Symbolic potential ($U:K^{\mathbb{N}}\to \mathbb{R}$)

A symbolic potential assigns a real weight to an infinite symbolic sequence. Its orbit sum gives the energy in a weighted partition function. Agreement on the first $r$ symbols defines the variation ${\mathrm{var}}_{r}U$; summable variations control sensitivity to a remote history. Geometric potentials encode interval contraction, while other potentials can assign different statistical weights to the same symbolic dynamics.