arXiv:2603.17849v3 Announce Type: replace Abstract: Port-Hamiltonian energy balance and Koopman approximation describe different aspects of nonlinear dynamics. This paper distinguishes exact observable identities, measure-dependent Galerkin projections, constrained surrogate identification, and certification of a controller on the nonlinear plant. Conservative weak skewness requires compatible measure and boundary conditions. Dissipative positivity belongs to the metric bracket; it does not generally make the projected first-order dissipative derivation positive. We give an explicit fixed-metric compatibility condition and a constant--Hamiltonian dictionary obstruction, and state the assumptions under which surrogate certificates imply plant bounds. A reproduced pendulum experiment illustrates the practical gap: its learned controller admits an exact shaped-energy certificate on the nonlinear plant, although its fitted quadratic storage is not a plant Lyapunov function and its archived residual-norm diagnostic is uninformative. These conclusions use the specified plant structure; they are not generic finite-data guarantees. The bracket, projection, and Lyapunov arguments are positioned relative to existing structured identification and residual-aware control methods.