Result 314, Topology

Cyclic length and chromatic fixed-point loss

Determines the optimal chromatic loss from geometric H-fixed points to geometric G-fixed points for every subgroup H of a finite p-group G. At every nonnegative height, the loss equals the shortest subnormal-chain length from H to G with cyclic quotients. Each quotient counts once regardless of order, and finite spectra witness sharpness.

Proof

The bigger picture

Why it matters

In stable topology, geometric fixed points extract information associated with a chosen symmetry subgroup. The manuscript reports exactly how much chromatic information, organized into heights describing periodic structure, can be lost when passing from a subgroup to a larger group.

What changes?

For any subgroup H of a finite p-group G, whose order is a power of a prime p, the claimed optimal loss from geometric H-fixed points to geometric G-fixed points equals the shortest subnormal chain length with cyclic quotients. Such a chain runs from H to G, each group normal in the next, with each quotient generated by one element. The equality holds at every prime and nonnegative chromatic height; each quotient counts once, regardless of order.

What does that help mathematicians do?

The formula distinguishes the number of cyclic stages from their sizes: a cyclic quotient of large prime-power order still contributes only one unit of loss. Researchers can therefore determine the sharp bound by studying subgroup chains, rather than estimating it solely from group orders. The supplied summary says finite spectra witness sharpness, so even these finite objects of stable topology prevent any smaller universal bound.

Are there practical applications?

Its immediate value is foundational: it connects a group-theoretic measure of how H sits inside G to a precise limit on symmetry-based constructions in stable topology. This supplies a sharp constraint for arguments comparing chromatic information across subgroups. The supplied sources do not establish a direct practical application.

This section was generated by GPT-6 Astra Medium. This explanation is based on the result summary and manuscript abstracts below. This context is separate from OpenAI's source text.

Manuscript

Cyclic length and chromatic fixed-point loss

September 24, 2026 30 pages

For a finite p-group G and a subgroup H, we prove that the optimal chromatic fixed-point loss equals the shortest length of a subnormal chain from H to G with cyclic quotients. The equality holds at every prime and every nonnegative height, resolving positively the equality proposed by Kuhn and Lloyd.

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Cite (BibTeX) 
@misc{OAI:Cyclic-Length-and-Chromatic-Fixed-Point-Loss-September-24-2026,
  author = {{OpenAI}},
  title = {{Cyclic length and chromatic fixed-point loss}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Cyclic-Length-and-Chromatic-Fixed-Point-Loss-September-24-2026/Cyclic-Length-and-Chromatic-Fixed-Point-Loss-September-24-2026.pdf}{OAI:Cyclic-Length-and-Chromatic-Fixed-Point-Loss-September-24-2026}},
  year = {2026}
}

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