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Math-2545-A1 · revision r1 · on Math-2545

OpenAI claim summary for Math-2545

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, 2026-10-05): 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. Generalized-canonical flip termination on Kähler fourfolds
  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

Generalized-canonical flip termination on Kähler fourfolds

OpenAI's claim

Generalized-canonical flip sequences with fixed rational nef data on compact Kähler fourfolds are claimed to terminate even when exceptional discrepancies equal one. The cited manuscript argues for that result; its relation to this problem is specified below.

Mathematical statement of the claimed result

Mathematical claim 1.
Start with a normal irreducible globally Weil rational-factorial compact Kähler fourfold $X_0$, effective rational boundary $B_0$ of coefficients below one, and a fixed rational nef b-divisor $\mathbf M$ represented on a projective birational compact Kähler model. Require generalized log discrepancies positive for every prime and at least one for exceptional primes. For strict transforms $B_i$, traces $M_i$ and rational Cartier $D_i=K_{X_i}+B_i+M_i$, OpenAI claims every sequence of projective small nonisomorphic bimeromorphic diagrams with connected fibers \[ X_i\xrightarrow{f_i}Z_i\xleftarrow{f_i^+}X_{i+1},\qquad -D_i\text{ relatively ample for }f_i,\quad D_{i+1}\text{ relatively ample for }f_i^+, \] is finite, provided every $Z_i$ is normal compact Kähler, all models are normal globally Weil rational-factorial compact Kähler fourfolds, and canonical representatives are compatible. Small means no exceptional prime divisor. This theorem has a generalized-canonical discrepancy hypothesis, not unrestricted generalized log canonicity.

Outline of the argument

Manuscript 1 (family 056, 2026-10-05): A corrected finite count of low-discrepancy places first gives ordinary terminality along general points of flipped surfaces. Fixed denominator gaps and analytic surface-cycle comparisons then force a finite decrease, including zero exceptional boundary coefficients and nonnormal boundary components.

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