EXACT LEAKAGE OF GRADED POLYNOMIAL KOOPMAN DICTIONARIES: GEOMETRIC DECAY OF THE EDMD RESIDUAL AND ITS BASIS DEPENDENCE

A finite-dimensional Koopman-invariant subspace yields an exact linear predictor of a nonlinear system, but the graded polynomial dictionaries used by EDMD (all monomials up to degree d) never provide one when the vector field has degree p ≥ 2: the Lie derivative always raises the degree (Proposition 1). This qualitative obstruction is essentially known; we make it quantitative. For every polynomial vector field in any dimension we give an exact closed-form expression for the population EDMD residual R(d) on the cube [-1,1]^n in terms of Legendre coefficients of monomials (Theorem 1), which collapses to a diagonal sum of squares when p ≤ 3. We prove that R(d) decays geometrically at exactly the rate 2^(-d) for every p ≥ 2, with polynomial prefactor Θ(d^((n+1)/2)) only for p ∈ {2, 3} (for p ≥ 4 the prefactor is bracketed between d^((n+1)/2) and d^((n+1)/2 + floor((p-2)/2)); Theorem 2), and we compute the exact asymptotic constants for the scalar cubic model, the damped double-well Duffing oscillator, the Van der Pol oscillator and the Lorenz system (Theorem 3); every formula agrees with independent quadrature computations to better than 1e-12 (typically 1e-14). We then show that this geometric decay is a property of the monomial basis, not of the subspace: the basis-independent (Hilbert-Schmidt) leakage of V_d grows at least linearly in d for every such field (Proposition 2). An EDMD experiment on the Duffing oscillator confirms both statements quantitatively: the monomial residual falls while the residual in a whitened basis rises, matching the theory to within 1% at small time step for d ≤ 10 (the normalisations and the Δt → 0 comparison are derived explicitly). We discuss what this implies for residual certification and for the choice of dictionary, comment on the scaling with the dimension n, and list open problems.