An AI-Discovered Classification of Equal-Statistics Nontransitive Dice
Year: 2026
Type: preprint
Abstract: Open release of a mathematical result discovered by an AI system and independently verified. On 7 November 2026, Chris Heikell prompted OpenAI's ChatGPT to 'discover something that hasn't been discovered yet'; the discovery was derived in a ChatGPT session using GPT-6 Astra Pro, with same-model parallel agents and exact computational verification, and the proof artifact was finalized November 7, 2026, 9:31:01 PM PST. GPT-6 Astra Pro originated the problem, the classification, the proof, and the verifier code. The theorem: for three fair n-sided dice partitioning the integers 1..3n, with equal mean, equal population variance, and equal median, a strict nontransitive cycle exists if and only if n is divisible by 6 and n >= 12. The release document gives the full AI-produced proof text reproduced verbatim, an independent verification log (exact recomputation of the n=12 example, the exhaustive n=6 exclusion, and generated examples up to n=120), and a literature/priority search summary: no exact prior statement was found as of 7-8 November 2026 (confidence moderate-to-high; this is a negative search result, not proof of non-existence). Related work that must be distinguished: Schaefer & Schweig (2017) and Kim et al. (2024) on balanced dice in the partition model (equal mean only), and Yakusheva (2022) on dice with equal means and variances in a different label model without a median condition.
Author: Heikell, Chris
Language: English
Subfield: Statistics and Probability
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7225587456