Lemma: Positive finite-sensing position transfer

If $\mu >0$, ${ϵ}_{\mathrm{m}}>0$, and $\tau >0$, then $r_x>0$. If ${ϵ}_{\mathrm{m}}=0$, then $r=0$.

Proof. The unconditional double commutator with $x^2$ is averaging over a classical white stochastic quadratic Hamiltonian. Its phase-space dynamics is a random linear symplectic evolution. The homogeneous response of $v_x(u)$ to an initial increment in $v_x$, with the other second moments unchanged, is $\mathbb{E}[T_{11}(u{)}^2]\ge 0$. This is also the $11$-entry of $e^{B_{\mathrm{s}}u}$, by the same closed moment equations. It equals one at $u=0$ and is continuous. Its integral is strictly positive. Multiplication by ${\eta }_{\tau }/\tau >0$ proves the claim. Zero monitoring efficiency makes ${\eta }_{\tau }=0$.