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Math-26-A2 · revision r1 · on Math-26

OpenAI claim summary for Math-26

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): Termination of every maximal ordinary klt rational program on Q-factorial compact Kähler fourfolds; even where arbitrary choices are included, the theorem is restricted to fourfolds and rational boundary, while the catalogue flip-termination target has arbitrary dimension and its broader allowed boundary category.

Manuscript 2 (family 056, 2026-09-24): Termination of every permitted rational lc projective fourfold program in characteristic zero; even where arbitrary choices are included, the theorem is restricted to fourfolds and rational boundary, while the catalogue flip-termination target has arbitrary dimension and its broader allowed boundary category.

Manuscript 3 (family 056, 2026-10-05): Termination of generalized-canonical rational flip sequences on Q-factorial compact Kähler fourfolds; even where arbitrary choices are included, the theorem is restricted to fourfolds and rational boundary, while the catalogue flip-termination target has arbitrary dimension and its broader allowed boundary category.

Manuscript 4 (family 056, 2026-10-05): Termination of every existing generalized lc rational flip sequence on Q-factorial Kähler fourfolds with fixed nef b-data; even where arbitrary choices are included, the theorem is restricted to fourfolds and rational boundary, while the catalogue flip-termination target has arbitrary dimension and its broader allowed boundary category.

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

Manuscript

Contents
  1. Termination of Flips Conjecture in the Minimal Model Program
  2. OpenAI's claim
  3. Mathematical statement of the claimed result
  4. Mathematical claim 1.
  5. Mathematical claim 2.
  6. Mathematical claim 3.
  7. Mathematical claim 4.
  8. Outline of the argument
  9. Formal verification
  10. Status caveat
  11. Sources

Termination of Flips Conjecture in the Minimal Model Program

OpenAI's claim

Manuscript 1 (family 056, 2026-10-05): Every maximal ordinary negative-ray program for the stated rational compact Kähler klt fourfold pairs is claimed to terminate at a nef model or Mori fiber space according to pseudo-effectivity; Manuscript 2 (family 056, 2026-09-24): 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. Manuscript 3 (family 056, 2026-10-05): Generalized-canonical flip sequences with fixed rational nef data on compact Kähler fourfolds are claimed to terminate even when exceptional discrepancies equal one; Manuscript 4 (family 056, 2026-10-05): Every existing generalized log-canonical flip sequence in the specified globally factorial compact Kähler fourfold category is claimed to terminate, without scaling or pseudo-effectivity assumptions.

Mathematical statement of the claimed result

Mathematical claim 1.
Let $X$ be normal globally Weil rational-factorial compact Kähler of dimension four, and $\Delta\ge0$ rational with $(X,\Delta)$ klt. Global Weil rational-factoriality means every global Weil divisor has a positive Cartier multiple. Fix compatible canonical Weil representatives, or the intrinsic canonical rational line bundle. OpenAI claims every maximal ordinary negative-extremal-ray program beginning exactly on $X$ terminates in finitely many divisorial contractions and flips. Models remain in the same category; contractions are projective with relative line-bundle rank and Bott--Chern dimension one. For $D=K_X+\Delta$, \[ D\text{ pseudo-effective}\ \Longrightarrow\ D_n\text{ nef},\qquad D\text{ not pseudo-effective}\ \Longrightarrow\ X_n\to S\text{ a Mori fiber contraction}. \] The latter has connected fibers, normal compact Kähler base of dimension below four, relative Picard number one and relatively ample $-D_n$. In the first case no discrepancy decreases, while each original contracted prime has a strictly larger discrepancy on the endpoint, and the composite extracts no prime. Global strong rational-factoriality is also preserved when initially imposed.
Mathematical claim 2.
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.
Mathematical claim 3.
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.
Mathematical claim 4.
Let $X_0$ be normal irreducible globally Weil rational-factorial compact Kähler of dimension four, with effective rational finite-support $B_0$ of coefficients in $[0,1]$ and a fixed analytically nef rational b-divisor represented on a projective bimeromorphic compact Kähler model. Suppose $D_0=K_{X_0}+B_0+M_0$ rational Cartier and all generalized log discrepancies nonnegative. With strict transforms, the same b-divisor and compatible canonical representatives, OpenAI claims finiteness of every sequence \[ X_i\xrightarrow{f_i}Z_i\xleftarrow{f_i^+}X_{i+1},\qquad -D_i\text{ }f_i\text{-ample},\quad D_{i+1}\text{ }f_i^+\text{-ample}, \] with every $Z_i$ normal compact Kähler, whose maps are projective small nonisomorphic bimeromorphic maps with connected fibers and whose models remain normal irreducible globally Weil rational-factorial compact Kähler fourfolds with rational Cartier adjoints. The coefficients and discrepancy bound permit generalized log canonical, not only generalized canonical, pairs.

Outline of the argument

Manuscript 1 (family 056, 2026-10-05): Auxiliary generalized programs supply the comparison with the ordinary sequence. Descent of numerically trivial actual lines preserves Cartier indices, while corrected discrepancy counts and analytic surface homology prove terminal fourfold termination even with real boundary, allowing arbitrary ordinary ray choices. Manuscript 2 (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.

Manuscript 3 (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. Manuscript 4 (family 056, 2026-10-05): Relative programs and special termination reduce the sequence to controlled terminal pieces while preserving the nef datum and actual adjoints. Analytic cycle ranks remove divisorial losses, and strict mixed-Hodge weight comparisons bound the residual surface changes, leaving only finitely many possible steps.

Formal verification

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

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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