The Hovey–Strickland conjecture

The Hovey–Strickland conjecture predicts that, at every prime p and height n, the thick tensor ideals of the category of dualizable K(n)-local spectra are exactly the n + 2 ideals generated by localized finite spectra of each chromatic type, so that its Balmer spectrum is a chain. A manuscript dated 24 September 2026 in OpenAI's public mathematics collection (github.com/openai/math) claims a proof, through a classification of the ideals of the Lubin–Tate deformation ring that are invariant under the Morava stabilizer group, which it describes as Chai's invariant-ideal assertion. This page summarizes the published claims and lists publicly available follow-up work. Statements are the respective authors' claims; the publication of a paper, code or a formal proof is not presented here as independent verification.

Background

• As recounted in the introduction of the manuscript:
  • Hopkins and Smith classified the thick subcategories of finite spectra by chromatic type. Hovey and Strickland proposed the dualizable K(n)-local counterpart in their study of Morava K-theory and localization, and their Problem 16.8 asks directly about stabilizer-invariant prime and radical ideals of the deformation ring.
  • Chai studied the action of the stabilizer on the closed fiber of Lubin–Tate space and proposed that invariant irreducible formal loci should be height loci.
  • Barthel, Heard and Naumann proved the Hovey–Strickland conjecture at height two for every prime, and showed that Chai's invariant-ideal assertion, in their finite-residue-field formulation (Hope 4.1), implies the classification at every height.

The original result

• Invariant ideals and thick tensor ideals (24 September 2026). The manuscript states that for every prime p, every height n and every open subgroup U of the Morava stabilizer group, the proper prime ideals of the Lubin–Tate ring E₀ stabilized by U are exactly 0 and the height ideals I₁, …, I_n, and that the invariant radical ideals are these together with E₀. Combined with the Barthel–Heard–Naumann support comparison, it concludes that the dualizable K(n)-local category has exactly n + 2 thick tensor ideals and that its Balmer spectrum is a chain of n + 1 points.
  • Method, as described: the main step concerns the height strata in characteristic p: at a point of height between 1 and n − 1, a nonzero analytic germ on the corresponding height locus cannot vanish on the stabilizer orbit of the point. Vectors of the étale Tate module are specialized at nearby parameters to give analytic test functions whose parameter differential is invertible, and these are compared along stabilizer-orbit points converging to the given point.
• Sources. Commit 1581d4c of the openai/math repository, folder Stabilizer orbits and thick tensor ideals of dualizable K(n)-local spectra.
• Formalization. The collection has no Lean scope note in lean/docs/ for this family.

Public follow-up work

Date	Work	Relation	Public claim	Public status (checked 17 October 2026)
2026-10-07	Chenglong Ma, arXiv:2610.10072 (v1), titled Invariant Primes in Lubin-Tate Space and Hovey-Strickland at Every Height	Parallel work, with a statement on provenance	For every height and every prime, the paper proves that the prime ideals of the special fiber of the Lubin–Tate ring that are stable under an open subgroup of the stabilizer are exactly the height ideals, and that a stable prime of the deformation ring avoiding p is zero, which implies the Hovey–Strickland classification through the forward implication of Barthel–Heard–Naumann. Its introduction states that the same result was proved independently and by a different method by Ningyi Li, and that the result is also contained in the OpenAI manuscript released on 6 October 2026, "in which the proof strategy and specific choices of notation are identical to a preliminary version of the present paper that was uploaded to ChatGPT on September 8, 2026".	arXiv v1. The statement is the author's; this page does not assess it.
2026-10-16	J. Jardine, Lattice24, referee review in JJardine919/openai-math-audit (archived at Zenodo 10.5281/zenodo.23422995)	Review (AI-assisted audit)	The referee review reports no gap after reading all six sections. Its one minor note says that a cited comparison of morphisms between Banach–Colmez sheaves is stated in a stronger form than checked, while only the standard linear form is used.	Part of a completed audit of the collection's manuscripts without Lean; the reviews are published under CC BY 4.0.

Notes

• The list covers follow-ups found in public sources up to the date above; it is not a priority or completeness claim.