OpenAI · 314-01 · 2026-09-24 · Topology
Cyclic length and chromatic fixed-point loss
preprints/Cyclic-Length-and-Chromatic-Fixed-Point-Loss-September-24-2026
Family 314. The chromatic Smith fixed-point problem for finite -groups.
The authors are OpenAI. This companion does not verify the proof and does not adopt it as a theorem of Ngo Cheung’s series. It is not a premise. O.1 stays NOT_CLOSED. Anti-de Sitter results are not a de Sitter state, not Em02-E’s relational lapse, not XVI-P’s QdS scheme, and not XVI-Q’s sigma-image record. No quantum coefficient is admitted. No compilation is asserted.
Read against the standing problems
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
Determines the optimal chromatic loss from geometric -fixed points to geometric -fixed points for every subgroup of a finite -group . At every nonnegative height, the loss equals the shortest subnormal-chain length from to with cyclic quotients. Each quotient counts once regardless of order, and finite spectra witness sharpness.
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.
Preprint source and Lean scope
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No Lean scope note is in the repository for this family. The preprint source below is still the full manuscript, not only the abstract. No compilation is asserted.