We establish an absolute geometric lower bound on the local dissipation rate within open, discrete relational networks whose coupling coefficients undergo Fibonacci-modulated scaling. Representing the network states as a transfer-matrix track over a unimodular lattice, we demonstrate that the system's global phase balance maps identically onto a non-linear trace recurrence. By applying Hurwitz's theorem on Diophantine approximations, we prove that the golden ratio φ = (1+5) /2 acts as an absolute, irremovable boundary barrier against external stochastic dephasing. Because φ possesses the slowest converging continued fraction expansion among all real numbers, a network parameterized precisely to this fixed point maximizes its topological distance from environmental rational resonances. This result provides a simple, self-contained number-theoretic mechanism explaining the universal emergence of golden-ratio scaling in stable, non-equilibrium physical and biological architectures, independent of empirical tuning parameters. Pipeline Disclosure: The core conceptual formulation—structuring the trace-map recurrence matrix parameters into the formalisms of adelic product formulas, non-equilibrium steady states, and Hurwitz continued-fraction minima—was fully authorized and directed by the author. Initial layout organized via Grok (xAI) ; rigorous mathematical validation, domain confinement tracking, and production-ready LaTeX typesetting finalized via Gemini (Google).
Daphne Garrido (2026) studied this question.