A declared eliminated-cavity sensing model

During either sensing stage use the continuous measurement model

with record

Here $\mu$ has units ${\mathrm{m}}^{-4}{\mathrm{s}}^{-1}$, and $0\le {ϵ}_{\mathrm{m}}\le 1$ is monitoring efficiency. The dynamics includes the common mean sensor curvature, force noise, and gravity during sensing.

Equation (eq:SME) is a specified reduced sensing model, consistent in form with continuous squared-position monitoring. It is not a proof of cavity elimination for every apparatus. Assumption ass:admitted requires its instrument realization. The reference uses additional approximations to obtain its split filter Pino2018; we do not use those approximations to declare finite sensing impulsive.

For a duration $\tau$, define the recorded number

In the frozen-position efficient limit this gives $Z={\eta }_{\tau }{X}^2+N$, $N\sim N(0,1/4)$, and the earlier filter. For finite motion, it has the exact mean

The expectation is unconditional, so it uses the complete sensing instrument, not an accepted normalized branch.