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

A coarse-graining collapses many microstates into one macrostate — all the molecular arrangements we call “a gas at 300 K.” Statistical mechanics assumes such a collapse is available and reads the second law off it, but not every collapse is physically admissible, and which one is correct is usually left to convention. This paper fixes the criterion. It imports the finite-Markov machinery unchanged — strong lumpability, the coarsest lumping by partition refinement, the monotonicity of relative entropy to a stationary distribution — and adds a derivation. La Profilée fixes, from a pre-physical root, which distinctions are constitutive for the system whose persistence is in question: what must be held (), what may vary (), which transitions are admissible (), and which existence/identity verdict () must survive. A macropartition is then persistence-sufficient and dynamically closed exactly when it preserves those constitutive labels and is strongly lumpable — and that lumpability condition, independent of the representative of , is the classical-Markov form of the dynamic-congruence criterion that decides the physical system cut in the quantum case (LP-PHY-05). One admissible-reduction structure appears in two domains; the recurrence is a datum for the universality claim, not a metaphor. On that basis the paper proves the necessary-and-sufficient criterion, the existence and construction of a unique coarsest admissible partition, and a distributional H-theorem: relative entropy to any named stationary distribution is monotone, while approach to a unique equilibrium requires a further mixing premise. The second law is not reinvented: it is made precise exactly where a coarse-graining describes the persistence of the claimed system, and shown to be inapplicable as an autonomous macro-law where the coarse-graining discards continuation-relevant information. What is not delivered is stated throughout — monotonicity of a single microtrajectory, the low-entropy initial condition, a universal entropy functional, and the continuum limit are all outside the finite-Markov sector that is closed here.