Toda's Gepner conjecture for the quintic threefold

Let $X\subset\mathbb P^4$ be a smooth complex quintic threefold and $\Phi=T_{\mathcal O_X}\circ(-\otimes\mathcal O_X(1))$ , the composite of tensoring by the hyperplane bundle and the Seidel-Thomas spherical twist at $\mathcal O_X$ , which satisfies $\Phi^5\simeq[2]$ . Physics predicts a stability condition at the Gepner point of the stringy Kahler moduli space fixed by this symmetry. Toda (2013) conjectured, in normalized form, that there is a Bridgeland stability condition $\sigma=(Z,\mathcal P)$ on $D^b(\mathrm{Coh}\,X)$ with $Z(\Phi E)=e^{2\pi i/5}Z(E)$ and $\Phi\mathcal P(\varphi)=\mathcal P(\varphi+2/5)$ , and reduced it to a strong Bogomolov-Gieseker type inequality. Does every smooth quintic threefold admit such a Gepner-type stability condition?

Result: Proved(see note)
Status: Candidate (review pending)
AI contribution: AI-discovered
Method: Construction
Field: Algebraic geometry; Bridgeland stability conditions on Calabi-Yau threefolds
Posed by: Yukinobu Toda
Year posed: 2013
Years open: 13y
Solved: 2026-09-24
Model: Unreleased internal OpenAI model
Vendor: OpenAI
Collaborators: —
Verification: Unreviewed
Publication: Announced
Collection: OpenAI math release (October 2026), version adc7f12
Significance: 38 / 100
Disclosed cost: —
Wikipedia: No dedicated article

What was actually shown

Theorem 1.1: every smooth complex quintic threefold has a numerical Bridgeland stability condition, with support property on $K_{\mathrm{num}}(X)_{\mathbb R}$ , on which $\Phi$ acts by rotating the charge by $e^{2\pi i/5}$ and shifting phases by $2/5$ , normalized by $Z(\mathcal O_x)=-1$ . The construction is a double tilt starting at slope $-1/2$ , refined to a periodic grid of aisles. It relies on Xu's 2026 inequality. It does not prove Toda's general strong Bogomolov-Gieseker conjecture, treat other Calabi-Yau hypersurfaces, or describe the full stability manifold near the Gepner point.

What the AI did

The release README says every result in it was produced by an unreleased internal OpenAI model with a fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result, and that some outputs build on earlier model results. This result is not among the README's exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region). The manuscript is authored 'OpenAI' and names no human author. The principal manuscript (September 24, 2026) is self-contained apart from a cited inequality of Xu (arXiv 2511.21288v2); the companion on large-volume charges (same date) is a separate result.

Verification

No independent mathematician has checked this yet. Checked here: Theorem 1.1 of the principal manuscript was read against Toda's conjecture; it gives a numerical Bridgeland stability condition with the support property on the full numerical Grothendieck group, $Z(\Phi E)=e^{2\pi i/5}Z(E)$ , $\Phi\mathcal P(\varphi)=\mathcal P(\varphi+2/5)$ and $Z(\mathcal O_x)=-1$ , on every smooth quintic, and identifies the charge with Toda's by uniqueness of the normalized eigenform. A reader should know that the slope-sheaf input is taken from Xu's stronger Bogomolov-Gieseker inequality on the quintic (arXiv 2511.21288v2, 2026), an external preprint not checked here. Not refereed. No Lean formalization in the release.

Toda's Gepner conjecture for the quintic threefold

What was actually shown

What the AI did

Verification

Sources

Discussion