Result 235, Probability and statistical mechanics

Limiting random SAT thresholds, sharp variance and computability

For random k-SAT with independent uniformly signed proper clauses sampled with replacement, proves finite positive limiting thresholds and hitting-time variance Θk(n)\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.

Lean formalization Proof

The bigger picture

Why it matters

Random SAT asks whether true-or-false choices can satisfy a growing list of randomly chosen 'or' rules. These manuscripts describe both a limiting rules-per-variable boundary between satisfiability and unsatisfiability and the scale of fluctuations in where that boundary appears.

What changes?

The manuscripts report finite, positive limiting thresholds for every fixed clause size k at least 3. Clauses use k distinct variables, each independently negated with probability one-half; clauses are sampled independently with replacement. The hitting time, the number of clauses at the first unsatisfiable prefix, has variance bounded above and below by positive constants times n, where n counts variables and constants may depend on k. For 3-SAT specifically, they also report that the threshold is a computable real.

What does that help mathematicians do?

The variance bounds identify the standard deviation of the hitting time as proportional to the square root of n, for fixed k. This pins down its fluctuation scale, not merely whether the rules-per-variable boundary stabilizes. For 3-SAT, the upper bound removes an earlier logarithmic gap. The family supplies an alternative threshold-existence proof, crediting Gaia Carenini with priority, while adding these fluctuation bounds and computability.

Are there practical applications?

The immediate value is foundational for random constraint problems. For 3-SAT, the reported computability result means one finite deterministic procedure can approximate the threshold to any requested accuracy. Its search uses lower and upper certificates and halts without needing a computable finite-size convergence rate. This establishes algorithmic accessibility, not an efficient numerical method or a faster SAT solver.

This section was generated by GPT-6 Astra Medium. This explanation is based on the result summary and manuscript abstracts below. This context is separate from OpenAI's source text.

4 manuscripts

A Limiting Satisfiability Threshold for Every Fixed Clause Size

September 25, 2026 21 pages

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.

PDF LaTeX source
Cite (BibTeX) 
@misc{OAI:A-Limiting-Satisfiability-Threshold-for-Every-Fixed-Clause-Size-September-25-2026,
  author = {{OpenAI}},
  title = {{A Limiting Satisfiability Threshold for Every Fixed Clause Size}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/A-Limiting-Satisfiability-Threshold-for-Every-Fixed-Clause-Size-September-25-2026/article.pdf}{OAI:A-Limiting-Satisfiability-Threshold-for-Every-Fixed-Clause-Size-September-25-2026}},
  year = {2026}
}

Linear Variance of the Random 3-SAT Hitting Time

October 5, 2026 10 pages

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 Θ(n)\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.

PDF LaTeX source
Cite (BibTeX) 
@misc{OAI:Linear-Variance-of-the-Random-3-SAT-Hitting-Time-October-5-2026,
  author = {{OpenAI}},
  title = {{Linear Variance of the Random 3-SAT Hitting Time}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Linear-Variance-of-the-Random-3-SAT-Hitting-Time-October-5-2026/linear-variance-of-the-random-3-sat-hitting-time.pdf}{OAI:Linear-Variance-of-the-Random-3-SAT-Hitting-Time-October-5-2026}},
  year = {2026}
}

Variance of the Random k-SAT Hitting Time

September 27, 2026 37 pages

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 Var(Hn)=Θk(n)\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 nlog⁡nn\log n; the companion paper on random 3-SAT sharpens this to Θ(n)\Theta(n). The same upper bounds proved here hold after clipping at any fixed positive multiple of n.

PDF LaTeX source
Cite (BibTeX) 
@misc{OAI:Variance-of-the-Random-k-SAT-Hitting-Time-September-27-2026,
  author = {{OpenAI}},
  title = {{Variance of the Random $k$-SAT Hitting Time}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Variance-of-the-Random-k-SAT-Hitting-Time-September-27-2026/article.pdf}{OAI:Variance-of-the-Random-k-SAT-Hitting-Time-September-27-2026}},
  year = {2026}
}

Computing the Random 3-SAT Threshold

September 27, 2026 48 pages

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.

PDF LaTeX source
Cite (BibTeX) 
@misc{OAI:Computing-the-Random-3-SAT-Threshold-September-27-2026,
  author = {{OpenAI}},
  title = {{Computing the Random 3-SAT Threshold}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Computing-the-Random-3-SAT-Threshold-September-27-2026/article.pdf}{OAI:Computing-the-Random-3-SAT-Threshold-September-27-2026}},
  year = {2026}
}

Lean formalization

OpenAI's note on what the formalization covers, from lean/docs/235.md.

Limiting random SAT thresholds, sharp variance and computability

The following describes the scope of the Lean formalization related to the following accompanying paper(s):

Scope

The random kk-SAT threshold problem asks whether satisfiability changes at one limiting clause density. The formalization proves that every fixed integer k≥3k\ge3 has a finite positive threshold αk\alpha_k: for every fixed density c<αkc<\alpha_k, the satisfiability probability tends to one as the number of variables grows, while for every c>αkc>\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 HnH_n be the first unsatisfiable prefix length in random kk-SAT with independent uniformly signed clauses on kk distinct variables, sampled with replacement. The formalization proves Var(Hn)=Θk(n)\mathrm{Var}(H_n)=\Theta_k(n) for every fixed k≥4k\ge4. For k=3k=3, it proves a positive linear lower bound and an O(nlog⁡n)O(n\log n) upper bound. The same upper bounds hold after clipping at any fixed positive multiple of nn, 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=3k=3 variance bound is necessary.

The formalization proves that the limiting threshold α\alpha for uniform random 33-SAT is a computable positive real. Satisfiability tends to one at every fixed density below α\alpha and to zero at every fixed density above it. One deterministic machine produces a rational number qrq_r for every requested precision rr with ∣qr−α∣≤2−r|q_r-\alpha|\le2^{-r}. No computable rate of finite-size convergence is assumed in the statement.

Comparator links

Result Comparator statement
Limiting satisfiability threshold for every fixed k≥3k\ge3 FixedClauseThreshold.lean
Sharpness of random kk-SAT lifetime and replacement bounds SATSharpness.lean
Variance bounds for the random kk-SAT hitting time SATVariance.lean
Computability of the random 33-SAT threshold SATComputability.lean

Data from github.com/openai/math at commit adc7f12, committed October 6, 2026 at 21:58 UTC, last checked for changes about 8 hours ago. Titles, subjects, summaries, abstracts and Lean notes are OpenAI's; page counts are read from the PDFs. The map, related results, search, kinds of results and the named-problem index are Emergent Mind's, built with text embeddings and an LLM, and may contain errors.

An Emergent Mind Labs project. Emergent Mind is not affiliated with OpenAI. None of these results has been peer reviewed. Cite the manuscripts themselves, using the BibTeX on each result's page.