Thermodynamic completeness in quantum and classical Markovian dynamics

We develop a path-space action formulation for quantum and classical Markovian thermodynamics that addresses a reconstruction problem: which thermodynamic observables can be inferred from state trajectories alone, and which require additional current or measurement record? The formulation treats the state trajectory and the thermodynamic record as distinct components of a Markovian path. In quantum systems, the record is specified by a quantum instrument; in the commutative classical representation, it is a density current constrained by a continuity equation. The main result is a thermodynamic completeness test. It identifies current or measurement-record perturbations that do not change the state trajectory and shows that any observable changed by such a perturbation cannot be reconstructed from state data alone. Hence two Markovian models can have the same state generator, stationary state, linear response, and local state geometry, but different thermodynamic current means or current noise. For quantum Markovian dynamics, an unconditioned density-matrix generator therefore does not determine heat-current, particle-transfer, photon-counting, spin-transfer, or continuous-measurement statistics; the quantum instrument and the thermodynamic increment assigned to each record outcome must also be specified. For classical density-current dynamics, the same test identifies hidden exchange, reaction, transport, and kinetic current records that are eliminated by the projection to the state trajectory. We further show that this incompleteness has geometric and topological origins: the current-space Hessian projects to a quotient state geometry, while graph cycles, divergence-free currents, and harmonic currents span directions invisible to state observations.