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Counterexamples to the Hahn-Wilson conjecture at height two
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Abstract
We disprove the Hahn–Wilson conjecture at height two. For every sufficiently large prime p, we construct a connective p-complete spectrum X of exact fp-type two outside the ordinary thick subcategory generated by the specified standard form of $\mathrm{BP}\langle2\rangle_p^\wedge$. The same spectrum satisfies both localization comparisons $L_2^fX\simeq L_2X$ and $L_{T(2)}X\simeq L_{K(2)}X$.
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Collected automatically by AI tools (Claude Opus 5.5 and GPT-6 Astra, 2026-10-21) and not checked by a person. May be wrong or incomplete. Report an error.
Follow-up works
- acprk (GitHub repository openai-math-remarks), Remark R7: Hahn–Wilson counterexamples for every prime p ≥ 293 (GitHub, undated; literature search stated up to 2026-10-09). Re-selects the parameters of the paper's operator construction (conditions
op:parametersandop:correction-bound) to claim the counterexample for every prime p ≥ 293, versus the paper's explicit condition first holding at p = 17,073,437; it states its checks are computational, "not human peer review", and do not re-verify the rest of the paper. [AI-generated per source]
Possibly relevant literature (found by the agent, not checked)
- Christian Carrick and Michael A. Hill, On higher real K-theories and finite spectra (arXiv:2507.07051, July 2025). Shows that the higher real K-theories eo_h are fp spectra of type h, and its Question (1) in Section 2.4 asks whether certain fp spectra BP^((G))⟨m⟩^H lie in the thick subcategory generated by BP⟨h⟩, "consistent with the Hahn–Wilson conjecture".
Digests and expositions
- Latent Seminar, Hahn–Wilson finite generation: a gluing obstruction (7 October 2026). Explains the manuscript’s operator construction, gluing and finite-tower obstruction, preserving the sufficiently-large-prime restriction and both localization comparisons, and explicitly disclaims independent verification of the complete proof. [family-level]
Elsewhere
- VibeMathed: Problem record
- Emergent Mind: family 319
Independent sites; their content is not reviewed here.
Cite
@misc{OAI:Counterexamples-to-the-Hahn-Wilson-conjecture-at-height-two-September-26-2026,
author = {{OpenAI}},
title = {{Counterexamples to the Hahn--Wilson conjecture at height two}},
howpublished = {OpenAI Math Release preprint
\href{https://github.com/openai/math/blob/main/preprints/Counterexamples-to-the-Hahn-Wilson-conjecture-at-height-two-September-26-2026/paper.pdf}{OAI:Counterexamples-to-the-Hahn-Wilson-conjecture-at-height-two-September-26-2026}},
year = {2026}
}Family 319: Counterexamples to finite generation at chromatic height two
Refutes the Hahn–Wilson conjecture at chromatic height two. For every sufficiently large prime p, constructs a connective p-complete spectrum of exact fp-type two that cannot be built from completed $\mathrm{BP}\langle2\rangle$ by finitely many sums, shifts, cones and retracts. The examples nevertheless satisfy the finite and telescopic localization comparisons.
This is the only paper in this family.