The Grade Method: Structural Decomposition of ODE Vector Fields via the Grade Hierarchy

We present the grade decomposition method as a general-purpose structural analysis tool for smooth dynamical systems. Any analytic vector field \(F\) decomposes into interaction grades \(F = \sum_{k=0}^{\infty} A^{(k)}\) with exponential suppression \(\|A^{(k)}\| \leq C_0/\rho^k\), where \(\rho\) is the analyticity radius. A four-step protocol — (1) identify the state space, (2) compute the grade decomposition, (3) measure \(\rho\), (4) determine the effective grade \(k_{\text{eff}}\) — makes the method applicable to any domain. We demonstrate it on epidemiology (SIR is exactly grade-2; \(R_0\) is the grade-2/grade-1 ratio), ecology (Lotka-Volterra is grade-2; higher-order interactions in microbial communities are grade-3), neuroscience (Wilson-Cowan has vanishing grade-2 at E-I balance; Hodgkin-Huxley requires grade-3), gene regulation (Hill-function \(\rho\) controls switch sharpness; Boolean networks are the \(\rho \to 0\) limit), climate (Lorenz is grade-2; \(\rho = \text{Ra}_c/\text{Ra}\)), chemistry (mass-action kinetics is grade-2; \(K_\text{eq}\) is a grade ratio), stochastic dynamics (additive noise preserves grades; multiplicative noise shifts thresholds via Itô correction), and economics (Solow model has \(\rho = k^*\); IS-LM is grade-1). The method subsumes normal form theory and Carleman linearization while requiring exponentially fewer parameters. SINDy with a polynomial library recovers the grade spectrum from data. The single parameter \(\rho\) universally measures distance to criticality across all domains. All 99 structural theorems are machine-verified (46 proof strategy engine + 53 proof kernel, 0 errors). Software implementing the method is provided.