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The Exponential Time Hypothesis is False

OpenAI

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.

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Sources for this record

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.

This revision has no four-dimension assessment. Read the criteria.

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.

Technical verification

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
Evidence source

Checking a formal statement does not establish that it matches the original problem.

Correction and publication history

2026-10-29 · verification

New independent build, axiom and elaborated-statement comparison of the retained ETH target; toy-scale checks retain their separate limits.

Read this published revision