Emergent Schrödinger Dynamics from the Universal Ψ Lagrangian - Non-Relativistic Coherent Limit, Effective Hamiltonian, and Wave Evolution

We derive the Schr¨odinger equation directly from the Universal Ψ Lagrangian through a firstordernon-relativistic functional expansion around the stationary solution Ψ0. No quantum structureis postulated: the linear evolution operator, the temporal–spatial separation, the effectiveHamiltonian, and the emergent reduced Planck constant all arise uniquely from the coherent lowenergysector of the universal variational principle. The derivation contains no adjustable parameters,is fully self-contained, and shows that non-relativistic quantum mechanics is the uniquecoherent limit of the fundamental action.