OpenAI claims the abundance conjecture; Jihao Liu raises priority

Your AI and this story

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 OpenAI claims a Millennium Prize problem). 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’