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Math-2298-A1 · revision r1 · on Math-2298

OpenAI claim summary for Math-2298

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.

Scalar-bounded Ricci-flow singularity: A finite maximal smooth existence time with uniformly bounded scalar curvature has bounded scalar maximum at that singular time. The record states no dimension restriction; this closed high-dimensional example refutes its universally stated blowup question.

Four-dimensional Ricci-flow extension: The scalar-blowup corollary proves the recorded blowup conclusion for closed four-manifolds only. The stated assertion has no dimension restriction.

The claimed counterexample lies within the question’s hypotheses and contradicts its universal conclusion.

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

Manuscript

Contents
  1. Finite traceless-Ricci budget and Ricci-flat bubbles under bounded scalar curvature
  2. OpenAI's claim
  3. Mathematical statement of the claimed result
  4. Mathematical claim 1.
  5. Mathematical claim 2.
  6. Outline of the argument
  7. Formal verification
  8. Status caveat
  9. Sources

Finite traceless-Ricci budget and Ricci-flat bubbles under bounded scalar curvature

OpenAI's claim

Scalar-bounded Ricci-flow singularity: In sufficiently high dimension, the paper claims closed Ricci flows whose full curvature blows up at a finite maximal time even though scalar curvature stays bounded; in one fixed high dimension, arbitrarily large power-law blowup exponents are possible.

Four-dimensional Ricci-flow extension: The paper claims smooth continuation of a Ricci flow on a closed four-manifold through any finite time at which scalar curvature stays uniformly bounded; the continuation occurs on the same manifold.

Mathematical statement of the claimed result

Mathematical claim 1.
There exist an integer $q\ge10$, $T>0$ for which a smooth solution of $\partial_tg=-2\operatorname{Ric}_g$ exists on $M=S^2\times S^{q+1}$, a connected closed manifold for $0\le t<T$ with \[\sup_{M\times[0,T)}|\operatorname{Scal}_{g(t)}|<\infty,\qquad\lim_{t\uparrow T}\max_M|\operatorname{Rm}_{g(t)}|=\infty.\] Its initial metric is smooth and $T$ is the finite maximal smooth existence time. This example is in dimension $q+3$, rather than four.
Mathematical claim 2.
On a closed connected smooth four-manifold consider a Ricci flow $g(t)$ whose initial metric and time evolution are smooth for $0\le t<T<\infty$. The claimed implication is \[\sup_{M\times[0,T)}|\operatorname{Scal}_{g(t)}|<\infty\quad\Longrightarrow\quad\exists\epsilon>0:\ g(t)\text{ extends smoothly to }[0,T+\epsilon).\] In particular at a finite maximal four-dimensional existence time, $\limsup_{t\uparrow T}\max_M\operatorname{Scal}_{g(t)}=+\infty$. No corresponding extension conclusion in arbitrary dimension is asserted.

Outline of the argument

Scalar-bounded Ricci-flow singularity: Two sphere factors are warped over an interval, with one factor collapsing at the endpoints and the other shrinking at the singular time. The parabolic scale approximates a Ricci-flat cone, while a smaller cap scale controls smoothness at the pole. One-sided cone comparison, logarithmic-slope monotonicity, and scalar estimates keep scalar curvature bounded without suppressing the smaller-scale full-curvature blowup.

Four-dimensional Ricci-flow extension: Outermost concentration scales produce buffered time patches and finite Ricci-flat bubble trees. Parabolic annular estimates and a time maximal function convert spacetime Ricci energy into the static weighted error of a companion elliptic theorem. Comparing positive functional variation with static smallness forces the root energy to vanish, contradicting the selected change of concentration scale.

Formal verification

Scalar-bounded Ricci-flow singularity: The pinned catalogue lists no Lean formalization for this result.

Four-dimensional Ricci-flow extension: 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