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

Abstract

The Kaleidoscope Yang-Baxter equation of Qiu, Guan, and Yu (2026) is the consistency condition of multiple scattering in Gaudin's kaleidoscope models. At an even order $N$ it involves two matrices: a shift, and a square-zero matrix with one complex parameter. They conjectured that every chained word, the square-zero matrix alternating with integer powers of the shift, vanishes once the number of powers reaches $N/2$, verified through order ten. We prove it for every even order and every parameter where the matrices exist. The bound is sharp. One step below it, a word is nonzero exactly when no factor's power times its position is divisible by half the order, and then has rank two. The proof rests on one flag that every factor lowers by a step. The algebra the two matrices generate is a monotone-path algebra: its basis paths multiply only while their level keeps direction. It is independent of the parameter and has dimension $N^2-N+2$. Its radical, generated by the square-zero matrix, is nilpotent of index one more than half the order, and its representation type is infinite. Four conditions over an arbitrary field reproduce the threshold and the radical. None can be dropped, and the root-of-unity model is one instance.