The exact mask family

Let ${\Pi }_M=I_{\{{\mathsf{N}}_g+{\mathsf{N}}_{\varphi }\le M\}}$ and $Q_{\varphi ,M}=I_{\{{\mathsf{N}}_{\varphi }\le M\}}$. Writing

define

It is bounded self-adjoint for finite $R,M$. The scalar free term has its own number cutoff. Replacing it by a total-number compression would introduce a false gravitational dependence at $\lambda =0$.