Exact Progression Lengths in Finite Interval-Digit Sets

For integers b >= 2, n >= 1 and 0 <= r <= b-2, let D consist of the nonnegative integers with at most n base-b digits, all in {0,...,r}. We prove that the longest nonconstant arithmetic progression in D has exactly (r+1) times the largest divisor a of b^(n-1) satisfying (b-r-1)a < b terms, with the singleton case assigned length one. An explicit progression attains the bound. This finite-depth refinement of Walker and Walker's orbit argument also gives the exact infinite-length limit, its first attaining depth, and a sharp least-prime-divisor criterion for the minimum possible value r+1. We determine the longest progression in every iterated sumset of a finite interval-digit cube, including the carry regime. The contribution is a prescribed-alphabet and finite-depth refinement; known one-missing-digit and prime-base constructions are credited. This is an AI-assisted, self-audited, unrefereed preprint. It does not resolve the entire multi-part AIM-COMBINATORICS-0025 record, and no independent review, formal verification or certified novelty is claimed.