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The Hahn-Wilson conjecture: fp-type at most n spectra are exactly the thick subcategory generated by BP<n>

Fix a prime pp. A bounded-below pp-complete spectrum XX is fp if H∗(X;Fp)H^*(X;\mathbb F_p) is finitely presented over the Steenrod algebra; it has fp-type at most nn if π∗(V∧X)\pi_*(V\wedge X) is finite in total for a finite spectrum VV of type n+1n+1 (Mahowald-Rezk). Hahn-Wilson redshift shows K(BP⟨n⟩)K(\mathrm{BP}\langle n\rangle) has fp-type n+1n+1, and Lee-Pstragowski proved the statement below at height one. The Hahn-Wilson conjecture asks whether finite presentation plus a height bound characterizes finite constructions from truncated Brown-Peterson spectra: is {X: X has fp-type at most n}=Thick(BP⟨n⟩p∧)\{X:\ X\text{ has fp-type at most }n\}=\mathrm{Thick}(\mathrm{BP}\langle n\rangle_p^\wedge), the closure under finite sums, shifts, cofiber sequences and retracts, for every nn and pp?

Result
Disproved(see note)
Status
Candidate (review pending)
AI contribution
AI-discovered
Method
Construction
Field
Chromatic homotopy theory; stable homotopy theory
Posed by
Jeremy Hahn and Dylan Wilson (reported from 2021); written as Conjecture 1.1 in David Jongwon Lee and Piotr Pstragowski, The monochromatic Hahn-Wilson conjecture, Inventiones Mathematicae (2026)
Year posed
2021
Years open
5y
Solved
2026-09-26
Model
Unreleased internal OpenAI model
Vendor
OpenAI
Collaborators
—
Verification
Unreviewed
Publication
Announced
Collection
OpenAI math release (October 2026), version adc7f12
Significance
18 / 100
Disclosed cost
—
Wikipedia
No dedicated article

What was actually shown

Claims that for every sufficiently large prime pp there is a connective pp-complete spectrum XX of exact fp-type two with X∉Thick(BP⟨2⟩p∧)X\notin\mathrm{Thick}(\mathrm{BP}\langle2\rangle_p^\wedge) for a standard form of BP⟨2⟩\mathrm{BP}\langle2\rangle, refuting the universally quantified conjecture at height two. The same XX satisfies L2fX≃L2XL_2^fX\simeq L_2X and LT(2)X≃LK(2)XL_{T(2)}X\simeq L_{K(2)}X, so the obstruction lies in how the chromatic pieces are glued, not in a telescope-type failure; the monochromatic form proved by Lee-Pstragowski is untouched. It does NOT treat small primes, heights above two, or the height-one case, which Lee-Pstragowski proved.

What the AI did

The release README says all results were produced by an unreleased internal OpenAI model, the vast majority by one fixed procedure using on average about three hours of ChatGPT Pro thinking compute per result. Its named exceptions (the Hodge conjecture for CM abelian varieties, and the Re(s) > 11/12 zero-free region, whose write-up was human edited for readability) do not concern this family. The manuscript is credited to OpenAI with no human author named. No Lean formalization of this result is in the release.

Verification

No independent mathematician has checked this yet. The abstract, introduction and main theorem were read against the conjecture as the manuscript quotes it from Lee-Pstragowski; that paper was not opened here. The counterexample is for every sufficiently large prime (no explicit bound read here) and uses one specific regular quotient form of BP⟨2⟩p∧\mathrm{BP}\langle2\rangle_p^\wedge; the paper relies on Lee (2026), with Angeltveit-Lind, for independence of the underlying spectrum from the choice of generators at these primes. No Lean formalization exists for this family.

Sources

Changelog1 change

Discussion