Notes

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.

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.