Inverse-Limit Formulas and Stable-Range Rigidity for Cyclotomic Sums

TL;DR

This paper develops a formal framework for analyzing cyclotomic sums and cosine evaluations, proving a stable range rigidity theorem and exploring polynomiality and symmetric families in the context of cyclotomic and cosine point evaluations.

Contribution

It introduces an inverse limit formalism for cyclotomic sums, establishes a stable range rigidity theorem, and extends the framework to include multiplicative invariants and symmetric polynomial families.

Findings

Stable range rigidity theorem for cosine point evaluations.

Eventual polynomiality in n for symmetric families of bounded degree.

Coefficient extraction yields bounded degree symmetric families.

Abstract

We study truncation compatible families F = (F_m)_{m>=1} over Q[z] through an inverse limit formalism, and we evaluate them at the punctured cyclotomic cosine points alpha_{k,n} = cos(2 pi k/n) with the specialization z equals n-1. For symmetric families of uniformly bounded total degree in x <= d, we prove a stable range rigidity theorem: for all n >= d+2, the cosine point evaluation factors through the finitely many punctured cosine power sums the finitely many power sums P1(n) through Pd(n). In the purely polynomial case this implies eventual polynomiality in n. We then extend the framework to include fixed product factors and package their cosine point contribution in multiplicative invariants MQ(n). In the stable range, the bounded degree symmetric part collapses as before; any remaining cyclotomic dependence occurs only through these explicit product terms. Finally, we show that coefficient extraction from such products produces further bounded degree symmetric families, and we apply this to complete symmetric functions h_r evaluated at cosine points.