Finite-Horizon Structures VIII develops the rigidity and completion layer of the finite-horizon programme from the admissible propagation structures introduced in FHS VII. Starting from a chosen smooth positive representative of an underlying projective Y-structure and its coherence one-form, the article studies what becomes geometrically forced once admissible local transport has been fixed on the regular locus. It classifies local propagation cone configurations, recasts finite-speed admissibility as a quantitative rigidity condition, establishes front rigidity at regular threshold interfaces, and derives a corresponding rigidity theory for maintained superlevel domains and the propagated local Y-measure class. The paper also introduces directional comparison structures based on reachability and propagation cost, derives local normal forms in adapted logarithmic coordinates, and formulates compatibility, gluing, and obstruction criteria for assembling local rigid propagation data into coherent completed regimes. The result is a purely structural and pre-metric theory of rigidity, comparison, and structural completion for admissible propagation, positioned between the propagative layer of FHS VII and any later metric, Lorentzian, quadratic, or field-theoretic reconstruction. This article forms part of the Ranesis framework developed by Alexandre Ramakers.
Alexandre Ramakers (2026) studied this question.