Existence of minimal models and Mori fibre spaces for projective log canonical pairs, in generalized-pair form
The minimal model conjecture in its existence form asserts that every projective log canonical pair over an algebraically closed field of characteristic zero has a minimal model (a birational model with nef and no worse discrepancies) when is pseudo-effective, and a Mori fibre space otherwise. Birkar-Cascini-Hacon-McKernan proved this for klt pairs with big boundary or big adjoint; the general case was open in dimension at least five, and for generalized pairs (Birkar-Zhang) carrying a fixed nef part. Does every projective (generalized) log canonical -pair admit a minimal model or a Mori fibre space?
- Result
- Proved(see note)
- Status
- Candidate (review pending)
- AI contribution
- AI-discovered
- Method
- Argument
- Field
- Birational geometry; minimal model program
- Posed by
- Central conjecture of the minimal model program; the manuscript cites its statement as Conjecture 1.1 of Gongyo (2011); generalized pairs of Birkar and Zhang
- Year posed
- —
- Years open
- —
- Solved
- 2026-09-24
- Model
- Unreleased internal OpenAI model
- Vendor
- OpenAI
- Collaborators
- —
- Verification
- Unreviewed
- Publication
- Announced
- Collection
- OpenAI math release (October 2026), version adc7f12
- Significance
- 68 / 100
- Disclosed cost
- —
- Wikipedia
- No dedicated article
What was actually shown
Theorem 1.1: for a projective generalized lc -pair over an algebraically closed field of characteristic zero there is a birational map extracting no divisors, with generalized lc and log discrepancies not decreasing (strictly on contracted divisors), such that is nef if is pseudo-effective, and otherwise has a Mori fibre space structure with relative Picard number one. With this gives the ordinary lc case. Companion: smooth projective varieties with have -factorial terminal minimal models. Not shown: termination of arbitrary flip sequences, real coefficients, abundance (claimed in a separate release paper), or Q-factorial models in general.
What the AI did
The release README says the results were produced by an unreleased internal OpenAI model with a fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result. This result is not among the README's exceptions (the Hodge conjecture for CM abelian varieties, and the Re(s) > 11/12 zero-free region whose write-up was human-edited). The manuscripts are authored 'OpenAI' and name no human author. The principal manuscript and the numerical-dimension-one companion are both dated September 24, 2026. The release's log abundance manuscript (already in the catalog) uses this paper's ordinary-pair corollary.
Verification
No independent mathematician has checked this yet. Checked here: introduction and Theorem 1.1 of 'Minimal models and Mori fibre spaces for generalized log canonical Q-pairs' read against the existence conjecture as the paper cites it (Gongyo, Conjecture 1.1). No Lean formalization. The paper states its scope: rational boundary and rational nef data, no Q-factoriality, existence of some terminating program only (not termination of every MMP), and no semiampleness or nonvanishing. The terminating transfer is credited to Tsakanikas-Xie and earlier NQC MMP results; the new part is a local phase theorem for cyclic Gorenstein klt germs and a global contradiction for an infinite MMP.