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Counting ad infinitum is the holographic observable to a statistical dynamicswith finite states under independent repeated sampling. Entropy provides theinfinitesimal probability for an observed frequency $\hat{\boldsymbol{\nu}}$w.r.t. a probability prior ${\bf p}$. Following Callen's postulate and throughLegendre-Fenchel transform, without help from mechanics, we show an internalenergy $\boldsymbol{\mu}$ emerges; it provides a linear representation ofreal-valued observables with full or partial information. Gibbs' fundamentalthermodynamic relation and theory of ensembles follow mathematically.$\boldsymbol{\mu}$ is to $\hat{\boldsymbol{\nu}}$ what $\omega$ is to $t$ inFourier analysis.

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