Finite-Sample Reconstruction from Carry Propagation in Canonical Beta Numeration

This paper develops a finite-sample reconstruction theory for carry propagation in canonical $\beta$-numeration. Building on the exact inverse results of Paper V, it asks how much of the canonical decoder can be recovered when only finitely many carry observations are available. The analysis isolates a geometric decoder model in which a geometric reset depth is observed through a monotone additive decoder. Within this model, the paper proves deterministic reconstruction from finite exact spectra, a quantitative rigidity theorem for the inverse map, and a certified nearest-neighbour algorithm that recovers the first $m$m decoder digits with probability at least $1-\delta$ from $O_q\!\left(q^{-m}\log(m/\delta)\right)$ observations. A matching information-theoretic lower bound shows that the exponential dependence on reconstruction depth is unavoidable. The paper also explains how finite-prefix recovery yields complete reconstruction for bounded Parry systems and presents numerical experiments confirming both the $q^{-m}$ scaling and the necessity of certification.