Bivariate Scaling Relationship Between Stochastic Perturbation and Critical Proximity in Nonlinear Dynamical Systems

This document records a mathematical scaling relationship describing how stochastic perturbations interact with proximity to critical transitions in nonlinear dynamical systems. An empirically observed bivariate scaling form is stated, along with a derived perturbation threshold relationship and a hub-dependent geometric correction factor. A validation taxonomy is provided to delineate confirmed empirical baselines from theoretical predictions. This memorandum places the mathematical relationships into the public scientific record to establish definitive prior art and ensure open accessibility.