The declared gravitational contribution

For intrinsic mass measure ${\mu }_{\mathrm{m}}$ of total mass $M$, Gaussian smearing of Cartesian standard deviation $r_{\mathrm{c}}>0$ gives

We use the decoherence-only generator

Here $k_e$ is the component along the retained motion. This is the normalization corresponding to

The alternative adds $\lambda {\mathcal{G}}_{r_{\mathrm{c}}}$, $\lambda \ge 0$.

Since $|{\hat{\mu }}_{\mathrm{m}}|\le M$,

The kick generator is bounded on trace class, with induced norm at most twice that rate. Its exponential is the Poisson mixture of the normalized random unitary kick channel. Thus its decoherence dynamics is well defined independently of the unbounded Hamiltonian; adding it to an admitted baseline is the ordinary bounded perturbation problem.

Define

All kick moments needed here are finite for $r_{\mathrm{c}}>0$. The quadratic comparison is