Birkar's Stein-degree conjecture for log Calabi-Yau pairs: the contraction-to-a-point case for ordinary pairs
For a proper integral variety over a field the Stein degree is . Birkar conjectured that boundary components of log Calabi-Yau fibrations with coefficient bounded below have uniformly bounded Stein degree; Birkar and Qu proved normalized bounds over algebraically closed fields of characteristic zero. For a contraction to a point over any field of characteristic zero: given and , is there such that for every projective lc pair of dimension with and , every prime component of with coefficient at least has Stein degree at most ?
- Result
- Proved(see note)
- Status
- Partial result
- AI contribution
- AI-discovered
- Method
- Argument
- Field
- Birational geometry; boundedness
- Posed by
- Caucher Birkar, Stein degree of proper morphisms (arXiv 2603.23939, 2026), Conjecture 11.1
- Year posed
- 2026
- Years open
- 0y
- Solved
- 2026-09-25
- Model
- Unreleased internal OpenAI model
- Vendor
- OpenAI
- Collaborators
- —
- Verification
- Contested
- Publication
- Announced
- Collection
- OpenAI math release (October 2026), version adc7f12
- Significance
- 10 / 100
- Disclosed cost
- —
- Wikipedia
- No dedicated article
What was actually shown
Theorem 1.1: for every field of characteristic zero and every normal integral projective lc -pair of dimension with , and , every prime component of with coefficient at least satisfies , where is the algebraic closure of in ; this bounds the Stein degree of . It covers only the contraction to and ordinary pairs; the general fibration form of the conjecture and generalized pairs are not treated.
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 was read against the conjecture as the manuscript describes it; the paper itself says it proves only the contraction-to-a-point formulation for ordinary Q-pairs, hence partial. Birkar's paper was not opened here. The proof (following Birkar-Qu with an added arithmetic argument on Galois orbits of valuations) was not refereed. No Lean formalization. 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.