  Counting ad infinitum is the holographic observable to a statistical dynamics
with finite states under independent repeated sampling. Entropy provides the
infinitesimal probability for an observed frequency $\hat{\boldsymbol{\nu}}$
w.r.t. a probability prior ${\bf p}$. Following Callen's postulate and through
Legendre-Fenchel transform, without help from mechanics, we show an internal
energy $\boldsymbol{\mu}$ 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.
$\boldsymbol{\mu}$ is to $\hat{\boldsymbol{\nu}}$ what $\omega$ is to $t$ in
Fourier analysis.
