Technical Derivation of SM and GR Pillars UCL

This paper provides a mathematically closed derivation of the Einstein--Hilbert action and the full Standard Model action as conditional infrared universality classes arising from the Universal Correlational Lattice (UCL) framework. All structural assumptions are stated explicitly, all limiting procedures are controlled, and all identifications are performed via spectral-geometric and operator-theoretic constructions. No geometric, gauge-theoretic, or field-theoretic structures are postulated.