Admissible Coarse-Graining and the Second Law - When a macrostate is a physical system, and when entropy is its entropy

A coarse-graining identifies fine states. Its physical admissibility requires that the identification preserve the constituted bearer, the retained persistence verdicts and the relevant continuation. In the finite Markov sector we prove that label-faithfulness and strong lumpability are necessary and sufficient for an autonomous, verdict-preserving macroprocess for every fine initial distribution. Partition refinement computes its unique coarsest realization. On that quotient relative entropy to a stationary distribution is non-increasing; convergence, Shannon-entropy increase and phase-volume entropy each require their own stated physical premises. State autonomy does not establish thermodynamic process completeness. We derive the exact stationary entropy-production loss under aggregation: internal dissipation plus a relative-entropy mismatch between forward and reverse channel allocations. A driven six-state ring supplies an exact autonomous parity process with zero apparent production and arbitrarily large hidden production at fixed macro rates. Retaining clockwise and counterclockwise channel marks restores its stationary production. Thus a complete thermodynamic reduction must preserve the relevant process information as well as the state dynamics. A finite-horizon theorem bounds the distribution and relative-entropy errors of approximately lumpable reductions. A separate bounded-stock theorem shows when continued load requires sustained removal, and when a physically established entropy cost turns that requirement into a dissipation bound. Together these results locate the second law on the correct physical quotient without confusing distributional relaxation, path irreversibility, persistence and maintenance. All central constructions, witnesses and bounds are derived below.