Internal Energy, Fundamental Thermodynamic Relation, and Gibbs' Ensemble Theory as Emergent Laws of Statistical Counting

Statistical counting ad infinitum is the holographic observable to a statistical dynamics with finite states under independent and identically distributed N sampling. Entropy provides the infinitesimal probability for an observed empirical frequency nu<^> with respect to a probability prior p, when nu<^>not equal p as N ->infinity. Following Callen's postulate and through Legendre-Fenchel transform, without help from mechanics, we show that an internal energy u emerges; it provides a linear representation of real-valued observables with full or partial information. Gibbs' fundamental thermodynamic relation and theory of ensembles follow mathematically. u is to nu<^> what chemical potential mu is to particle number N in Gibbs' chemical thermodynamics, what beta=T-1 is to internal energy U in classical thermodynamics, and what omega is to t in Fourier analysis.