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Math-2458 · mathematics · Topology

Realizing Chromatic Types at Nested Fixed Points

Status Partially solved · unsolvedmath.comLLM status Open unverified

Statement

Definitions and mathematical statement

Shared notation.
$\mathbb N$, $\mathbb Z$, $\mathbb Q$, $\mathbb R$, and $\mathbb C$ denote the natural numbers, integers, rationals, reals, and complex numbers, respectively. All additional parameters and hypotheses are specified in the question.
Statement recorded in the supplied source.
Let $H<G$ be subgroups of a finite $p$-group. For which pairs $(m,n)$ does there exist a finite $G$-spectrum $X$ such that $\Phi^H X$ has chromatic type $m$ and $\Phi^G X$ has chromatic type $n$?

Formulation references: [S1], [S2].

Short English statement

Let $H<G$ be subgroups of a finite $p$-group. For which pairs $(m,n)$ does there exist a finite $G$-spectrum $X$ such that $\Phi^H X$ has chromatic type $m$ and $\Phi^G X$ has chromatic type $n$?

Sources

Catalogue entry

Statement licence: CC-BY-SA-4.0 · Source file

Contributions 1

Math-2458-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-2458

The claimed sharp fixed-point loss determines the finite chromatic type pairs: a finite $G$-spectrum with $H$-type $m$ and $G$-type $n$ exists exactly when $m\le n+\ell(G,H)$. The loss bound excludes larger $m$. For $n<m\le n+\ell(G,H)$, a sharpness witness can be smashed with an inflated type-$n$ spectrum and wedged with an inflated type-$m$ spectrum to obtain these exact types. For $m\le n$, an orbit induced from $H$, together with inflated spectra of those types, gives the required pair. This is a consequence of the claimed sharpness theorem and the standard existence of finite spectra of every type.

Realizing Chromatic Types at Nested Fixed Points

OpenAI's claim

For every subgroup of a finite p-group, the optimal chromatic loss under fixed points is claimed to equal its shortest subnormal-chain length with cyclic quotients, independently of height. The cited manuscript argues for that result; its relation to this problem is specified below.

Mathematical statement of the claimed result

Mathematical claim 1.
Fix a prime $p$, finite $p$-group $G$, subgroup $H$ and integer $n\ge0$. Let $r_n(G,H)$ be the least $r\ge0$ such that for every finite $p$-local genuine $G$-spectrum $X$, \[ K(n+r)_*(\Phi^HX)=0\quad\Longrightarrow\quad K(n)_*(\Phi^GX)=0. \] Here $\Phi$ denotes geometric fixed points and $K(0)=H\mathbb Q$. Let $\ell(G,H)$ be the smallest number of steps among subnormal chains from $H$ to $G$ having a cyclic quotient at each step, and $\ell(G,G)=0$. OpenAI claims $r_n(G,H)=\ell(G,H)$ for all these parameters. Minimality includes finite witness spectra ruling out every smaller loss.

Outline of the argument

Manuscript 1 (family 314, 2026-09-24): Character evaluations construct a pro-p comparison group whose finite sections need few generators. Groupoid integration supplies subgroup-count congruences that force a sufficiently short closed series with Z_p factors. Finite chromatic fracture translates this arithmetic bound into fixed-point vanishing, and tensor-prime separation provides witnesses showing that no smaller loss can work.

Formal verification

Family 314 has no Lean scope page in this snapshot; no formalization of these manuscript results is listed.

This site has not executed the Lean files or checked their dependencies. These descriptions report the scope of the linked declarations, rather than a new verification of the proofs.

Status caveat

OpenAI attributes these results to an internal model. Its release notice cautions that results lacking full formalization can contain mistakes. This site has not independently assessed the proofs; a solved label records the scope of the reported claim.

Sources

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