Regulated generators and coherent correspondence

Define on $\mathrm{Dom}\mathsf{N}\subset {\mathcal{H}}_3$

The sector proof of Lemma lem:quadratic2 gives

It is symmetric. Expanding the normally ordered stress and the response formula shows on ${\mathcal{D}}_3$ that its matrix elements equal (eq:jform) with the cutoff scalar field. The factors two above count the two mixed polarized terms. Fubini is justified by the locally integrable transfer bounds.

Thus

is bounded self-adjoint, even though ${\Pi }_M$ need not have finite rank. The momentum-kernel estimate, not number compression alone, supplies boundedness.

Figure fig:model3-form-hierarchy distinguishes the unregulated response form from the regulated measured operator.