Structural Compatibility I: Linearity as a Rigidity Normal Form on the Persistence Leaf under Finite-Horizon Admissibility

This article initiates the Structural Compatibility series within the broader finite-horizon programme. It investigates the structural conditions under which aggregation on a normalized persistence leaf admits a linear normal form. Starting from finite-horizon persistence as a minimal admissibility condition for identifiable regimes, the paper introduces maintainability under perturbations and combinability as structural requirements for internal reorganization. It then restricts attention to an enriched branch in which the normalized persistence leaf is assumed to admit an affine model of internal reorganization. Within this explicitly selected affine setting, the main result is a conditional rigidity theorem: under affine-leaf regularity, scale invariance, and a global bi-affine regularity hypothesis, any admissible aggregation law on the persistence leaf reduces to a linear normal form in affine coordinates. The paper does not claim to derive the full quantum formalism. Rather, it isolates a minimal linear backbone that may underlie richer realizations. Standard quantum theory is therefore discussed only as a compatible enriched realization built on top of this structural pattern, not as a result derived in full. The transition from weaker infinitesimal compatibility conditions to global bi-affinity is identified as the main open integration problem. In this sense, the article should be read as a foundational structural preprint and as the entry point of an enriched branch developed further in the subsequent papers of the series.