# Cyclic-chain length is computable for bounded finite examples

Family 314; scoped reading addendum, 9 October 2026 Australia/Brisbane. Source revision `fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb`.

## Source finding

The introduction defines cyclic length as the shortest subnormal subgroup chain with cyclic successive quotients and claims equality with chromatic fixed-point loss for every finite p-group, embedded subgroup and nonnegative height. A cyclic quotient of order p squared counts once. The source gives two differently embedded subgroups in C_(p squared) times C_p with losses two and one.

## Applicability boundary

Abstract subgroup isomorphism type is insufficient. Exhaustive subgroup enumeration is a finite reference method, not a polynomial algorithm. The spectrum hypotheses and proof remain outside a subgroup calculator. No family formal-scope document is available.

## Business decision

A small exact subgroup-chain utility is a viable research/teaching component; no lucrative standalone market has been established.

## Exact sections read

- [preprints/Cyclic-Length-and-Chromatic-Fixed-Point-Loss-September-24-2026/build/sections/01-introduction.tex](https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/Cyclic-Length-and-Chromatic-Fixed-Point-Loss-September-24-2026/build/sections/01-introduction.tex).

These readings establish recorded source scope, not proof correctness. No independent Lean verification was run. Companion manuscripts and construction effectiveness remain separately tracked. See [source metadata](source.json).