Exact $1/4$ mean-rung population and $\sqrt{2}$ continuum revival-time ratio in the anti-Jaynes--Cummings vertex at $f_\star=\sqrt{\bar n+1}$

At atomic resonance the Jaynes--Cummings (JC) vertex of a single bosonic mode coupled to a two-level emitter is blind to $f=ω/λ$ at zero detuning and Kerr; the anti-Jaynes--Cummings (AJC) vertex is not. We show that the mean-rung excited population of the AJC block equals exactly $1/4$ at $f_\star=\