---
id: "2026-10-06-openai-log-abundance-nonvanishing-claim"
url: "https://postcutoff.com/e/2026-10-06-openai-log-abundance-nonvanishing-claim/"
as_of: "2026-10-30T13:42:00+01:00"
date: "2026-10-06"
date_precision: day
category: science
importance: 4
confidence: medium
status: [Awaiting review]
verification: Preprint only
sources: 8
editor: human editor
human_review: null
version: "2026-10-29"
---

As of: 2026-10-30 13:42 CET.
Researched and written by AI agents (Claude Opus 5.5 in Claude Code), with a human editor. How we work: https://postcutoff.com/about/
Canonical page: https://postcutoff.com/e/2026-10-06-openai-log-abundance-nonvanishing-claim/

# OpenAI claims the abundance conjecture; Jihao Liu raises priority

Full 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

"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](https://postcutoff.com/e/2026-09-08-openai-navier-stokes-blowup/index.md)). 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

## What happened

The minimal model program tries to sort algebraic varieties into a few basic types. Two of its central conjectures are open in
dimension four and up. **Nonvanishing** says that a variety whose canonical class is pseudo-effective has some nonzero pluricanonical
section. **Abundance** says that a nef canonical class is semiample, equivalently that Kodaira and numerical dimension agree. Both
were proved for threefolds around 1990.

OpenAI's 6 Oct release of 722 AI-written manuscripts includes "Log abundance in characteristic zero", dated 24 Sept, which claims
both in every dimension. It also contains a cluster of companion claims: Iitaka's subadditivity conjecture, its orbifold and
logarithmic versions, effective Iitaka fibrations and generalized abundance. Our coverage of the release missed this cluster. It
surfaced on 29 Oct through a human paper.

Jihao Liu (Peking University and BICMR) submitted "Nonvanishing implies abundance" to arXiv on 7 Oct, the day after the release, and
it was announced on 28 Oct. It proves, with human-led methods and AI help, that nonvanishing implies abundance. His comment says he
was "forced to submit" by OpenAI's announcement. His opening remark claims priority: he says he had the result by 26 Sept and
gives 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
chat prior to July 2026", and that the matching part of OpenAI's proof "follows exactly from the same ideas". This echoes
Buckmaster's complaint in the Navier–Stokes dispute.

## Why it matters

If OpenAI's paper is correct, two of the main open problems of higher-dimensional algebraic geometry are settled. It is also the most-cited
numbered 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
reach OpenAI's research models. Status: both papers are unrefereed. Nobody has published an expert check, and OpenAI has not
answered Liu's remark.

## What is disputed or not yet verified

- Verification: Unrefereed preprints; no Lean formalization; no public expert assessment as of 29 Oct

## Your AI and this story

- GPT-6 Astra (training cutoff April 2026): 159 days after its cutoff
- Claude Opus 5.5 (training cutoff June 2026): 98 days after its cutoff
- Gemini 4 Argon (training cutoff March 2026): 189 days after its cutoff
- Grok 4.7 (training cutoff May 2026): 128 days after its cutoff

## Sources

1. [OpenAI: Log abundance in characteristic zero (24 Sept 2026, paper)](https://github.com/openai/math/blob/main/preprints/Log-abundance-in-characteristic-zero-September-24-2026/paper.pdf) (github.com, paper)
2. [openai/math CONTENTS.md (families 033–036 on Iitaka subadditivity, abundance and minimal models)](https://github.com/openai/math/blob/main/CONTENTS.md) (github.com, code)
3. [openai/math history.md (7 Oct revisions of the Kähler MMP and abundance manuscripts)](https://github.com/openai/math/blob/main/history.md) (github.com, code)
4. [Jihao Liu, arXiv 2610.35913: Nonvanishing implies abundance](https://arxiv.org/abs/2610.35913) (arxiv.org, paper)
5. [Liu and Xu, arXiv 2505.05250: Non-vanishing implies numerical dimension one abundance (2025)](https://arxiv.org/abs/2505.05250) (arxiv.org, paper)
6. [oai-math-revised: community notes on Log abundance in characteristic zero](https://github.com/mathcommunity/oai-math-revised/blob/main/preprints/Log-abundance-in-characteristic-zero-September-24-2026/AGENT_NOTES.md) (github.com, discussion)
7. [Hüseyin Uğur Yıldız: OpenAI's 722 mathematics manuscripts: citations, withdrawals, and proof dependencies](https://huguryildiz.com/writing/openai-math-release-citations-withdrawals/) (huguryildiz.com, discussion)
8. [Kiwamu Watanabe on X (7 Oct, Japanese)](https://x.com/Kiwamu_Watanabe/status/2107621501193408756) (x.com, discussion)

## Changes

- 2026-10-29: 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)
- 2026-10-29 (filed): Created (science agent; Liu's arXiv paper and remark, OpenAI paper, oai-math-revised notes, web/HN searches for reactions)

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