The Cone Theorem for Effective Fourfold Pairs in Characteristic
Extending the minimal model program beyond threefolds in positive characteristic is a standing goal of birational geometry. Assuming the log resolution conjecture for all log pairs birational to , the cone theorem holds for projective log canonical, -factorial fourfold pairs with , over bases of positive and mixed characteristic .
- Result
- Proved
- Status
- Partial result
- AI contribution
- AI co-developed
- Method
- Argument
- Field
- Birational geometry (positive characteristic)
- Posed by
- The char-p minimal model program (Birkar, Hacon, Xu, Waldron and others)
- Year posed
- —
- Years open
- —
- Solved
- 2026-08-14
- Model
- ChatGPT 5.6 Sol, Codex
- Vendor
- OpenAI
- Collaborators
- Joe Waldron
- Verification
- Unreviewed
- Publication
- Preprint
- Significance
- 25 / 100
- Disclosed cost
- —
- Wikipedia
- No dedicated article
What the AI did
The paper's AI statement, in the author's words: he worked out the outline with all essential ingredients in Spring 2025, but one gap remained that he could not repair; he gave the unfinished proof to ChatGPT 5.6 Sol in Summer 2026 asking it to fill the gap, and it modified the approach to avoid the issue, which after further editing by hand and with Codex became the current version.
Verification
Checked by this site on 17 August 2026 against the paper's LaTeX (arXiv:2608.14236, Waldron, Michigan State): the AI statement is verbatim as quoted - among the frankest in the catalog, a professional birational geometer crediting the model with repairing a gap he could not. The result is conditional on the log resolution conjecture and is recorded as Partial for that reason. The proof was not checked here; days-old preprint, no independent review.