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Math-2307-A1 · revision r1 · on Math-2307

OpenAI claim summary for Math-2307

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 at unsolvedmath.com: partially solved. 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.

Finite-time Calabi-flow singularity: The counterexample supplies finite-time Calabi behavior on a toric projective tenfold; the bundled source also specifically asks long-time existence on toric surfaces, which is not resolved.

This page records partial support. Complete resolution of the catalogue question is not established by this record.

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

Manuscript

Contents
  1. Long-time existence and behavior of the Calabi flow
  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

Long-time existence and behavior of the Calabi flow

OpenAI's claim

Finite-time Calabi-flow singularity: The manuscript claims a smooth Kähler metric in the standard class on $\mathbb {CP}^{10}$ whose scalar curvature becomes unbounded during Calabi evolution before a finite time. Thus the failure can occur in a class that also contains a constant-scalar-curvature metric.

Mathematical statement of the claimed result

Mathematical claim 1.
The claimed initial Kähler form $\omega_0$ lies in the Fubini--Study class on $\mathbb{CP}^{10}$, is smooth, and is invariant under $U(10)$ whose maximal smooth Calabi flow has finite lifespan $0<T_*<\infty$. At some fixed point $p$ and with $a>0$, \[S(\omega(t))(p)\sim a(T_*-t)^{-1/2}\quad(t\uparrow T_*).\] Here the flow is $\partial_t\omega=i\partial\bar\partial S(\omega)$, and $S(\omega)=\operatorname{tr}_{\omega}\operatorname{Ric}(\omega)$, using the complex Kähler trace convention. This refutes universal smooth long-time existence.

Outline of the argument

Finite-time Calabi-flow singularity: A radial momentum ansatz reduces the geometric flow to a fourth-order scalar equation with smooth endpoint conditions. An exact shrinking solution of the flat model is constructed first. A correction on the compact projective manifold preserves the shrinking mechanism and the positivity of the metric, giving the finite-time curvature singularity.

Formal verification

Finite-time Calabi-flow singularity: The pinned catalogue lists no Lean formalization for this result.

This site has not run the Lean development. These links describe OpenAI’s selected formal statements and should be read with their stated scope.

Status caveat

OpenAI reports that an internal model produced the manuscripts. Their arguments have not been independently verified as proofs by this site. Any portion outside the documented formal scope may contain mistakes, and the catalogue label reports the announced mathematical claim.

Sources