family stringlengths 3 3 | title stringlengths 23 85 | claim stringlengths 209 497 | abstracts listlengths 1 3 | papers listlengths 1 3 | lean_files listlengths 1 5 | lean_statements listlengths 1 5 | formalization_scope stringlengths 295 1.65k | judge_qwen3_8_27b dict | judge_neohorse_code dict |
|---|---|---|---|---|---|---|---|---|---|
012 | Independent largest prime factors of consecutive integers | Independent largest prime factors of consecutive integers. Resolves the Erdős–Pomerance joint Dickman conjecture: the logarithmic sizes of the largest prime factors of <i>n</i> and $`n+1`$ are asymptotically independent in ordinary natural density. In particular, the integers satisfying $`P^+(n)\lt P^+(n+1)`$ have den... | [
"Let $P^+(n)$ denote the largest prime factor of $n$. We prove that $\\log P^+(n)/\\log n$ and $\\log P^+(n+1)/\\log n$ are asymptotically independent in ordinary natural density, with Dickman marginals. This resolves the Erd\\H{o}s--Pomerance joint Dickman conjecture positively and implies that the ordering $P^+(n... | [
"https://github.com/openai/math/tree/main/preprints/The-joint-Dickman-law-for-consecutive-integers-September-24-2026"
] | [
"https://github.com/openai/math/blob/main/lean/ComparatorChallenges/JointDickman.lean"
] | [
"-- ComparatorChallenges/JointDickman.lean\nimport Mathlib\n\nnamespace OAI\n\n/-! # The joint Dickman law for consecutive integers -/\n\nnamespace Erdos970.NumberTheoryLean.DelayConstruction\n\nnoncomputable def kernel (f : ℝ → ℝ) (t : ℝ) : ℝ := f (t - 1) / max 1 t\n\nnoncomputable def stepApprox (σ : ℝ) : ℕ → ℝ →... | Let $P^+(n)$ be the largest prime factor of $n$. The formalization proves the joint Dickman law in ordinary natural density: for every $0<a,b<1$, the density of integers satisfying $P^+(n)\le n^a$ and $P^+(n+1)\le n^b$ tends to $\rho(1/a)\rho(1/b)$, where $\rho$ is the Dickman function.
It also proves that each of the... | {
"score": 5,
"reason": "{\"reason\": \"The Lean definitions implement ordinary natural density and the Dickman function via its delay equation. `joint_law` states the Erdős–Pomerance rectangle law: for 0<a,b<1, the density of n with P^+(n)≤n^a and P^+(n+1)≤n^b tends to ρ(1/a)ρ(1/b), which is equivalent to asymptot... | {
"score": 5,
"reason": "{\"reason\": \"The Lean formalization accurately captures the main mathematical claim. The theorem `joint_law` expresses the asymptotic independence of the normalized largest prime factors via the factorization of their joint distribution function into the product of Dickman functions, matc... |
013 | Ostmann’s inverse Goldbach conjecture | Ostmann’s inverse Goldbach conjecture. Proves that no finite modification of the primes can be written as $`A+B`$ with $`A,B\subseteq\mathbb Z_{\ge0}`$ each containing at least two elements. This resolves Ostmann's inverse Goldbach conjecture on additive indecomposability. | [
"We prove Ostmann's inverse Goldbach conjecture: no set differing from the primes by finitely many elements can be written as $A+B$, where $A$ and $B$ are sets of nonnegative integers with at least two elements each."
] | [
"https://github.com/openai/math/tree/main/preprints/the-additive-indecomposability-of-the-primes-September-24-2026"
] | [
"https://github.com/openai/math/blob/main/lean/ComparatorChallenges/OstmannComplete.lean",
"https://github.com/openai/math/blob/main/lean/ComparatorChallenges/OstmannPrimes.lean"
] | [
"-- ComparatorChallenges/OstmannComplete.lean\nimport Mathlib\n\nnamespace OAI\n\nopen scoped Pointwise symmDiff\n\nnamespace Ostmann\n\ndef primes : Set ℕ := {n | Nat.Prime n}\n\nabbrev sumset (A B : Set ℕ) : Set ℕ := A + B\n\ndef EventuallyPrimeSumset (A B : Set ℕ) : Prop :=\n ∃ N : ℕ, ∀ n, N ≤ n → (n ∈ sumset A... | The formalization proves Ostmann's inverse Goldbach conjecture. For any two sets $A,B$ of nonnegative integers, each containing at least two elements, the symmetric difference between their sumset $A+B$ and the set of primes is infinite. Thus no set differing from the primes by only finitely many elements can be decomp... | {
"score": 5,
"reason": "{\"reason\": \"The main Lean statements `inverseGoldbach` and `main` say that for all A,B ⊆ ℕ with at least two elements (`Nontrivial`), the symmetric difference of A+B and the set of primes is infinite. This is equivalent to the claim that no set differing from the primes by only finitely ... | {
"score": 5,
"reason": "{\"reason\": \"The Lean formalization faithfully expresses the main claim. The theorem `main` (and equivalently `InverseGoldbach`) states that for any nontrivial sets A and B (each containing at least two elements), the symmetric difference between their sumset A+B and the primes is infinit... |
020 | Squarefree quartics and power-free polynomial values | Squarefree quartics and power-free polynomial values. Proves the squarefree-values conjecture for irreducible integer quartics with no fixed prime-square divisor: squarefree values on positive integers have the predicted positive Euler-product density. More generally, establishes the $`(d-2)`$-power-free density for i... | [
"We prove that every irreducible integer quartic with no fixed prime-square divisor takes squarefree values with the predicted positive Euler-product density. More generally, we obtain the corresponding $(d-2)$-power-free density in degrees $4\\le d\\le8$. The proof combines number-field factorization, determinant ... | [
"https://github.com/openai/math/tree/main/preprints/Squarefree-values-of-quartics-and-power-free-values-of-polynomials-September-24-2026"
] | [
"https://github.com/openai/math/blob/main/lean/ComparatorChallenges/PowerFreeValues.lean"
] | [
"-- ComparatorChallenges/PowerFreeValues.lean\nimport Mathlib\n\nnamespace OAI\n\n\n\nopen Filter Asymptotics\nopen scoped Topology\n\nnamespace QuarticPowerFree\n\ndef PowerFree (k : ℕ) (a : ℤ) : Prop :=\n ∀ p : ℕ, p.Prime → ¬ (p : ℤ) ^ k ∣ a\n\ndef rho (f : Polynomial ℤ) (q : ℕ) : ℕ :=\n ((Finset.range q).filte... | 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... | {
"score": 5,
"reason": "{\"reason\": \"The Lean statement quantifies over integer polynomials of degree at least 4, assumes irreducibility over Q and local admissibility for k = d-2, and concludes a positive convergent Euler-product constant with count = C X + o(X). This captures the squarefree quartic case (d=4) ... | {
"score": 5,
"reason": "{\"reason\": \"The Lean theorem statement faithfully expresses the paper's main claim. It captures: (1) irreducibility over ℚ (equivalent to ℤ for primitive polynomials), (2) degree ≥ 4 covering all cases mentioned (4–8 directly, plus higher degrees via Browning's theorem as noted in the ab... |
021 | A quadratic bound for Jacobsthal’s function | A quadratic bound for Jacobsthal’s function. 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 <i>k</i> distinct prime divisors, for an absolute constant <i>C</i>. The bound is uniform over prime se... | [
"Let $h(k)$ be the least integer such that every interval of $h(k)$ consecutive integers contains an integer coprime to any prescribed positive integer having at most $k$ distinct prime divisors. We prove $h(k)\\ll k^2/(\\log\\log(3k))^2$, giving an affirmative answer to Jacobsthal's quadratic-bound question."
] | [
"https://github.com/openai/math/tree/main/preprints/A-quadratic-bound-for-Jacobsthals-function-September-25-2026"
] | [
"https://github.com/openai/math/blob/main/lean/ComparatorChallenges/Jacobsthal.lean",
"https://github.com/openai/math/blob/main/lean/ComparatorChallenges/JacobsthalImproved.lean"
] | [
"-- ComparatorChallenges/Jacobsthal.lean\nimport Mathlib\n\nnamespace OAI\n\nnamespace Erdos970\n\nnamespace NumberTheoryLean.Targets\n\nopen Filter\nopen scoped Topology\n\ndef IsJacobsthalBound (k m : ℕ) : Prop :=\n ∀ n : ℕ, 0 < n → n.primeFactors.card ≤ k →\n ∀ a : ℤ, ∃ i : ℕ, i < m ∧ (a + i).natAbs.Coprime ... | 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 s... | {
"score": 5,
"reason": "{\"reason\": \"The definition IsJacobsthalBound correctly encodes the uniform Jacobsthal property: for every positive n with at most k distinct prime factors and every integer starting point a, one of a, ..., a+m-1 is coprime to n, using natAbs to handle negative integers. The quadratic the... | {
"score": 5,
"reason": "{\"reason\": \"The Lean formalization faithfully expresses the main claim. The `IsJacobsthalBound` definition correctly captures the requirement that every interval of `m` consecutive integers contains an integer coprime to any `n` with at most `k` distinct prime factors. The `JacobsthalQua... |
025 | Short Egyptian fractions | Short Egyptian fractions. Every rational $`a/b`$ with $`1\le a\lt b`$ is a sum of $`O(\log\log b)`$ distinct positive unit fractions. The worst-case minimum number of terms has the same order, resolving Erdős’s conjecture on short Egyptian fractions. | [
"We prove a conjecture of Erd\\H{o}s: for every sufficiently large integer $b$, every rational number $a/b$ with $1\\le a<b$ is a sum of $O(\\log\\log b)$ distinct positive unit fractions, with an absolute implied constant. This order is best possible when the numerator varies. We also show that both the number of ... | [
"https://github.com/openai/math/tree/main/preprints/Short-Egyptian-fractions-September-25-2026"
] | [
"https://github.com/openai/math/blob/main/lean/ComparatorChallenges/EgyptianFractions.lean",
"https://github.com/openai/math/blob/main/lean/ComparatorChallenges/ShortEgyptianFractions.lean"
] | [
"-- ComparatorChallenges/EgyptianFractions.lean\nimport Mathlib\n\nnamespace OAI\n\nnoncomputable section\n\nopen scoped BigOperators\n\nnamespace Problem337\n\ndef IsEgyptianExpansion {k : ℕ} (x : ℚ) (n : Fin k → ℕ) : Prop :=\n (∀ i, 2 ≤ n i) ∧ StrictMono n ∧\n (∑ i : Fin k, (1 : ℚ) / (n i : ℚ)) = x\n\ndef egy... | An Egyptian-fraction expansion writes a rational number as a sum of distinct unit fractions. The formalization proves that every $a/b$ with $1\le a<b$ has such an expansion and that the largest minimum length at denominator $b$ is $\Theta(\log\log b)$. If $F(k)$ counts exact $k$-term expansions of $1$, it also proves $... | {
"score": 5,
"reason": "{\"reason\": \"The ShortEgyptianFractions.main theorem directly states that for all 1≤a<b there is a strictly increasing list of denominators ≥2 whose reciprocal sum is a/b, and that the maximum over a of the minimum such length is Θ(log log b) for large b. This matches the main Erdős claim... | {
"score": 5,
"reason": "{\"reason\": \"The Lean statements faithfully formalize the main claim. The `ShortEgyptian` file's `theorem main` directly asserts the existence of expansions for all valid $a/b$ and proves the tight asymptotic bound $\\Theta(\\log\\log b)$ on the worst-case minimum length, matching the cla... |
028 | Uniformly bounded components of Gaussian-prime graphs | Uniformly bounded components of Gaussian-prime graphs. Proves the Gaussian moat conjecture: no infinite walk through distinct Gaussian primes can have uniformly bounded steps. More strongly, for every distance bound <i>D</i>, the graph joining Gaussian primes at distance at most <i>D</i> has uniformly bounded finite c... | [
"We prove the Gaussian moat conjecture: no infinite walk through distinct Gaussian primes can have bounded steps. More strongly, for each fixed finite step bound, the connected components of the Gaussian-prime graph have uniformly bounded size. This bound applies to every starting prime, including primes on the coo... | [
"https://github.com/openai/math/tree/main/preprints/Bounded-Step-Walks-on-Gaussian-Primes-September-26-2026"
] | [
"https://github.com/openai/math/blob/main/lean/ComparatorChallenges/GaussianMoat.lean"
] | [
"-- ComparatorChallenges/GaussianMoat.lean\nimport Mathlib\n\nnamespace OAI\n\nnamespace GaussianMoat\n\nabbrev PrimeVertex := {z : GaussianInt // Irreducible z}\n\nnoncomputable def primeGraph (D : ℝ) : SimpleGraph PrimeVertex where\n Adj p q := p ≠ q ∧ dist (p.val : ℂ) (q.val : ℂ) ≤ D\n symm := by\n construc... | The Gaussian moat problem asks whether an infinite walk through distinct Gaussian primes can have bounded step lengths. The formalized result gives a negative answer for every real step bound $D$. More strongly, one finite bound depending only on $D$ limits the size of every connected component of the bounded-step grap... | {
"score": 5,
"reason": "{\"reason\": \"The Lean statement defines the Gaussian-prime graph for each real D on irreducible Gaussian integers with Euclidean distance at most D, and UniformEndpoint asserts a D-dependent natural bound on all component cardinalities, matching the stronger uniform-component claim. MainE... | {
"score": 5,
"reason": "{\"reason\": \"The Lean formalization accurately captures both parts of the paper's claim. `MainEndpoint` corresponds to the non-existence of an infinite walk through distinct Gaussian primes with uniformly bounded steps, and `UniformEndpoint` corresponds to the uniform boundedness of the s... |
073 | The Falconer distance conjecture | The Falconer distance conjecture. Resolves the Falconer distance conjecture in every dimension <i>d</i> ≥ 2: 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. | [
"We resolve the Falconer distance conjecture in every dimension. For every integer \\(d\\ge2\\), a compact subset of \\(\\R^d\\) with Hausdorff dimension greater than \\(d/2\\) determines a set of Euclidean distances of positive Lebesgue measure."
] | [
"https://github.com/openai/math/tree/main/preprints/The-Falconer-distance-conjecture-in-all-dimensions-September-23-2026"
] | [
"https://github.com/openai/math/blob/main/lean/ComparatorChallenges/FalconerAllDimensions.lean",
"https://github.com/openai/math/blob/main/lean/ComparatorChallenges/PlanarFalconer.lean"
] | [
"-- ComparatorChallenges/FalconerAllDimensions.lean\nimport Mathlib\n\nnamespace OAI\n\nnamespace Falconer\nopen MeasureTheory\nopen scoped ENNReal\n\n/-- A compact subset of Euclidean space with Hausdorff dimension greater than\nhalf the ambient dimension determines a distance set of positive Lebesgue measure. -/\... | 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... | {
"score": 5,
"reason": "{\"reason\": \"The first Lean theorem states exactly the full Falconer distance conjecture: for every natural d ≥ 2, every compact E ⊆ ℝ^d with dimH E > d/2 has a distance set of positive Lebesgue measure. The types, quantifiers, compactness hypothesis, Hausdorff dimension threshold, and po... | {
"score": 5,
"reason": "{\"reason\": \"The Lean statement faithfully captures the Falconer distance conjecture for all dimensions d ≥ 2. It correctly uses EuclideanSpace ℝ (Fin d) for ℝ^d, requires IsCompact E, imposes the strict Hausdorff dimension condition dimH E > d/2 via (d : ℝ≥0∞) / 2 < dimH E, and concludes... |
099 | The sharp exponential scale of edit-distance distortion | The sharp exponential scale of edit-distance distortion. Determines the least distortion of embedding edit distance on words of length at most <i>d</i> into real <i>ℓ</i><sub>1</sub>: it is $`\exp(\Theta(\sqrt{\log d\,\log\log d}))`$. Insertions, deletions and substitutions have unit cost. The constants are uniform ov... | [
"We determine the exponential scale of the least $\\ell_1$ distortion of unit-cost edit distance on all strings of length at most $d$. For every sufficiently large $d$, uniformly over finite alphabets of size at least two, the distortion lies between $\\exp(c\\sqrt{\\log d\\,\\log\\log d})$ and $\\exp(C\\sqrt{\\log... | [
"https://github.com/openai/math/tree/main/preprints/Edit-Distance-in-l1-Matching-Bounds-up-to-Constants-in-the-Exponent-September-27-2026",
"https://github.com/openai/math/tree/main/preprints/Finite-Circle-Obstructions-Binary-Codes-and-Histogram-Embeddings-for-Edit-Distance-September-27-2026",
"https://github.c... | [
"https://github.com/openai/math/blob/main/lean/ComparatorChallenges/BinaryEditLower.lean",
"https://github.com/openai/math/blob/main/lean/ComparatorChallenges/EditDistance.lean",
"https://github.com/openai/math/blob/main/lean/ComparatorChallenges/FiniteCircle.lean",
"https://github.com/openai/math/blob/main/l... | [
"-- ComparatorChallenges/BinaryEditLower.lean\nimport Mathlib\n\nnamespace OAI\n\nnamespace TreeEdit\n\nuniverse u\n\n/-- A single unit-cost insertion, deletion, or substitution at any position. -/\ninductive EditStep {α : Type u} : List α → List α → Prop\n | insert (p q : List α) (a : α) : EditStep (p ++ q) (p ++... | 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{\l... | {
"score": 5,
"reason": "{\"reason\": \"The EditDistance.MainClaim theorem states the full two-sided exp(sqrt(log d log log d)) bound for the least injective ℓ₁ distortion of unit-cost edit distance on all words of length ≤ d, uniformly over all finite alphabets of size at least two, and it also requires a finite b... | {
"score": 5,
"reason": "{\"reason\": \"The Lean formalization faithfully captures the main claim. The `edit_distance_main` theorem establishes uniform two-sided bounds on the least ℓ₁ distortion of unit-cost edit distance on words of length at most d, with constants c, C > 0 and threshold d₀, matching the exp(Θ(√(... |
106 | Hardness of coloring three-colorable graphs | Hardness of coloring three-colorable graphs. It is NP-hard to color a three-colorable graph using any fixed number <i>c</i> ≥ 3 of colors. More strongly, for every fixed $`0\lt \delta\lt 1/3`$, a deterministic polynomial-time reduction from 3SAT produces simple unweighted graphs that are three-colorable in the satisfi... | [
"We prove that, for every fixed $0<\\delta<1/3$, it is NP-hard to distinguish three-colorable graphs from graphs in which every independent set has fewer than $\\delta$ times the number of vertices. Consequently, for every fixed integer $c\\ge3$, finding a proper $c$-coloring of a three-colorable graph is NP-hard."... | [
"https://github.com/openai/math/tree/main/preprints/Hardness-of-finding-large-independent-sets-in-three-colorable-graphs-September-24-2026"
] | [
"https://github.com/openai/math/blob/main/lean/ComparatorChallenges/IndependentSets.lean"
] | [
"-- ComparatorChallenges/IndependentSets.lean\nimport Mathlib\n\nnamespace OAI\n\nnamespace LargeIndependentSets\n\nstructure Literal where\n name : ℕ\n positive : Bool\n deriving DecidableEq\n\nstructure Formula where\n clauses : List (List Literal)\n width : ∀ C ∈ clauses, C.length ≤ 3\n\ndef Formula.Satisfi... | The formalized result shows hardness of finding large independent sets even under a three-colorability promise. For every fixed $0<\delta<1/3$, a deterministic polynomial-time reduction maps binary 3SAT formulas to nonempty finite simple graphs. Satisfiable formulas produce three-colorable graphs; unsatisfiable formula... | {
"score": 5,
"reason": "{\"reason\": \"The Lean statement formalizes the stronger gap reduction: for every real δ with 0<δ<1/3 there exists a deterministic polynomial-time reduction from encoded 3SAT formulas to finite simple unweighted graphs, with satisfiable formulas mapped to 3-colorable graphs and unsatisfiab... | {
"score": 5,
"reason": "{\"reason\": \"The Lean statement faithfully formalizes the core technical result of the main claim: the existence of a deterministic polynomial-time reduction from 3SAT that produces three-colorable graphs for satisfiable instances and graphs with independence number < δn for unsatisfiable... |
108 | A cubic permanent–determinant lower bound | A cubic permanent–determinant lower bound. Proves an $`\Omega(n^3)`$ lower bound for the border determinantal complexity of the $`n\times n`$ permanent over ℂ. Even coefficientwise limits of determinants of affine-linear matrices require matrix size at least $`cn^3`$, for an absolute <i>c</i> > 0 and all sufficient... | [
"We prove that the complex border determinantal complexity of the $m\\times m$ permanent is $\\Omega(m^3)$, allowing arbitrary affine-linear determinant representations and coefficientwise limits. It also gives cubic lower bounds for exact determinantal complexity and for the numbers of vertices and edges in affine... | [
"https://github.com/openai/math/tree/main/preprints/A-cubic-lower-bound-for-border-determinantal-complexity-of-the-permanent-September-24-2026"
] | [
"https://github.com/openai/math/blob/main/lean/ComparatorChallenges/PermanentCubic.lean",
"https://github.com/openai/math/blob/main/lean/ComparatorChallenges/SmoothInitialForm.lean"
] | [
"-- ComparatorChallenges/PermanentCubic.lean\nimport Mathlib\n\nnamespace OAI\n\nnamespace PermanentBorder\n\nopen Filter\nopen scoped Topology\n\nabbrev Variables (m : ℕ) := Fin m × Fin m\nabbrev Poly (σ : Type*) := MvPolynomial σ ℂ\n\nnoncomputable def permanentPolynomial (m : ℕ) : Poly (Variables m) :=\n ∑ τ : ... | 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 $n\ge m^3/(5529600e)$.
A general supporting theorem applies ... | {
"score": 5,
"reason": "{\"reason\": \"The main Lean theorem exactly formalizes the stated border and exact cubic lower bounds for the permanent: if an affine-linear determinant representation or coefficientwise limit of size n exists, then n is at least a positive constant times m^3 for all m≥1408. The definition... | {
"score": 5,
"reason": "{\"reason\": \"The Lean formalization faithfully expresses the main claim. The `permanent_cubic_lower_bounds` theorem directly addresses the Ω(m³) lower bound for the border and exact determinantal complexity of the permanent over ℂ, with the correct asymptotic form n ≥ cm³ (where c = 1/(55... |
118 | Bin packing and unbounded configuration-LP gaps | Bin packing and unbounded configuration-LP gaps. Disproves the modified integer round-up conjecture of Scheithauer and Terno: the integral bin-packing optimum can exceed its configuration linear-programming value by an arbitrarily large additive constant. Approximating the optimum within any fixed additive constant is... | [
"We disprove the Modified Integer Round-Up Conjecture for bin packing by constructing instances with arbitrarily large additive gaps between the configuration-LP value and the integral optimum. We also prove that, for every fixed nonnegative integer $c$, distinguishing instances that fit in $B$ bins from those requ... | [
"https://github.com/openai/math/tree/main/preprints/Additive-hardness-and-unbounded-configuration-gaps-in-bin-packing-September-24-2026"
] | [
"https://github.com/openai/math/blob/main/lean/ComparatorChallenges/BinPackingGap.lean"
] | [
"-- ComparatorChallenges/BinPackingGap.lean\nimport Mathlib\n\nnamespace OAI\n\nnoncomputable section\n\nnamespace BinPackingGames.Foundations.Complexity\n\ndef encodeWord (n : Nat) : List Bool := List.replicate n true ++ [false]\n\nend BinPackingGames.Foundations.Complexity\n\nnamespace BinPackingGames.Foundations... | 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 th... | {
"score": 5,
"reason": "{\"reason\": \"The first conjunct gives arbitrarily large gaps: for every c there is an instance with 5B items, sizes >1/6, both individual and type configuration-LP values equal to B, and integral optimum > B+c. The second conjunct formalizes NP-hardness of the B vs B+c bin-packing gap for... | {
"score": 5,
"reason": "{\"reason\": \"The Lean formalization faithfully captures all components of the paper's main claim. The `main_results` theorem asserts: (1) for every c, there exist instances with n=5B items, each size in (1/6, 1), where opt I > B + c while both individualLP and typeLP equal B (disproving t... |
125 | The metric <i>k</i>-median approximation threshold and recovery | The metric <i>k</i>-median approximation threshold and recovery. Gives a deterministic polynomial-time $`(1+2/e+\varepsilon)`$-approximation for finite rational metric <i>k</i>-median with specified candidate facilities, for every fixed <i>ε</i> > 0. Assuming $`P\ne NP`$, the optimal infimum approximation factor is... | [
"We give an exact-budget recovery algorithm for metric $k$-median with single-exponential dependence on the number of comparison clusters without accurate, distinct proxies in a supplied anchor solution. On positive integral metrics of polynomially bounded diameter, a sufficiently small total proxy error and logari... | [
"https://github.com/openai/math/tree/main/preprints/Single-Exponential-Recovery-and-Bounded-Price-Strictness-for-Metric-k-Median-September-24-2026",
"https://github.com/openai/math/tree/main/preprints/The-Approximation-Threshold-for-Metric-k-Median-September-24-2026"
] | [
"https://github.com/openai/math/blob/main/lean/ComparatorChallenges/KMedianRecovery.lean",
"https://github.com/openai/math/blob/main/lean/ComparatorChallenges/KMedianRefinedRecovery.lean",
"https://github.com/openai/math/blob/main/lean/ComparatorChallenges/KMedianThreshold.lean",
"https://github.com/openai/ma... | [
"-- ComparatorChallenges/KMedianRecovery.lean\nimport Mathlib\n\nnamespace OAI\n\nnamespace MetricKMedianRecovery\n\nstructure RationalMetricInput where\n pointCount : ℕ\n distance : Fin pointCount → Fin pointCount → ℚ\n clients : Finset (Fin pointCount)\n facilities : Finset (Fin pointCount)\n budget : ℕ\n f... | 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 rel... | {
"score": 5,
"reason": "{\"reason\": \"The statements main_finite_work and threshold together capture the catalogue claim: for every ε>0 there is a deterministic finite-alphabet polynomial-time algorithm outputting a feasible set of at most k candidate facilities with cost ≤(1+2/e+ε)OPT, and under a formal P≠NP th... | {
"score": 5,
"reason": "{\"reason\": \"The Lean formalization faithfully captures both parts of the main claim: (1) the deterministic polynomial-time (1+2/e+ε)-approximation is expressed by `main_finite_work` in `MetricKMedian.lean`, which asserts existence of a deterministic poly-time algorithm achieving the appr... |
134 | Generalized star height at most three | Generalized star height at most three. Every regular language over a finite alphabet has a generalized regular expression with at most three nested Kleene stars, allowing union, concatenation and complement over the same alphabet. This establishes an absolute bound independent of automaton size, resolving the uniform-... | [
"Every regular language over a finite alphabet has a generalized regular expression of star height at most thirteen over that same alphabet. We prove this uniform bound by representing finite monoid computations as affine updates and recovering them through twelve successive split constructions.",
"Every regular ... | [
"https://github.com/openai/math/tree/main/preprints/Finite-Monoid-Computations-and-a-Uniform-Generalized-Star-Height-Bound-September-25-2026",
"https://github.com/openai/math/tree/main/preprints/Generalized-Star-Height-at-Most-Four-September-25-2026",
"https://github.com/openai/math/tree/main/preprints/Generali... | [
"https://github.com/openai/math/blob/main/lean/ComparatorChallenges/GeneralizedStarHeight.lean"
] | [
"-- ComparatorChallenges/GeneralizedStarHeight.lean\nimport Mathlib\n\nnamespace OAI\n\nnamespace GeneralizedStarHeight\n\nuniverse u\n\ninductive Expression (Alphabet : Type u) : Type u\n | zero : Expression Alphabet\n | one : Expression Alphabet\n | letter : Alphabet → Expression Alphabet\n | union : Expressi... | 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 thirtee... | {
"score": 5,
"reason": "{\"reason\": \"The Lean statement quantifies over every finite alphabet and every regular language L, and asserts the existence of an expression whose denoted language is exactly L and whose star height is at most 3. The Expression language includes union, concatenation, complement, and Kle... | {
"score": 5,
"reason": "{\"reason\": \"The Lean theorem `main` precisely formalizes the main claim. It asserts that for any finite alphabet and any regular language over it, there exists an expression in the defined `Expression` type with star height at most 3 (measured by the `height` function, where `star` adds ... |
139 | Subpolynomial query complexity for log-concave sampling | Subpolynomial query complexity for log-concave sampling. For <i>C</i><sup>2</sup> 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 <i>ε</i> > 0. The... | [
"For every fixed $\\varepsilon>0$, we give a sampling algorithm using at most $C_\\varepsilon d^\\varepsilon$ exact first-order queries on every execution for $C^2$ potentials on $\\R^d$ with a known minimizer and Hessian between $I_d$ and $2I_d$. The output has total-variation distance at most $1/10$ from the targ... | [
"https://github.com/openai/math/tree/main/preprints/Subpolynomial-query-complexity-for-well-conditioned-log-concave-sampling-September-26-2026"
] | [
"https://github.com/openai/math/blob/main/lean/ComparatorChallenges/LogConcaveQuery.lean"
] | [
"-- ComparatorChallenges/LogConcaveQuery.lean\nimport Mathlib\n\nnamespace OAI\n\nnoncomputable section\n\nopen MeasureTheory Filter\nopen scoped ENNReal NNReal Topology\n\nnamespace LogConcaveSampling\n\nabbrev Point (d : ℕ) := EuclideanSpace ℝ (Fin d)\nabbrev Reply (d : ℕ) := ℝ × Point d\nabbrev Transcript (d q :... | 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_\varepsi... | {
"score": 5,
"reason": "{\"reason\": \"The Lean statement formalizes adaptive randomized algorithms with a fixed deterministic query budget, exact value-and-gradient replies, admissible C^2 potentials with Hessian bounds, and TV error 1/10. It states the desired C_ε d^ε upper bound via the minimal feasible query b... | {
"score": 5,
"reason": "{\"reason\": \"The Lean formalization faithfully captures the main claim. The `Admissible` structure correctly encodes C² potentials with a known minimizer (at 0) and Hessian bounded between I and 2I. The `CanSample` definition properly models randomized adaptive algorithms with a fixed que... |
158 | The Euclidean plane cannot be colored with five colors | The Euclidean plane cannot be colored with five colors. Proves that every five-coloring of the Euclidean plane has a monochromatic pair at distance one, with no restriction on the color classes. This advances the Hadwiger–Nelson problem: together with the classical seven-coloring, only six and seven remain possible ch... | [
"We prove that every coloring of the Euclidean plane with five colors has a monochromatic unit-distance pair, with no regularity assumption on the color classes. Consequently, the chromatic number of the plane is either six or seven."
] | [
"https://github.com/openai/math/tree/main/preprints/The-Euclidean-plane-is-not-five-colorable-September-23-2026"
] | [
"https://github.com/openai/math/blob/main/lean/ComparatorChallenges/EuclideanFiveColor.lean",
"https://github.com/openai/math/blob/main/lean/ComparatorChallenges/PlaneColoring.lean"
] | [
"-- ComparatorChallenges/EuclideanFiveColor.lean\nimport Mathlib\n\nnamespace OAI\n\nnamespace EuclideanFiveColor\n\ndef ProperColoring (colorCount : ℕ) (coloring : ℂ → Fin colorCount) : Prop :=\n ∀ point otherPoint : ℂ, ‖point - otherPoint‖ = 1 → coloring point ≠ coloring otherPoint\n\ntheorem no_proper_five_colo... | 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 assump... | {
"score": 5,
"reason": "{\"reason\": \"The theorem no_proper_five_coloring states that there is no map ℂ → Fin 5 such that all unit-distance pairs receive different colors, which is exactly the claim that every five-coloring of the Euclidean plane has a monochromatic unit-distance pair. The definition of ProperCol... | {
"score": 5,
"reason": "{\"reason\": \"The Lean statement in EuclideanFiveColor.lean precisely formalizes the main claim: it asserts that no proper 5-coloring of the Euclidean plane exists (¬∃ coloring : ℂ → Fin 5, ProperColoring 5 coloring), which is logically equivalent to 'every five-coloring has a monochromati... |
162 | Counterexamples to Ryser’s covering conjecture | Counterexamples to Ryser’s covering conjecture. Disproves Ryser's covering conjecture by constructing intersecting $`(q+1)`$-partite, $`(q+1)`$-uniform hypergraphs with covering number $`q+1`$, rather than the predicted bound <i>q</i>, for every sufficiently large prime <i>q</i>. A separate construction over extension... | [
"For every sufficiently large prime $s\\equiv2\\pmod3$ and every sufficiently large odd integer $n$, with the threshold depending on $s$, we construct an intersecting $(s^n+1)$-partite $(s^n+1)$-uniform hypergraph with covering number $s^n+1$. This disproves Ryser's covering conjecture in its intersecting case.",
... | [
"https://github.com/openai/math/tree/main/preprints/A-Counterexample-to-Rysers-Covering-Conjecture-September-23-2026",
"https://github.com/openai/math/tree/main/preprints/Balanced-Counterexamples-to-Rysers-Conjecture-at-Prime-Orders-September-27-2026"
] | [
"https://github.com/openai/math/blob/main/lean/ComparatorChallenges/BalancedRyser.lean",
"https://github.com/openai/math/blob/main/lean/ComparatorChallenges/RyserCovering.lean",
"https://github.com/openai/math/blob/main/lean/ComparatorChallenges/RyserOddExtensions.lean"
] | [
"-- ComparatorChallenges/BalancedRyser.lean\nimport Mathlib\n\nnamespace OAI\n\nnamespace Balanced\nopen scoped _root_.Balanced\n\nstructure PartiteHypergraph (r n : ℕ) where\n edges : Finset (Fin r → Fin n)\n\nnamespace PartiteHypergraph\n\nvariable {r n : ℕ}\n\nabbrev Vertex (r n : ℕ) := Fin r × Fin n\n\ndef Cov... | Ryser's covering conjecture predicts that an intersecting $r$-partite hypergraph has a vertex cover of size at most $r-1$. The formalization proves that every sufficiently large prime $q$ has a finite intersecting $(q+1)$-partite, $(q+1)$-uniform hypergraph with covering number $q+1$ and exactly $q+1$ nonisolated verti... | {
"score": 5,
"reason": "{\"reason\": \"The core theorem BalancedRyser.main_result exactly formalizes the main Ryser counterexample: for all sufficiently large primes q it asserts a finite (q+1)-partite, (q+1)-uniform hypergraph, encoded as a Finset of functions Fin (q+1) → Fin (q+1), that is intersecting, has all ... | {
"score": 5,
"reason": "{\"reason\": \"The Lean theorem statements faithfully express the main claim and the supporting constructions described in the abstracts. The `Balanced.lean` file correctly formalizes the main claim: for every sufficiently large prime $q$, there exists an intersecting $(q+1)$-partite $(q+1)... |
173 | Seymour’s second-neighborhood conjecture | Seymour’s second-neighborhood conjecture. Proves Seymour's second-neighborhood conjecture: every nonempty finite oriented graph has a vertex with at least as many vertices at directed distance exactly two as at directed distance one. Oriented graphs may be arbitrary apart from the exclusion of loops and oppositely dir... | [
"We prove that every nonempty finite oriented graph has a vertex with at least as many vertices at directed distance two as at directed distance one. This resolves Seymour's second neighborhood conjecture positively."
] | [
"https://github.com/openai/math/tree/main/preprints/A-proof-of-Seymours-second-neighborhood-conjecture-September-23-2026"
] | [
"https://github.com/openai/math/blob/main/lean/ComparatorChallenges/SeymourSecondNeighborhood.lean"
] | [
"-- ComparatorChallenges/SeymourSecondNeighborhood.lean\nimport Mathlib.Data.Fintype.Card\nimport Mathlib.Data.Finset.Union\n\nnamespace OAI\n\nnamespace SeymourSecondNeighborhood\n\nvariable {V : Type*}\n\nstructure IsOriented (r : V → V → Prop) : Prop where\n loopless : ∀ v, ¬ r v v\n asymmetric : ∀ {u v}, r u ... | 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 ... | {
"score": 5,
"reason": "{\"reason\": \"The Lean theorem quantifies over an arbitrary finite nonempty type V and an arbitrary relation r satisfying loopless and asymmetric, which exactly captures finite oriented graphs. firstNeighbors is the out-neighborhood, and secondNeighbors is the set of vertices distinct from... | {
"score": 5,
"reason": "{\"reason\": \"The Lean statement accurately formalizes Seymour's second-neighborhood conjecture. The definitions of `firstNeighbors` and `secondNeighbors` correctly capture the sets of vertices at directed distance exactly one and exactly two, respectively, with `secondNeighbors` properly ... |
181 | The Erdős–Gallai cycle-decomposition conjecture | The Erdős–Gallai cycle-decomposition conjecture. Proves that the edges of every finite simple undirected graph on <i>n</i> vertices can be partitioned into at most $`Cn`$ simple cycles and single edges, for an absolute constant <i>C</i>. This resolves the Erdős–Gallai cycle-decomposition conjecture, bounding the numbe... | [
"We prove that every finite simple undirected graph on \\(n\\) vertices has an edge partition 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 positively."
] | [
"https://github.com/openai/math/tree/main/preprints/A-linear-cycle-and-edge-decomposition-of-every-graph-September-24-2026"
] | [
"https://github.com/openai/math/blob/main/lean/ComparatorChallenges/CycleDecomposition.lean"
] | [
"-- ComparatorChallenges/CycleDecomposition.lean\nimport Mathlib\n\nnamespace OAI\n\nnoncomputable section\n\n\nopen Filter Asymptotics Real\nopen scoped Topology\n\nopen MeasureTheory ProbabilityTheory Finset\n\nnamespace ErdosGallai\n\ndef CycleOrSingleEdge {V : Type} (G : SimpleGraph V)\n (s : Set (Sym2 V)) :... | 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... | {
"score": 5,
"reason": "{\"reason\": \"The Lean statement quantifies over all simple graphs on Fin n and asserts the existence of a positive real constant C, independent of the graph, such that the edge set can be covered by k pairwise disjoint sets, each either the edge set of a cycle walk or a singleton edge, wi... | {
"score": 5,
"reason": "{\"reason\": \"The Lean formalization faithfully captures the Erdős–Gallai cycle-decomposition conjecture. The definition `CycleOrSingleEdge` correctly characterizes a set of edges as either a simple cycle or a single edge. The `EdgeDecomposition` definition properly encodes an edge partiti... |
187 | Snaky in 21 Maker moves | Snaky in 21 Maker moves. Settles the Snaky achievement problem: Maker can force the six-cell Snaky shape within 21 of its own moves on the initially empty infinite square board. Maker moves first, each player claims one free cell per turn, and translations, rotations and reflections count as wins. | [
"We prove that Maker can achieve the Snaky hexomino within 21 actual Maker moves against arbitrary legal Breaker play on the initially empty infinite square board. The same bound holds on a $17\\times17$ square; in fact, Maker can confine its claims to a fixed $251$-cell board."
] | [
"https://github.com/openai/math/tree/main/preprints/Snaky-in-21-Maker-moves-September-25-2026"
] | [
"https://github.com/openai/math/blob/main/lean/ComparatorChallenges/SnakyCertificate.lean",
"https://github.com/openai/math/blob/main/lean/ComparatorChallenges/SnakyConditional.lean",
"https://github.com/openai/math/blob/main/lean/ComparatorChallenges/SnakyTwentyOne.lean"
] | [
"-- ComparatorChallenges/SnakyCertificate.lean\nimport Mathlib\n\nnamespace OAI\n\nnamespace SnakyCertificate\n\ninstance rotationModulus_neZero : NeZero 8 := @Nat.instNeZeroSucc 7\n\ninstance digitModulus_neZero : NeZero 10 := @Nat.instNeZeroSucc 9\n\nabbrev Cell := ℤ × ℤ\n\ndef origin : Cell := (0, 0)\n\ndef snak... | 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. T... | {
"score": 5,
"reason": "{\"reason\": \"The 21-move theorem exactly asserts a Maker strategy on ℤ² such that, for every Breaker sequence whose first 20 replies are legal, after 21 Maker moves the Maker set contains a translated/rotated/reflected copy of the six-cell snaky. The definitions of Cell, snaky, orient, pl... | {
"score": 5,
"reason": "{\"reason\": \"The main claim asserts that Maker can force the Snaky shape within 21 of its own moves. The Lean file `SnakyTwentyOne.lean` contains the theorem `snaky_winning_strategy_21_with_legal_states`, which formally states the existence of a strategy `σ` such that for any legal Breake... |
196 | A counterexample to Kaplansky’s zero-divisor conjecture | A counterexample to Kaplansky’s zero-divisor conjecture. Constructs a finitely presented torsion-free group <i>G</i> 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. | [
"We disprove Kaplansky's zero-divisor conjecture by constructing a finitely presented torsion-free group \\(G\\) for which \\(\\mathbb F_2[G]\\) has nonzero zero divisors. The group admits a finite two-dimensional classifying space."
] | [
"https://github.com/openai/math/tree/main/preprints/A-Torsion-Free-Group-Algebra-with-Zero-Divisors-September-23-2026"
] | [
"https://github.com/openai/math/blob/main/lean/ComparatorChallenges/TorsionFreeZeroDivisors.lean"
] | [
"-- ComparatorChallenges/TorsionFreeZeroDivisors.lean\nimport Mathlib\n\nnamespace OAI\n\nnamespace TorsionFreeZeroDivisors\n\n/-- Ordinary group torsion-freeness, without requiring uniqueness of all roots. -/\ndef TorsionFree (G : Type) [Group G] : Prop :=\n ∀ (g : G) (n : ℕ), 0 < n → g ^ n = 1 → g = 1\n\n/-- A c... | 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.
The same group a... | {
"score": 5,
"reason": "{\"reason\": \"The Lean statement asserts the existence of a group G that is finitely presented, torsion-free, has a finite 2-dimensional CW complex with fundamental group isomorphic to G and a contractible covering space (equivalently a K(G,1)), and has nonzero elements α,β in MonoidAlgebr... | {
"score": 5,
"reason": "{\"reason\": \"The Lean formalization accurately captures all components of the mathematical claim. The `MainTheorem` asserts the existence of a group G that is finitely presented, torsion-free, admits a finite two-dimensional classifying space (formalized via a finite 2-dimensional K(G,1) ... |
198 | A counterexample to finitistic-dimension finiteness | A counterexample to finitistic-dimension finiteness. Constructs a finite-dimensional complex algebra whose finite-dimensional modules have unbounded finite projective dimensions. This disproves the little finitistic-dimension conjecture. | [
"We construct a finite-dimensional complex algebra whose finite-dimensional modules have unbounded finite projective dimensions. This disproves the little finitistic-dimension conjecture."
] | [
"https://github.com/openai/math/tree/main/preprints/An-algebra-of-infinite-little-finitistic-dimension-September-23-2026"
] | [
"https://github.com/openai/math/blob/main/lean/ComparatorChallenges/FinitisticAsymmetry.lean",
"https://github.com/openai/math/blob/main/lean/ComparatorChallenges/LittleFinitistic.lean"
] | [
"-- ComparatorChallenges/FinitisticAsymmetry.lean\nimport Mathlib\n\nnamespace OAI\n\nnoncomputable section\nopen CategoryTheory\n\nnamespace LittleFinitistic\nuniverse u\nvariable (A : Type u) [Ring A]\n\nattribute [-instance] CompletePartialOrder.toSupSet in\ndef littleFinitisticDimension : WithBot ℕ∞ :=\n ⨆ (M ... | 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... | {
"score": 5,
"reason": "{\"reason\": \"The definition of littleFinitisticDimension matches the usual supremum over finitely generated modules with finite projective dimension, and for a finite-dimensional complex algebra finite generation is equivalent to finite-dimensionality. The theorem exists_counterexample di... | {
"score": 5,
"reason": "{\"reason\": \"The Lean statement `exists_counterexample` in `LittleFinitistic.lean` faithfully formalizes the main claim. It asserts the existence of a finite-dimensional complex algebra $A$ such that `littleFinitisticDimension A = ⊤` (meaning the supremum of finite projective dimensions o... |
213 | Critical percolation on every quasi-transitive graph | Critical percolation on every quasi-transitive graph. Resolves the Benjamini–Schramm criticality conjecture for bond percolation on every infinite connected locally finite quasi-transitive graph with $`p_c\lt 1`$: at the critical probability, there is almost surely no infinite cluster. The family also establishes this... | [
"We prove that nearest-neighbor Bernoulli bond and site percolation on $\\mathbb Z^3$ have no infinite cluster at their respective critical parameters. The proof combines a finite connection inequality for independent hyperedges with a finite-scale extension estimate and an adaptive exploration.",
"We prove that ... | [
"https://github.com/openai/math/tree/main/preprints/Critical-bond-and-site-percolation-on-the-cubic-lattice-September-24-2026",
"https://github.com/openai/math/tree/main/preprints/No-percolation-at-criticality-on-quasi-transitive-graphs-September-24-2026"
] | [
"https://github.com/openai/math/blob/main/lean/ComparatorChallenges/CriticalPercolation.lean",
"https://github.com/openai/math/blob/main/lean/ComparatorChallenges/CriticalZ3.lean"
] | [
"-- ComparatorChallenges/CriticalPercolation.lean\nimport Mathlib\n\nnamespace OAI\n\nopen Set MeasureTheory ProbabilityTheory\n\nnamespace CriticalPercolation\n\n/-- An undirected graph with labelled bonds, including loops and parallel bonds. -/\nstructure BondGraph (V E : Type*) where\n ends : E → Sym2 V\n\nname... | 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 belo... | {
"score": 5,
"reason": "{\"reason\": \"The general theorem exactly formalizes the quasi-transitive bond percolation claim: it assumes an infinite, connected, locally finite, quasi-transitive graph with critical probability below one and concludes that the critical Bernoulli measure of the event of an infinite open... | {
"score": 5,
"reason": "{\"reason\": \"The Lean formalization faithfully expresses the main claim. The first file's theorem states that for any infinite connected locally finite quasi-transitive graph with critical probability less than 1, the probability of percolation (existence of at least one infinite cluster)... |
214 | The Benjamini–Schramm nonuniqueness conjecture | The Benjamini–Schramm nonuniqueness conjecture. Proves $`p_c\lt p_u`$ for Bernoulli bond percolation on every infinite connected locally finite nonamenable quasi-transitive graph, resolving the Benjamini–Schramm nonuniqueness conjecture. Thus there is a nonempty range of probabilities with infinitely many infinite clu... | [
"We prove the bond-percolation nonuniqueness conjecture of Benjamini and Schramm: every infinite connected, locally finite, nonamenable quasi-transitive graph has a nonempty interval of parameters for which Bernoulli bond percolation almost surely has infinitely many infinite clusters. We also prove Hutchcroft's st... | [
"https://github.com/openai/math/tree/main/preprints/Nonuniqueness-of-percolation-on-nonamenable-quasi-transitive-graphs-September-24-2026"
] | [
"https://github.com/openai/math/blob/main/lean/ComparatorChallenges/BenjaminiSchramm.lean",
"https://github.com/openai/math/blob/main/lean/ComparatorChallenges/CayleyPercolation.lean"
] | [
"-- ComparatorChallenges/BenjaminiSchramm.lean\nimport Mathlib\n\nnamespace OAI\n\nopen MeasureTheory ProbabilityTheory\nopen scoped ENNReal NNReal\n\nnamespace Percolation\nuniverse u v\nvariable {V : Type u} {E : Type v}\n\nabbrev BondGraph (V : Type u) (E : Type v) := E → Sym2 V\n\ndef Adj (G : BondGraph V E) (x... | The formalization proves the nonuniqueness phase conjecture for Bernoulli bond percolation on infinite connected locally finite nonamenable quasi-transitive graphs. It establishes $p_c<p_{2\to2}\le p_u$ and a common nonempty coupled interval with infinitely many infinite clusters almost surely. For Cayley graphs, the r... | {
"score": 5,
"reason": "{\"reason\": \"The general theorem full_main has the intended hypotheses: infinite, connected, locally finite, quasi-transitive, and nonamenability encoded as a positive vertex-boundary Cheeger constant, which is equivalent under these assumptions. Its conclusion gives pc < ptwo ≤ pu, hence... | {
"score": 5,
"reason": "{\"reason\": \"The Lean formalization faithfully expresses the main claim. The `full_main` theorem takes a connected, locally finite, quasi-transitive graph with positive isoperimetric constant (equivalent to nonamenability for quasi-transitive graphs) and proves `MainConclusion G`, which i... |
241 | Rigidity of the Turing degrees | Rigidity of the Turing degrees. Every order automorphism of the Turing degrees is the identity, resolving their rigidity problem. Thus no nontrivial relabeling of degrees preserves the ordering by relative computability. | [
"We prove that every order automorphism of the full partial order of Turing degrees is the identity, resolving the rigidity conjecture for the Turing degrees positively."
] | [
"https://github.com/openai/math/tree/main/preprints/Rigidity-of-the-Turing-degrees-September-24-2026"
] | [
"https://github.com/openai/math/blob/main/lean/ComparatorChallenges/DegreeRigidity.lean"
] | [
"-- ComparatorChallenges/DegreeRigidity.lean\nimport Mathlib\n\nnamespace OAI\n\nnamespace TuringRigidity\n\nabbrev Oracle := ℕ → Bool\n\ndef oracleFunction (A : Oracle) : ℕ →. ℕ := fun n => Part.some (if A n then 1 else 0)\n\ndef Reduces (A B : Oracle) : Prop := TuringReducible (oracleFunction A) (oracleFunction B... | 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 ... | {
"score": 5,
"reason": "{\"reason\": \"The Lean code defines a type `Degree` as the antisymmetrization of all oracles `ℕ → Bool` under the relation `TuringReducible` of their characteristic functions, which is the standard quotient defining Turing degrees. `MainTheorem` then states that every order isomorphism `De... | {
"score": 5,
"reason": "{\"reason\": \"The Lean statement `MainTheorem` formalizes exactly the main claim: every order automorphism of the Turing degrees is the identity. The type `Degree ≃o Degree` represents order isomorphisms (automorphisms) of the partial order of Turing degrees, and the conclusion `π a = a` s... |
245 | Weak normalization implies strong normalization in pure type systems | Weak normalization implies strong normalization in pure type systems. Proves that weak normalization implies strong normalization for every pure type system: if every legal expression in every valid context has a <i>β</i>-normal form, every <i>β</i>-reduction sequence terminates. This resolves the <i>β</i>-Barendregt–... | [
"We prove that every weakly $\\beta$-normalizing pure type system is strongly $\\beta$-normalizing. Both properties quantify over all legal expressions in all valid contexts, and reduction acts inside type annotations. No functionality hypothesis is required. This resolves the $\\beta$-Barendregt--Geuvers--Klop con... | [
"https://github.com/openai/math/tree/main/preprints/Weak-and-strong-normalization-in-pure-type-systems-September-25-2026"
] | [
"https://github.com/openai/math/blob/main/lean/ComparatorChallenges/TypeSystemNormalization.lean"
] | [
"-- ComparatorChallenges/TypeSystemNormalization.lean\nimport Mathlib\n\nnamespace OAI\n\n/-!\nFull annotated PTS syntax, modulo alpha-renaming via de Bruijn indices.\nContexts are stored newest-first, each declaration in its prefix context.\nNo functionality assumptions on axioms or product rules.\n-/\n\nnamespace... | 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 ... | {
"score": 5,
"reason": "{\"reason\": \"The theorem quantifies over an arbitrary PTS specification with no functionality and over all valid contexts and legal expressions, matching the paper's scope. Weak normalization is exactly existence of a β-normal reduct, and strong normalization is Acc of the converse β rela... | {
"score": 5,
"reason": "{\"reason\": \"The Lean formalization accurately captures the main claim. The definitions of `WeaklyNormalizing` (existence of a normal form) and `StronglyNormalizing` (termination of all reduction sequences via `Acc`) match the mathematical concepts. The system-wide predicates correctly qu... |
253 | An infinite finitely presented simple amenable group | An infinite finitely presented simple amenable group. Constructs an infinite finitely presented simple amenable group, answering the longstanding question of whether these properties can occur simultaneously. | [
"We construct an infinite finitely presented simple amenable group, giving a positive answer to the finite-presentation existence question for simple amenable groups."
] | [
"https://github.com/openai/math/tree/main/preprints/An-Infinite-Finitely-Presented-Simple-Amenable-Group-September-23-2026"
] | [
"https://github.com/openai/math/blob/main/lean/ComparatorChallenges/SimpleAmenable.lean"
] | [
"-- ComparatorChallenges/SimpleAmenable.lean\nimport Mathlib\n\nnamespace OAI\n\nopen scoped symmDiff\n\nnamespace SimpleAmenable\n\nnoncomputable def FolnerAmenable (G : Type*) [Group G] : Prop := by\n classical\n exact ∀ K : Finset G, ∀ ε : ℝ, 0 < ε →\n ∃ D : Finset G, D.Nonempty ∧ ∀ g ∈ K,\n (((D.image... | 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.
The later general claims about central kernels and enlargements across families are not included. | {
"score": 5,
"reason": "{\"reason\": \"The Lean theorem asserts the existence of a group G that is infinite, finitely presented, simple, and satisfies the standard Følner condition for every finite subset K and every ε > 0. This Følner condition is the usual equivalent characterization of amenability for discrete ... | {
"score": 5,
"reason": "{\"reason\": \"The Lean statement faithfully expresses the main claim. It asserts the existence of a group G that is infinite, finitely presented, simple, and amenable (via the Folner condition). All four properties from the paper's claim are present with correct mathematical meaning: 'Infi... |
274 | Parity is not in QAC<sup>0</sup> | Parity is not in QAC<sup>0</sup>. 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,... | [
"We prove that constant-depth quantum circuits with arbitrary one-qubit and unbounded-arity Toffoli gates cannot compute parity with any fixed positive worst-case advantage using polynomially many qubits. Ancillas start in zero, only one output qubit is measured, and all final garbage is unrestricted. This resolves... | [
"https://github.com/openai/math/tree/main/preprints/Product-projection-localization-and-the-QAC0-parity-lower-bound-September-24-2026",
"https://github.com/openai/math/tree/main/preprints/Regular-trajectories-pruning-and-quantum-parity-September-24-2026"
] | [
"https://github.com/openai/math/blob/main/lean/ComparatorChallenges/QACParity.lean",
"https://github.com/openai/math/blob/main/lean/ComparatorChallenges/RegularParity.lean"
] | [
"-- ComparatorChallenges/QACParity.lean\nimport Mathlib\n\nnamespace OAI\n\nnoncomputable section\nopen scoped BigOperators Matrix.Norms.L2Operator\n\nnamespace QAC\n\nabbrev Bit := Fin 2\nabbrev Word (N : ℕ) := Fin N → Bit\nabbrev Operator (N : ℕ) := Matrix (Word N) (Word N) ℂ\nabbrev QubitOperator := Matrix Bit B... | 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 meas... | {
"score": 5,
"reason": "{\"reason\": \"The `ParityStatement` theorem captures the measured-output parity claim: for fixed depth, polynomial total-qubit bound, and any fixed positive advantage ε, no choice of output qubit can have worst-case success probability at least 1/2+ε. The definitions initialize ancillas to... | {
"score": 5,
"reason": "{\"reason\": \"The Lean formalization accurately captures the main claim. It defines the circuit model (constant-depth, one-qubit + Toffoli gates), the input/ancilla setup (ancillas zero-initialized), the measurement model (summing over garbage for a single output qubit), and the success cr... |
279 | Exact quantum factoring over a fixed finite gate set | Exact quantum factoring over a fixed finite gate set. Gives a polynomial-time uniform quantum circuit family that outputs the complete prime factorization of every integer with probability one. Both gate count and qubit count are polynomial in the input length, and one fixed finite gate set suffices. | [
"We give a polynomial-time uniform quantum circuit family that outputs the complete prime factorization of every integer $N\\ge2$ with probability one. A fixed finite set of bounded-arity gates suffices, and both the gate count and the number of qubits have polynomial worst-case bounds in the input length."
] | [
"https://github.com/openai/math/tree/main/preprints/Exact-quantum-factoring-over-a-fixed-finite-gate-set-September-25-2026"
] | [
"https://github.com/openai/math/blob/main/lean/ComparatorChallenges/ExactQuantumFactoring.lean"
] | [
"-- ComparatorChallenges/ExactQuantumFactoring.lean\nimport Mathlib\n\nnamespace OAI\n\n\n\nnamespace ExactQuantumFactoring\n\nopen scoped BigOperators\n\n/-- The five primitive gates exactly as in Section 2. -/\ninductive Primitive where\n | not | cnot | toffoli | hadamard | phase\n deriving DecidableEq\n\ndef P... | 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 ... | {
"score": 5,
"reason": "{\"reason\": \"The Lean statement asserts a length-uniform family over a fixed finite 20-gate set, with a single polynomial bound on qubits and instruction count, and for every N≥2 of bit length ℓ it requires the total Born probability of basis states whose designated output wires encode a ... | {
"score": 5,
"reason": "{\"reason\": \"The Lean statement faithfully captures all key aspects of the main claim: a uniform quantum circuit family (via the Uniform definition using polynomial-time TM2 generation), polynomial bounds on both qubit count and gate/instruction count (via PolynomialResources and the expl... |
288 | Kadison's similarity conjecture | Kadison's similarity conjecture. Proves Kadison's similarity conjecture: every bounded complex-linear unital algebra homomorphism from a unital complex <i>C</i><sup>∗</sup>-algebra to operators on a Hilbert space becomes a $`*`$-homomorphism after conjugation by a bounded invertible operator. | [
"We prove that every bounded complex-linear unital algebra homomorphism from a unital complex $C^*$-algebra into the bounded operators on an arbitrary Hilbert space is similar to a $*$-homomorphism. This resolves Kadison's similarity conjecture positively. We also obtain one universal hyperreflexivity constant for ... | [
"https://github.com/openai/math/tree/main/preprints/Kadisons-similarity-theorem-through-uniform-derivation-estimates-September-23-2026"
] | [
"https://github.com/openai/math/blob/main/lean/ComparatorChallenges/KadisonSimilarity.lean",
"https://github.com/openai/math/blob/main/lean/ComparatorChallenges/UniformCommutator.lean"
] | [
"-- ComparatorChallenges/KadisonSimilarity.lean\nimport Mathlib\n\nnamespace OAI\n\nnoncomputable section\n\nnamespace KadisonSimilarity\n\nuniverse u v\n\nsection Homomorphisms\n\nvariable (A H : Type*) [CStarAlgebra A]\n [NormedAddCommGroup H] [InnerProductSpace ℂ H] [CompleteSpace H]\n\nabbrev BoundedUnitalHom ... | 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... | {
"score": 5,
"reason": "{\"reason\": \"The Lean `similarityTheorem` quantifies over a C*-algebra `A`, a Hilbert space `K`, and a continuous (hence bounded) complex algebra homomorphism `π : A → B(K)`, and asserts the existence of a unit `S` in the bounded operators such that `S π(star a) S⁻¹ = star(S π(a) S⁻¹)` fo... | {
"score": 5,
"reason": "{\"reason\": \"The Lean statement `similarityTheorem` faithfully expresses Kadison's similarity conjecture. It quantifies over all unital C*-algebras A and Hilbert spaces K, and asserts that every bounded unital algebra homomorphism π from A to B(K) is similar to a *-homomorphism (defined v... |
295 | The Kadison–Ringrose cohomology conjecture | The Kadison–Ringrose cohomology conjecture. Proves that every bounded Hochschild cocycle of degree at least two on a complex von Neumann algebra, with coefficients in the algebra itself, has a bounded primitive. Equivalently, all higher bounded Hochschild cohomology groups vanish, resolving the Kadison–Ringrose conjec... | [
"We prove that every bounded Hochschild cocycle of degree at least two on a complex von Neumann algebra, with values in the algebra itself, has a bounded primitive. Together with the established degree-one inner-derivation theorem, this resolves the Kadison--Ringrose cohomology conjecture positively."
] | [
"https://github.com/openai/math/tree/main/preprints/Vanishing-of-higher-bounded-Hochschild-cohomology-September-23-2026"
] | [
"https://github.com/openai/math/blob/main/lean/ComparatorChallenges/KadisonRingrose.lean"
] | [
"-- ComparatorChallenges/KadisonRingrose.lean\nimport Mathlib\n\nnamespace OAI\n\n/-! Every ordinary bounded higher Hochschild cocycle of a complex von Neumann\nalgebra is the coboundary of a bounded cochain with values in the same algebra. -/\n\nnoncomputable section\n\nnamespace BoundedHochschild\n\nabbrev Cochai... | 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 res... | {
"score": 5,
"reason": "{\"reason\": \"The Lean theorem quantifies over a W*-algebra M (the formalization of a complex von Neumann algebra), an arbitrary n, and a continuous multilinear Hochschild cocycle f of degree n+2 ≥ 2 with values in M. The differential definition matches the standard Hochschild differential... | {
"score": 5,
"reason": "{\"reason\": \"The Lean statement faithfully formalizes the Kadison–Ringrose conjecture. The type `Cchain` uses `ContinuousMultilinearMap`, which for Banach spaces (and thus von Neumann algebras) corresponds exactly to bounded multilinear maps, satisfying the 'bounded' requirement of the cl... |
322 | Tingley’s sphere-isometry problem | Tingley’s sphere-isometry problem. Resolves Tingley's problem: every surjective isometry between the unit spheres of nonzero real Banach spaces extends uniquely to a surjective real-linear isometry of the whole spaces. No dimension restriction is imposed, so the metric geometry of the unit sphere determines the Banach... | [
"Every surjective isometry between the unit spheres of real Banach spaces extends uniquely to a surjective real-linear isometry, giving an affirmative solution to Tingley's problem. If distances between different radii are not preserved, we realize the positive maximal defect in a possibly enlarged pair of Banach s... | [
"https://github.com/openai/math/tree/main/preprints/A-positive-solution-to-Tingleys-problem-September-23-2026"
] | [
"https://github.com/openai/math/blob/main/lean/ComparatorChallenges/TingleySphereIsometry.lean"
] | [
"-- ComparatorChallenges/TingleySphereIsometry.lean\nimport Mathlib.Analysis.Normed.Module.RCLike.Basic\nimport Mathlib.Topology.MetricSpace.Isometry\n\nnamespace OAI\n\nnoncomputable section\n\nnamespace Tingley\n\nuniverse u v\n\nabbrev UnitSphere (X : Type u) [NormedAddCommGroup X] := {x : X // ‖x‖ = 1}\n\nvaria... | 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, o... | {
"score": 5,
"reason": "{\"reason\": \"The Lean theorem quantifies over arbitrary nontrivial complete real normed spaces, i.e. nonzero real Banach spaces, and a surjective metric isometry between their unit spheres. It concludes that there exists a real-linear isometry equivalence extending the sphere map, with th... | {
"score": 5,
"reason": "{\"reason\": \"The Lean statement faithfully expresses the main claim. It correctly formalizes that every surjective isometry between unit spheres of nonzero real Banach spaces extends uniquely to a surjective real-linear isometry of the whole spaces. The hypotheses (complete normed spaces,... |
324 | Lipschitz equivalent Banach spaces need not be linearly isomorphic | Lipschitz equivalent Banach spaces need not be linearly isomorphic. Constructs separable real Banach spaces that are globally bi-Lipschitz equivalent but not linearly isomorphic, resolving the separable Lipschitz-isomorphism problem negatively. Thus even the complete metric structure up to bi-Lipschitz equivalence doe... | [
"We construct a separable real Banach space \\(Z\\) that contains no linear copy of \\(c_0\\), yet is bi-Lipschitz equivalent to \\(Z\\oplus_\\infty c_0\\). The same space contains a bi-Lipschitz image of every separable metric space.",
"There are separable real Banach spaces that are globally bi-Lipschitz equiva... | [
"https://github.com/openai/math/tree/main/preprints/Bi-Lipschitz-Absorption-of-c0-Without-a-Linear-Copy-of-c0-September-26-2026",
"https://github.com/openai/math/tree/main/preprints/Lipschitz-Equivalent-Separable-Banach-Spaces-Need-Not-Be-Linearly-Isomorphic-September-24-2026"
] | [
"https://github.com/openai/math/blob/main/lean/ComparatorChallenges/C0Absorption.lean",
"https://github.com/openai/math/blob/main/lean/ComparatorChallenges/LipschitzEquivalence.lean"
] | [
"-- ComparatorChallenges/C0Absorption.lean\nimport Mathlib\n\nnamespace OAI\n\nnamespace C0Absorption\n\nopen scoped ZeroAtInfty\n\nabbrev C0 := C₀(ℕ, ℝ)\n\ndef BiLip {X Y : Type*} [MetricSpace X] [MetricSpace Y] (f : X → Y) : Prop :=\n ∃ c C : ℝ, 0 < c ∧ c ≤ C ∧\n ∀ x y, c * dist x y ≤ dist (f x) (f y) ∧ dist ... | 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 ... | {
"score": 5,
"reason": "quivalent. Good.\n\nPotential issue: The first's `MainConclusion` does not assert `Z` is a `Type` with `MetricSpace` but NormedAddCommGroup gives. Good.\n\nNow, let's consider if the second's `MainClaim` could be considered not faithful because it does not assert the two spaces are \"Lipsch... | {
"score": 5,
"reason": "{\"reason\": \"Both Lean files faithfully express the main claim. The first file (C0Absorption) directly formalizes the abstract's construction: a separable Banach space Z with no linear copy of c₀ that is bi-Lipschitz equivalent to Z ⊕_∞ c₀, plus metric universality and non-linear-isomorph... |
325 | The complete Crouzeix conjecture | The complete Crouzeix conjecture. Resolves the complete Crouzeix conjecture: for every bounded operator <i>A</i> on a complex Hilbert space and every finite matrix-valued polynomial <i>P</i>, 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 sh... | [
"We give a direct proof of the sharp constant-two numerical-range inequality for matrix-valued polynomials in all finite base and coefficient dimensions. This resolves the complete Crouzeix conjecture in its matrix formulation, including matrices whose numerical ranges are points or line segments.",
"We resolve t... | [
"https://github.com/openai/math/tree/main/preprints/A-direct-proof-of-the-complete-Crouzeix-inequality-September-26-2026",
"https://github.com/openai/math/tree/main/preprints/The-complete-Crouzeix-theorem-September-23-2026"
] | [
"https://github.com/openai/math/blob/main/lean/ComparatorChallenges/CompleteCrouzeix.lean",
"https://github.com/openai/math/blob/main/lean/ComparatorChallenges/CrouzeixHilbert.lean",
"https://github.com/openai/math/blob/main/lean/ComparatorChallenges/DirectCrouzeix.lean",
"https://github.com/openai/math/blob/... | [
"-- ComparatorChallenges/CompleteCrouzeix.lean\nimport Mathlib\n\nnamespace OAI\n\nnoncomputable section\nopen scoped BigOperators Matrix.Norms.L2Operator ComplexOrder\n\nnamespace CompleteCrouzeix\n\ndef numericalRange {n : ℕ} (A : Matrix (Fin n) (Fin n) ℂ) : Set ℂ :=\n {z | ∃ x : EuclideanSpace ℂ (Fin n), ‖x‖ = ... | 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 n... | {
"score": 5,
"reason": "{\n \"reason\": \"The Hilbert-space file states the full infinite-dimensional claim: for every complete complex inner-product space and bounded operator, all matrix sizes and polynomial degrees satisfy the norm inequality with constant 2, and it includes a sharpness example attaining ratio... | {
"score": 5,
"reason": "{\"reason\": \"The Lean formalization faithfully expresses the main claim. The theorem `hilbert` in CrouzeixHilbert.lean states that for every bounded operator A on a complex Hilbert space (complete inner product space), the inequality ‖polynomialEval A B‖ ≤ 2 * supNorm (numericalRange A) (... |
328 | Nonexpansive fixed points in reflexive Banach spaces | Nonexpansive fixed points in reflexive Banach spaces. Resolves Kirk's reflexive-space fixed-point problem: every nonexpansive selfmap of a nonempty closed bounded convex subset of a real reflexive Banach space has a fixed point. The result uses the original norm, without assuming uniform convexity. | [
"Every nonexpansive selfmap of a nonempty closed bounded convex subset of a real reflexive Banach space has a fixed point. This resolves the reflexive-space fixed point problem for the given norm."
] | [
"https://github.com/openai/math/tree/main/preprints/Fixed-Points-of-Nonexpansive-Maps-in-Reflexive-Banach-Spaces-September-24-2026"
] | [
"https://github.com/openai/math/blob/main/lean/ComparatorChallenges/ReflexiveFixedPoints.lean"
] | [
"-- ComparatorChallenges/ReflexiveFixedPoints.lean\nimport Mathlib\n\nnamespace OAI\n\nnoncomputable section\n\nnamespace ReflexiveFixedPoints\n\nuniverse u\n\nvariable (X : Type u) [NormedAddCommGroup X] [NormedSpace ℝ X]\n\n/-- Surjectivity of the canonical map into the continuous bidual. -/\ndef CanonicallyRefle... | 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 nonseparabl... | {
"score": 5,
"reason": "{\"reason\": \"The Lean theorem quantifies over a real normed space with CompleteSpace and surjectivity of the canonical map into the continuous bidual, which matches a real reflexive Banach space. It assumes C is nonempty, closed, bounded (via the normed-space bornology), and convex, and F... | {
"score": 5,
"reason": "{\"reason\": \"The Lean statement faithfully captures the main claim. It assumes X is a real Banach space (CompleteSpace, NormedSpace ℝ) that is reflexive (CanonicallyReflexive, which for Banach spaces is equivalent to the canonical embedding into the double dual being surjective), C is a n... |
329 | A counterexample to metric-entropy duality | A counterexample to metric-entropy duality. Disproves Pietsch's dimension-free duality conjecture for metric entropy. Origin-symmetric convex bodies violate every proposed choice of universal constants in the conjectured comparison between covering numbers and those of the polar bodies, even when the covering body is ... | [
"We disprove Pietsch's dimension-free duality conjecture for metric entropy. For every proposed pair of universal constants, we construct origin-symmetric convex bodies that violate the corresponding covering-entropy inequality, already when the covering body is a cube."
] | [
"https://github.com/openai/math/tree/main/preprints/Counterexamples-to-the-duality-conjecture-for-metric-entropy-September-24-2026"
] | [
"https://github.com/openai/math/blob/main/lean/ComparatorChallenges/MetricEntropyDuality.lean"
] | [
"-- ComparatorChallenges/MetricEntropyDuality.lean\nimport Mathlib.Analysis.Convex.Topology\nimport Mathlib.Analysis.Normed.Group.Constructions\nimport Mathlib.Order.Lattice.Nat\nimport Mathlib.Analysis.SpecialFunctions.Log.Basic\n\nnamespace OAI\n\nuniverse u\n\nnoncomputable section\n\nnamespace MetricEntropyDual... | 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^\... | {
"score": 5,
"reason": "{\"reason\": \"The theorem quantifies over arbitrary constants a,b in the natural regime a,b≥1 and produces a positive dimension and an origin-symmetric compact convex body K. Its definitions of cube, polar, and finite covering number match the usual notions, and the conclusion asserts the ... | {
"score": 5,
"reason": "{\"reason\": \"The Lean theorem faithfully expresses the main claim. It states that for any proposed universal constants a, b ≥ 1, there exists a finite-dimensional origin-symmetric convex body K such that the covering-number inequality is violated, with the covering body being a cube (as r... |
330 | A uniformly discrete counterexample to bounded approximation in Lipschitz-free spaces | A uniformly discrete counterexample to bounded approximation in Lipschitz-free spaces. Constructs a countable uniformly discrete metric space whose real Lipschitz-free Banach space has the approximation property but not the bounded approximation property, answering Kalton's question negatively. Finite-rank operators a... | [
"We construct a countable uniformly discrete metric space whose real Lipschitz-free space has the approximation property but fails the bounded approximation property, answering Kalton's question negatively. The identity can be approximated on every compact set by finite-rank operators, but no uniform bound on their... | [
"https://github.com/openai/math/tree/main/preprints/Failure-of-Bounded-Approximation-in-a-Lipschitz-Free-Space-over-a-Uniformly-Discrete-Metric-Space-September-26-2026"
] | [
"https://github.com/openai/math/blob/main/lean/ComparatorChallenges/DiscreteLipschitzFree.lean",
"https://github.com/openai/math/blob/main/lean/ComparatorChallenges/RealL1Renorming.lean"
] | [
"-- ComparatorChallenges/DiscreteLipschitzFree.lean\nimport Mathlib\n\nnamespace OAI\n\n/-!\n# Real Lipschitz-free spaces and approximation properties\n-/\n\nnoncomputable section\nopen scoped BigOperators ENNReal\n\nnamespace DiscreteFree\n\nvariable {M : Type*} [MetricSpace M]\n\ndef lipSubmodule (o : M) : Submod... | 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 fai... | {
"score": 5,
"reason": "{\"reason\": \"The first file's MainStatement asserts the claimed counterexample: a countable uniformly discrete metric space (with the stronger extra properties of being unbounded and non-proper) whose real Lipschitz-free space FreeSpace o has HasAP and fails HasBAP for every bound Λ≥1. Th... | {
"score": 5,
"reason": "{\"reason\": \"The Lean formalization faithfully captures the main claim. The HasAP and HasBAP definitions match the standard mathematical definitions of the approximation property and bounded approximation property. The FreeSpace construction (as the closed span of evaluation functionals) ... |
343 | Symplectic ball packing in higher dimensions | Symplectic ball packing in higher dimensions. 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 <i>R</i> exactly when $`\sum_iR_i^n\lt R^n`$ and $`R_i+R_j\lt R`... | [
"We prove that, for all integers \\(n\\ge3\\) and \\(k\\ge1\\) and all positive real capacities \\(R_1,\\ldots,R_k\\), the closed standard symplectic \\(2n\\)-balls of these capacities embed disjointly into the interior of a ball of capacity \\(R>0\\) if and only if \\[ \\sum_{i=1}^k R_i^n<R^n, \\qquad R_i+R_j<R\\q... | [
"https://github.com/openai/math/tree/main/preprints/Symplectic-Ball-Packings-in-Higher-Dimensions-September-23-2026"
] | [
"https://github.com/openai/math/blob/main/lean/ComparatorChallenges/BallPacking.lean",
"https://github.com/openai/math/blob/main/lean/ComparatorChallenges/BallPackingNecessity.lean"
] | [
"-- ComparatorChallenges/BallPacking.lean\nimport Mathlib\n\nnamespace OAI\n\nnamespace HigherDimensionalBallPacking\nopen scoped ContDiff\n\nabbrev Phase (n : ℕ) := Fin n → ℂ\n\nnoncomputable def capacity {n : ℕ} (z : Phase n) : ℝ :=\n Real.pi * ∑ j, Complex.normSq (z j)\n\ndef closedBall (n : ℕ) (r : ℝ) : Set (P... | 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
$\sum_i R_i^n<R^n$ and $R_i+R_j<R$ fo... | {
"score": 5,
"reason": "{\"reason\": \"The Lean statement uses C^n as Phase n, defines capacity as π∑|z_j|^2, and formalizes closed source balls and the open target ball exactly as in the abstract. HasPacking requires each map to be a C∞ symplectic embedding on an open neighborhood of the closed source ball and to... | {
"score": 5,
"reason": "{\"reason\": \"The Lean formalization faithfully captures the main claim. The definitions of capacity, closedBall, openBall, SymplecticOn, HasPacking, and PackingInequalities all match the mathematical statement: capacity is π times the squared Euclidean radius, embeddings are symplectic (p... |
362 | Global smoothness for relativistic Vlasov–Maxwell | Global smoothness for relativistic Vlasov–Maxwell. Proves large-data global existence and uniqueness for the three-dimensional, one-species relativistic Vlasov–Maxwell system. Smooth admissible initial data may be arbitrary provided the particle density is compactly supported and the electromagnetic fields have finite... | [
"We prove global existence and uniqueness for arbitrary smooth admissible initial data in the three-dimensional, one-species relativistic Vlasov--Maxwell system. The particle density is initially compactly supported, and the electromagnetic fields have finite energy and bounded derivatives of all orders. The soluti... | [
"https://github.com/openai/math/tree/main/preprints/Global-classical-solutions-of-the-three-dimensional-relativistic-Vlasov-Maxwell-system-September-23-2026"
] | [
"https://github.com/openai/math/blob/main/lean/ComparatorChallenges/VlasovMaxwell.lean"
] | [
"-- ComparatorChallenges/VlasovMaxwell.lean\nimport Mathlib\n\nnamespace OAI\n\nnoncomputable section\nopen MeasureTheory Set\nopen scoped ContDiff\nnamespace RVM\n\nabbrev Vec := EuclideanSpace ℝ (Fin 3)\nabbrev Phase := Vec × Vec\nabbrev Particle := ℝ × Phase → ℝ\nabbrev Field := ℝ × Vec → Vec\n\ndef q (v : Vec) ... | 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... | {
"score": 5,
"reason": "ns:\n5 = the same claim as the paper's main claim (or stronger)\n4 = essentially the same claim, with only minor technical differences in how it is stated\n3 = a substantial special case or a clearly weaker version of the claim\n2 = only loosely related to the claim, or much weaker\n1 = doe... | {
"score": 5,
"reason": "{\"reason\": \"The Lean formalization faithfully captures the main claim. The `Admissible` predicate correctly encodes the initial data conditions: smooth particle density with compact support, electromagnetic fields with finite energy (L²) and bounded derivatives of all orders (via `Bounde... |
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