For twenty-five years, a large part of theoretical computer science has rested on one assumption: that 3-SAT, the basic problem of finding values that satisfy a logic formula, cannot be solved much faster than trying every combination. It is called the Exponential Time Hypothesis, and Russell Impagliazzo and Ramamohan Paturi stated it in 2001. Anthropic’s paper counts more than 800 papers with at least one result that rests only on it or on a stronger version. On October 26 and 27, 2026, two AI labs published papers saying it is false. OpenAI’s internal model and Anthropic’s internal Claude model each found, on the same day, October 16, a deterministic algorithm that solves 3-SAT in subexponential time, and each lab released a Lean formalization. Impagliazzo and Paturi, with Ryan Williams, are leading the examination of OpenAI’s proof, stating that their involvement is not certification. Both papers are explicit that nothing changes in practice: by Anthropic’s estimate, the general bound beats brute force only when the number of variables has a hundred million digits. Two independent machines reaching the same refutation, and releasing it for checking, is the amplifying signal. Understanding it is now the field’s work.
3-SAT asks whether some setting of true and false values makes a formula of three-literal clauses true. Trying every setting of n variables takes 2ⁿ steps. Decades of work shaved the base of that exponent; Anthropic’s paper gives the prior records as 1.328ⁿ for deterministic algorithms and 1.308ⁿ for randomized ones.[3] The Exponential Time Hypothesis, stated by Impagliazzo and Paturi in 2001, says no algorithm can do fundamentally better: 3-SAT needs time exponential in n.[3][8] Many results lean on it. In a literature search carried out by Claude, Anthropic counted more than 800 papers that prove at least one result of their own only under ETH or a hypothesis that implies it.[3]
OpenAI’s paper, “The Exponential Time Hypothesis is False,” dated October 17 and revised October 25, gives “a deterministic algorithm for exactly counting the satisfying assignments of an arbitrary k-CNF formula” in time roughly 2 to the power n divided by the square root of log n.[1] It states that the result also refutes the strong, gap and nondeterministic versions of the hypothesis.[1] Anthropic’s paper, “A 2^o(n) Time Algorithm for 3-SAT,” credited to “Claude, Anthropic,” reaches a bound of the same shape.[3] Its provenance note: “The core algorithm was found on October 16 by an internal Claude research model with 14M output tokens and no human input,” and “Independently and concurrently, an internal model at OpenAI claims to have proven a similar result.”[3]
Both labs say practice is untouched. Anthropic: “Our result is purely asymptotic.” For typical formulas its bound does not improve on exhaustive search unless n exceeds 10 to the power 10,000, and in general not unless n has about a hundred million digits; brute force is already impractical at n = 100.[3] The repository README puts it in one line: the result “says nothing about solving formulas in practice.”[4] What changes is the map of what is believed to be hard.
The release process is the second half of the story. OpenAI published its paper and Lean proof on GitHub and asked outside experts to act as reviewers; for this result, Impagliazzo, Paturi and Williams lead an independent working group, and at their request the materials were made public immediately.[2] They wrote that their involvement “should not be taken as certification of the claim.”[6] Anthropic’s Lean proof was re-checked by a second, independently written kernel, and its README is candid about the remaining gap: “Lean checks the proofs, but no machine checks that this file says what the paper says.”[4]
One machine-made proof invites the question of whether it was checked. Two independent ones, by different systems at different companies, reaching the same refutation on the same day, add a different kind of evidence — not a review, but a convergence. It is not proof that either is right. It does make a single hidden error less likely to explain both.
The way the hypothesis’s authors responded matters as much. Impagliazzo and Paturi stated ETH; Williams has long worked on its consequences. They accepted the role of examining the refutation, asked for the raw materials to be released, and said in advance that taking part does not mean agreeing.[2][6] That is the field’s own checking process, run in the open and at speed.
The release came with two other claims. OpenAI says its models proved Sogge’s local smoothing conjecture in every dimension, which would also settle the Kakeya conjecture there, and existence results in four-dimensional quantum field theory including quantum chromodynamics; the physics group led by David Gross chose not to release the AI-produced material for now.[2][5] This case follows the computer-science result, the one both labs claim.
Paper and Lean proof on GitHub; Impagliazzo, Paturi and Williams to examine it.[2]
ReleaseWith at least one result proved only under ETH or a stronger hypothesis.[3]
Lean checks the proofs, but no machine checks that this file says what the paper says. — Anthropic, ETH repository README [4]
| Dimension | Evidence |
|---|---|
| Quality (D5) Origin · 88 | Subexponential deterministic algorithms for 3-SAT from OpenAI and Anthropic, both found October 16.[1][3]Two Independent Refutations |
| Operational (D6) L1 · 80 | Papers and Lean proofs public; a second kernel; materials opened at the reviewers’ request.[2][4]Released to Be Checked |
| Employee (D2) L1 · 76 | Impagliazzo, Paturi and Williams lead the examination: involvement “should not be taken as certification.”[6]Its Authors Review It |
| Customer (D1) L2 · 72 | More than 800 papers prove at least one result only under ETH or a hypothesis implying it.[3]800 Papers on One Premise |
| Regulatory (D4) L2 · 70 | “No machine checks that this file says what the paper says” — acceptance still runs through people.[4]How a Field Accepts |
The cascade originates in D5 (Quality): a claimed subexponential algorithm for 3-SAT from two independent systems.[1][3] It propagates to D6 (Operational), the release machinery — public papers, Lean proofs, a second checking kernel, materials opened at the reviewers’ request[2][4] — and to D2 (Employee), the researchers now examining a refutation of a hypothesis two of them stated.[6] It lands in D4 (Regulatory), the norms by which a field accepts machine-produced proofs, and in D1 (Customer), the community whose 800-plus conditional results now rest on a premise both labs say is false.[3] D3 (Revenue) is not scored.
It follows UC-329, where a claimed improvement was too small for any computer to notice; this one is larger in theory and just as invisible in practice. It follows UC-333, where a machine check covered less than the paper claimed — Anthropic names that gap itself. And it follows UC-334, where the author of a problem worked on the AI’s answer. The honest limits: no outside review has reported; the two papers are claims, not confirmations; and the result changes what is believed to be hard, not what anyone can compute.
-- UC-339: Two Machines, One Hypothesis: 6D Amplifying Cascade
-- ETH stated by Impagliazzo and Paturi 2001. Prior 3-SAT records 1.328^n deterministic, 1.308^n randomized. OpenAI, The Exponential Time Hypothesis is False, dated Oct 17 revised Oct 25 2026, 67 pages, released Oct 26 with Lean: deterministic exact counting of k-CNF in time about 2^(n/sqrt(log n)) up to lower-order terms, poly space; also refutes SETH, Gap-ETH, NSETH. Reviewers Impagliazzo, Paturi, Williams; involvement not certification; materials public at their request. Anthropic, A 2^o(n) Time Algorithm for 3-SAT, by Claude, dated Oct 26, released Oct 27, 49 pages: core algorithm found Oct 16 by internal Claude model, 14M output tokens, no human input; #3-SAT in 2^O(n loglog n / sqrt(log n)); purely asymptotic; beats exhaustive search only if n above 10^10000 for m=5n, above 10^(10^8) in general; more than 800 papers rely only on ETH or a stronger hypothesis; Lean re-checked by second kernel nanoda; no machine checks that the file says what the paper says. Same release: local smoothing all dimensions (Kakeya), QCD existence (physics material withheld).
FORAGE two_machines_one_hypothesis
WHERE independent_labs_same_result = true
AND lean_formalization_public = true
AND hypothesis_authors_reviewing = true
ACROSS D5, D6, D2, D1, D4
DEPTH 3
SURFACE two_machines_one_hypothesis
DIVE INTO convergent_refutation
WHEN external_review_reported = false
AND practical_speedup = false
TRACE refutation_acceptance_cascade
EMIT two_machines_one_hypothesis_signal
DRIFT two_machines_one_hypothesis
METHODOLOGY 90
PERFORMANCE 40
FETCH two_machines_one_hypothesis
THRESHOLD 1000
ON WATCH CHIRP high 'Internal models at OpenAI and Anthropic each found, on October 16, 2026, a subexponential algorithm for 3-SAT that refutes the 2001 Exponential Time Hypothesis. Both released papers and Lean proofs; the hypothesis's authors lead the review and say their involvement is not certification. The result is purely asymptotic: no effect in practice.'
SURFACE analysis AS jsonRuntime: @stratiqx/cal-runtime · Spec: cal.semanticintent.dev · DOI: 10.5281/zenodo.18905193
Two systems, two companies, one day, one refutation. Not a review — but harder to explain away than one claim.[1][3]
The general bound beats brute force only for numbers of variables with about a hundred million digits. Nothing gets faster.[3]
Those who stated ETH lead the examination of its refutation, and said first that taking part is not agreeing.[6]
“No machine checks that this file says what the paper says.” The last check is still a reader.[4]
Primary sources are both papers, read in full where it matters, and both GitHub repositories, including OpenAI’s release notes and Anthropic’s README. Reviewer statements come from press coverage of their open letter. No outside review has reported. Source 10 is an interactive companion made for this case.
This case follows a field that built hundreds of results on one hypothesis — and the moment two machines said it was false. Every organisation has premises like that. Book a discovery call to map what yours are holding up.