The Hahn-Wilson conjecture on fp-spectra

Let $p$ be a prime, and let a spectrum of fp-type $n$ mean a bounded below, $p$ -complete fp-spectrum whose associated thick subcategory of finite spectra has type at least $n+1$ . Let $\mathrm{BP} \langle n \rangle$ denote the truncated Brown-Peterson spectrum, and let the thick subcategory generated by it be taken inside the category of $p$ -complete spectra. Hahn-Wilson conjecture. Spectra of fp-type $n$ are exactly the thick subcategory of $p$ -complete spectra generated by $\mathrm{BP} \langle n \rangle$ . The conjecture is known in the cases $n=-1$ and $n=0$ ; the stated general structure result remains open. It would identify, up to thick subcategories, all fp-spectra with the expected chromatic examples.

Claimed by OpenAI. Claims height-two Hahn–Wilson counterexamples at every sufficiently large prime: connective p-complete spectra of exact fp-type two outside the specified BP-truncation thick subcategory despite both stated localization comparisons.

Scope relative to this problem: The source constructs connective p-complete spectra of exact fp-type 2 outside Thick(BP<2>_p-complete) at every sufficiently large prime, using the standard BP-module quotient form. The same objects satisfy the two stated finite/ordinary and telescope/MoravaK localization comparisons. This is a height 2 negative instance, not failure at every prime or every height; global total-finiteness fp conventions differ from merely degreewise local finiteness.

GitHub repository: https://github.com/openai/math

Manuscript: https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Counterexamples-to-the-Hahn-Wilson-conjecture-at-height-two-September-26-2026/paper.pdf

The Hahn-Wilson conjecture on fp-spectra

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