Finite-Horizon Structures VII: Propagation, Reachability, and Maintained Domains

Finite-Horizon Structures VII develops the propagation layer of the finite-horizon programme from a chosen smooth positive representative of an underlying projective Y-structure. Building on the compatible projective dynamics established in FHS V, and keeping distinct the supplementary admissibility layer introduced in FHS VI, the article defines local admissible transport families, admissible propagation directions, forward and backward propagation cones, reachability relations, propagation envelopes, maintained domains, and structural fronts on the regular locus. The core geometric object is the coherence one-form associated with the chosen representative, whose sign structure yields a projectively invariant first-order classification of admissible transport. The paper also introduces finite-speed admissibility as an additional bounded-transmissibility layer, studies the propagation of coherence geometry and of the local Y-measure class under admissible transport, and analyses the interaction between propagation and regular superlevel domains. The result is a purely structural and pre-metric theory of propagation, reachability, threshold interfaces, and maintained regions, positioned between compatible projective dynamics and later rigidity questions in the Finite-Horizon Structures series.This article forms part of the Ranesis framework developed by Alexandre Ramakers.