{"schema":"postcutoff/event@1","as_of":"2026-10-30T13:42:00+01:00","url":"https://postcutoff.com/e/2026-10-06-openai-log-abundance-nonvanishing-claim/","md":"https://postcutoff.com/e/2026-10-06-openai-log-abundance-nonvanishing-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-log-abundance-nonvanishing-claim","date":"2026-10-06","date_precision":"day","short_title":"OpenAI claims the abundance conjecture; Jihao Liu raises priority","deck":null,"takeaway":"A manuscript in OpenAI's 6 Oct 2026 math release, 'Log abundance in characteristic zero' (dated 24 Sept), claims the log abundance conjecture in every dimension in characteristic zero, and with it canonical nonvanishing for smooth projective varieties. Both are central open problems of the minimal model program, proved only up to dimension three. Companion papers claim Iitaka's subadditivity conjecture and related results. On 7 Oct Jihao Liu (Peking University) submitted a human-led proof that nonvanishing implies abundance. arXiv announced it only on 28 Oct. His comment line says he was 'forced to submit here due to an announcement of the proof of the abundance conjecture by OpenAI'. In a remark he suggests that an earlier draft, sent to 'codex chat' before July 2026, may have fed OpenAI's work. No birational geometer has assessed the claims publicly, there is no Lean proof, and OpenAI has not responded.","category":"science","category_label":"Science & math","importance":4,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":[{"n":1,"title":"OpenAI: Log abundance in characteristic zero (24 Sept 2026, paper)","url":"https://github.com/openai/math/blob/main/preprints/Log-abundance-in-characteristic-zero-September-24-2026/paper.pdf","type":"paper","group":"primary","domain":"github.com"},{"n":2,"title":"openai/math CONTENTS.md (families 033–036 on Iitaka subadditivity, abundance and minimal models)","url":"https://github.com/openai/math/blob/main/CONTENTS.md","type":"code","group":"primary","domain":"github.com"},{"n":3,"title":"openai/math history.md (7 Oct revisions of the Kähler MMP and abundance manuscripts)","url":"https://github.com/openai/math/blob/main/history.md","type":"code","group":"primary","domain":"github.com"},{"n":4,"title":"Jihao Liu, arXiv 2610.35913: Nonvanishing implies abundance","url":"https://arxiv.org/abs/2610.35913","type":"paper","group":"primary","domain":"arxiv.org"},{"n":5,"title":"Liu and Xu, arXiv 2505.05250: Non-vanishing implies numerical dimension one abundance (2025)","url":"https://arxiv.org/abs/2505.05250","type":"paper","group":"primary","domain":"arxiv.org"},{"n":6,"title":"oai-math-revised: community notes on Log abundance in characteristic zero","url":"https://github.com/mathcommunity/oai-math-revised/blob/main/preprints/Log-abundance-in-characteristic-zero-September-24-2026/AGENT_NOTES.md","type":"discussion","group":"reactions","domain":"github.com"},{"n":7,"title":"Hüseyin Uğur Yıldız: OpenAI's 722 mathematics manuscripts: citations, withdrawals, and proof dependencies","url":"https://huguryildiz.com/writing/openai-math-release-citations-withdrawals/","type":"discussion","group":"reactions","domain":"huguryildiz.com"},{"n":8,"title":"Kiwamu Watanabe on X (7 Oct, Japanese)","url":"https://x.com/Kiwamu_Watanabe/status/2107621501193408756","type":"discussion","group":"reactions","domain":"x.com"}],"official":5,"filed":"2026-10-29","updated":"2026-10-29","orgs":["OpenAI","Peking University"],"title":"OpenAI's release claims the abundance and nonvanishing conjectures of birational geometry; Jihao Liu posts 'Nonvanishing implies abundance' and says his draft, sent to Codex before July, may have fed OpenAI's proof","summary":"\"Log abundance in characteristic zero\" (OpenAI, 24 Sept 2026, released 6 Oct, family 034): \"every nef ℚ-Cartier log canonical divisor on a projective log canonical pair with effective rational boundary is semiample\", and \"over ℂ, the proof also establishes canonical nonvanishing for smooth projective varieties in every dimension\". Both conjectures were known only up to dimension 3 (Miyaoka, Kawamata). Companion papers claim ordinary Iitaka subadditivity (C_{n,m}) in characteristic zero, Campana's orbifold version, the effective Iitaka fibration conjecture and generalized abundance. Jihao Liu's arXiv 2610.35913 proves that the nonvanishing conjecture implies abundance (good minimal models or Mori fibre spaces for lc pairs). Its Remark 0.1 says the \"nonvanishing implies abundance\" part of OpenAI's proof \"follows exactly from the same ideas\". Unrefereed, no formalization.","key_facts":["Claim (OpenAI abstract, verbatim): 'We prove the rational-boundary log abundance conjecture in every dimension over algebraically closed fields of characteristic zero: every nef Q-Cartier log canonical divisor on a projective log canonical pair with effective rational boundary is semiample. Over C, the proof also establishes canonical nonvanishing for smooth projective varieties in every dimension.' About 278k characters of text; proof by induction on dimension, nonvanishing first in each dimension, then 'good log minimal models'","Prior state: nonvanishing and abundance hold in dimension ≤ 3 (Miyaoka 1988, Kawamata 1992; lc and slc cases by Keel–Matsuki–McKernan 1994 and Fujino 2000) and are 'widely open in dimension ≥ 4' (Liu). Demailly–Hacon–Păun (2013) showed nonvanishing plus dlt extension imply abundance; Liu and Zheng Xu (arXiv 2505.05250, 2025) showed nonvanishing implies abundance in numerical dimension ≤ 1. OpenAI's paper cites Liu–Xu ('Recent work of Liu and Xu treats numerical dimension at most one … The induction here has no numerical-dimension restriction')","Companion claims in the release: 'Projective Hodge lines and ordinary Iitaka subadditivity' (27 Sept: κ(X) ≥ κ(F) + κ(Z) in characteristic zero), Campana's orbifold Iitaka subadditivity (family 033), the effective Iitaka fibration conjecture (3 Oct), the Generalised Abundance Conjecture up to numerical equivalence and minimal models for generalized lc pairs (family 036), termination of fourfold MMPs (family 056). On 7 Oct OpenAI revised six 'Kähler minimal model programs and abundance' manuscripts ('expanded positivity and contraction arguments and clarified which results are used as inputs')","Jihao Liu, 'Nonvanishing implies abundance' (arXiv 2610.35913, 48 pp., submitted 7 Oct 2026 15:16 UTC, announced 28 Oct). arXiv comment, verbatim: 'Still under revision for writing but forced to submit here due to an announcement of the proof of the abundance conjecture by OpenAI'. Theorem 1.3: if nonvanishing holds in dimension d, every projective lc pair of dimension ≤ d has a good minimal model or a Mori fibre space, so abundance holds in dimension ≤ d. Method: Kawamata's approach, root stacks and mixed Hodge modules to lift sections through thickenings","Liu's Remark 0.1 (verbatim): 'We get the result no later than September 26th.' An earlier manuscript was 'timestamped no later than September 28th' with a published SHA-256 hash. 'An earlier incomplete draft of the paper, timestamped back to Novemeber 2025, was submitted to codex chat prior to July 2026. Considering that the internal model of OpenAI which completely solved the abundance conjecture started training by the end of August 2026, there is a possibility that the ideas in this paper were adopted by OpenAI to obtain their results.'","Liu's AI declaration (verbatim): 'We have used Deepseek, Fable, GLM, Chatgpt, the Danus system [Liu+ 26], and the Rethlas system [Ju+ 26] in the preparation of the work.' (Danus and Rethlas are maths-agent systems described in arXiv 2607.06447, with Liu among the authors, and arXiv 2604.03789.) He thanks Zheng Xu, who 'voluntarily declined to have the authorship'","Precedent: in the Navier–Stokes dispute (Sept), Tristan Buckmaster raised the possibility that his Codex drafts fed OpenAI's work, and OpenAI's eventual denial cited 'no user inputs past July 3rd' (see 2026-09-08-openai-navier-stokes-blowup). Liu dates his Codex upload to before July. No OpenAI response to Liu found as of 29 Oct","Reactions: none found from birational geometers (Hacon, McKernan, Birkar, Kollár, Chenyang Xu) on the web, HN or Chinese-language press as of 29 Oct. On 7 Oct Kiwamu Watanabe called the release's algebraic-geometry list, log abundance included, 'quite amazing' (translated from Japanese) but said he could not judge it. Community notes (oai-math-revised, 21 Oct): 'Comments and discussion: None found'. In a citation map of the release, Hüseyin Uğur Yıldız (10 Oct) finds Log abundance cited most often among the six numbered papers of the largest cluster, adding that 'a citation is not a proof dependency'","Formal status: no Lean formalization of the abundance or nonvanishing claims. The only Lean scope note in the neighbouring Iitaka family (033) covers one branch of a logarithmic Kodaira additivity paper"],"key_numbers":[],"tags":["math","algebraic-geometry","birational-geometry","minimal-model-program","abundance-conjecture","nonvanishing-conjecture","iitaka-conjecture","openai-math-release","priority-dispute","codex"],"science":{"field":"mathematics","subfield":"algebraic geometry (birational geometry, minimal model program)","problem":"Abundance conjecture (κ = ν; nef lc adjoints are semiample) and the nonvanishing conjecture, open in dimension ≥ 4; also Iitaka's subadditivity conjecture C_{n,m}","result":"OpenAI claims log abundance for rational-boundary lc pairs and canonical nonvanishing in every dimension in characteristic zero. Liu (human-led, AI-assisted) proves nonvanishing ⇒ abundance for lc pairs","open_since":"1980s (abundance in its standard form; threefold case finished in 1992)","ai_system":["OpenAI internal model (math release)","Liu: DeepSeek, Fable, GLM, ChatGPT, Danus, Rethlas"],"human_role":"OpenAI: autonomous AI output released by OpenAI. Liu: human-led paper written with several AI models and agent systems","verification":{"key":"preprint","label":"Preprint only","text":"Unrefereed preprints; no Lean formalization; no public expert assessment as of 29 Oct"},"status":"pending","shock":"Liu's comment and remark turn a claimed solution of one of birational geometry's central conjectures into a priority question, the second time a mathematician has suggested that work stored in Codex could have reached OpenAI's model"},"body_md":"## What happened\n\nThe minimal model program tries to sort algebraic varieties into a few basic types. Two of its central conjectures are open in\ndimension four and up. **Nonvanishing** says that a variety whose canonical class is pseudo-effective has some nonzero pluricanonical\nsection. **Abundance** says that a nef canonical class is semiample, equivalently that Kodaira and numerical dimension agree. Both\nwere proved for threefolds around 1990.\n\nOpenAI's 6 Oct release of 722 AI-written manuscripts includes \"Log abundance in characteristic zero\", dated 24 Sept, which claims\nboth in every dimension. It also contains a cluster of companion claims: Iitaka's subadditivity conjecture, its orbifold and\nlogarithmic versions, effective Iitaka fibrations and generalized abundance. Our coverage of the release missed this cluster. It\nsurfaced on 29 Oct through a human paper.\n\nJihao Liu (Peking University and BICMR) submitted \"Nonvanishing implies abundance\" to arXiv on 7 Oct, the day after the release, and\nit was announced on 28 Oct. It proves, with human-led methods and AI help, that nonvanishing implies abundance. His comment says he\nwas \"forced to submit\" by OpenAI's announcement. His opening remark claims priority: he says he had the result by 26 Sept and\ngives a hash for a draft dated no later than 28 Sept. He also says an incomplete draft from November 2025 had been \"submitted to codex\nchat prior to July 2026\", and that the matching part of OpenAI's proof \"follows exactly from the same ideas\". This echoes\nBuckmaster's complaint in the Navier–Stokes dispute.\n\n## Why it matters\n\nIf OpenAI's paper is correct, two of the main open problems of higher-dimensional algebraic geometry are settled. It is also the most-cited\nnumbered paper in the release's largest citation cluster. If Liu's account holds, it raises again the question of whether users' work in Codex can\nreach OpenAI's research models. Status: both papers are unrefereed. Nobody has published an expert check, and OpenAI has not\nanswered Liu's remark.","disputed":[{"label":"Verification","text":"Unrefereed preprints; no Lean formalization; no public expert assessment as of 29 Oct","sources":[]}],"related":[{"id":"2026-10-29-ai-assisted-results-october-29-listing","url":"https://postcutoff.com/e/2026-10-29-ai-assisted-results-october-29-listing/","date":"2026-10-29","date_precision":"day","short_title":"AI-assisted results in the 29 Oct 2026 sweep","deck":"Counterexamples to Wilking's 2007 question found by GPT-6 Astra, an LLM-written Plateau paper by De Lellis and Glaudo, explicit Reed–Solomon codes with optimal list size (Guruswami et al.), rank-four quiver mutation decided by GPT Astra, and a dozen named conjectures","takeaway":"The 29 Oct sweep covered the arXiv announcement of 28 Oct, which the previous run had read only through keyword queries. This time 43 papers with an AI statement surfaced, and a PDF scan of a further 405 new and cross-listed submissions in math, cs.CC, cs.DS and cs.DM found about 180 more with an AI statement, most of them describing more than editing help. Four results have their own entries: OpenAI's abundance claim and Jihao Liu's priority remark, OpenAI's Lean-checked L = RL = BPL, R(3,3,3,3) ≤ 61 and the sharp Banach–Mazur exponent. This page lists the rest. GPT-6 Astra found the counterexamples to Wilking's 2007 question 'with minimal human input' and discovered the main results of a rank-four quiver paper. Claude found examples in opinion dynamics and proposed a minimal-surface construction. An LLM supplied 'the main idea' of a paper by Camillo De Lellis, a member of the IAS advisory group on AI and mathematics, and wrote it.","category":"science","category_label":"Science & math","importance":2,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":23,"official":23,"filed":"2026-10-29","updated":"2026-10-29","orgs":["OpenAI","Anthropic","various researchers"]},{"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":"2026-10-07-chenglong-ma-hovey-strickland-openai-chatgpt-upload-provenance","url":"https://postcutoff.com/e/2026-10-07-chenglong-ma-hovey-strickland-openai-chatgpt-upload-provenance/","date":"2026-10-07","date_precision":"day","short_title":"Hovey–Strickland: author says OpenAI's paper matches his ChatGPT-uploaded draft","deck":null,"takeaway":"OpenAI's 6 Oct maths release includes a paper dated 24 Sept that proves Chai's invariant-ideal conjecture and, through Barthel–Heard–Naumann, the Hovey–Strickland classification of thick tensor ideals of dualizable K(n)-local spectra (family 311). The next day Copenhagen mathematician Chenglong Ma posted his own proof on arXiv. In it he writes that OpenAI's 'proof strategy and specific choices of notation are identical to a preliminary version of the present paper that was uploaded to ChatGPT on September 8, 2026.' OpenAI's paper does not mention him. He also reports an independent proof by Ningyi Li. We found no response from OpenAI as of 29 Oct. The allegation has not been examined publicly beyond a GitHub label and a one-comment Hacker News thread.","category":"science","category_label":"Science & math","importance":3,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":6,"official":3,"filed":"2026-10-29","updated":"2026-10-29","orgs":["OpenAI","University of Copenhagen"]},{"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":"2026-09-08-openai-navier-stokes-blowup","url":"https://postcutoff.com/e/2026-09-08-openai-navier-stokes-blowup/","date":"2026-09-08","date_precision":"day","short_title":"OpenAI claims a Millennium Prize problem","deck":"10,000 AI agents prove forced Navier–Stokes blow-up; priority dispute erupts","takeaway":"It is the first credible AI claim on a Clay Millennium Prize problem, even if only a technically permitted variant.","category":"science","category_label":"Science & math","importance":5,"confidence":"medium","status":{"key":"disputed","labels":["Disputed"]},"sources":42,"official":4,"filed":"2026-09-29","updated":"2026-10-29","orgs":["OpenAI"]},{"id":"2026-08-19-yau-tian-donaldson-cscK-disproof-claim","url":"https://postcutoff.com/e/2026-08-19-yau-tian-donaldson-cscK-disproof-claim/","date":"2026-08-19","date_precision":"day","short_title":"Peking University preprint claims an AI-found disproof of the Yau–Tian–Donaldson conjecture for constant scalar curvature metrics","deck":null,"takeaway":"The Yau–Tian–Donaldson programme is central to modern complex geometry, and the paper is explicit that its main result is AI-generated.","category":"science","category_label":"Science & math","importance":4,"confidence":"medium","status":{"key":"pending","labels":["Awaiting review"]},"sources":2,"official":2,"filed":"2026-09-30","updated":"2026-10-07","orgs":["Peking University","Jihao Liu","Anthropic","OpenAI"]}],"people":[],"posts":[],"videos":[],"models":[],"changes":[{"date":"2026-10-29","type":"edit","text":"Related: another credit claim on OpenAI's release, Chenglong Ma's statement that the Hovey–Strickland paper matches the draft he uploaded to ChatGPT (own entry)"},{"date":"2026-10-29","type":"filed","text":"Created (science agent; Liu's arXiv paper and remark, OpenAI paper, oai-math-revised notes, web/HN searches for reactions)"}],"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/openai-abundance-conjecture"}