We construct a $P(\phi)_2$ Gibbs state on infinite volume periodic surfaces (namely, with discrete ``time translations'') by analogy with 1-dimensional spin chains and establish the mass gap for our Gibbs state, there are no phase transitions. We also derive asymptotic properties of the $P(\phi)_2$ partition function on certain towers of cyclic covers of large degrees that converge to the periodic surface in some appropriate sense. This gives the first construction of an interacting Quantum Field Theory on surfaces of infinite genus with a mass gap. The main ingredient in our approach is to reconcile the so-called $P(\phi)_2$ model from classical constructive quantum field theory (CQFT) with Riemannian version of the axioms proposed by G. Segal in the 90's. We show the $P(\phi)_2$ model satisfies these axioms, appropriately adjusted. One key ingredient in our proof is to use what we call ``the Bayes principle'' of conditional probabilities in the infinite dimensional setting. We also give a precise statement and full proof of the locality of the $P(\phi)_2$ interaction.