume ratio in (0,1), and euclideanDimension n k + 1 independent harmonic functions of growth ≤ k: a single higher-dimensional counterexample, as the docs say. The summary's claim (complete metrics on ℝ³ with ≥ (k+2)² harmonic functions of growth ≤ k for every sufficiently large integer k) is not implied. The challenge file also contains `axiom mainStatement : MainClaim` and `theorem main := mainStatement` instead of `sorry`, a non-standard stub shape; no `axiom` exists in the solution cone.", "definitions_to_check": ["`axiom mainStatement : MainClaim` in the challenge file (the only challenge here with an axiom and no `sorry`)", "Hand-written Riemannian geometry on ℝⁿ: `ricci`, `laplaceBeltrami`, `distance := sInf of C¹ path lengths`, `Complete`, `PositiveSubunitAVR` (ball volume/(ω_n R^n) → a ∈ (0,1)), `HarmonicGrowth` (|u| ≤ C(1+d(0,x))^k) and `euclideanDimension n k := C(n+k-1,k) + C(n+k-2,k-1)` (dimension of Euclidean harmonic polynomials of degree ≤ k)"], "external_packages": [], "cone_lines_max": 36732, "machine_check": "comparator-sandboxed-pass", "lab_scope_note": "The formalized result disproves the proposed Euclidean upper comparison for dimensions of harmonic functions with integer polynomial growth. It constructs one complete smooth metric with nonnegative Ricci curvature on an even-dimensional Euclidean space, Euclidean near the origin and with asymptotic volume ratio strictly between zero and one, admitting more independent harmonic functions of the prescribed growth degree than the Euclidean count. The construction uses dimension $16$ and degree $50000$. Separate tangent-cone and cone-spectrum conclusions are not included.", "lab_docs_url": "https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/docs/361.md", "overview_entry": "family 361 in overview.pdf / CONTENTS.md", "papers": [{"dir": "A-counterexample-to-integer-degree-harmonic-dimension-comparison-September-25-2026", "title": "A counterexample to integer-degree harmonic dimension comparison", "url": "https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-counterexample-to-integer-degree-harmonic-dimension-comparison-September-25-2026"}], "audit_part": 3, "second_reader": "", "referee_verdict": ""}, {"family": "362", "title": "Global smoothness for relativistic Vlasov--Maxwell", "subject": "Partial differential equations", "headline": "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.", "verdict": "full", "challenges": ["VlasovMaxwell"], "review_note": "`global_classical_solution`: for every admissible datum (f₀ ≥ 0 smooth compactly supported in ℝ³×ℝ³; E₀, B₀ C^∞ with bounded derivatives of all orders, in L², div E₀ = ρ[f₀], div B₀ = 0) there is a solution of the full relativistic system (v̂ = v/√(1+|v|²), force E + v̂×B, ∂ₜE - curl B = -j, ∂ₜB + curl E = 0, Gauss laws) that is C^∞ on every [0,T], and every C¹ solution in the class (nonnegative, L²-continuous fields, compact phase support on finite horizons) coincides with it for t ≥ 0; no smallness, symmetry or neutrality assumption.", "definitions_to_check": ["`def Classical (d) (s)` inside namespace `RVM` shadows the `Classical` namespace name (harmless); the Vlasov, Maxwell and Gauss equations are written out by hand with `derivWithin (Ici 0)` at t = 0"], "external_packages": [], "cone_lines_max": 89993, "machine_check": "comparator-sandboxed-pass", "lab_scope_note": "The formalized result gives global existence and uniqueness for the three-dimensional one-species relativistic Vlasov–Maxwell system. The initial particle density is nonnegative, smooth, and compactly supported; the initial fields have bounded derivatives of all orders, finite energy, and satisfy both Gauss constraints. The solution is smooth on every finite time interval, has compact particle phase support there, and is unique among classical solutions. No smallness, symmetry, or neutrality assumption is imposed.", "lab_docs_url": "https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/docs/362.md", "overview_entry": "family 362 in overview.pdf / CONTENTS.md", "papers": [{"dir": "Global-classical-solutions-of-the-three-dimensional-relativistic-Vlasov-Maxwell-system-September-23-2026", "title": "Global classical solutions of the three-dimensional relativistic Vlasov–Maxwell system", "url": "https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Global-classical-solutions-of-the-three-dimensional-relativistic-Vlasov-Maxwell-system-September-23-2026"}], "audit_part": 3, "second_reader": "", "referee_verdict": ""}, {"family": "363", "title": "Boltzmann nonuniqueness with exact local conservation", "subject": "Partial differential equations", "headline": "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 $L^1$ and satisfy exact local conservation of mass, momentum and kinetic energy.", "verdict": "weaker-statement", "challenges": ["BoltzmannNonuniqueness"], "review_note": "`nonuniqueness` gives bounded-velocity-support data f₀ and two `AdmissibleGlobal` renormalized solutions F, G (all renormalizations β, gain/loss integrability, weak continuity, local mass conservation as a weak identity, global momentum conservation, global energy inequality ≤ and entropy-dissipation inequality) that are strongly L¹-continuous with integrable gain/loss on a common [0,T] only and differ at some t ≤ T. The summary's exact local conservation of momentum and kinetic energy, and strong L¹ continuity for all times, are not stated (docs: 'later local-conservation refinements are outside it').", "definitions_to_check": ["Hand-built kinetic setup: `gain`/`loss` as Bochner integrals over ℝ³×S² with `sphereArea := volume.toSphere` (the `collision_fibers` hypotheses prevent silent zero values of non-integrable integrals), `extendedEntropyProduction` (lower semicontinuous extension of (A-B)log(A/B)), `entropyDissipation`, `relativeEntropy` against a normalized Maxwellian; a statement-level `run_cmd Lean.modifyEnv ... auxLemmasExt` meta command"], "external_packages": [], "cone_lines_max": 159510, "machine_check": "none", "lab_scope_note": "The formalization proves nonuniqueness for the periodic hard-sphere Boltzmann equation by constructing two distinct global renormalized solutions on $\\mathbb T^3\\times\\mathbb R^3$ with the same nonnegative initial density. The data have bounded velocity support and finite mass, energy, and absolute entropy. Both solutions conserve local mass and total momentum and satisfy the global energy and entropy-dissipation inequalities.\n\nOn a common initial interval they are strongly continuous in $L^1$, with integrable collision gains and losses. These are the conditions of the selected periodic result; later local-conservation refinements are outside it.", "lab_docs_url": "https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/docs/363.md", "overview_entry": "family 363 in overview.pdf / CONTENTS.md", "papers": [{"dir": "Nonuniqueness-for-the-periodic-hard-sphere-Boltzmann-equation-September-23-2026", "title": "Nonuniqueness for the periodic hard-sphere Boltzmann equation", "url": "https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Nonuniqueness-for-the-periodic-hard-sphere-Boltzmann-equation-September-23-2026"}], "audit_part": 3, "second_reader": "", "referee_verdict": ""}, {"family": "365", "title": "Joint metric and connection recovery from one boundary patch", "subject": "Partial differential equations", "headline": "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.", "verdict": "partial", "challenges": ["Conductivity"], "review_note": "`main_nonuniqueness` covers only the contrast claim: on the ball B(0,3) ⊂ ℝ³ there are two bounded measurable conductivities in [c, C], both equal to 1 near the boundary, differing on a positive-measure set, with one and the same weak Dirichlet-to-Neumann operator Λ (`IsDirichletToNeumann`: for every trace a unique harmonic H¹ extension with Λ f (trace v) = ∫γ∇u·∇v). The summary's first-named result, recovery of a smooth metric and unitary connection on a rank-two Hermitian bundle over manifolds of dimension ≥ 3 from zero-frequency data on one boundary patch (up to diffeomorphism and gauge), has no statement; the docs scope also only describes the conductivity result.", "definitions_to_check": ["H¹(B) is hand-built as the closure of smooth jets (f, ∇f) in L²(B; ℝ⁴), H¹₀ as the closure of compactly supported jets, `TraceSpace := H1 ⧸ H10`, and the DN operator is a continuous bilinear map TraceSpace → TraceSpace*; `EqualOneNearBoundary` is only a.e. in r < ‖x‖ < 3"], "external_packages": [], "cone_lines_max": 75520, "machine_check": "comparator-sandboxed-pass", "lab_scope_note": "The Calderón inverse problem asks whether boundary measurements determine an interior conductivity. The formalized result gives nonuniqueness for bounded measurable scalar conductivities on the ball $B(0,3)\\subset\\mathbb R^3$: two uniformly positive conductivities differ on a set of positive volume, equal $1$ near the boundary, and have the same full weak Dirichlet-to-Neumann operator. Weak solutions exist uniquely for every boundary trace.", "lab_docs_url": "https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/docs/365.md", "overview_entry": "family 365 in overview.pdf / CONTENTS.md", "papers": [{"dir": "Nonuniqueness-for-Bounded-Measurable-Scalar-Conductivities-in-Three-Dimensions-September-23-2026", "title": "Nonuniqueness for bounded measurable scalar conductivities in three dimensions", "url": "https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Nonuniqueness-for-Bounded-Measurable-Scalar-Conductivities-in-Three-Dimensions-September-23-2026"}], "audit_part": 3, "second_reader": "", "referee_verdict": ""}, {"family": "367", "title": "The critical dimension for the one-phase Bernoulli problem", "subject": "Partial differential equations", "headline": "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.", "verdict": "partial", "challenges": ["BernoulliNonflatInSeven"], "review_note": "`nonflat_in_seven`: there is u : ℝ⁷ → ℝ that is a nonnegative global minimizer of ∫ |∇u|² + 1_{u>0} (ballwise comparison with H¹ competitors agreeing on the boundary, all balls), nonzero, 1-homogeneous and not a half-space solution max(x·e,0): the existence half of 'seven is the first dimension'. The other half (flatness of such minimizers in dimensions ≤ 6) and the consequences (smooth free boundaries through dimension six, singular set dimension ≤ n-7) are not stated (docs: 'outside this selected statement').", "definitions_to_check": ["`LocalMinimizer`/`GlobalMinimizer` written with hand-built H¹, H¹₀ (L²-approximation by compactly supported smooth functions) and the exact energy `∫ ‖G‖² + if 0 < u x then 1 else 0` with an explicit weak-gradient representative G; flatness, homogeneity and nonvanishing are a.e. statements"], "external_packages": [], "cone_lines_max": 13545, "machine_check": "comparator-sandboxed-pass", "lab_scope_note": "The paper identifies seven as the critical dimension for nonflat one-homogeneous global minimizers of the one-phase Bernoulli problem. The linked formalization proves the existence side: in dimension seven there is a nonzero one-homogeneous global minimizer that is not a half-space solution.\n\nThe flatness classification in dimensions at most six and the resulting regularity and singular-set bounds are outside this selected statement.", "lab_docs_url": "https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/docs/367.md", "overview_entry": "family 367 in overview.pdf / CONTENTS.md", "papers": [{"dir": "The-critical-dimension-for-one-phase-Bernoulli-minimizers-September-24-2026", "title": "The critical dimension for one-phase Bernoulli minimizers", "url": "https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-critical-dimension-for-one-phase-Bernoulli-minimizers-September-24-2026"}], "audit_part": 3, "second_reader": "", "referee_verdict": ""}, {"family": "369", "title": "The hot spots conjecture for simply connected planar domains", "subject": "Partial differential equations", "headline": "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.", "verdict": "full", "challenges": ["HotSpots"], "review_note": "`main_theorem`: on every nonempty bounded open simply connected planar domain with smooth boundary, every nonzero u in the first positive Neumann eigenspace (smooth up to the boundary, mean zero, weak eigen-equation against all H¹ test pairs with eigenvalue = inf of the mean-zero Rayleigh quotient) has ∇u ≠ 0 at every interior point and inf_∂Ω u < u(x) < sup_∂Ω u inside, i.e. no interior critical point and strict hot spots; eigenvalue multiplicity is allowed because the statement is for any element of the eigenspace.", "definitions_to_check": ["`firstPositiveNeumannValue Ω := sInf (rayleighValues Ω)` with hand-built `HasH1Gradient` and `SmoothBoundary` (local defining functions ρ with nonzero gradient); `InFirstNeumannEigenspace` requires `ContDiffOn ℝ ∞ u (closure Ω)`"], "external_packages": [], "cone_lines_max": 29338, "machine_check": "comparator-sandboxed-pass", "lab_scope_note": "The strict hot-spots conjecture asks whether extrema of a first nonconstant Neumann eigenfunction occur only on the boundary. The formalization proves the stronger interior statement on every nonempty smooth bounded simply connected planar domain: every nonzero eigenfunction in the first positive Neumann eigenspace has nonvanishing gradient throughout the interior. Consequently all global maxima and minima lie on the boundary. The conclusion applies to every eigenfunction even when the eigenvalue is multiple.", "lab_docs_url": "https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/docs/369.md", "overview_entry": "family 369 in overview.pdf / CONTENTS.md", "papers": [{"dir": "Strict-hot-spots-and-absence-of-interior-critical-points-on-smooth-simply-connected-planar-domains-September-24-2026", "title": "Strict hot spots and absence of interior critical points on smooth simply connected planar domains", "url": "https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Strict-hot-spots-and-absence-of-interior-critical-points-on-smooth-simply-connected-planar-domains-September-24-2026"}], "audit_part": 3, "second_reader": "", "referee_verdict": ""}, {"family": "370", "title": "The subcritical Lane--Emden and H\\'enon--Lane--Emden conjectures", "subject": "Partial differential equations", "headline": "Resolves the subcritical Lane--Emden conjecture and its weighted H\\'enon extension. For $n\\ge2$, $p,q>0$ and real $A,B$, the system $-\\Delta u=|x|^A v^p$, $-\\Delta v=|x|^B u^q$ has no positive entire solution when $(n+A)/(p+1)+(n+B)/(q+1)>n-2$, 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 $n\\ge3$ and $A,B>-2$.", "verdict": "full", "challenges": ["HenonEmden"], "review_note": "`main_nonexistence`: for n ≥ 2, p, q > 0, real A, B and (n+A)/(p+1) + (n+B)/(q+1) > n-2 there are no positive u, v continuous on ℝⁿ and C² off the origin with -Δu = |x|^A v^p and -Δv = |x|^B u^q for x ≠ 0, with no symmetry or growth assumption; the radial existence criterion mentioned in the summary is background and not stated.", "definitions_to_check": [], "external_packages": [], "cone_lines_max": 12709, "machine_check": "comparator-sandboxed-pass", "lab_scope_note": "The formalized result proves nonexistence of positive entire solutions to the subcritical Hénon–Lane–Emden system. For $n\\ge2$, $p,q>0$, and real $A,B$ with $(n+A)/(p+1)+(n+B)/(q+1)>n-2$, there are no positive functions, continuous everywhere and $C^2$ off the origin, satisfying $-\\Delta u=|x|^A v^p$ and $-\\Delta v=|x|^B u^q$ away from the origin. It includes the globally $C^2$ unweighted case for $n\\ge3$. The critical equality case is not included.", "lab_docs_url": "https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/docs/370.md", "overview_entry": "family 370 in overview.pdf / CONTENTS.md", "papers": [{"dir": "The-Subcritical-Henon-Lane-Emden-Conjecture-September-24-2026", "title": "The Subcritical Hénon–Lane–Emden Conjecture", "url": "https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-Subcritical-Henon-Lane-Emden-Conjecture-September-24-2026"}], "audit_part": 3, "second_reader": "", "referee_verdict": ""}, {"family": "371", "title": "Stable blowup for a defocusing Schr\\\"odinger equation", "subject": "Partial differential equations", "headline": "For a sufficiently large odd nonlinearity power, constructs a nonempty open set of initial data in $H^k(\\mathbb T^{12})$, with $k>8$, whose solutions of the scalar defocusing nonlinear Schr\\\"odinger equation blow up in finite time. Thus finite-time blowup is stable under Sobolev perturbations in this supercritical regime.", "verdict": "full", "challenges": ["DefocusingNLS"], "review_note": "`stable_blowup`: for every p₀ there are an odd p ≥ max(p₀,3), an integer k > 8 and a nonempty open U ⊆ H^k(𝕋¹²) (as weighted ℓ² Fourier data) such that every f ∈ U has a maximal classical solution of i∂ₜu = -Δu + |u|^(p-1)u (interaction-picture ODE with sobolevOddPower) that blows up self-similarly at a finite time T, ‖u(t)(x)‖ ~ c(T-t)^(-1/(p-1)); the quantifier gives arbitrarily large odd powers rather than visibly all large ones, and the Gaussian-data corollaries are extra. The solution cone uses the external package FixedPointTheorems.", "definitions_to_check": ["The challenge file contains `by sorry` inside a definition: `sobolevProduct k _hk f g : FourierL2 := ⟨fun n => weight * convolution coefficient, by sorry⟩` (the ℓ² membership proof of the Sobolev-algebra product; it is a Prop field so it cannot change the function, but it is a definition-level sorry)", "The PDE is encoded in Fourier space: `lowerSobolevInclusion`/`lowerSobolevGenerator` contraction multipliers, `maximalSobolevInteractionFlow` defined by `Classical.choose` of a patch with fallback 0 and `maximalSobolevInteractionDomain` as a union of patches, `HasFiniteTimeSelfSimilarBlowup` (pointwise value at a torus point via `sobolevTorusFunction`, sup of the domain = T, discontinuity at T); about a dozen statement-level `run_cmd Lean.modifyEnv ... auxLemmasExt` meta commands and `attribute [local irreducible]`"], "external_packages": ["FixedPointTheorems"], "cone_lines_max": 180149, "machine_check": "none", "lab_scope_note": "The formalization constructs stable self-similar finite-time blowup for defocusing nonlinear Schrödinger equations on the twelve-dimensional torus. For every prescribed lower bound on the power, it selects an odd power at least that large and a Sobolev index $k>8$ for which a nonempty open set of $H^k$ initial data has the stated classical blowup behavior. The selected quantifier gives arbitrarily large odd powers, rather than every sufficiently large odd power.\n\nFor each such construction, Gaussian Fourier data with decay exponent $\\alpha>k+6$ assign positive probability to the blowup set, and almost-sure global existence fails. The exponent $\\alpha$ may be arbitrarily large.", "lab_docs_url": "https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/docs/371.md", "overview_entry": "family 371 in overview.pdf / CONTENTS.md", "papers": [{"dir": "Stable-Self-Similar-Blowup-for-a-Supercritical-Defocusing-Schrodinger-Equation-on-the-Torus-September-24-2026", "title": "Stable self-similar blowup for a supercritical defocusing Schrödinger equation on the torus", "url": "https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Stable-Self-Similar-Blowup-for-a-Supercritical-Defocusing-Schrodinger-Equation-on-the-Torus-September-24-2026"}], "audit_part": 3, "second_reader": "", "referee_verdict": ""}, {"family": "372", "title": "Global uniqueness in smooth isotropic elasticity", "subject": "Partial differential equations", "headline": "Proves that full static boundary displacement-to-traction data determine both smooth real Lam\\'e moduli on every bounded connected smooth domain in $\\mathbb R^3$, provided $\\mu>0$ and $3\\lambda+2\\mu>0$ on the closure. Neither analyticity, proximity to constant coefficients nor prior knowledge near the boundary is required.", "verdict": "full", "challenges": ["ElasticityUniqueness"], "review_note": "`MainClaim`: for every open connected bounded smooth domain Ω ⊂ ℝ³ and two pairs of Lamé moduli smooth in a neighbourhood of the closure with μ > 0 and 3λ+2μ > 0 on the closure, equality of the full displacement-to-traction pairing `DN` as a function on all pairs of boundary data forces λ₁ = λ₂ and μ₁ = μ₂ on Ω; no analyticity, near-constant or near-boundary assumption. The `Classical.choose` in `dirichletSolution` picks the weak Dirichlet solution but only enters `DN` through energy against it, which is well defined because that solution is a weak solution (unique under the stated positivity); the fallback `f.out` is used only when no solution exists.", "definitions_to_check": ["`dirichletSolution Ω lam mu f := if h : ∃ u, trace u = f ∧ WeakSolution u then Classical.choose h else f.out` and `DN f g := energy (dirichletSolution f) g.out` (Quot.out representatives of the trace classes); H¹ and traces hand-built as closures of smooth jets in L² and `Quot (SameTrace)`, strain and divergence from jets, energy = ∫ λ div u div v + 2μ ε(u):ε(v)"], "external_packages": [], "cone_lines_max": 31645, "machine_check": "comparator-sandboxed-pass", "lab_scope_note": "The formalization proves global uniqueness in the three-dimensional static isotropic elasticity inverse problem. On a bounded connected smooth domain, let two pairs of smooth real Lamé moduli satisfy $\\mu>0$ and $3\\lambda+2\\mu>0$ on the closure. If their full displacement-to-traction maps agree, then both Lamé moduli agree throughout the domain. The selected statement is uniqueness; it does not supply a reconstruction algorithm.", "lab_docs_url": "https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/docs/372.md", "overview_entry": "family 372 in overview.pdf / CONTENTS.md", "papers": [{"dir": "Global-Uniqueness-for-the-Smooth-Isotropic-Elasticity-Inverse-Problem-September-24-2026", "title": "Global Uniqueness for the Smooth Isotropic Elasticity Inverse Problem", "url": "https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Global-Uniqueness-for-the-Smooth-Isotropic-Elasticity-Inverse-Problem-September-24-2026"}], "audit_part": 3, "second_reader": "", "referee_verdict": ""}, {"family": "373", "title": "Nonattainment in three-marginal Coulomb transport", "subject": "Partial differential equations", "headline": "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.", "verdict": "partial", "challenges": ["CoulombCounterexample"], "review_note": "`coulomb_counterexample_and_equal_infima`: there is a smooth compactly supported probability density ρ on ℝ³ (with smooth compactly supported √ρ) whose three-marginal Kantorovich problem for the cost Σ 1/|x_i-x_j| with identical marginals has finite value attained by some coupling, no pair of measurable measure-preserving maps (T₂,T₃) attains it, yet the Monge infimum equals the Kantorovich value and finite-cost Monge maps approach it. The summary's extension to every inverse-power Riesz exponent in each dimension ≥ 2 is not stated (docs: 'outside this statement').", "definitions_to_check": ["`invDistance x y := (ofReal ‖x - y‖)⁻¹` in ℝ≥0∞ (value ⊤ on the diagonal), `kantorovichValue`/`mongeValue` as ⨅ over couplings / measure-preserving map pairs; all standard"], "external_packages": [], "cone_lines_max": 10668, "machine_check": "comparator-sandboxed-pass", "lab_scope_note": "The Monge ansatz asks whether an optimal multi-marginal transport plan can be induced by maps from one marginal. For the three-marginal Coulomb cost in $\\mathbb R^3$, the formalization constructs a smooth compactly supported probability density, with smooth compactly supported square root, for which no pair of measure-preserving Borel maps attains the Kantorovich minimum.\n\nNevertheless, the Monge and Kantorovich infima are equal: preserving maps with finite costs approach the minimum. This is the three-dimensional Coulomb result; the paper's inverse-power extensions in every dimension are outside this statement.", "lab_docs_url": "https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/docs/373.md", "overview_entry": "family 373 in overview.pdf / CONTENTS.md", "papers": [{"dir": "A-counterexample-to-the-Monge-ansatz-for-the-three-marginal-Coulomb-cost-September-25-2026", "title": "A counterexample to the Monge ansatz for the three-marginal Coulomb cost", "url": "https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/A-counterexample-to-the-Monge-ansatz-for-the-three-marginal-Coulomb-cost-September-25-2026"}], "audit_part": 3, "second_reader": "", "referee_verdict": ""}, {"family": "374", "title": "Sharp one-third stability of optimal transport maps", "subject": "Partial differential equations", "headline": "For uniform source measure $\\rho$ on a compact convex body with interior in dimension at least two, quadratic optimal transport maps satisfy $\\|T_\\mu-T_\\nu\\|_{L^2(\\rho)}\\le C W_2(\\mu,\\nu)^{1/3}$ uniformly over targets in a fixed compact set. The exponent is sharp, even for three-atom targets, disproving Letrouit's conjectured square-root bound.", "verdict": "full", "challenges": ["Brenier"], "review_note": "All 12 `definition_names` were read: `uniformMeasure` ((vol S)⁻¹·vol|_S), `IsCoupling`, `quadraticPlanCost` (∫⁻ ofReal ‖x-y‖²), `wasserstein2` (√ toReal sInf of coupling costs), `graphPlan`, `IsQuadraticOptimalMap`, `IsUniqueQuadraticOptimalMap` (optimal and every coupling of cost ≤ it equals the graph plan), `mapL2Dist`, `cube`, `HasExactlyThreeAtoms`, `IsSupported`, `E`; all standard. `one_third_stability` gives, for d ≥ 2, any compact convex source with interior and any compact nonempty container, one constant with ‖T_μ - T_ν‖_{L²(uniform)} ≤ C·W₂(μ,ν)^(1/3) for all probability targets in the container (unique optimal maps exist), and `one_third_exponent_is_sharp` shows for every exponent > 1/3 and constant, three-atom targets in the cube violate the bound (so the conjectured square root fails).", "definitions_to_check": ["`wasserstein2` and `mapL2Dist` use `ENNReal.toReal`/Bochner integrals (value 0 if infinite or non-integrable), harmless here because targets lie in a compact container so all quantities are finite"], "external_packages": [], "cone_lines_max": 33020, "machine_check": "comparator-sandboxed-pass", "lab_scope_note": "The formalization proves one-third Hölder stability of Brenier maps from the uniform measure on a compact convex body with nonempty interior in $\\mathbb R^d$, for every $d\\ge2$. For targets supported in one fixed nonempty compact set, the $L^2$ distance between the unique quadratic optimal maps is at most a constant times the one-third power of the targets' $2$-Wasserstein distance. The constant is uniform over those targets.\n\nIt also proves optimality of the exponent: on a fixed cube, pairs of three-atom target measures violate every analogous bound with exponent greater than $1/3$. In particular, the conjectured uniform square-root estimate fails.", "lab_docs_url": "https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/docs/374.md", "overview_entry": "family 374 in overview.pdf / CONTENTS.md", "papers": [{"dir": "Sharp-One-Third-Stability-of-Brenier-Maps-September-25-2026", "title": "Sharp One-Third Stability of Brenier Maps", "url": "https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Sharp-One-Third-Stability-of-Brenier-Maps-September-25-2026"}], "audit_part": 3, "second_reader": "", "referee_verdict": ""}, {"family": "376", "title": "Universal computation in forced Navier--Stokes flows", "subject": "Partial differential equations", "headline": "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.", "verdict": "full", "challenges": ["BalancedBoxRouting", "BalancedThreeStack", "ForcedNavierStokesComputation", "NavierStokesAlternating", "NavierStokesVelocity", "SolenoidalSheetPrograms"], "review_note": "`NavierStokesAlternating.alternating` (fixed computable ν > 0) gives computable compilers (Machine × input ↦ triple of `Nat.Partrec.Code` programs returning approximations, moduli and rapid-decay bounds) and, for every well-formed Turing machine and input, a smooth force f = residual ν U supported in one compact K, with U the unique energy-class solution from rest (zero pressure), such that the machine halts iff the particle started at (-1,0,0) reaches {x₁ > 0}. ForcedNavierStokesComputation and BalancedThreeStack (one theorem stated twice: all ν > 0, particle from (4,0,0) into (-1,2)³) and NavierStokesVelocity (torus and ℝ²×𝕋 detection) are variants. The forcing is not unconstrained: it must be uniformly computable in (M, w) (`ProgramFor`/`EffectiveFamily`/`Represents`), which excludes a force that already knows the halting answer.", "definitions_to_check": ["The flow is prescribed and the force is derived from it: `f = residual ν U := ∂ₜU + U·∇U - νΔU` (Alternating) or `f₀ := inertialCoefficient U`, `f₁ := viscousCoefficient U`, `affineForce f₀ f₁ ν` with pressure ≡ 0 (Forced/BalancedThreeStack); legitimate for forced Navier-Stokes since solutions are shown unique in an energy comparison class, but it means the force is only constrained through smoothness, compact support, computability and uniqueness", "NavierStokesVelocity: `def compile (B) (M) (w) : Program b N := ⟨B, M, w⟩` is trivial packaging; the effectiveness is carried by the fixed interpreter `evaluate := Nat.rfindOpt (valueTrial ..)` and the clause `Represents`, i.e. 'terminating compiler' there means the interpreter always halts with an ε-approximation (the genuinely computable compiler is `Computable compileForce` in NavierStokesAlternating)", "Three different machine models: `Machine := ℕ × ℕ × ℕ × List ℕ × List (Option Rule)` (Alternating), `FiniteMachine` with `Option Transition` (Forced/BalancedThreeStack), Mathlib `Turing.TM0.Machine` (Velocity), and 'halts' is `∃ n, isHalting (run n)`, `Halts` (reachable configuration without a transition) and `(TM0.eval M w).Dom` respectively; BalancedThreeStack and ForcedNavierStokesComputation share the theorem name `OAI.BalancedTransport.balanced_three_stack_realization` and differ only by a `Fintype Direction` instance"], "external_packages": [], "cone_lines_max": 22602, "machine_check": "comparator-sandboxed-pass", "lab_scope_note": "The formalization realizes finite families of prescribed positive diagonal maps on coding sheets by smooth incompressible flows on a flat three-torus. At every positive computable viscosity, it supplies a solution starting from rest with zero pressure and a force that is divergence free, has spatial mean zero, and is one-periodic from time zero. Each map acts on every point of its source sheet, and the construction provides effective bounds for all mixed derivatives.\n\nThis selected statement covers the sheet programs. The paper's fixed-particle halting detector and its reciprocal maps on solid boxes are outside its scope.\n\nThe formalization proves that a fluid particle can detect arbitrary finite-machine halting using alternating-coordinate memory in smooth incompressible Navier–Stokes flows whose velocity and force decay faster than every inverse power of time, including every mixed derivative. For each fixed positive computable viscosity, the construction starts from rest and provides effective forces, unique material trajectories, and an exact equivalence between entering a fixed open detector and halting.\n\nThe velocity and force share one compact support for every machine and input. The solution-uniqueness assertion uses the stated energy comparison class.\n\nThe formalization constructs smooth forced three-dimensional Navier–Stokes flows from rest whose velocity fields detect whether a prescribed machine halts, at any positive computable viscosity. On the unit flat torus, halting is equivalent to the vertical velocity exceeding $1/2$ somewhere in a fixed observation strip. On $\\mathbb R^2\\times\\mathbb T$, it is equivalent to the integral of the nonnegative vertical velocity over the observation half-plane times the circle exceeding $1/2$.\n\nThe torus force is confined to a fixed horizontal region. The cylindrical force has globally bounded mixed derivatives and compact horizontal support on every finite time interval. Both constructions have effective descriptions and uniqueness within their respective smooth solution-comparison classes.\n\nThe formalization realizes finitely many positive diagonal affine maps of determinant one between rational solid boxes by a smooth compactly supported divergence-free flow. Source boxes are pairwise disjoint and target boxes are pairwise disjoint, while overlap between the two families is allowed. Each map holds on a neighborhood of the entire source box, and the velocity is supported in the middle half of the time interval.\n\nIt also encodes every finite machine and input in a smooth, compactly supported incompressible Navier–Stokes flow on $\\mathbb R^3$ starting from rest: the particle initially at $(4,0,0)$ enters the fixed open box $(-1,2)^3$ exactly when the machine halts. One velocity works for every viscosity $\\nu>0$, with zero pressure and force $f_0+\\nu f_1$. Velocity and force have bounded mixed derivatives and are one-periodic after time one; the repeated velocity depends only on the machine. The force is computable for computable $\\nu$. Velocity is unique among smooth zero-data solutions with $u\\in C_tH_x^2\\cap C_t^1L_x^2$, bounded $u$ and $\\nabla u$ on finite time intervals, and pressure modulo a function of time in $C_tH_x^1$.", "lab_docs_url": "https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/docs/376.md", "overview_entry": "family 376 in overview.pdf / CONTENTS.md", "papers": [{"dir": "Geometric-Programs-for-Solenoidal-Forcing-September-27-2026", "title": "Geometric Programs for Solenoidal Forcing", "url": "https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Geometric-Programs-for-Solenoidal-Forcing-September-27-2026"}, {"dir": "Computation-under-Rapidly-Vanishing-Navier-Stokes-Forcing-September-27-2026", "title": "Computation under Rapidly Vanishing Navier–Stokes Forcing", "url": "https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Computation-under-Rapidly-Vanishing-Navier-Stokes-Forcing-September-27-2026"}, {"dir": "Velocity-Field-Detection-of-Computation-in-Forced-Navier-Stokes-Flows-September-27-2026", "title": "Velocity-Field Detection of Computation in Forced Navier–Stokes Flows", "url": "https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Velocity-Field-Detection-of-Computation-in-Forced-Navier-Stokes-Flows-September-27-2026"}, {"dir": "Incompressible-Box-Transport-and-Finite-Computation-September-27-2026", "title": "Incompressible Box Transport and Finite Computation", "url": "https://github.com/openai/math/tree/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Incompressible-Box-Transport-and-Finite-Computation-September-27-2026"}], "audit_part": 3, "second_reader": "", "referee_verdict": ""}], "challenges": [{"challenge": "AbhyankarSathaye", "family": "049", "family_verdict": "partial", "result_label": "Noncoordinate polynomial in every dimension at least four", "theorem_names": ["OAI.AbhyankarSathaye.exists_noncoordinate_polynomial"], 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