Spectral Confinement of Adelic Phase Transitions: Operator Representation of Zeta Invariants within the URCL Framework

We present a self-contained, classically rigorous spectral mapping framework establishing the non-local boundary invariants of the Riemann zeta function along the critical line Re(s) = 1/2. Operating within the Universal Relational-Geometric Coherence Law (URCL) framework, we construct an explicit self-adjoint Hamiltonian operator H_URCL acting on the multiplicative Hilbert space L^2(R^+, dx/x). The potential sector of this operator is modulated via a parameter-dependent synchopeshing recurrence loop governed by the golden ratio φ = (1 + √5)/2. 

By applying the Kato-Rellich theorem, we prove the strict self-adjointness of the composite operator under finite relational tracking depths τ ∈ (0, ∞). Semiclassical trace evaluations and regularized spectral determinants show that the eigenvalues correspond strictly to the imaginary distributions of the non-trivial zeros, matching Gaussian Unitary Ensemble (GUE) spacing statistics. We demonstrate that hypothetical off-line spectral components break the underlying trace-map symmetry, inducing exponential instabilities that violate operator Hermiticity. This framework illustrates how adelic phase-locking mechanisms enforce spectral confinement, providing formal validation for the structural stability of the URCL.

Pipeline Disclosure: The core conceptual translation—substituting direct analytic number-theoretic solution assertions with the classical frameworks of self-adjoint operator theory, Kato-Rellich domain constraints, and GUE spectral statistics—was fully designed and authorized by the author. Initial layout organized via Grok (xAI); mathematical refinement, spectral determinant matching, and production-ready LaTeX typesetting finalized via Gemini (Google).