The Kuhn-Lloyd question: does the chromatic fixed-point loss of a p-group equal the shortest cyclic subnormal chain length?

For a finite $p$ -group $G$ , a subgroup $H$ and $n\ge0$ , let $r_n(G,H)$ be the least $r$ such that $K(n+r)_*(\Phi^HX)=0$ implies $K(n)_*(\Phi^GX)=0$ for every finite $p$ -local genuine $G$ -spectrum $X$ , where $K(i)$ is Morava K-theory. This controls the inclusions among primes in the Balmer spectrum of finite $G$ -spectra. Let $\ell(G,H)$ be the shortest length of a subnormal chain from $H$ to $G$ with cyclic quotients; then $r_n\le\ell$ . The abelian case was settled by Barthel et al., and Kuhn and Lloyd proved equality for central extensions of elementary abelian 2-groups by $C_2$ . They expressed hope that $r_n(G,H)=\ell(G,H)$ in general and asked whether the loss is independent of the height $n$ . Is $r_n(G,H)=\ell(G,H)$ for all finite $p$ -groups?

Result: Proved(see note)
Status: Candidate (review pending)
AI contribution: AI-discovered
Method: Argument
Field: Equivariant stable homotopy theory, chromatic homotopy
Posed by: Nicholas J. Kuhn and Christopher J. R. Lloyd
Year posed: 2024
Years open: 2y
Solved: 2026-09-24
Model: Unreleased internal OpenAI model
Vendor: OpenAI
Collaborators: —
Verification: Unreviewed
Publication: Announced
Collection: OpenAI math release (October 2026), version adc7f12
Significance: 24 / 100
Disclosed cost: —
Wikipedia: No dedicated article

What was actually shown

Theorem 1.1: for every prime $p$ , finite $p$ -group $G$ , subgroup $H$ and $n\ge0$ , $r_n(G,H)=\ell(G,H)$ , the shortest cyclic subnormal chain length; sharpness is witnessed by finite spectra, and the case $n=0$ includes rational detection. Combined with Balmer-Sanders reductions, a corollary determines the finite-height prime inclusions in the Balmer spectrum of finite genuine $G$ -spectra for every finite group $G$ . Not shown: anything about infinite height beyond what Balmer-Sanders already give, or non-finite spectra.

What the AI did

The release README says every result in it was produced by an unreleased internal OpenAI model following a fixed procedure, on average about three hours of ChatGPT Pro thinking compute per result. This result is not among the README's two exceptions (the Hodge conjecture for CM abelian varieties and the Re(s) > 11/12 zero-free region, whose write-up was human-edited). The manuscripts are authored 'OpenAI' and name no human author. The family is a single manuscript (September 24, 2026).

Verification

No independent mathematician has checked this yet. Checked here: the abstract, introduction and Theorem 1.1 were read against the Kuhn-Lloyd question as the manuscript quotes it; the theorem states the hoped-for equality for every prime, every finite $p$ -group and every height, hence height independence. The proof (profinite character sources, relative integration, telescopic estimates) was not refereed. No Lean formalization accompanies this manuscript.

The Kuhn-Lloyd question: does the chromatic fixed-point loss of a p-group equal the shortest cyclic subnormal chain length?

What was actually shown

What the AI did

Verification

Sources

Discussion