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Brennan's Conjecture and the Universal Integral Means Spectrum

  • Ilia Binder
  • Oleg Ivrii
  • Istvan Prause
  • Stanislav Smirnov
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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 t2/4 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.''}

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Cite

BibTeXRIS
BibTeX
@misc{binder2026brennans,
  title = {{Brennan's Conjecture and the Universal Integral Means Spectrum}},
  author = {Ilia Binder and Oleg Ivrii and Istvan Prause and Stanislav Smirnov},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.23853},
  eprint = {2610.23853},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.23853},
}
RIS
TY  - UNPB
TI  - Brennan's Conjecture and the Universal Integral Means Spectrum
AU  - Ilia Binder
AU  - Oleg Ivrii
AU  - Istvan Prause
AU  - Stanislav Smirnov
PY  - 2026
DA  - 2026/10/16
PB  - arXiv
DO  - 10.48550/arxiv.2610.23853
UR  - https://arxiv.org/abs/2610.23853
AB  - 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.''}
ER  -