Every string has probabilistic automatic complexity at most three

Gill (arXiv:2402.13376) introduced the probabilistic automatic complexity $A_P(w)$ of a finite string $w$: the least number of states of a probabilistic finite automaton (PFA) for which $w$ is the unique most probably accepted string of its length.

He asked whether $A_P$ is unbounded, noting that no string with $A_P > 3$ was known (Question 4.14 of that paper).

We answer the question by proving that $A_P(w)\le 3$ for every string $w$ over every finite alphabet.

The witnessing three-state automaton is explicit: its reduced dynamics tracks the pair $(u,u^2)$, where $u$ is the reversed base-$b$ value of the input, and its acceptance functional is a downward parabola peaked at the value of the target string.

Combined with Gill's classification of the binary strings with $A_P=2$, this completely determines $A_P$ on binary strings.