Lucian Method: Mono-Variable Extreme Scale Analysis for Revealing Geometric Structure in Nonlinear Coupled Equation Systems

Description/Abstract: This paper formalizes a mathematical methodology for detecting geometric structure in nonlinear coupled equation systems that is invisible at any single scale of analysis. The method isolates a single variable, extends it across extreme orders of magnitude — far beyond the range of any single physical domain — and observes the geometric behavior of the coupled variables across the full range. The output is not numerical solutions but geometric morphology: the shapes, symmetries, and structural invariances that the equation system produces under extreme-range variation. Applied to Einstein's field equations, the method revealed fractal geometric classification hidden for 111 years. Applied to the Yang-Mills equations, it revealed the same. The method does not solve equations. It reveals what equations are.