The compress-with-another threshold of Szykuła's Figure 3 family

We give a self-contained pair-automaton proof of the exact compress-with-another threshold of the corrected Figure 3 family from a recent survey of open problems in synchronizing automata. For every $p\ge 3$, the automaton has $n=3p$ states and $\mu (q_0)=4p=4n\mathrm{/}3$. The word $(ba{)}^p(ab{)}^p$ attains this value. Two entrance potentials and an excluded region yield the lower bound, with the endpoint exception at $p=3$ treated explicitly. A separate reset construction proves $\mathrm{rt}(A_p)\le 3p^2+4p-1$ for every parameter. Reproducible computations verify the transition and entrance identities; the all-parameter results follow from the explicit proofs.