Hopkins' chromatic splitting conjecture (weak and odd-prime strong forms)

Fix a prime $p$ , let $L_{n-1}$ be localization at $E(n-1)$ and $L_{K(n)}$ localization at Morava $K$ -theory. Hopkins' chromatic splitting conjecture, recorded by Hovey (1995, Conjecture 4.2), describes the chromatic overlap $L_{n-1}L_{K(n)}S_p^\wedge$ . Its weak form asserts that for the $p$ -completion $X$ of a finite spectrum the canonical map $L_{n-1}X\to L_{n-1}L_{K(n)}X$ has a homotopy retraction (finite-input form: Beaudry-Goerss-Henn, Conjecture 1.1.11). Its strong form prescribes an explicit splitting of $L_{n-1}L_{K(n)}S_p^\wedge$ into suspended localized spheres indexed by an exterior algebra; Beaudry disproved this at $n=p=2$ , and Barthel-Beaudry (2019, Conjecture 6.3) retain it at odd primes. Both forms were known through height two at odd primes. Does the canonical map split for all $n$ and $p$ , and does the overlap decompose as predicted at odd primes?

Result: Disproved(see note)
Status: Candidate (review pending)
AI contribution: AI-discovered
Method: Argument
Field: Stable homotopy theory: chromatic homotopy
Posed by: M. J. Hopkins, recorded by M. Hovey, Bousfield localization functors and Hopkins' chromatic splitting conjecture (1995), Conjecture 4.2; refined by Barthel-Beaudry (2019) and Beaudry-Goerss-Henn
Year posed: 1995
Years open: 31y
Solved: 2026-09-27
Model: Unreleased internal OpenAI model
Vendor: OpenAI
Collaborators: —
Verification: Unreviewed
Publication: Announced
Collection: OpenAI math release (October 2026), version adc7f12
Significance: 40 / 100
Disclosed cost: —
Wikipedia: No dedicated article

What was actually shown

Weak form: for the derived $p$ -completion $X=S_p^\wedge$ , the canonical map $L_{n-1}X\to L_{n-1}L_{K(n)}X$ has no homotopy retraction at height $n=p$ for every $p\ge5$ and at $n=p+1$ for every $p\ge7$ ; the class $\beta_1^{(p-1)^2}$ is a nonzero kernel element (at $p=5$ , $\beta_1^{16}$ in degree 608). Strong form: for $p\ge5$ the map $L_0L_{K(3)}S\to L_0L_{K(2)}L_{K(3)}S$ is nonzero on $\pi_{-3}$ , so the height-three splitting fails even as an equivalence of $E(2)$ -local spectra. Companions: at $n=p=3$ the overlap is not in the thick subcategory generated by $L_0S,L_1S,L_2S$ ; positively, for $p>n+1$ the overlap has a $2^n$ -stage filtration with the predicted cofibers (nonsplit), with explicit attachments at height three. Heights below $p-1$ remain open.

What the AI did

The OpenAI math release (github.com/openai/math, commit adc7f12) states that its results were produced by an unreleased internal OpenAI model under one fixed procedure, averaging about three hours of ChatGPT Pro thinking compute per result. This result is not among the README's stated exceptions (the Re(s) > 11/12 zero-free region write-up and the Hodge conjecture for CM abelian varieties). The manuscript is credited to OpenAI alone and names no human author. The family has five manuscripts dated 25 and 27 September 2026: two counterexample papers, one negative finite-assembly result, and two positive filtration results. The rational-obstruction manuscript ships a small exact-arithmetic support script whose coverage note says it checks only Chevalley-Eilenberg matrix arithmetic, not the main theorem.

Verification

No independent mathematician has checked this yet. Checked here: the abstracts, introductions and main theorems of the TeX sources of all five manuscripts, read against Hovey's Conjecture 4.2 and the Barthel-Beaudry and Beaudry-Goerss-Henn formulations they cite; the proofs were not refereed. The weak-splitting counterexamples (principal manuscript) use the derived p-completion of the sphere, i.e. finite input, as the conjecture requires; they rely on the Hopkins-Miller computation recorded by Heard and on Devinatz-Hopkins nilpotence. The strong-form disproof at height three uses the solid Lubin-Tate theory of Barthel, Mann, Ray, Schlank, Senger, Weinstein and Zhou (cited as 2026). The disproofs cover specific ranges of p and n only; the weak form at heights below p - 1 and at small primes is not addressed. No Lean formalization is supplied for this family.

Hopkins' chromatic splitting conjecture (weak and odd-prime strong forms)

What was actually shown

What the AI did

Verification

Sources

Discussion