Exact $1\mathrm{/}4$ 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 $f=\omega \mathrm{/}\lambda$ 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 $1\mathrm{/}4$ at $f_{\star }=\sqrt{\bar{n}+1}$, while the JC population at the same point is $1\mathrm{/}2$. The Poisson-averaged packet value at $\bar{n}=16$ is $0.24650$, within $3.5\times {10}^{-3}$ of the exact value, with the deviation reproduced by $-\bar{n}\mathrm{/}[16(\bar{n}+1{)}^2]$. At $f_{\star }$ the modulation depth of the AJC inversion is $1\mathrm{/}2$ and the relative sensitivity of the continuum revival-time ratio $r=t_R^{\mathrm{A}\mathrm{J}\mathrm{C}}\mathrm{/}{t}_R^{\mathrm{J}\mathrm{C}}$ is maximal with value $1\mathrm{/}(2\sqrt{\bar{n}+1})$. Full diagonalization on $2(n_{\max }+1)$ states confirms the analytic values. The $1\mathrm{/}4$ value can be generated by Floquet-engineered counter-rotating coupling provided that the static transverse rate satisfies $g_0\ll g_1$. Parameter translations for an ideal trapped-ion blue sideband and a flux-tunable circuit-QED coupler are reported.