What was already known
Impagliazzo and Paturi formulated ETH; Impagliazzo, Paturi and Zane developed its subexponential reductions. Earlier SAT algorithms improved the exponential base while retaining exponential worst-case dependence on n.
AI involvement
OpenAI is credited as author and explicitly attributes the proof to its internal AI system. The disclosure does not establish task-only autonomy.
Importance
Would refute ETH and its stronger variants, affecting a standard foundation of conditional hardness results.
Limitations
The assessed bound is the weaker one explicitly covered by the Lean statement. The revised manuscript gives sharper general and bounded-density exponents; those sharper bounds are outside this checking assessment. This does not settle P versus NP, supply a practical SAT solver, or automatically make every NP problem subexponential. The announced expert working group has not yet completed its review. The independent re-check leaves standard-model simulation and encoding correspondence open. McElveen’s exact toy computations illustrate finite cases and do not verify the asymptotic saving; the reported audit was agent-run.
Technical verification
OpenAI reports Lean verification on 2026-10-17. The standalone target and matching proof entry specify the assessed exact counting machine, canonical encoding, time bound and polynomial visited space. The manuscript’s sharper exponent is not part of this selected check. This is source-reported checking; no independent replay was performed here. ZHAO Han’s archived check at the current OpenAI commit reports all 3,977 modules built, only the three standard Lean axioms, and 69 identical elaborated constants including definition bodies; a changed-runtime negative control is rejected. The build was completed in segments in one workspace on Apple M4. This was not a sandboxed Comparator or second-kernel replay. The archived heuristic scan flags benign notation options and comment text; the report explains those flags. Standard-machine simulation and input-encoding fidelity remain a human review task. The statement readers were Claude models.
Independent checking
ZHAO Han’s archived check at the current OpenAI commit reports all 3,977 modules built, only the three standard Lean axioms, and 69 identical elaborated constants including definition bodies; a changed-runtime negative control is rejected. The build was completed in segments in one workspace on Apple M4. This was not a sandboxed Comparator or second-kernel replay. The archived heuristic scan flags benign notation options and comment text; the report explains those flags. Standard-machine simulation and input-encoding fidelity remain a human review task. The statement readers were Claude models.
- Checker
- ZHAO Han: source-reported Lean build and statement-closure audit
- Scope
- For each fixed k≥2, one deterministic finite-tape machine exactly counts satisfying assignments of every canonically encoded k-CNF formula on n variables. With encoding length L, its time is (L+1)^{O_k(1)} 2^{O_k(n(log log n)²/√log n)} and its visited tape space is polynomial in L at every execution prefix. In particular, fixed-width 3-SAT has a subexponential-time decision algorithm.
- Basis
- Reported by the linked checker
- Check date
- 2026-10-28
Formal checking
OpenAI reports the completed Lean check on 2026-10-17. The selected exact target is linked separately; checking does not establish human understanding.
- Checker
- OpenAI: Lean kernel
- Scope
- For each fixed k≥2, one deterministic finite-tape machine exactly counts satisfying assignments of every canonically encoded k-CNF formula on n variables. With encoding length L, its time is (L+1)^{O_k(1)} 2^{O_k(n(log log n)²/√log n)} and its visited tape space is polynomial in L at every execution prefix. In particular, fixed-width 3-SAT has a subexponential-time decision algorithm.
- Basis
- Reported by the linked checker
- Check date
- 2026-10-17
Checking a formal statement does not establish that it matches the original problem.
Human understanding
Few is the default when evidence for a higher level is insufficient. Few remains the documentation default. McElveen provides an AI-assisted executable exposition at toy scale, while the independent statement readers were Claude models. A completed named human explanation of the full asymptotic mechanism or review by the announced expert group is not documented.
Human follow-up work
Independent build and statement-closure check of the ETH formalization
28 October 2026A segmented Lean build, axiom report and definition-body comparison cover the current formal target. The audit uses one kernel in one workspace; standard-model correspondence remains open.
ZHAO Han, with Claude Code tooling; the statement readers were Claude models.Measurement-Indexed Counting: an executable study
28 October 2026Exact toy models illustrate the counting identities and interpolation. Finite tests do not establish the asymptotic saving; the reported audit was agent-run.
Chelsea A. McElveen; implementation and drafting assisted by Claude.
Correction and publication history
New independent build, axiom and elaborated-statement comparison of the retained ETH target; toy-scale checks retain their separate limits.
Read this published revisionNewly reviewed exact-counting algorithm and reported Lean check.
Read this published revision