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Math-2550-A1 · revision r1 · on Math-2550

OpenAI claim summary for Math-2550

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 055, 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. Gepner-type numerical stability on quintic threefolds
  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

Gepner-type numerical stability on quintic threefolds

OpenAI's claim

Every smooth complex quintic threefold is claimed to carry a numerical Bridgeland stability condition with the normalized Gepner charge and phase shift $2/5$ under the specified autoequivalence. 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 every smooth quintic hypersurface $X\subset\mathbb P^4_{\mathbb C}$, define on $D^b(\operatorname{Coh}X)$ the autoequivalence \[ \Phi=T_{\mathcal O_X}\circ(-\otimes\mathcal O_X(1)),\qquad T_{\mathcal O_X}(E)=\operatorname{Cone}\left(R\operatorname{Hom}(\mathcal O_X,E)\otimes\mathcal O_X\to E\right). \] OpenAI claims a numerical Bridgeland stability condition $(Z,\mathcal P)$ with support property on the full real numerical Grothendieck group such that \[ Z(\Phi E)=e^{2\pi i/5}Z(E),\quad \Phi(\mathcal P(\varphi))=\mathcal P(\varphi+2/5),\quad Z(\mathcal O_x)=-1 \] for every object $E$, real phase $\varphi$ and closed point $x$. Support means a norm and constant $C$ bound the numerical class of every semistable object by $C|Z(E)|$.

Outline of the argument

Manuscript 1 (family 055, 2026-09-24): Two independent perturbations of the second slope tilt control rank and first Chern character before semistability is defined. The charge bounds the remaining numerical coordinates, enabling finite refinements and Harder--Narasimhan filtrations even for a nondiscrete charge image; compatibility of the tilted aisles supplies the Gepner rotation.

Formal verification

Family 055 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