Geometric Boundaries on Entropy Production Rates in Open Systems: A Formal Extension to Non-Equilibrium Thermodynamics via the URCL

We present a self-contained geometric boundary framework establishing a strict lower bound on entropy production rates in open, non-equilibrium relational dynamical systems. Operating within the Universal Relational-Geometric Coherence Law (URCL) framework, we model the exchange of a system's internal coherence budget across an adele ring topology. When the network's boundary states are modulated via a parameter-dependent, Fibonacci-scaled geometric protection factor, the underlying transfer-matrix dynamics are governed by a discrete trace-map recurrence. By applying the properties of Hurwitz's theorem on continued fractions, we prove that when the Relational Bio-Seismograph Index (RBSI) crosses the critical threshold dictated by the golden ratio φ = (1 + √5)/2, the local dissipation rate experiences non-linear suppression. Semiclassical fluctuation-dissipation analysis shows that the golden ratio acts as a universal stable fixed-point attractor that minimizes destructive phase interference across the network boundaries. This mechanism establishes a formal extension to classical non-equilibrium thermodynamics, providing a quantitative, falsifiable model for sustained order and structural self-reinforcement in open systems. Pipeline Disclosure: The core conceptual translation—structuring the custom thermodynamic protection statements into the formalisms of non-equilibrium steady states (NESS), fluctuation-dissipation theorems, and Hurwitz continued-fraction minima—was fully authorized and directed by the author. Initial layout organized via Grok (xAI); rigorous mathematical validation and production-ready LaTeX typesetting finalized via Gemini (Google).