Histogram Approximation for Expander Walks with Arbitrary Label Distributions

This manuscript proves a total variation bound of min{1, 6 lambda sqrt(q)} between the q-label histogram of a stationary expander walk and the multinomial law with the same label probabilities. The estimate is uniform in walk length and the minimum positive label mass. Transitions may vary with time and need not be reversible, provided they preserve a common stationary distribution, start from that distribution, and satisfy the stated singular-norm contraction. The paper includes complete proofs, a matching worst-case alphabet lower bound, and a rare-label example excluding a uniform chi-square analogue. Generative AI assistance in mathematical development, proof examination, computational checks, drafting, and typesetting is disclosed in the manuscript. No pseudorandom-generator seed-length improvement is claimed.