Derivation of a Self-Contained Relational Universe from Nonseparable Time–Space Coupling

This study investigates whether the structure obtained after nonseparable time–space coupling closes into a self-contained dynamical universe. Starting from the filtration of the coupling algebra and integral arithmetic residues, macroscopic observables are constructed from microscopic incidence, and the reaction–spatial bridge fixes the relation between clean dyadic refinement and reaction steps. The canonical core reaction is then represented as an isometry on a counting Hilbert space, and the Wold decomposition separates rooted temporal and recurrent sectors. In the clean dyadic branch, a three-dimensional regular bulk is constructed, and an exact scale ratio of 2 together with inverse-square spectral rescaling is derived along the unit-edge base direction. Finally, several natural field realizations are shown to implement the same relational trajectory, thereby separating the nonuniqueness of physical field representation from the relational dynamics itself. The result is a self-contained relational universe with a state space, autonomous reaction, intrinsic temporal orientation, a derived spatial sector, ancestry structure, and a scale law.