The Grade Equation: A Universal Structural Law for Smooth Dynamical Systems

We prove that every analytic dynamical system \(\dot{\mathbf{x}} = F(\mathbf{x})\) satisfies a universal structural law — the Grade Equation — which decomposes the dynamics into a hierarchy of interaction grades \(F = \sum_{k=0}^{\infty} A^{(k)}\) with exponential suppression \(\|A^{(k)}\| \leq C_0 / \rho^k\), where \(\rho > 1\) is the analyticity radius. We show that this decomposition, combined with domain-specific symmetry constraints, recovers the major field equations of physics as special cases: (i) the Einstein field equations emerge as grade-0 + grade-2 constraints on spacetime dynamics with \(\rho = M_P/H_0\); (ii) the Moment Hypothesis for the Riemann zeta function is a grade bound on the moment-generating function with \(\rho\) determined by the Euler product; (iii) the Mixture Collapse theorem for portfolio distributions is an eigenvalue-grade truncation with \(\rho = \lambda_1/\lambda_2\); (iv) the Koopman spectral decomposition of Hamiltonian mechanics is the dynamical grade hierarchy with \(\rho\) from the nearest phase-space singularity. We prove a Grade–Symmetry Correspondence: a field equation is uniquely determined by its Grade Equation (the decay rate) plus its symmetry class (the algebraic constraints on \(A^{(k)}\)). We establish two structural theorems: the Grade Product Theorem (the product of grade-\(j\) and grade-\(k\) quantities is grade-\((j+k)\), with \(\rho\)-decay preserved) and the Self-Consistency Theorem (the analyticity radius \(\rho\) is determined by the dynamics, which is determined by \(\rho\) — the system of equations is closed). We apply the framework to derive the double grade seesaw for the cosmological constant (\(\rho_\Lambda = M_{\text{eff}}^8 / (\sqrt{3} M_P^4)\), within 0.11% of observation) and to explain why the Riemann Hypothesis is equivalent to a grade bound on \(|\zeta(1/2+it)|^2\). All structural theorems are formalized in Lean 4. We discuss the honest limitations: the framework does not derive gauge invariance from analyticity alone, does not determine \(\rho\) without solving the dynamics, and provides necessary but not sufficient conditions for specific field equations.