{"schema":"postcutoff/event@1","as_of":"2026-10-30T13:42:00+01:00","url":"https://postcutoff.com/e/2026-10-06-openai-sidorenko-conjecture-counterexample-claim/","md":"https://postcutoff.com/e/2026-10-06-openai-sidorenko-conjecture-counterexample-claim/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":"2026-10-06-openai-sidorenko-conjecture-counterexample-claim","date":"2026-10-06","date_precision":"day","short_title":"Sidorenko's conjecture disproved? (OpenAI, Lean)","deck":null,"takeaway":"A 51-page manuscript credited to 'OpenAI' (dated 23 Sept, public on 6 Oct) claims that Sidorenko's 1986 conjecture is false. Sidorenko predicted that every bipartite graph H appears in any host graph at least as often as in a random graph of the same edge density. The counterexample H is explicit: the incidence graph of 22 triples on 13 points, with 35 vertices and 66 edges. The host graph is shown to exist by sampling from a finite weighted kernel but is not written down. Lean proves the existence statement and it passed a community comparator run. The same H also refutes the Skokan–Thoma forcing conjecture.","category":"science","category_label":"Science & math","importance":4,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":[{"n":1,"title":"OpenAI (PDF): A counterexample to Sidorenko's conjecture (23 Sept 2026)","url":"https://github.com/openai/math/blob/main/preprints/A-counterexample-to-Sidorenkos-conjecture-September-23-2026/paper.pdf","type":"paper","group":"primary","domain":"github.com"},{"n":2,"title":"OpenAI Lean scope note for family 161","url":"https://github.com/openai/math/blob/main/lean/docs/161.md","type":"code","group":"primary","domain":"github.com"},{"n":3,"title":"oai-math-revised comparator results (SidorenkoCounterexample accepted)","url":"https://github.com/mathcommunity/oai-math-revised/blob/main/checks/comparator-2026-10-26/results.tsv","type":"code","group":"primary","domain":"github.com"},{"n":4,"title":"Wikipedia: Sidorenko's conjecture","url":"https://en.wikipedia.org/wiki/Sidorenko%27s_conjecture","type":"docs","group":"primary","domain":"en.wikipedia.org"},{"n":5,"title":"Carlos Toledo: OpenAI's math release, what it claims beyond what it checks (finding G10)","url":"https://carlostoledo.co/reports/openai-math.html","type":"discussion","group":"reactions","domain":"carlostoledo.co"},{"n":6,"title":"Hamed Hatami on X (6 Oct)","url":"https://x.com/HamedHa26534517/status/2107621391616876611","type":"discussion","group":"reactions","domain":"x.com"},{"n":7,"title":"Gil Kalai: Updates: Sharing AI progress on mathematics (amazing!) (7 Oct)","url":"https://gilkalai.wordpress.com/2026/10/07/updates-sharing-ai-progress-on-mathematics-amazing-and-my-lecture-plans/","type":"discussion","group":"reactions","domain":"gilkalai.wordpress.com"},{"n":8,"title":"oai-math-revised AGENT_NOTES for the Sidorenko manuscript","url":"https://github.com/mathcommunity/oai-math-revised/blob/main/preprints/A-counterexample-to-Sidorenkos-conjecture-September-23-2026/AGENT_NOTES.md","type":"discussion","group":"reactions","domain":"github.com"}],"official":4,"filed":"2026-10-29","updated":"2026-10-29","orgs":["OpenAI"],"title":"OpenAI's math release claims a counterexample to Sidorenko's conjecture: a 35-vertex, 66-edge bipartite graph with fewer copies in some host graph than in a random graph of the same density; it also refutes the forcing conjecture; Lean proves the host exists, but none is published","summary":"\"A counterexample to Sidorenko's conjecture\" (OpenAI, 23 Sept 2026, 51 pages; family 161 of the 6 Oct release): \"We disprove Sidorenko's conjecture with a bipartite graph on 35 vertices and 66 edges: its homomorphism density in some finite simple graph is smaller than the conjectured lower bound. The same connected graph also disproves the forcing conjecture.\" H is the incidence graph of 22 triples on 13 points in which each of the 33 point pairs that occur lies in exactly two triples; a host G with t(H,G) < p(G)^66 is obtained by sampling from a finite kernel with a strict density gap. Lean (challenge SidorenkoCounterexample) proves that such a host exists and was accepted in the oai-math-revised comparator run; the forcing consequence is outside it. No explicit host graph has been published, and no extremal combinatorialist has published a check.","key_facts":["Manuscript (abstract, verbatim): 'We disprove Sidorenko's conjecture with a bipartite graph on 35 vertices and 66 edges: its homomorphism density in some finite simple graph is smaller than the conjectured lower bound. The same connected graph also disproves the forcing conjecture: at one fixed density, asymptotically matching the edge and pattern densities does not imply quasirandomness.' 51 pages, dated 23 Sept 2026","Construction (paper): H is the point–triple incidence graph of a system of 22 triples on 13 points in which each of the 33 point pairs used lies in exactly two triples; 'We first construct a strict density gap for a finite nonnegative kernel and then retain it when sampling a finite simple host. This route specifies H explicitly and proves existence of G.' Tensor powers make the ratio t(H,F)/p(F)^66 arbitrarily small","Background: Sidorenko posed the conjecture in 1986; Wikipedia calls it 'a major conjecture in the field of extremal graph theory'. It was known for many classes, including trees, even cycles and complete bipartite graphs. The forcing conjecture (Skokan–Thoma) says a graph is forcing, i.e. its density controls quasirandomness, exactly when it is bipartite and not a tree","Lean (OpenAI scope note lean/docs/161.md): 'The formalization disproves this for the paper's fixed bipartite graph with 35 vertices and 66 edges: it constructs a nonempty finite simple host graph with t(H,G) < t(K₂,G)^66. The separate forcing-conjecture consequence in the paper is outside this statement.' Challenge 'accepted' in the oai-math-revised comparator run of 25–26 Oct","Witness not published: Carlos Toledo's audit records H as given in the paper's Table 1 but 'Host: NOT PUBLISHED', and notes that 'Where the Lean is kernel-checked (Hadwiger, Sidorenko, Ryser) the existence is formally proved; the gap is that nobody can look at the object.' An AI agent's public repository (piiiico/openai-math-counterexamples) recomputed H and the paper's clauses on the triple system without constructing the host","Reactions: Hamed Hatami (X, 6 Oct, in a reply): 'Not surprising, but the counterexample seems complicated and not just a numerical solution to some optimization problem. Maybe there is something interesting there.' Gil Kalai (7 Oct): 'The announced counterexample has 35 vertices and 66 edges and also disproves the related forcing conjecture.' Wikipedia's article now records the claim. No dedicated Hacker News thread found"],"key_numbers":[],"tags":["math","extremal-combinatorics","graph-theory","sidorenko-conjecture","graph-homomorphisms","quasirandomness","counterexample","openai-math-release","lean","formal-verification"],"science":{"field":"mathematics","subfield":"extremal combinatorics (graph homomorphism densities, quasirandomness)","problem":"Sidorenko's conjecture (1986): every bipartite graph H satisfies t(H,G) ≥ p(G)^|E(H)| in every host graph G; also the Skokan–Thoma forcing conjecture","result":"Claimed counterexample: an explicit 35-vertex, 66-edge bipartite incidence graph H and a (non-explicit) finite host G with t(H,G) < p(G)^66; the same H is not forcing","open_since":"1986","ai_system":["OpenAI internal model (math release)"],"human_role":"Released as model output under OpenAI's fixed procedure, with no human author named; no human exposition or check yet","verification":{"key":"peer-review","label":"Peer-reviewed","text":"Lean formalization of the existence of a violating host graph, accepted by an independent comparator run; no explicit host published; manuscript unrefereed"},"status":"pending","shock":"A central conjecture of extremal graph theory is claimed false with a small explicit pattern graph, yet the host graph that witnesses the failure exists only as a sampling argument."},"body_md":"## What happened\n\nSidorenko's conjecture says that random graphs minimize the number of copies of any bipartite graph H, given the edge density. It had\nbeen proved for many families of H, and its failure for some H would mean that some pattern can be suppressed below its random count.\n\nOpenAI's 6 Oct release claims such an H. It is small and explicit: the points and triples of a system of 22 triples on 13 points, in which each\npair of points that occurs lies in exactly two triples, give a bipartite graph with 35 vertices and 66 edges. The manuscript builds a weighted \"kernel\" on which H is strictly\nrarer than the edge density predicts, then samples a finite graph that keeps the gap. Lean proves that a finite host graph exists, and a\ncommunity comparator run accepted the proof. The same H shows that matching edge and H-densities does not force quasirandomness, against\nthe forcing conjecture.\n\n## Why it matters\n\nIf correct, a well-known conjecture of extremal combinatorics is false, and the many known positive cases become a boundary to map\nrather than steps toward a theorem. The formal proof makes the existence claim strong, but no one can yet inspect a host graph. The one\nexpert remark found so far calls the result unsurprising and the construction possibly interesting.","disputed":[{"label":"Verification","text":"Lean formalization of the existence of a violating host graph, accepted by an independent comparator run; no explicit host published; manuscript unrefereed","sources":[]}],"related":[{"id":"2026-10-24-oai-math-revised-community-hub","url":"https://postcutoff.com/e/2026-10-24-oai-math-revised-community-hub/","date":"2026-10-24","date_precision":"day","short_title":"oai-math-revised: mathematicians launch a community hub to revise OpenAI's AI-written math manuscripts (guest post on Tao's blog)","deck":null,"takeaway":"This is the first organized attempt by research mathematicians to digest and credit the OpenAI catalogue.","category":"research","category_label":"Research","importance":3,"confidence":"high","status":{"key":"confirmed","labels":["Confirmed"]},"sources":6,"official":5,"filed":"2026-10-25","updated":"2026-10-27","orgs":["OpenAI","mathcommunity"]},{"id":"2026-10-06-openai-math-release-722-manuscripts","url":"https://postcutoff.com/e/2026-10-06-openai-math-release-722-manuscripts/","date":"2026-10-06","date_precision":"day","short_title":"OpenAI releases 722 AI-written math manuscripts claiming hundreds of open problems","deck":"Including quasi-Riemann, Unique Games, Hodge for CM abelian varieties and free group factors","takeaway":"If even a fraction of these results hold up, this is the largest single jump in mathematical knowledge on record, produced by an AI system in about six weeks.","category":"science","category_label":"Science & math","importance":5,"confidence":"high","status":{"key":"pending","labels":["Event confirmed","Awaiting review"]},"sources":79,"official":13,"filed":"2026-10-07","updated":"2026-10-29","orgs":["OpenAI"]},{"id":"2026-10-06-openai-hadwiger-conjecture-counterexample-claim","url":"https://postcutoff.com/e/2026-10-06-openai-hadwiger-conjecture-counterexample-claim/","date":"2026-10-06","date_precision":"day","short_title":"Hadwiger's conjecture disproved? (OpenAI, Lean)","deck":null,"takeaway":"A 104-page manuscript credited to 'OpenAI' (dated 23 Sept, public on 6 Oct) claims arbitrarily large graphs whose chromatic number exceeds their largest clique minor. That would refute Hadwiger's 1943 conjecture, a far-reaching generalization of the four colour theorem known to hold only up to K6. The examples have no three pairwise non-adjacent vertices and also beat the fractional weakening. A Lean formalization of the ordinary counterexample, added in OpenAI's 7 Oct update, was accepted in a community comparator run, and an independent Lean project derived the fractional version. The graphs come from a random construction; no explicit example has been published, and no expert has published an assessment.","category":"science","category_label":"Science & math","importance":5,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":13,"official":10,"filed":"2026-10-29","updated":"2026-10-29","orgs":["OpenAI"]}],"people":[],"posts":[{"id":"x-hamedha26534517-2107621391616876611","title":"“Not surprising, but the counterexample seems complicated and not just a numerical…”","url":"https://postcutoff.com/p/x-hamedha26534517-2107621391616876611/"}],"videos":[],"models":[],"changes":[{"date":"2026-10-29","type":"filed","text":"Created (CONTENTS entry, PDF (51 pp), Lean scope note, comparator results, Hatami's post via fxtwitter, Kalai, oai-math-revised notes, Wikipedia)"}],"provenance":{"agents":[{"model":"Claude Opus 5.5","maker":"Anthropic","tool":"Claude Code"}],"filed":"2026-10-29","run":null,"sources_read":null,"updated":"2026-10-29","human_review":null,"version":{"date":"2026-10-29"}},"gaps":[{"model_id":"gpt-6-astra","name":"GPT-6 Astra","cutoff":"2026-04","days_after":159,"in_training_data":false},{"model_id":"claude-opus-5-5","name":"Claude Opus 5.5","cutoff":"2026-06","days_after":98,"in_training_data":false},{"model_id":"gemini-4-argon","name":"Gemini 4 Argon","cutoff":"2026-03","days_after":189,"in_training_data":false},{"model_id":"grok-4-7","name":"Grok 4.7","cutoff":"2026-05","days_after":128,"in_training_data":false}],"short_url":"https://postcutoff.com/s/sidorenko-conjecture-disproved"}