Proposition: Smooth soft-density control

On a compact parameter set with $p$ bounded away from zero and one, define

Then $r_{\ell }$ has a positive minimum, and

Moreover,

The ramp observables in Theorem thm:accuracystability are Lipschitz in $(d_e,f)$ on such compact sets, with constants obtained from (eq:parameterstability), (eq:surrogatestability), and (eq:softderivative).

Proof. Implicit differentiation of (eq:softcount) gives the formula for $q_{\ell }^{\prime }$. Each summand $s(1-s)$ is positive, and continuity on a compact interval gives a positive minimum for $r_{\ell }$. Differentiating the sigmoid and using $s(1-s)\le 1/4$ proves (eq:softderivative). The stability estimates then give the stated Lipschitz bound.