Notes

We prove that the nonvanishing conjecture implies the abundance conjecture.

Status: Verified by a human; the writing is awaiting polishing.

A klt del Pezzo surface of Picard number one over an algebraically closed field of characteristic three has at most seven singular points. Over the complex numbers the bound is four (Belousov), and Bernasconi’s example with seven singular points shows that seven is sharp in characteristic three. The proof follows the hunt of Keel–M^cKernan and combines it with rulings of small degree on the exceptional divisor, local adjunction, and comparisons with open del Pezzo pairs and canonical del Pezzo surfaces carrying anticanonical curves of arithmetic genus one.

The question was contributed by Paolo Cascini to FrontierMath Open Problems, which asked for a surface with at least eight singular points; the theorem shows that no such surface exists. The main result was obtained using the Danus system in July 2026.

Karol Palka has informed me that he and Tomasz Pełka had already proved this bound earlier, as part of their complete classification of del Pezzo surfaces of rank one, which has been announced but is not yet public; their work is not used in this note. For this reason the note is posted here for the time being and is not submitted to arXiv.

Status: AI-generated; verified by a human.

On August 1, 2026, OpenAI released the report Ten Advances in Mathematics and Theoretical Computer Science, in which an internal version of its Astra model proved the inequality part of Ehrhart’s volume conjecture (1964), the eighth of the ten results. The conjecture concerns a very intuitive statement: an n-dimensional convex body whose barycenter is its unique interior lattice point has volume at most (n+1)ⁿ/n!. OpenAI proved that the volume indeed cannot exceed this number; which bodies attain the maximal volume — the equality part of the conjecture — was explicitly left open in the report.

We noticed that the core idea of OpenAI’s proof transports Fujita’s proof of the volume upper bound for K-semistable Fano varieties to convex geometry, and that the same paper of Fujita also settled the equality case on the Fano side: only the projective space attains the bound. The core of Fujita’s equality analysis is the Seshadri constant: equality forces the Seshadri constants to equal n+1 at every point, and the projective space attains this value exactly. The convex-geometric counterpart of the Seshadri constant is the lattice width: Seshadri = n+1 corresponds to lattice width at least n+1 in every integer direction, and the extremal simplex attains the bound exactly, its minimal lattice width being n+1. The plan of attack therefore centered on the width saturation.

This idea was given to our Danus system. Running with 8 agents, Danus carried out the first half of the proof along this correspondence; the second half has no complex-geometric counterpart, and Danus found it on its own. The total run took 3 hours and 16 minutes. Combined with the result of OpenAI, both the inequality part and the equality part of Ehrhart’s volume conjecture are now resolved.

The manuscript has passed the author’s preliminary verification and is posted here as a note. A more detailed revision is in progress, and the note will be posted to arXiv upon its completion. Comments are very welcome.

Update (August 1, 2026): Following discussions with Kewei Zhang, the note now includes a dictionary of the Fano–convex correspondences behind the proof, relating its first half to Zhang’s IMRN 2022 and Proc. AMS 2025 papers on the Fano side; each section is now annotated with its Fano-side counterpart, and the second half of the proof has no Fano-side antecedent. I thank Kewei Zhang for pointing out these references and for useful discussions on the correspondences.

Update (August 2, 2026): A revised version of this note has now been posted to arXiv as arXiv:2608.01040.

Version 1 of Supravat Sarkar’s paper brought Question 1.1 to my attention. Here “independently” refers specifically to the construction: the counterexample in this PDF was independently generated using the Danus system and sent to Supravat Sarkar on 14 July 2026, before I saw or received a description of the distinct counterexample or proof later included in version 2 of Sarkar’s paper. Since Sarkar’s v2 now gives a different counterexample to the same question, I will not submit this note to arXiv. I am instead making the PDF available on my personal webpage as a dated note, for the benefit of the community and for future reference.

Disclaimer: This is an AI-generated manuscript. The proof has not been verified by a human; readers should independently check every mathematical claim before relying on or citing it.

This note proves the conjecture of Kotěšovec on the factorial growth of the Matryoshka numbers (OEIS A177384), which arise from the combinatorics of the cosmohedron: the limit S = lim a_n/(n+4)! exists and is finite and positive, so a_n ∼ S n! n⁴, and S is certified to lie in an explicit interval of width below 10⁻⁹. The proof is elementary and independent of the resurgence approach, and the note was produced by an automated proof system. It is made available here following a suggestion of Professor Federico Ardila.

Status: Verified by a human before July 7, 2026. This note will not be posted to arXiv.

This note disproves a conjecture of Esser and Totaro on the relative primeness of two explicitly constructible integers. As a consequence, the large-index conjecture of Esser, Totaro, and Wang fails in dimension 159: the prime 53 divides both E₁₅₉ and m′₁₅₉. The note will not be posted to arXiv.

Status: I have been informed by Burt Totaro that this is essentially a counterexample, and that it has been verified by Louis Esser.