Diabatic Model Reduction Theory (Part II): Projection Operator Identities and Finite-Time Error Budgets

Reduced diabatic models are ubiquitous in nonadiabatic chemical dynamics and in open quantum systems: a small electronic subspace is coupled to many nuclear and environmental degrees of freedom, yet only a low-dimensional effective model is tractable for dynamics and observables. This paper develops a \emph{Diabatic Model Reduction Theory} (DMRT) that organizes the reduction from a microscopic Hamiltonian to a diabatic system--bath model as a sequence of explicit approximations, each assigned a distinct finite-time error contribution at the level of observables. Two complementary projection-operator identities underpin the framework: (i) the Feshbach (Schur-complement) identity for Hilbert-space subspace elimination and energy-dependent self-energies, and (ii) the Nakajima--Zwanzig identity for reduced density-operator dynamics with memory kernels. The result is a rigorous ``accounting'' decomposition of model error into adjacent-stage discrepancies (truncation, diabatic closure, coordinate coarse-graining, bath restriction, and computational closure), together with a set of diagnostics and assumptions that determine which terms are expected to dominate in a given application. The canonical diabatization inputs and topology diagnostics used in the diabatic-closure stage are provided by the companion Part~I diabatic gauge theory framework.