Mathematical Foundations of the Universal Relational-Geometric Coherence Law: Ergodic Stabilization and Trace-Map Convergence

We introduce the formal axiomatic foundations of the Universal Relational-Geometric Coherence Law (URCL). By constructing a linear operator framework over a separable Hilbert space, we model the evolution of structural states under a bounded tracking loop governed by a Fibonacci-modulated trace-map recurrence. We prove that the introduction of a non-linear geometric protection threshold forces the spectral dynamics to possess a unique, globally attractive invariant manifold governed by the irrational golden constant ϕ = (1 + √5)/2. Finally, we evaluate the ergodic properties of the system, proving that the coupled tracking mechanics satisfy strict contractive convergence conditions that exponentially suppress localized stochastic perturbations.