The Hahn–Wilson conjecture
From OpenAI Aftermath
The Hahn–Wilson conjecture proposes that a p-complete spectrum has fp-type at most n exactly when it can be built from the completed truncated Brown–Peterson spectrum BP⟨n⟩p∧ by finitely many sums, shifts, cofiber sequences and retracts. Here a bounded-below p-complete spectrum is fp if its mod-p cohomology is finitely presented over the Steenrod algebra, and it has fp-type at most n if its smash product with a finite spectrum of type n + 1 has finitely many nonzero homotopy groups, each finite. A manuscript dated 26 September 2026 in OpenAI's public mathematics collection (github.com/openai/math) claims counterexamples at height two for every sufficiently large prime. This page summarizes the published claim and lists publicly available follow-up work. Statements are the respective authors' claims; this page does not certify them.
Background
- As recounted in the introduction of the manuscript:
- Mahowald and Rezk introduced fp-spectra, showed that the cohomological condition is equivalent to the finiteness test above for some nonzero finite spectrum, and asked whether finite and Johnson–Wilson localization agree on fp-spectra. Burklund, Hahn, Levy and Schlank later disproved the unrestricted telescope conjecture at every prime and every height at least two; Lee and Pstrągowski note that the relevant homotopy of those counterexamples is bounded but not finite in total, so they are not fp counterexamples.
- The written formulation of the conjecture appears in Lee and Pstrągowski's The monochromatic Hahn–Wilson conjecture (arXiv:2410.08029), Conjecture 1.1 and Note 1, who report learning it from Dylan Wilson in 2021. Hahn and Wilson's redshift theorem supplies examples: the algebraic K-theory of suitable forms of BP⟨n⟩p∧ has fp-type n + 1.
- The cases of fp-type at most −1 and 0 are classical, and Lee and Pstrągowski prove the conjecture at height one. After K(n)-localization they classify the locally fp spectra at every height as the thick subcategory generated by Morava E-theory, and they suggest combining a height decomposition with this local theorem to prove global generation for fp-spectra that satisfy the telescope comparison.
The original result
- Claim. Theorem thm:main of Counterexamples to the Hahn–Wilson conjecture at height two states that for every sufficiently large prime p there is a connective p-complete spectrum X of exact fp-type two that does not lie in the thick subcategory generated by a standard form of BP⟨2⟩p∧, although X satisfies both localization comparisons L2fX ≃ L2X and LT(2)X ≃ LK(2)X. The manuscript states that this shows that the additional telescope hypothesis suggested by Lee and Pstrągowski does not suffice at height two. It gives no explicit range of primes.
- Method, as described: at large primes the example is built in an algebraic model, a periodic derived category of sheaves on the stack of formal groups of height at most two, through Pstrągowski's chromatic algebraicity comparison. A cone on one completed operation is glued to a discrete lattice of kernel and cokernel directions on the height-two overlap, and the resulting complex is transferred to spectra.
- Sources. At commit
1581d4c: Counterexamples to the Hahn–Wilson conjecture at height two. - Formalization. The repository's Lean documentation has no scope note for this family at that commit.
Public follow-up work
| Date | Work | Relation | Public claim | Public status (checked 24 October 2026) |
|---|---|---|---|---|
| 2026-10-10 | GitHub account acprk, acprk/hahn-wilson-height-two-explicit-prime, note An explicit prime range for the height-two counterexamples to the Hahn–Wilson conjecture with exact computations, at commit f15be63 | Extends (explicit range) | States that the manuscript's parameter conditions first hold at p = 17,073,437, and that with one parameter of its operator section chosen differently and the argument otherwise unchanged, the construction works for every prime p ≥ 293, the least prime this argument can reach; a sharper reading of one step adds 199 and 211. The finite numerical conditions are checked in exact rational arithmetic. | Unrefereed note, not formalized; it describes itself as a re-tuning of constants that assumes the manuscript's correctness, written by AI agents under the direction of the repository owner, whose name the note leaves as a placeholder. The repository includes an AI-written adversarial review. |
| 2026-10-16 | J. Jardine, Lattice24, referee review rank_133 of the manuscript in JJardine919/openai-math-audit | Review (AI-assisted audit) | Reports no gap and no minor issue. It reads the main text, the obstruction, gluing and transfer sections in full and most of the operator section, and checks by hand the finite-tower criterion, the growth and dual-ghost arguments, the index and valuation arithmetic and the operator parameter conditions. It does not re-derive the geometry and comparison sections or the fracture propositions, and takes the cited external theorems as given. | Part of an audit of the collection's manuscripts that the audit describes as complete. |
Notes
- The earlier results are due to the authors named in the Background section, as recounted by the manuscript.
- The list covers follow-ups found in public sources up to the date above; it is not a priority or completeness claim.
