Inverse-Limit Formulas and Stable-Range Rigidity for Cyclotomic Sums
Juan D. Velez, Carlos Cadavid
TL;DR
This work develops an inverse-limit framework for n-dependent, symmetry-bounded cyclotomic sums evaluated at punctured cosine points. By packaging level-specific data into polynomial formulas with a fixed x-degree, the authors prove stable-range rigidity: for n large enough, evaluations depend only on a finite set of additive invariants $P_h(n)$ and any residual cyclotomic variation occurs via explicit multiplicative factors $M_Q(n)$. In the purely polynomial case this yields eventual polynomiality in n, and more generally the framework produces systematic constructions of admissible families, notably via coefficient extraction from fixed products. The complete symmetric sums $h_r$ at cosine points reveal a Catalan-hypergeometric structure in stable range, with explicit closed forms and generating-function identities. The results provide a structural, verifiable route to global identities across all cyclotomic levels, with potential extensions to Galois-invariant or twisted evaluations.
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.