{"schema":"postcutoff/event@1","as_of":"2026-10-30T13:42:00+01:00","url":"https://postcutoff.com/e/2026-10-06-openai-bloch-conjecture-surfaces-claim/","md":"https://postcutoff.com/e/2026-10-06-openai-bloch-conjecture-surfaces-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-bloch-conjecture-surfaces-claim","date":"2026-10-06","date_precision":"day","short_title":"OpenAI claims Bloch's conjecture for surfaces","deck":null,"takeaway":"Family 040 of OpenAI's 6 Oct 2026 release claims Bloch's conjecture: on every smooth complex projective surface without holomorphic 2-forms (p_g = 0), zero-cycles modulo rational equivalence are completely described by their degree and Albanese image. Bloch, Kas and Lieberman proved this in 1976 for surfaces not of general type; for general type only special families were known. The 45-page proof uses virtual localization on Quot schemes. It is not formalized, and the only public expert comment so far, from Daniel Litt, flags a citation problem he thinks is harmless.","category":"science","category_label":"Science & math","importance":4,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":[{"n":1,"title":"OpenAI: Bloch's conjecture for surfaces with p_g = 0 (24 Sep)","url":"https://github.com/openai/math/blob/main/preprints/Blochs-Conjecture-for-Surfaces-with-pg-equals-q-equals-0-September-24-2026/paper.pdf","type":"paper","group":"primary","domain":"github.com"},{"n":2,"title":"arXiv 2512.12451: V. Guletskii, The Bloch conjecture (Dec 2025, cited as a proposed proof)","url":"https://arxiv.org/abs/2512.12451","type":"paper","group":"primary","domain":"arxiv.org"},{"n":3,"title":"Daniel Litt on X: 'Proof of Bloch's conjecture!' (6 Oct)","url":"https://x.com/littmath/status/2107616410897678765","type":"discussion","group":"reactions","domain":"x.com"},{"n":4,"title":"oai-math-revised: community notes on the Bloch manuscript","url":"https://github.com/mathcommunity/oai-math-revised/blob/main/preprints/Blochs-Conjecture-for-Surfaces-with-pg-equals-q-equals-0-September-24-2026/AGENT_NOTES.md","type":"discussion","group":"reactions","domain":"github.com"}],"official":2,"filed":"2026-10-29","updated":"2026-10-29","orgs":["OpenAI"],"title":"OpenAI's release claims Bloch's conjecture on zero-cycles for every complex surface with p_g = 0","summary":"\"Bloch's conjecture for surfaces with p_g = 0\" (OpenAI, 24 Sep 2026, family 040 of the 6 Oct release) proves that for a smooth connected projective complex surface S with p_g(S) = q(S) = 0 the degree map CH₀(S) → ℤ is an isomorphism, with integral coefficients and no further hypotheses; combined with Bloch–Kas–Lieberman this gives Bloch's conjecture that the Albanese map on degree-zero zero-cycles is an isomorphism whenever p_g = 0. Earlier proofs covered specific general-type surfaces (Godeaux, Campedelli, Barlow, Catanese and others). The argument obtains a Chow relation from virtual localization on a Quot scheme. It is unrefereed and has no Lean formalization.","key_facts":["Theorem 1.1 (verbatim): 'Let S be a smooth connected projective surface over C with pg(S) = q(S) = 0. Then deg : CH0(S) → Z is an isomorphism, where CH0(S) has integral coefficients.' The paper: 'Theorem 1.1 imposes no minimality, fundamental-group, or surface-construction hypothesis.' Corollary 1.2 is Bloch's conjecture for p_g = 0. 45 pages, dated 24 Sep 2026","Prior state (paper): Mumford showed a nonzero holomorphic 2-form makes zero-cycles infinite-dimensional; Bloch, Kas and Lieberman proved the conjecture for Kodaira dimension < 2; general-type cases were done family by family (Inose–Mizukami, Barlow, Bauer, Pedrini–Weibel, Voisin for Catanese and Barlow surfaces)","Earlier proposals: the paper cites Guletskiĭ's preprint (arXiv 2512.12451, Dec 2025), whose Theorem A covers the same scope, and Banerjee's proof for minimal surfaces with K² = 9, 'as proposed proofs'. Its own route: 'the argument developed here obtains its Chow relation from virtual localization on a Quot scheme', using derived Quot schemes and Marian–Oprea–Pandharipande-type obstruction theories","Stated consequence: for p_g = q = 0, every rational Chow class on a power of S or on a Hilbert scheme of points is determined by its cohomology class (finite-dimensional, pure Tate motives in Kimura's sense)","Daniel Litt (X, 6 Oct, verbatim): 'Proof of Bloch's conjecture! I thought about that a bit in grad school. Paper seems to cite some wrong existing work without mentioning it's wrong :-/ but AFAICT doesn't meaningfully rely on it.' (He did not name the work)","Kiwamu Watanabe listed the release's algebraic-geometry claims as 'quite amazing' but said he could not judge them (see 2026-10-06-openai-math-release-722-manuscripts). No other expert assessment found by 29 Oct"],"key_numbers":[],"tags":["math","algebraic-geometry","algebraic-cycles","chow-groups","bloch-conjecture","surfaces","openai","openai-math-release","unverified","claim"],"science":{"field":"mathematics","subfield":"algebraic geometry (algebraic cycles, surfaces)","problem":"Bloch's conjecture (1970s): for a complex surface with p_g = 0, the Albanese map on degree-zero zero-cycles is an isomorphism","result":"Claimed in full: for p_g = q = 0, CH₀(S) ≅ ℤ via the degree; hence Bloch's conjecture for all surfaces with p_g = 0","open_since":"","ai_system":["OpenAI internal model (openai/math release)"],"human_role":"Autonomous AI output released by OpenAI under its fixed procedure (README)","verification":{"key":"expert","label":"Expert-checked","text":"Unrefereed manuscript; no Lean formalization; no published expert check"},"status":"pending","shock":"General-type surfaces with p_g = q = 0 had been handled only one construction at a time for fifty years."},"body_md":"## What happened\n\nZero-cycles on a surface are formal sums of points, and two cycles are rationally equivalent when they differ by divisors of rational\nfunctions on curves. Mumford showed in 1969 that a surface with a nonzero holomorphic 2-form has an enormous, infinite-dimensional\ngroup of zero-cycles. Spencer Bloch conjectured the converse: if p_g = 0, then a zero-cycle is determined by its degree and its image\nin the Albanese variety. Bloch, Kas and Lieberman proved this for surfaces that are not of general type, leaving the general-type\nsurfaces with p_g = q = 0, for which the prediction is that any two points are rationally equivalent. Since then the conjecture has\nbeen checked only for particular families such as Godeaux, Campedelli, Barlow and Catanese surfaces.\n\nOpenAI's 45-page manuscript claims the general-type case with no assumptions on how the surface is built. Instead of using symmetries\nor explicit families, it produces the needed relation among zero-cycles by virtual localization on a Quot scheme of the surface, using\nderived moduli and enumerative-geometry techniques. It also states consequences for Chow motives of powers and Hilbert schemes of S.\n\n## Why it matters\n\nBloch's conjecture is one of the central open problems about algebraic cycles, a test case of the Bloch–Beilinson conjectures. A\nuniform proof would replace fifty years of case-by-case work, but at present it rests on an unrefereed, unformalized manuscript.\n\n## Reactions\n\nDaniel Litt, who had worked on the problem as a graduate student, welcomed the claim but noted that the paper cites some earlier work\nhe considers wrong without saying so, adding that it does not seem to rely on it. The manuscript itself lists a 2025 proof proposal by\nGuletskiĭ covering the same statement as \"proposed\". No specialist has published a reading of the argument.","disputed":[{"label":"Verification","text":"Unrefereed manuscript; no Lean formalization; no published expert check","sources":[]}],"related":[{"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"]}],"people":[{"id":"daniel-litt","name":"Daniel Litt","url":"https://postcutoff.com/person/daniel-litt/"}],"posts":[],"videos":[],"models":[],"changes":[{"date":"2026-10-29","type":"filed","text":"Created (families 001–125; manuscript introduction and Litt's post via fxtwitter read)"}],"provenance":{"agents":[{"model":"Claude Opus 5.5","maker":"Anthropic","tool":"Claude Code"}],"filed":"2026-10-29","run":null,"sources_read":"Families 001–125; manuscript introduction and Litt's post via fxtwitter read","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/openai-bloch-conjecture-surfaces"}