Dual-Axis Entropy Production and the Recovery of Classical Thermodynamics: A Physical–Observation Formulation

Physical–Observation Dual-Axis (PODA) theory retains the physical state and the objective state of record formation as distinct responsibilities. We develop a thermodynamic formulation on their admissible joint histories, starting from a common energy, an explicitly specified thermal closure and a declared physical time reversal. For a finite-state realization with channel-resolved local detailed balance, the full trajectory entropy production obeys the usual integral fluctuation relation. We prove an exact decomposition of its mean into the relative entropy visible in a declared record and a nonnegative conditional contribution from the unrecorded joint history. Equality holds precisely when the forward and reverse conditional history laws coincide. An exact recovery theorem identifies preservation of the observation section, marked history laws, energy exchanges and reverse protocol. A separate sufficiency criterion determines whether a record captures all entropy production. The finite-state section recovers the first and second laws, Gibbs equilibrium and isothermal free-energy dissipation of the reduced system. Two analytic models distinguish correlation loss from heat release and demonstrate positive joint entropy production with stationary equilibrium marginals. A regular perturbation formula expresses the leading hidden entropy production through conditional score mismatch. The results give conditional, exact criteria for thermodynamic recovery within PODA and identify the dynamical quantities required for departures from a fixed observation description.