Exact Chained-Word Threshold and Monotone-Path Structure in the Even Root-of-Unity Kaleidoscope Yang–Baxter Algebra

Abstract

We prove the chained-word vanishing conjecture of Qiu, Guan, and Yu in the even root-of-unity Kaleidoscope Yang–Baxter representation. For every even N ≥ 4, every complex parameter u with u ≠ 0 and u^N ≠ 1, and every integer exponent tuple, the exact universal vanishing threshold is N/2. We completely classify the last nonzero layer and extend the analysis to cyclic functional calculus, obtaining exact rank formulas. A common invariant flag and spectral decomposition yield a parameter-independent presentation of the generated algebra, its full radical filtration, center, Cartan rank, global dimension, and representation type. An arbitrary-field formulation on exhaustive flags of unrestricted dimensions isolates the hypotheses responsible for sharpness and radical identification, providing a reusable algebraic framework beyond the original representation.