Brennan's Conjecture and the Universal Integral Means Spectrum

Abstract

In October 2026, OpenAI released two preprints \cite{openai-brennan,openai-kraetzer}, establishing Brennan's celebrated conjecture~\cite{brennan,brennan2} and disproving Kraetzer's more general $t^2/4$ conjecture \cite{kraetzer, kraetzer-thesis}. While validating these results, we observed that much of their argument is redundant and arrived at a drastically shorter proof of a significantly stronger statement. The goal of this note is to give a short and concise presentation of the reborn proof, sacrificing some of the context and consequences for brevity, and relegating them to a follow-up work. In this particular case the human-AI interaction was fruitful and quite satisfactory, and hoping for a similar future in other subjects, we would paraphrase Rick Blaine: {\em ``AI, we hope this is the beginning of a beautiful friendship.''}