UED Reconstruction Programme — Paper 5: Persistence and Matter

Paper 5 of the UED Reconstruction Programme reconstructs matter as a regime of persistent physical configuration rather than as an assumed inventory of fundamental particles. It asks what measurable properties must coexist before a configuration can function as matter: localization, structural identity, persistence, restoring stability, finite disruption resistance, positive inertial response, and reproducible interaction thresholds. A coarse-grained configuration descriptor is introduced for structural comparison without identifying it with a wavefunction, particle density, electromagnetic field, or universal SVE amplitude. Localization is characterized through geometric moments, while structural identity is quantified through rigid-motion-optimized correlation and an operational persistence time. Where reversible work supports an effective configurational-energy description, local stability is determined through the second variation and finite transformation resistance through a configuration-space barrier. A central mathematical result is the Finite-Scale Persistence Condition: a finite localized configuration cannot be stabilized when all nonzero positive energy contributions vary monotonically with the same sign under uniform dilation. In particular, a positive local excess-energy term together with a positive gradient penalty is insufficient by itself to stabilize a finite static localized configuration in three dimensions. A viable persistent matter mechanism therefore requires an independently measurable competing scale dependence, constraint, topology, nonlocal coupling, additional state component, dynamical mechanism, or other justified stabilizing architecture. The paper also defines effective translational inertia as the low-velocity curvature of configuration energy with respect to translation velocity, without assuming a mass-energy equivalence law. Binding energy, disruption barriers, rotational inertia, internal response modes, and stable receiver thresholds are reconstructed as properties of persistent configuration. Paper 5 therefore establishes the minimal mathematical criteria for the matter regime and formulates the First Persistence Problem that subsequent atomic reconstruction must address.