{"schema":"postcutoff/event@1","as_of":"2026-10-27T13:45:00+01:00","url":"https://postcutoff.com/e/2026-10-27-anthropic-claude-eth-refutation-3sat-subexponential/","md":"https://postcutoff.com/e/2026-10-27-anthropic-claude-eth-refutation-3sat-subexponential/index.md","disclosure":{"written_by":"AI agents (Claude Opus 5.5 in Claude Code)","editor":"human editor","policy":"https://postcutoff.com/about/"},"license":null,"id":"2026-10-27-anthropic-claude-eth-refutation-3sat-subexponential","date":"2026-10-27","date_precision":"day","short_title":"Anthropic's Claude also refutes ETH","deck":null,"takeaway":"Hours after OpenAI's ETH announcement, Anthropic posted 'A 2^o(n) Time Algorithm for 3-SAT', credited to Claude. It is a deterministic #k-SAT algorithm built on tensor-network contraction, with a different approach from OpenAI's paper, a near-identical bound and a Lean 4 proof that uses only the standard axioms. The paper says the core algorithm was found on 16 Oct with 14M output tokens and no human input. OpenAI reports its model found its disproof that same day. Neither proof has had expert review.","category":"science","category_label":"Science & math","importance":5,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":[{"n":1,"title":"GitHub: anthropics/formal-math, eth/ (commit 16a5ce4, 27 Oct 2026)","url":"https://github.com/anthropics/formal-math/tree/16a5ce41304fcdaea4e0be7c132e1c361dc26da5/eth","type":"code","group":"primary","domain":"github.com"},{"n":2,"title":"Claude (Anthropic): A 2^o(n) Time Algorithm for 3-SAT (paper.pdf, 26 Oct 2026)","url":"https://github.com/anthropics/formal-math/blob/16a5ce41304fcdaea4e0be7c132e1c361dc26da5/eth/paper.pdf","type":"paper","group":"primary","domain":"github.com"},{"n":3,"title":"Hexagon 2610.00577v1: A 2^o(n) Time Algorithm for 3-SAT (contributed by Anthropic, submitted by Jelani Nelson)","url":"https://hexagonmath.org/abs/2610.00577v1","type":"paper","group":"primary","domain":"hexagonmath.org"},{"n":4,"title":"OpenAI: The Exponential Time Hypothesis is False (PDF)","url":"https://github.com/openai/math/blob/main/preprints/The-Exponential-Time-Hypothesis-is-False-October-25-2026/paper.pdf","type":"paper","group":"primary","domain":"github.com"},{"n":5,"title":"Levent Alpöge on X quoting the provenance paragraph (~94k views)","url":"https://x.com/__alpoge__/status/2114930095009968506","type":"discussion","group":"reactions","domain":"x.com"}],"official":4,"filed":"2026-10-27","updated":"2026-10-27","orgs":["Anthropic"],"title":"Anthropic publishes an independent refutation of the Exponential Time Hypothesis: an internal Claude research model found a 2^o(n) algorithm for #3-SAT on 16 Oct, the same day as OpenAI's model (Lean-formalized)","summary":"On 27 Oct 2026 (commit 16a5ce4, 03:07 UTC) Anthropic added an eth/ folder to github.com/anthropics/formal-math. It holds a 49-page paper dated 26 Oct, 'A 2^o(n) Time Algorithm for 3-SAT' by \"Claude (Anthropic)\", and a Lean 4 formalization written by AI models. The main theorem counts the satisfying assignments of a k-CNF deterministically in time 2^O(min{n log log n/√log n, m/√log m}) up to k-dependent factors, which refutes ETH and SETH. The provenance paragraph says the core algorithm \"was found on October 16 by an internal Claude research model with 14M output tokens and no human input\". It also notes that \"independently and concurrently, an internal model at OpenAI claims to have proven a similar result\". Levent Alpöge's post quoting that paragraph had ~94k views by 27 Oct.","key_facts":["README (verbatim): 'A deterministic algorithm solves 3-SAT on n variables in time 2^o(n), which refutes the Exponential Time Hypothesis (ETH). The algorithm in fact solves the harder problem #3-SAT, counting the satisfying assignments, in time 2^O(n log log n / √log n). The result is purely asymptotic and says nothing about solving formulas in practice. The result was discovered by an internal research Claude model. This folder holds the paper and its Lean 4 formalization, which was written by AI models.'","Provenance (paper, verbatim): 'The core algorithm was found on October 16 by an internal Claude research model with 14M output tokens and no human input. Anthropic researchers worked with Claude to improve the presentation, resulting in this paper. Independently and concurrently, an internal model at OpenAI claims to have proven a similar result [Ope26].'","Lean verification, as the README states it: MainResults.lean states ETHIsFalse and MainTheorem 'on Mathlib's Turing machines' and imports only Mathlib. PrintAxioms should give ''MainResults.ethIsFalse' depends on axioms: [propext, Classical.choice, Quot.sound]' (the same for mainTheorem), 'Lean's three standard axioms'. Lean and Mathlib v4.33.1 are pinned; the build takes about 8 minutes on 24 vCPUs. CI also runs Comparator against Challenge.lean and re-checks with the independent nanoda kernel","Caveat (README, verbatim): 'Lean checks the proofs, but no machine checks that this file says what the paper says; the file is kept short so that a reader can check this.' Appendix E of the paper prints the file and lists its differences from the printed statements; proof/PaperIndex.lean maps each numbered statement to its Lean counterpart","Method (paper keywords: 'exact algorithms, tensor networks, rank'): contraction of tensor networks on complete graphs K_K with λ bits per edge, a recursion e(9K¹⁶) ≤ 0.9·e(K) + O(1/K) obtained by writing swaps as I − 2·singlet and grouping by low-rank operators, then sparsification (Impagliazzo–Paturi–Zane) for the bound in n. The README gives a five-line 'condensed form' from which 'Claude Opus 5.5 at high effort can rederive the refutation' (only the 2^(δn) for every δ > 0 version)","Scope stated by the paper: refutes ETH and SETH and gives O(2^(δn)) algorithms for every δ > 0 for Independent Set, Clique, Vertex Cover and Dominating Set, and that E^NP lacks linear-size circuits; it 'does not refute the Orthogonal Vectors hypothesis'. A Claude literature search found 'more than 800 papers' with results proved only under ETH or a stronger hypothesis","Practicality (paper): 'for arbitrary 3-CNF formulas with m = 5n clauses, the bound we prove does not improve on the 2^n of exhaustive search unless n > 10^10 000', and the bound in n alone needs far larger n; 'our result says nothing about solving formulas in practice'","Comparison with OpenAI's paper (17/25 Oct, 67 pp.): both are deterministic #k-SAT algorithms with time 2^(n·log log n/√log n) up to constants (OpenAI: (n+m)^O_k(1)·2^(O_k(n log log n/√log n)), paper; its Lean covers (log log n)² in place of log log n). OpenAI's route goes through 'shared linear summaries', low-degree replacement forms and counting common zeros of sparse polynomial systems; Anthropic's goes through tensor-network contraction and rank bounds. OpenAI's paper also claims Gap-ETH and NSETH; Anthropic's says it does not reach Orthogonal Vectors. Anthropic's Lean covers its sharper bound, according to its README","Hexagon: posted 27 Oct as hexagon:2610.00577v1, 'Contributed by Anthropic', submitted by Jelani Nelson on Anthropic's behalf; the page tags it 'Primarily AI-generated text' and 'Human understanding: some parts', and its provenance note says an internal OpenAI model 'independently and concurrently claims a similar result'","Reactions on 27 Oct concerned OpenAI's paper (Williams, O'Donnell, Aaronson, Szeider; see the OpenAI entry); no complexity theorist had yet commented on the Anthropic paper's tensor-network method in public"],"key_numbers":[],"tags":["computer-science","complexity-theory","sat","exponential-time-hypothesis","anthropic","claude","lean","tensor-networks","unverified","claim"],"science":{"field":"computer-science","subfield":"computational complexity (exact exponential algorithms, SAT)","problem":"Exponential Time Hypothesis (Impagliazzo–Paturi 1999/2001) and SETH","result":"Claimed deterministic 2^O(n log log n/√log n)-time (fixed k) algorithm for #k-SAT, refuting ETH and SETH, found independently of OpenAI's","open_since":"1999","ai_system":["internal Claude research model (Anthropic)"],"human_role":"Autonomous discovery (14M output tokens, 'no human input'); Anthropic researchers worked with Claude on the presentation; Lean formalization written by AI models","verification":{"key":"lean","label":"Lean-verified","text":"Lean 4 proof of MainResults.ethIsFalse and mainTheorem with standard axioms only, plus Comparator and nanoda checks per the README; no machine check that the Lean statements match the paper; unrefereed, no expert review yet"},"status":"pending","shock":"Two labs' models refuted the same central complexity hypothesis on the same day, by different methods, both with formal proofs."},"body_md":"## What happened\n\nOpenAI announced its ETH refutation on the evening of 26 Oct (UTC). About five hours later Anthropic pushed a self-contained Lean\nproject to its formal-math repository with a paper credited simply to \"Claude (Anthropic)\". The paper says an internal research\nmodel found the core algorithm on 16 Oct, using 14M output tokens and no human input. OpenAI, according to press accounts of its\nblog, says its own model found its disproof on 16 Oct too and checked it in Lean the next day.\n\nThe algorithms differ. Claude's works with tensor networks. It shows that contracting networks on large complete graphs can be\nmade cheaper by a fixed factor through a recursion that splits swap gates into low-rank pieces, then reduces k-SAT to such\ncontractions. OpenAI's paper works with linear summaries and counting zeros of sparse polynomial systems. Both give a running\ntime of about 2^(n·log log n/√log n), far too slow to matter in practice. Anthropic's paper says it beats brute force only for\nastronomically large n.\n\nThe README is careful about what Lean shows: the proofs check, with only the three standard axioms, but whether MainResults.lean\nstates what the paper claims is left to human readers. That is why the file is kept short and is printed in Appendix E.\n\n## Why it matters\n\nAn independent second proof by a different method makes the result much more credible than either claim alone, though neither\nhas been reviewed by experts yet. Two labs reaching the same refutation of a 25-year-old hypothesis on the same day is also a\nstriking data point on how fast frontier systems are converging in research ability.","disputed":[{"label":"Verification","text":"Lean 4 proof of MainResults.ethIsFalse and mainTheorem with standard axioms only, plus Comparator and nanoda checks per the README; no machine check that the Lean statements match the paper; unrefereed, no expert review yet","sources":[]}],"related":[{"id":"2026-10-26-openai-eth-false-subexponential-sat-claim","url":"https://postcutoff.com/e/2026-10-26-openai-eth-false-subexponential-sat-claim/","date":"2026-10-26","date_precision":"day","short_title":"OpenAI claims ETH is false","deck":null,"takeaway":"Anthropic independently posted a different Lean-checked refutation the next day (separate entry). OpenAI's model output claims a deterministic algorithm that counts the satisfying assignments of any k-CNF in time 2^(n/(log n)^(1/2+o(1))). If correct, it refutes ETH, SETH, Gap-ETH and NSETH, the assumptions behind hundreds of fine-grained hardness results. OpenAI says the main theorem is formalized in Lean. A working group led by ETH's proposers, Russell Impagliazzo and Ramamohan Paturi, with Ryan Williams, will examine the proof. No expert has assessed it in public yet.","category":"science","category_label":"Science & math","importance":5,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":19,"official":7,"filed":"2026-10-26","updated":"2026-10-27","orgs":["OpenAI"]},{"id":"2026-10-26-openai-local-smoothing-kakeya-all-dimensions-claim","url":"https://postcutoff.com/e/2026-10-26-openai-local-smoothing-kakeya-all-dimensions-claim/","date":"2026-10-26","date_precision":"day","short_title":"OpenAI claims local smoothing and Kakeya in all dimensions","deck":null,"takeaway":"A 59-page OpenAI manuscript dated 25 Oct claims Sogge's local smoothing conjecture for the wave equation in every dimension n ≥ 2. The stated corollaries include the Bochner–Riesz conjecture, Fourier restriction throughout the interior of the conjectured range, the Kakeya and Nikodym maximal conjectures, and full Hausdorff dimension for every Euclidean Kakeya set. That last one is the Kakeya set conjecture, proved in ℝ³ only in 2025 (Wang–Zahl). OpenAI lists Lean comparator statements for the main estimate and the corollaries. No expert has assessed it yet.","category":"science","category_label":"Science & math","importance":5,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":8,"official":5,"filed":"2026-10-26","updated":"2026-10-27","orgs":["OpenAI"]},{"id":"2026-10-26-openai-yang-mills-qcd-existence-claim","url":"https://postcutoff.com/e/2026-10-26-openai-yang-mills-qcd-existence-claim/","date":"2026-10-26","date_precision":"day","short_title":"OpenAI claims 4D Yang–Mills and QCD existence","deck":null,"takeaway":"The physics third of OpenAI's 26 Oct announcement claims that certain interacting 4D quantum field theories exist mathematically, among them pure Yang–Mills and quantum chromodynamics. That is the existence half of the Yang–Mills Millennium problem, not the mass gap. A group led by David Gross, with Michael Douglas, Arthur Jaffe, Igor Klebanov and Nikita Nekrasov, is reviewing it. At the group's request the AI-produced material is not public while they write their own account. Nothing can be checked yet.","category":"science","category_label":"Science & math","importance":5,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":8,"official":1,"filed":"2026-10-27","updated":"2026-10-27","orgs":["OpenAI"]},{"id":"2026-10-16-yang-mills-claude-balaban-program-alpoge-sussman","url":"https://postcutoff.com/e/2026-10-16-yang-mills-claude-balaban-program-alpoge-sussman/","date":"2026-10-16","date_precision":"day","short_title":"Alpöge and Sussman say Claude has an (unreadable) existence argument for 4D Yang–Mills on ℝ⁴, by finishing Bałaban's 1980s program","deck":null,"takeaway":"A public repo of raw Claude output claims a construction of four-dimensional SU(2) Yang–Mills theory, the existence half of the Millennium problem but not the mass gap; its authors call it 'slop in progress' that nobody can yet read or check.","category":"science","category_label":"Science & math","importance":4,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":6,"official":4,"filed":"2026-10-18","updated":"2026-10-18","orgs":["Anthropic","Northwestern University"]}],"people":[{"id":"levent-alpoge","name":"Levent Alpöge","url":"https://postcutoff.com/person/levent-alpoge/"},{"id":"scott-aaronson","name":"Scott Aaronson","url":"https://postcutoff.com/person/scott-aaronson/"}],"posts":[{"id":"2026-10-27-alpoge-claude-eth-provenance","title":"Levent Alpöge quotes the provenance of Anthropic's ETH refutation: found on 16 Oct by an internal Claude model, 14M tokens, no human input","url":"https://postcutoff.com/p/2026-10-27-alpoge-claude-eth-provenance/"}],"videos":[],"models":[],"changes":[{"date":"2026-10-27","type":"edit","text":"Hexagon posting (submitted by Jelani Nelson, AI-generated-text tags); note that public reactions so far concern the OpenAI paper"},{"date":"2026-10-27","type":"filed","text":"Created (owner flagged the X posts; README, paper and commit read; fxtwitter for view counts)"}],"provenance":{"agents":[{"model":"Claude Opus 5.5","maker":"Anthropic","tool":"Claude Code"}],"filed":"2026-10-27","run":null,"sources_read":"Owner flagged the X posts; README, paper and commit read; fxtwitter for view counts","updated":"2026-10-27","human_review":null,"version":null},"gaps":[{"model_id":"gpt-6-astra","name":"GPT-6 Astra","cutoff":"2026-04","days_after":180,"in_training_data":false},{"model_id":"claude-opus-5-5","name":"Claude Opus 5.5","cutoff":"2026-06","days_after":119,"in_training_data":false},{"model_id":"gemini-4-argon","name":"Gemini 4 Argon","cutoff":"2026-03","days_after":210,"in_training_data":false},{"model_id":"grok-4-7","name":"Grok 4.7","cutoff":"2026-05","days_after":149,"in_training_data":false}],"short_url":"https://postcutoff.com/s/anthropic-claude-refutes-eth"}