Stability under the stated spatial regulator

Replace $\mathcal{N}$ by the Fourier multiplier

where the cutoff removes a neighborhood of $0$, has compact frequency support, and tends pointwise to $1$ as $ϵ\downarrow 0,\Lambda \uparrow \infty$. For the same representatives,

Their regulated gap is $-\lambda J_{ϵ,\Lambda }$, where

Indeed, the spatial Fourier transforms of $\xi$ have a uniform Schwartz bound on the compact time interval of $\mathrm{supp}f$; dominated convergence gives uniform convergence of the cutoff vectors there. The lower bound therefore tends to $|\lambda |J/2>0$.

This is a regularization of the declared auxiliary classical symbol. It is not a cone-wide inverse prescription, an operator regularization, or a field-measurement theorem.