UCQ — Canonical Equations and Formal Addendum _ A Closed Unification Framework for Relativity–Quantum Consistency

This work presents the canonical mathematical formulation of the Unifying Canonical Quadrature (UCQ) framework. The objective of UCQ is to establish a structurally consistent space in which the formalisms of General Relativity and Quantum Mechanics can coexist without contradiction. Rather than introducing new physical forces, particles, or phenomenological hypotheses, the UCQ framework defines a closed canonical structure ensuring compatibility between geometric relativistic descriptions and quantum operator dynamics. In this framework, both theories are interpreted as distinct projections of a shared invariant core, rather than competing or reducible descriptions. The canonical structure is defined within a quadrature space composed of three elements: the relativistic geometric manifold, the quantum operator space, and an invariant canonical core ensuring structural coherence. The central UCQ equation expresses a consistency constraint rather than a dynamical law, guaranteeing covariance in relativistic geometry, unitary invariance in quantum operators, and preservation of informational invariants. This document introduces the formal mathematical definitions required for the UCQ framework, including canonical variables, projection operators, invariance conditions, and structural closure principles. It also provides a formal addendum specifying the domain of validity of the framework, admissible extensions, and boundary regimes in which projections may lose direct physical interpretability while the canonical structure remains mathematically defined. The UCQ framework is explicitly non-predictive and structurally descriptive. Its role is not to replace existing physical theories but to provide a formal structure ensuring their compatibility within a unified canonical space. By establishing strict rules of canonical closure and projection consistency, this work defines the foundational reference for the broader UCQ corpus and its potential interdisciplinary applications across physics, complex systems, and structural modeling.