The laboratory-frame anti-Jaynes–Cummings (AJC) interaction retains a residual detuning 2f\lambda that is absent from the rotating-frame model. We map two proposed remedies—a Kerr shift \chi(\hat a^\dagger\hat a)^2 and collective Dicke coupling
The laboratory-frame anti-Jaynes–Cummings (AJC) interaction retains a residual detuning 2f\lambda that is absent from the rotating-frame model. We map two proposed remedies—a Kerr shift \chi(\hat a^\dagger\hat a)^2 and collective Dicke coupling of N two-level emitters—for a squeezed vacuum on that ladder (r=1, \langle n\rangle=\sinh^2 r\simeq 1.38). All quoted contrasts are the amplitude of the first turning point of the atomic ground-state population, which coincides with the two-level formula C=1/[1+(2f+\chi)^2] at r=0 to 10^{-9}. Three results follow. (i)~A single Kerr strength never restores unit contrast at r=1; the r=0 n-dependent shift \chi(2n+1) cannot cancel 2f\lambda on every occupied Fock component. (ii)~The exact r=1 contrast at \chi=0 is not reproduced by an incoherent sum \sum_n P_n(r)\,C_n built from the r=0 two-level formula; pointwise deviations are several tenths. (iii)~Collective coupling raises the contrast systematically. At f=5 one finds C=0.140 (N=1) and C=0.575 (N=8), above the unsqueezed value 8/[8+(2f)^2]=0.074. The N=16 point at this f remains truncation-limited and is not quoted to three digits. The same N\sim(2f)^2 estimate for C=1/2 places trapped-ion values f\sim 10^{2}–10^{3} outside the present construction. The relevant platform is ultrastrong circuit QED with f\sim 1–10. The calculation is a numerical control landscape, not a new solvable limit.