Almost-additive partition-function limit ($|a_{n+m}-a_n-a_m|\le D⟹\lim a_n/n\text{ exists}$)

The sequences $a_n+D$ and $a_n-D$ are respectively subadditive and superadditive. Apply Fekete's lemma to the first and its negative to the second; their normalized difference is $2D/n$, so their limits coincide. For full-shift weighted orbit sums, summable variations of the potential bound the energy change at a concatenation boundary, giving this estimate for the logarithm of the partition function. A fixed initial symbol or fixed remote tail changes only a bounded normalization and not the limit.