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Termination of flips for projective log canonical fourfolds with rational boundary

The Termination of Flips Conjecture of the minimal model program asserts that every sequence of flips (more generally, every program of KX+BK_X+B-negative divisorial contractions and flips) for a log canonical pair (X,B)(X,B) is finite. In dimension four it was known for terminal fourfolds (Kawamata), canonical pairs (Fujino), klt pairs with −(KX+B)-(K_X+B) numerically equivalent to an effective divisor (Alexeev-Hacon-Kawamata), pseudo-effective lc pairs (Chen-Tsakanikas, Moraga) and klt pairs with big boundary (Han-Liu-Zhuang). Does every minimal model program for a projective log canonical fourfold with rational boundary, over an algebraically closed field of characteristic zero, terminate for every choice of negative extremal rays, without pseudo-effectivity or bigness assumptions?

Result
Proved(see note)
Status
Partial result
AI contribution
AI-discovered
Method
Argument
Field
Birational geometry; minimal model program
Posed by
The Termination of Flips Conjecture of the minimal model program (Shokurov's difficulty method; the manuscript points to Chen-Tsakanikas for the statement)
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
34 / 100
Disclosed cost
—
Wikipedia
No dedicated article

What was actually shown

Theorem 1.1: for a projective lc pair (X,B)(X,B) of dimension four with rational boundary over an algebraically closed field of characteristic zero, every negative extremal ray at every stage has its contraction and, when birational, a positive model; every permitted program has finitely many birational steps for every sequence of choices; and every maximal program ends at a nef adjoint or a Mori fibre space. Proved over C\mathbb C by a weighted difficulty with a topological branch estimate, then transferred to all such fields. Not shown: real boundaries, generalized pairs in the projective setting, any dimension above four, or semiampleness of the nef endpoint without the release's log abundance result. The Kahler companions prove termination for generalized lc flips with projective small diagrams on globally Weil Q-factorial compact Kahler fourfolds, a four-dimensional rational form of Hacon-Xie's Conjecture 1.15.

What the AI did

The release README says the results were produced by an unreleased internal OpenAI model with one 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 manuscript is authored 'OpenAI' and names no human author. Four companion manuscripts (24 September and 5 October 2026) treat compact Kahler fourfolds: termination of generalized log canonical and generalized-canonical flips with projective small diagrams, and finite ordinary minimal model programs for klt Kahler fourfold pairs. The principal paper uses other release results (log abundance, complements) only for consequences at the endpoints.

Verification

No independent mathematician has checked this yet. Checked here: Theorem 1.1 of the principal manuscript was read against the conjecture; it is the projective four-dimensional lc case with rational boundary, for every permitted program from the given model (divisorial contractions, flips and mixed steps), with existence of the required contractions and flips included and no Q\mathbb Q-factoriality or pseudo-effectivity assumed. Real boundaries and dimensions five and up are not treated. The difficulty argument with a new branch estimate was not refereed here. No Lean formalization exists for this family.

Sources

Changelog1 change

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