Chebyshev quotients, Demazure multiplicities, and Dyck-path models

We study Chebyshev quotients that arise in the representation theory of Lie algebras, specifically within the theory of Demazure flags for fusion products of $\mathfrak{sl}_2[t]$-modules. Using a recent formula that expresses numerical Demazure multiplicities as coefficients of such quotients, we prove a general eventual non-negativity theorem for the same rational functions that compute these multiplicities: each quotient either terminates or has strictly positive coefficients for sufficiently large degrees, which we in turn interpret in terms of matchings and bounded walks. In several natural infinite families, these are unsigned bounded Dyck path models, giving both a structural explanation for the observed positivity phenomenon and concrete combinatorial models for key families of Demazure multiplicities. The theorems in this paper were autonomously produced and formalized in Lean/Mathlib by AxiomProver from natural-language statements.