Formal Canonical Mathematical Base UCL

We present a fully axiomatized, operator-theoretic, and spectral formulation of the Universal Correlational Lattice (UCL). The framework is developed as a self-contained mathematical structure, independent of any a priori geometric, field-theoretic, or continuum assumptions. We define its admissible configurations, equivalence classes, coarse-graining flows, spectral invariants, reconstruction procedures, and universality classes. This paper closes the formal foundations of UCL as a candidate fundamental theory, providing the complete technical arsenal required for all physical derivations.