On the Asymptotic Algebraic Entropy of Singular Words in Infinite Non-Abelian Braid Groups: A Refined Non-Trivial Limit via Dynamic Specialization

Abstract

We resolve the trivial algebraic collapse inherent in static root-of-unity specializations of the Lawrence-Krammer-Bigelow (LKB) representation by introducing a dynamic, double-variable asymptotic harmonic boundary condition. Let B_n be the Artin braid group on n strands. For each k >= 3, we analyze the action of a highly self-entangled deterministic structural word W_k in B_{2k} under the time-dependent cyclotomic parameters q_k = e^{i*pi*(1 - 1/k)/2} and t_k = e^{i*pi/k}. By avoiding the degenerative Hecke symmetries of static frameworks, the non-abelian quantum residuum is preserved. We define a re-scaled Asymptotic Algebraic Entropy Index E(k) and formulate the definitive open problem regarding its transfinite scaling limit, showcasing a rigorous interplay between the collapse of topological entropy and the growth of algebraic spectral radii