= Topological pressure
{title2=$P(\phi)=\lim_n n^{-1}\log\sum_{|w|=n}e^{S_n\phi(w\eta)}$}
For a finite <full shift> and a potential with summable variations, the displayed symbolic pressure exists and does not depend on the reference tail $\eta$. The <almost-additive partition-function limit> proves existence. The function $\gamma\mapsto P(\gamma U)$ is <convex>; finite-volume second derivatives are energy variances divided by block length. If $-B\leq U\leq-b<0$, its increments for $\delta>0$ lie between $-B\delta$ and $-b\delta$, so it is Lipschitz continuous, strictly decreasing, and has one positive zero.
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