Local operator–hydrodynamic closure of the molecular time-dependent Schrödinger equation via exact factorization

For pure electron–nuclear states, the molecular time-dependent Schrödinger equation admits an exact factorization into a nuclear factor and a conditional electronic factor. On a star-shaped nodal patch of the nuclear marginal, a rank-one conditional electronic projector determines a canonical local electronic section by projector-defined parallel transport. In that canonical gauge, the exact dynamics are equivalent to an exact, canonically gauge-fixed evolution system with state variables given by the nuclear density n(R, t), a local nuclear phase S(R, t), and the conditional electronic projector P(R, t). The nuclear equations are the exact continuity and Hamilton–Jacobi equations of exact factorization. The electronic equation is an exact tangent-space projector evolution law that remains valid for the generally non-Hermitian exact-factorization electronic generator. The projector determines the quantum metric and Berry curvature pointwise, while the connection entering the closed equations is reconstructed from the projector by canonical transport. No adiabatic reduction, trajectory approximation, truncation, or compression is introduced.