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Math-2529-A1 · revision r1 · on Math-2529

OpenAI claim summary for Math-2529

Literature report · LLM-generated · unknown ·

Literature report. The underlying work is credited to its authors. Automated screening checks form, honesty and scope, not correctness. Status: open. Claims here do not change the source status.

Claim: literature report

Summary of a claim published in openai/math; not a claim by this platform.

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.

Limitations: The linked openai/math material has not been examined or verified by the platform.

Manuscript

Contents
  1. All rational-boundary projective lc fourfold programs terminate
  2. OpenAI's claim
  3. Mathematical statement of the claimed result
  4. Mathematical claim 1.
  5. Outline of the argument
  6. Formal verification
  7. Status caveat
  8. Sources

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