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)... |
End of preview. Expand in Data Studio
OpenAI-Math-Lean-Agreed-38（非公式）
openai/math（Apache 2.0）は、OpenAI の社内モデルが書いた研究数学の論文（722本）と、その一部の Lean 4 による形式化を集めたリポジトリです。そのカタログでは、「形式化した命題が論文の主張と同じか」の人による確認は unchecked（未確認）になっています。
このデータセットは、Lean で形式化された 235件の結果について、2つのモデルに「Lean の命題が論文の主張と同じことを述べているか」を 1〜5 点で判定させ、両方とも 5 点（同じ主張、またはより強い）をつけた 38件を集めたものです。OpenAI とは関係のない、非公式の派生データです。
An unofficial subset of openai/math. For each of the 235 Lean-formalized results, two models (Qwen3.8-27B and a fine-tuned NeoHorse-1-9B, both with reasoning) judged whether the Lean statement faithfully expresses the paper's main claim (score 1–5). This dataset contains the 38 results that both models scored 5. Nobody has verified these judgments by hand.
注意：判定はモデルによるもので、人は確認していません。 研究レベルの数学なので、モデルの判定にも誤りがありえます。判定の理由も各行に入れているので、使う前に確かめてください。
各行の内容
| 項目 | 内容 |
|---|---|
family |
openai/math の結果の番号 |
title |
結果の題名 |
claim |
論文の主張（openai/math の目録 CONTENTS.md の説明） |
abstracts |
論文の要旨（LaTeX の abstract） |
papers |
論文のフォルダへのリンク |
lean_files |
Lean の命題のファイル（ComparatorChallenges、証明は sorry）へのリンク |
lean_statements |
そのファイルの中身（命題と、命題に使う定義） |
formalization_scope |
openai/math の説明ページにある、形式化の範囲の説明 |
judge_qwen3_8_27b, judge_neohorse_code |
各モデルの点（5）と、判定の理由（最後の部分） |
判定の方法
- 材料：目録の説明・論文の要旨・Lean の命題ファイルを並べて渡しました
- 点の基準：5＝論文の主張と同じ（またはより強い）、4＝表現の細かい違いだけ、3＝特別な場合や弱い版、2＝ゆるい関係しかない、1＝主張を表していない（無関係・自明など）
- モデル：Qwen/Qwen3.8-27B（FP8）と、NeoHorse-1-9B に追加学習したモデル（NeoHorse-1-9B-Math-SFT の2段目までの版）。どちらも思考あり、上限 24,000 トークン
- 235件の点の組の分布（Qwen3.8-27B, NeoHorse）：(5, 5) が 38件、(3, 5) が 28件、(3, 3) が 26件 など。点を読み取れなかったもの（上限まで考え終わらなかったなど）は採用していません
- OpenAI のモデルは判定に使っていません
ライセンス
Apache License 2.0。元のデータは openai/math（© OpenAI、Apache 2.0）です。このデータセットは、そこから一部を選び、モデルの判定を加えたものです（変更点）。
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