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

This preprint studies the represented algebra generated by the Kaleidoscope Yang–Baxter operators at even roots of unity. For every even integer N ≥ 4 and every admissible complex parameter u satisfying u ≠ 0 and u^N ≠ 1, it proves that all chained words vanish for every integer exponent tuple once the number of intervening powers is at least N/2. This proves the conjecture of Qiu, Guan, and Yu on its original unit-circle parameter domain, extends it to the full admissible complex domain, and shows that N/2 is the exact universal threshold by classifying the complete last nonzero layer. The paper further gives a functional-calculus extension and a parameter-independent monotone-path presentation of the represented algebra, yielding explicit descriptions of its radical filtration, center, Cartan rank, global dimension, and representation type.