A Stability Ratio Across Physical Systems

Many physical systems form stable patterns despite being continually disturbed by noise. This paper shows that, across diverse domains, the persistence of such patterns is governed by the same structural condition. In each system, the dynamics include a restoring process that reinforces the pattern and a disturbance process that disrupts it. Defining Λ_self as the restoring rate and Λ_env as the disturbance rate, a dimensionless ratio R = Λ_self / Λ_env determines whether a persistent interior pattern can be maintained. Four systems with experimentally known thresholds are examined: ferromagnetic iron at the Curie temperature, superconducting transmon qubits, helium–neon lasers, and nonlinear chemical oscillators of the Belousov–Zhabotinsky type. Despite their different mechanisms, scales, and equations, each system reduces to the inequality Λ_self ≥ R★ · Λ_env, with a system-specific threshold R★. Iron exhibits R★ ≈ 0.25–0.28, while qubits, lasers, and reaction–diffusion systems exhibit R★ ≈ 1. The universality of this inequality reflects a basic feature of linear stability analysis: any noisy dynamical system maintains a pattern only when its internal restoring modes dominate external disturbances. Part One presents this framework and the supporting calculations, showing that the same dimensionless ratio naturally emerges across multiple branches of physics and chemistry without requiring new assumptions or new physical laws.