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Limiting random SAT thresholds, sharp variance and computability

OpenAI groups related papers into a “family”. OpenAI’s catalogue describes this one:

For random k-SAT with independent uniformly signed proper clauses sampled with replacement, proves finite positive limiting thresholds and hitting-time variance \(\Theta_k(n)\) for every fixed k ≥ 3, and computability of the 3-SAT threshold. We credit Gaia Carenini with priority for resolving the threshold-existence conjecture in her concurrent ECCC TR26-229, made public October 5, 2026; this family supplies another proof and the sharper variance and computability results. (OpenAI’s Lean note)

Lean is a language for mathematical statements and proofs that a computer can verify. The assessments below ask whether those statements express the result OpenAI describes here.

The independent assessments find that the Lean statements express only part of the result described above, or a narrower statement. Read the assessments

The paper states

Theorem 1.1

In A Limiting Satisfiability Threshold for Every Fixed Clause Size, 25 September 2026

Possible match to OpenAI’s description, selected from the papers’ main statements because it shares the most wording. No review confirming the selection is recorded.

Three other possible matches are listed below the card, among them Theorem 1.1 in Computing the Random 3-SAT Threshold and Theorem 1.1 in Variance of the Random k-SAT Hitting Time.

For every fixed integer k\ge3, there is a constant \alpha_k\in(0,\infty) such that, for independent proper uniformly signed k-clauses sampled with replacement,

\lim_{n\to\infty}P_n(\lfloor cn\rfloor) = \begin{cases} 1,&0\le c<\alpha_k,\\ 0,&c>\alpha_k. \end{cases}

Three more possible matches

  • Theorem 1.1 in Computing the Random 3-SAT Threshold, 27 September 2026.
  • Theorem 1.1 in Variance of the Random k-SAT Hitting Time, 27 September 2026.
  • Theorem 1.1 in Linear Variance of the Random 3-SAT Hitting Time, 5 October 2026.
Reported proof verification
Four Lean statements in the release; independent groups report that Comparator accepted runs for all these files at the version shown here.
Papers and explanations
Four current papers, each readable here as a page beside its PDF.

Papers

Each paper opens as a reading page in the browser, with the PDF beside it. The statements each paper opens with are on its reading page.

A Limiting Satisfiability Threshold for Every Fixed Clause Size

25 September 2026, current edition.

Read it herePDF

For every fixed integer k ≥ 3, random k-SAT with independent uniformly signed clauses on distinct variables, sampled with replacement, has a finite positive limiting satisfiability threshold. We credit Gaia Carenini [[5]](https://eccc.weizmann.ac.il/report/2026/229/) with priority for resolving the satisfiability conjecture. This paper gives an alternative proof, using concentration of a capped last satisfiable index and a comparison between different system sizes.

Linear Variance of the Random 3-SAT Hitting Time

5 October 2026, current edition.

Read it herePDF

For random 3-SAT on n Boolean variables, with independent uniformly signed clauses on three distinct variables sampled with replacement, we prove that the first unsatisfiable prefix has variance \(\Theta(n)\). The upper bound removes the logarithmic loss in the earlier variance estimate; the matching lower bound follows from Wilson’s transition-width theorem.

Variance of the Random k-SAT Hitting Time

27 September 2026, current edition.

Read it herePDF

Let Hn be the index of the first unsatisfiable prefix in random k-SAT on n variables, with independent uniformly signed clauses using k distinct variables and sampled with replacement. For every fixed k ≥ 4, we prove \(\mathop{\mathrm{Var}}\nolimits (H_n)=\Theta_k(n)\). For k = 3, the variance is bounded below by a positive multiple of n and above by a constant multiple of \(n\log n\); the companion paper on random 3-SAT sharpens this to \(\Theta(n)\). The same upper bounds proved here hold after clipping at any fixed positive multiple of n.

Computing the Random 3-SAT Threshold

27 September 2026, current edition.

Read it herePDF

The limiting satisfiability threshold of uniform random 3-SAT is a computable real. We credit Gaia Carenini [[4]](https://eccc.weizmann.ac.il/report/2026/229/) with priority for resolving the satisfiability conjecture, which establishes the threshold’s existence. We prove that one finite deterministic machine can approximate it to any prescribed accuracy. A deletion estimate gives explicit lower certificates, while a finite hierarchical approximation of the soft pressure gives upper certificates at every larger rational density. A fair search through these certificates halts without requiring a computable rate of finite-size convergence.

Four Lean statements in this family

Choose one to see it beside the paper theorem it is paired with.

For a configured theorem, the Comparator program tests whether OpenAI’s separate proof proves the exact Lean statement using only the allowed axioms. Some statement files instead configure definitions for comparison by name and type. Acceptance does not establish that a formal statement means the same thing as the paper’s theorem.

FixedClauseThreshold

The paper states

The words they share: proper, clauses, integer, limiting, threshold, satisfiability, sampled, replacement, independent, signed, alpha.

Possible Lean counterpart

Selected from shared wording. No review of whether the statements mean the same thing is recorded.

Limiting satisfiability threshold for every fixed $k\ge3$

FixedClauseThreshold.lean, the release lists it as a supporting statement.

theorem main (k : ℕ) (hk : 3 ≤ k) : HasLimitingThreshold k := by  sorry

This file states the theorem with sorry, a placeholder, in place of its proof. OpenAI supplies a separate proof file. Comparator tests whether that proof proves the statement here.

three independent groups report that Comparator accepted runs for this Lean statement file: cert-machine; MathVet; Neal McBurnett, openai-math-lean-tooling. Who ran the proof-verification program

  • cert-machine records it as “CERTIFIED”.
  • MathVet records it as “comparator-sandboxed-pass”.
  • Neal McBurnett, openai-math-lean-tooling records it as “accepted”.

Read the complete Lean statement

SATComputability

The paper states

Theorem 1.1

In Computing the Random 3-SAT Threshold, 27 September 2026

There is a real number \alpha_3\in(0,\infty) such that, for every fixed real a\ge0,

\lim_{n\to\infty}p(n,\lfloor an\rfloor)= \begin{cases}1,&a<\alpha_3,\\0,&a>\alpha_3.\end{cases}

There is one finite deterministic Turing machine which, on input 1^r for any integer r\ge1, halts with a rational number q_r satisfying |q_r-\alpha_3|\le2^{-r}. Its only input is the unary precision 1^r; it uses no oracle, advice, or noncomputable real constant.

The words they share: deterministic, finite, fixed, integer, machine, number, precision, rational, real.

Possible Lean counterpart

Selected from shared wording. No review of whether the statements mean the same thing is recorded.

Computability of the random $3$-SAT threshold

SATComputability.lean, the release lists it as a supporting statement.

def MainStatement : Prop :=    ∃ (α : ℝ) (q : ℕ → ℚ) (c : Nat.Partrec.Code),      0 < α ∧      (∀ a : ℝ, 0 ≤ a → a < α →        Tendsto (fun n : ℕ => properSATProbability n 3 ⌊a*(n : ℝ)⌋₊) atTop (nhds 1)) ∧      (∀ a : ℝ, α < a →        Tendsto (fun n : ℕ => properSATProbability n 3 ⌊a*(n : ℝ)⌋₊) atTop (nhds 0)) ∧      Computable (fun r => Encodable.encode (q r)) ∧      (∀ r, c.eval r = Part.some (Encodable.encode (q r))) ∧      (∀ r, |(q r : ℝ)-α| ≤ (2^r : ℝ)⁻¹)theorem main : MainStatement := by  sorry

This file states the theorem with sorry, a placeholder, in place of its proof. OpenAI supplies a separate proof file. Comparator tests whether that proof proves the statement here.

two independent groups report that Comparator accepted runs for this Lean statement file: cert-machine; MathVet. Who ran the proof-verification program

  • cert-machine records it as “CERTIFIED”.
  • MathVet records it as “comparator-sandboxed-pass”.

Read the complete Lean statement

SATSharpness

The paper states

The words they share: clauses, fixed, independent, integer, replacement, sampled, signed, uniformly.

Possible Lean counterpart

Selected from shared wording. No review of whether the statements mean the same thing is recorded.

Sharpness of random $k$-SAT lifetime and replacement bounds

SATSharpness.lean, the release lists it as a supporting statement.

theorem sharpness (k : ℕ) (hk : 3 ≤ k) :    (∀ γ D A : ℝ, γ < (k : ℝ) / ((k-1 : ℕ) : ℝ) → 0 < D → 0 < A →      ∀ N : ℕ, ∃ u : ℕ, N ≤ u ∧ k ≤ u ∧ ∃ S : Finset (Assignment u),        S.Nonempty ∧ A * (u : ℝ) ^ (-((k-1 : ℕ) : ℝ)) ≤ blockKill u (k-1) S 1 ∧        D * (blockKill u (k-1) S 1) ^ (-γ) < lifetime u k S) ∧    (∃ u : ℕ → ℕ, ∃ S : ∀ g, Finset (Assignment (u g)),      (∀ g, 2 ≤ g → k ≤ u g ∧ (S g).Nonempty) ∧      Asymptotics.IsEquivalent Filter.atTop        (fun g => blockKill (u g) (k-1) (S g) 1 - blockKill (u g) k (S g) g)        (fun g => ((2 : ℝ)^k)⁻¹ * (g : ℝ)^(-((k-1 : ℕ) : ℝ)))) ∧    (¬ ∃ e : ℕ → ℝ,      Asymptotics.IsLittleO Filter.atTop e (fun g => (g : ℝ)^(-((k-1 : ℕ) : ℝ))) ∧      ∀ u, k ≤ u → ∀ S : Finset (Assignment u), S.Nonempty → ∀ g, 2 ≤ g →        blockKill u (k-1) S 1 - e g ≤ blockKill u k S g) := by  sorry

This file states the theorem with sorry, a placeholder, in place of its proof. OpenAI supplies a separate proof file. Comparator tests whether that proof proves the statement here.

three independent groups report that Comparator accepted runs for this Lean statement file: cert-machine; MathVet; Neal McBurnett, openai-math-lean-tooling. Who ran the proof-verification program

  • cert-machine records it as “CERTIFIED”.
  • MathVet records it as “comparator-sandboxed-pass”.
  • Neal McBurnett, openai-math-lean-tooling records it as “accepted”.

Read the complete Lean statement

SATVariance

The paper states

The words they share: sat, variance, theta, var.

Possible Lean counterpart

Selected from shared wording. No review of whether the statements mean the same thing is recorded.

Variance bounds for the random $k$-SAT hitting time

SATVariance.lean, the release lists it as a supporting statement.

theorem variance_main (k : ℕ) (hk : 3 ≤ k) :    (∃ C : ℝ, 0 ≤ C ∧ ∀ n : ℕ, k ≤ n →      variance (H : Stream n k → ℝ) (streamLaw n k) ≤ C * n * ell k n) ∧    (∀ B : ℝ, 0 < B → ∃ C : ℝ, 0 ≤ C ∧ ∀ n : ℕ, k ≤ n →      variance (T B : Stream n k → ℝ) (streamLaw n k) ≤ C * n * ell k n) ∧    (∃ c : ℝ, 0 < c ∧      (∃ N : ℕ, ∀ n : ℕ, N ≤ n → k ≤ n →        c * n ≤ variance (H : Stream n k → ℝ) (streamLaw n k)) ∧      (∀ B : ℝ, U k < B → ∃ N : ℕ, ∀ n : ℕ, N ≤ n → k ≤ n →        c * n ≤ variance (T B : Stream n k → ℝ) (streamLaw n k))) := by  sorry

This file states the theorem with sorry, a placeholder, in place of its proof. OpenAI supplies a separate proof file. Comparator tests whether that proof proves the statement here.

three independent groups report that Comparator accepted runs for this Lean statement file: cert-machine; MathVet; Neal McBurnett, openai-math-lean-tooling. Who ran the proof-verification program

  • cert-machine records it as “CERTIFIED”.
  • MathVet records it as “comparator-sandboxed-pass”.
  • Neal McBurnett, openai-math-lean-tooling records it as “accepted”.

Read the complete Lean statement

Other mathematical statements in OpenAI’s Lean library

Atlas found these propositions by scanning the text of the project’s Model.lean file. Each name links to its definition, with any associated statement file beside it. This list does not record running Lean to interpret or verify those definitions.

Independent comparisons and proof-verification reports

Does the Lean express the advertised result?

Neither group finds that the Lean statements express or imply the whole advertised result. They describe what the statements express in different terms.

MathVet and Carlos Toledo’s cert-machine independently assessed whether this family’s Lean statements express the whole result OpenAI describes in its catalogue. These are assessments of the family as a whole, not of each paper theorem separately.

MathVet finds that the Lean statements express part of the advertised result (their term: “partial”).

Their note: Stated: FixedClauseThreshold.main (k) (hk : 3 ≤ k) : HasLimitingThreshold k (∃ α > 0, P(SAT) → 1 for c < α and → 0 for c > α with m = ⌊c n⌋ i.i.d. uniform proper clauses), SATComputability.main (3-SAT threshold α with a Nat.Partrec.Code producing rationals q r, |q r - α| ≤ 2^-r) and variance_main. But the variance claim is only Θ(n) for k >= 4: for k = 3 the Lean gives Var(H) ≤ C n ell with ell = log(e n) (an n log n upper bound) and a linear lower bound, so the summary’s ‘hitting-time variance Θ_k(n) for every fixed k ≥ 3’ is not stated for k = 3 (docs: 'They do not show that the logarithmic factor in the k=3 variance bound is necessary').

“partial”, in their words: Only a special case is stated, or a co-equal headline claim has no challenge statement.

Their method: An independent, model-assisted, pre-referee reading of each family’s challenge statements against its headline claim under a published rubric (PROTOCOL.md), with its own Comparator runs per challenge.

Read the original assessment Read MathVet’s method

cert-machine finds that the Lean statements express a narrower statement than the advertised result (their term: “NARROWER — DECLARED”).

The gap they name: the k = 3 linear upper bound Var(H_n) = O(n) (Lean has O(n log n))

“NARROWER — DECLARED”, in their words: The challenge statements state less than the headline, and OpenAI’s Lean note says so.

Their method: Every Comparator challenge re-run on Linux with nanoda switched on, decided by a preregistered rule.

Read the original assessment Read cert-machine’s method

OpenAI’s own note on what its Lean covers

The random $k$-SAT threshold problem asks whether satisfiability changes at one limiting clause density. The formalization proves that every fixed integer $k\ge3$ has a finite positive threshold $\alpha_k$: for every fixed density $c<\alpha_k$, the satisfiability probability tends to one as the number of variables grows, while for every $c>\alpha_k$ it tends to zero. Clauses use distinct variables with independent uniform signs and are sampled independently with replacement. The statement makes no assertion at the threshold itself.

Let $H_n$ be the first unsatisfiable prefix length in random $k$-SAT with independent uniformly signed clauses on $k$ distinct variables, sampled with replacement. The formalization proves $\mathrm{Var}(H_n)=\Theta_k(n)$ for every fixed $k\ge4$. For $k=3$, it proves a positive linear lower bound and an $O(n\log n)$ upper bound. The same upper bounds hold after clipping at any fixed positive multiple of $n$, and linear lower bounds hold for sufficiently high clipping levels. The linked supplements prove sharpness of the separate survival-lifetime and clause-replacement estimates used in the argument. They do not show that the logarithmic factor in the $k=3$ variance bound is necessary. The formalization proves that the limiting threshold $\alpha$ for uniform random $3$-SAT is a computable positive real. Satisfiability tends to one at every fixed density below $\alpha$ and to zero at every fixed density above it. One deterministic machine produces a rational number $q_r$ for every requested precision $r$ with $|q_r-\alpha|\le2^{-r}$. No computable rate of finite-size convergence is assumed in the statement.

AI-drafted comparisons, with no recorded human review

  • Limiting satisfiability threshold for every fixed $k\ge3$ FixedClauseThreshold

    Unreviewed AI draft: states a paper’s theorem as the paper does. States the main theorem: for every k ≥ 3 a limiting satisfiability threshold α_k exists, with no assertion at the threshold itself.

  • Computability of the random $3$-SAT threshold SATComputability

    Unreviewed AI draft: states a paper’s theorem as the paper does. States the main theorem: the random 3-SAT threshold exists and is a computable real, approximable to precision 2^{-r} by one machine.

  • Sharpness of random $k$-SAT lifetime and replacement bounds SATSharpness

    Unreviewed AI draft: states a supporting result. States sharpness of auxiliary survival-lifetime and clause-replacement estimates used in the variance argument.

  • Variance bounds for the random $k$-SAT hitting time SATVariance

    Unreviewed AI draft: states a paper’s theorem as the paper does. States the variance theorem for the unsatisfiability hitting time: Θ_k(n) for k ≥ 4, and between linear and O(n log n) for k = 3. The linear k = 3 bound of the family’s later paper is not part of it.

Who ran the proof-verification program?

These groups report running Comparator with the release’s Lean statement files. For a configured theorem, it verifies OpenAI’s separate proof against the exact statement; some files instead configure definitions for comparison by name and type. Each row is the named group’s report. Atlas has not repeated the runs.

These reports do not list the software versions, proof-verification programs, axioms, time or memory in separate fields. Any such details in a report’s prose are reproduced below.

Limiting satisfiability threshold for every fixed $k\ge3$ FixedClauseThreshold

Three reports, all at the release’s recorded version.

Reported byTheir wordTheir accountDateOriginal report
cert-machine“CERTIFIED”Re-run through Comparator with the independent nanoda kernel switched on: Lean’s kernel and nanoda both accept OpenAI’s solution.10 October 2026Original report
MathVet“comparator-sandboxed-pass”Re-run through Comparator in a sandboxed virtual machine: accepted.8 October 2026Original report
Neal McBurnett, openai-math-lean-tooling“accepted”Re-run through Comparator with Lean’s kernel: accepted (the check took 57 s; peak memory 6.1 GiB).10 October 2026Original report

Computability of the random $3$-SAT threshold SATComputability

Two reports, all at the release’s recorded version.

Reported byTheir wordTheir accountDateOriginal report
cert-machine“CERTIFIED”Re-run through Comparator with the independent nanoda kernel switched on: Lean’s kernel and nanoda both accept OpenAI’s solution.10 October 2026Original report
MathVet“comparator-sandboxed-pass”Re-run through Comparator in a sandboxed virtual machine: accepted.8 October 2026Original report

Sharpness of random $k$-SAT lifetime and replacement bounds SATSharpness

Three reports, all at the release’s recorded version.

Reported byTheir wordTheir accountDateOriginal report
cert-machine“CERTIFIED”Re-run through Comparator with the independent nanoda kernel switched on: Lean’s kernel and nanoda both accept OpenAI’s solution.10 October 2026Original report
MathVet“comparator-sandboxed-pass”Re-run through Comparator in a sandboxed virtual machine: accepted.8 October 2026Original report
Neal McBurnett, openai-math-lean-tooling“accepted”Re-run through Comparator with Lean’s kernel: accepted (the check took 140 s; peak memory 6.2 GiB).10 October 2026Original report

Variance bounds for the random $k$-SAT hitting time SATVariance

Three reports, all at the release’s recorded version.

Reported byTheir wordTheir accountDateOriginal report
cert-machine“CERTIFIED”Re-run through Comparator with the independent nanoda kernel switched on: Lean’s kernel and nanoda both accept OpenAI’s solution.10 October 2026Original report
MathVet“comparator-sandboxed-pass”Re-run through Comparator in a sandboxed virtual machine: accepted.8 October 2026Original report
Neal McBurnett, openai-math-lean-tooling“accepted”Re-run through Comparator with Lean’s kernel: accepted (the check took 251 s; peak memory 6.3 GiB).11 October 2026Original report

What these papers cite in the release, and what cites them

These links come from the papers’ bibliographies. A citation records that one paper cites another; how each cited result is used has not been examined here.

Within this family

Papers elsewhere in the release that this family’s papers cite

Papers elsewhere in the release that cite this family’s papers

Explain a result, compare statements or propose a correction

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Offer to write an explanationCompare one Lean statement with its paper: FixedClauseThreshold, SATSharpness, SATVariance, SATComputabilitySuggest a source attributionSuggest a change

What people have said

Commentary on this family by others, dated and credited. It is commentary, not a finding.

  • An unsigned public tier list, the OpenAI Math Tier List, classes it as a proof and rates its significance, if it holds up, as “Major math journal”. The page says that nothing on it is verified.

    Author not stated. Source

Questions awaiting review

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Read the Lean statement against the paper

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  • An AI agent drafted an assessment of which part of the paper this Lean statement expresses. No named human review is recorded. Open the form
  • An AI agent drafted an assessment of which part of the paper this Lean statement expresses. No named human review is recorded. Open the form
  • An AI agent drafted an assessment of which part of the paper this Lean statement expresses. No named human review is recorded. Open the form

Identify the theorem the catalogue describes

  • Family 235 The theorem shown was selected from shared wording with OpenAI’s description. Atlas records no review confirming that it is the theorem the description means. Open the form

Compare the proposed paper and Lean matches

  • Family 235 Atlas records no review of whether each linked Lean statement and paper theorem mean the same thing. Open the form