The index conjecture for semi-log-canonical log Calabi-Yau pairs with finite rational coefficient sets

A projective log Calabi-Yau pair $(X,B)$ has $K_X+B\sim_{\mathbb Q}0$ , so some positive multiple $m(K_X+B)$ is Cartier and linearly trivial; the least such $m$ is the index. Gongyo showed such $m$ exists for slc pairs, but it may depend on the pair. The index conjecture asks for uniformity. In its finite-rational-coefficient slc form, fix $d$ and a finite set $\Phi\subset[0,1]\cap\mathbb Q$ : is there an integer $a(d,\Phi)$ such that $a(d,\Phi)(K_X+B)$ is Cartier and trivial for every connected projective slc log Calabi-Yau pair of dimension $d$ with coefficients in $\Phi$ ?

Result: Proved(see note)
Status: Candidate (review pending)
AI contribution: AI-discovered
Method: Argument
Field: Birational geometry; log Calabi-Yau pairs
Posed by: Index conjecture for log Calabi-Yau pairs; finite-rational-coefficient slc form stated by C. Jiang and H. Liu (Algebra and Number Theory, 2021), Conjecture 1.5
Year posed: —
Years open: —
Solved: 2026-10-05
Model: Unreleased internal OpenAI model
Vendor: OpenAI
Collaborators: —
Verification: Contested
Publication: Announced
Collection: OpenAI math release (October 2026), version adc7f12
Significance: 25 / 100
Disclosed cost: —
Wikipedia: No dedicated article

What was actually shown

Theorem 1.1: for every $d\ge4$ and finite $\Phi\subset[0,1]\cap\mathbb Q$ there is $a(d,\Phi)>0$ such that $a(d,\Phi)(K_X+B)$ is Cartier with trivial line bundle for every connected projective equidimensional slc log Calabi-Yau pair of dimension $d$ with coefficients in $\Phi$ , independently of the number of components. With Jiang-Liu's theorem for $d\le3$ this gives the conjecture in the cited form. Not shown: slc pairs with infinite DCC coefficient sets, or explicit values of $a(d,\Phi)$ .

What the AI did

The release README states that the vast majority of its results were produced by one fixed procedure with an unreleased internal OpenAI model, using on average about three hours of ChatGPT Pro thinking compute per result; roughly 4,000 problems were posed and the output was aggregated into result families and manuscripts, keeping those judged significant enough. Family 034 has fourteen manuscripts dated September 24 to October 5, 2026, which cite one another as inputs; this entry draws on the manuscript named as its source. The manuscript is credited to 'OpenAI' alone, names no human author and has no acknowledgements. The README's two exceptions to the fixed procedure (the zeta zero-free region work, whose Re(s) > 11/12 write-up was also human-edited, and the Hodge conjecture for CM abelian varieties) do not concern this family, so the result is presented as found and written up by the model. The release does not say how problems were chosen or how much human review happened before publication.

Verification

No independent mathematician has checked this yet. Checked here: Theorem 1.1 of 'Uniform indices for semi-log-canonical log Calabi-Yau pairs' was read against the conjecture as the manuscript quotes it; with the cited dimension-three theorem of Jiang-Liu it covers every dimension for finite rational coefficient sets. The original Jiang-Liu statement was not opened here. The proof was not refereed. No Lean formalization. It uses the normal lc index theorem of the effective Iitaka manuscript and the family's good-model theorem, both unreviewed. Listed as Contested because it depends on the log abundance entry, which the release itself is inconsistent about; see the claim issue.

Claim issue

This result uses the release's log abundance theorem as an input, and the release contradicts itself on whether that theorem is unconditional (24 September abstract) or a conditional resolution resting on logarithmic Iitaka subadditivity (5 October appendix). The entry is Contested with it.

The index conjecture for semi-log-canonical log Calabi-Yau pairs with finite rational coefficient sets

What was actually shown

What the AI did

Verification

Claim issue

Sources

Discussion