The Exponential Time Hypothesis
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ETH asks whether deciding satisfiability of 3-CNF Boolean formulas needs 2^{Ω(n)} time in the worst case, where n is the number of variables. Its refutation would require an algorithm whose exponential factor is 2^{o(n)} for all inputs.
Why it matters and what counts as progress
Why it matters: ETH supports conditional lower bounds across algorithms and complexity. A subexponential exact-counting algorithm would also decide satisfiability and overturn that assumption.
What counts as progress: Give a worst-case subexponential algorithm for fixed-width SAT with a precise machine model and justified time bound, or establish a rigorous obstruction. Exact counting and polynomial space are additional algorithmic targets; subexponential time alone does not settle P versus NP.
How the problem developed
- 2001 · Complexity framework [1]
Russell Impagliazzo, Ramamohan Paturi and Francis Zane Developed subexponential reductions connecting fixed-width SAT with other NP-complete problems, explaining the reach of the exponential-time hypothesis.
OpenAI Reported a deterministic exact-counting algorithm for each fixed clause width in subexponential time and polynomial space. The selected Lean target has a weaker exponent than the sharpened manuscript theorem.
- 26 Oct 2026 · AI algorithm and reported checking [5]
Claude (Anthropic) Reported a separately credited exact-counting algorithm with a sharper formalized exponent and a distinct finite-stack machine statement.
ZHAO Han; Chelsea A. McElveen An independent segmented Lean build and statement comparison supplement the original ETH check. An AI-assisted executable study illustrates finite counting identities. Neither completes the announced expert review or tests the full asymptotic saving.
AI contributions 2
A 2^{o(n)} Time Algorithm for 3-SAT
One deterministic finite-alphabet Mathlib TM2 machine exactly counts the satisfying assignments of k-CNF formulas with k≥1, m≥2 distinct clauses and n≥4 occurring variables in time 2^{O(min{k²n log log n/√log n, km/√log m})}, with base-two logarithms and an absolute hidden constant. A separately formalized decision statement refutes ETH.
The Exponential Time Hypothesis is False
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.
Sources
- Impagliazzo, Paturi and Zane: Which Problems Have Strongly Exponential Complexity?
- The Exponential Time Hypothesis is False
- OpenAI: result origins, checking dates and expert collaborations
- Scope of the reported Lean proof
- Claude: A subexponential-time algorithm for 3-SAT
- ZHAO Han: independent ETH build and statement audit
- McElveen: executable study of the counting pipeline
- Anthropic: provenance, checking and practical limits
- Exact counting and ETH statements
- Proof entry points
- Comparator target and permitted axioms
- Completed ETH verification job
- Standalone formal statement
- Comparator target and permitted axioms
- Completed proof entry point
Website timeline
Publication dates and editorial changes.