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Math-2529 · mathematics · Algebraic geometry

All rational-boundary projective lc fourfold programs terminate

Status OpenLLM status Open unverified

Statement

Definitions and mathematical statement

Statement recorded in the supplied source.
For a projective lc fourfold with effective rational boundary over an algebraically closed characteristic-zero field, without assumed Q-factoriality or pseudo-effectivity, do all permitted negative extremal-ray programs have the required contractions and flips and finitely many birational steps, regardless of choices? Must every maximal program end with nef adjoint or a lower-dimensional relative-Picard-one Mori fiber contraction?

Short English statement

For a projective lc fourfold with effective rational boundary over an algebraically closed characteristic-zero field, without assumed Q-factoriality or pseudo-effectivity, do all permitted negative extremal-ray programs have the required contractions and flips and finitely many birational steps, regardless of choices? Must every maximal program end with nef adjoint or a lower-dimensional relative-Picard-one Mori fiber contraction?

Sources

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-2529-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2529

Manuscript 1 (family 056, 2026-09-24): The problem selects the mathematical statement summarized here, including its stated hypotheses and parameter range. The cited theorem gives a claimed proof of that assertion in full.

All rational-boundary projective lc fourfold programs terminate

OpenAI's claim

Every permitted rational-boundary log-canonical fourfold program in characteristic zero is claimed to terminate, including mixed steps and arbitrary ray choices on initially nonfactorial models. The cited manuscript argues for that result; its relation to this problem is specified below.

Mathematical statement of the claimed result

Mathematical claim 1.
For a projective log canonical fourfold pair $(X,B)$ with rational boundary over a field that has characteristic zero and is algebraically closed, OpenAI claims termination of every permitted negative-extremal-ray program, without rational-factoriality or pseudo-effectivity assumptions. Its permitted birational steps are divisorial contractions, flips, or a divisorial contraction followed by a small positive modification; extra blowups and flops are excluded. It claims all required contractions and positive models exist at each stage, and every maximal program ends with \[ K_Y+B_Y\text{ nef}\quad\text{or}\quad Y\to T\text{ a Mori fiber space}. \] The fiber contraction has connected fibers and normal base of smaller dimension, with its relative Picard number equal to one and its negative adjoint relatively ample. Nef inputs give a zero-step program.

Outline of the argument

Manuscript 1 (family 056, 2026-09-24): Divisor-cycle ranks discard all steps that lose divisors, and the remaining small tail lifts to crepant terminal junctions. A fixed set of boundary labels is constrained by surface discrepancy ACC; mixed-Hodge support bounds and finite model comparisons make the remaining numerical changes finite.

Formal verification

Family 056 has no Lean scope page in this snapshot; no formalization of these manuscript results is listed.

This site has not executed the Lean files or checked their dependencies. These descriptions report the scope of the linked declarations, rather than a new verification of the proofs.

Status caveat

OpenAI attributes these results to an internal model. Its release notice cautions that results lacking full formalization can contain mistakes. This site has not independently assessed the proofs; a solved label records the scope of the reported claim.

Sources

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