Presentation defects before regulator removal

The identity separates:

1. the finite-cutoff comparison ${u_{\epsilon }}^{(P)}-P{u}_{\epsilon }$;

2. the residue comparison $r^{(P)}-Pr$;

3. the finite normalization left after both subtractions.

If both finite-cutoff and residue identities hold literally, then ${\mathcal{A}}_P=0$. Otherwise their failures must be computed in the specified presentation. A formal unregulated identity is not a substitute.

Transported differentiation is an immediate application: a fixed differential operator is continuous on distributions, so it transports an established finite part together with its residue. Cubic parenthesization is different. Intermediate multiplication of singular distributions is not automatically a defined continuous linear map $P$.

Tests on a product manifold must not be restricted to a selected family $f\otimes g\otimes f^{\prime }$ without a proved extension argument. Convergence on simple product tests alone is not the asserted distributional convergence on all compact joint tests. The correct organization of a future graph estimate may separate regular regions, transition regions, matching, and uniform control near an incidence intersection such as $\mathcal{R}\cap {\Delta }_{13}$. That organization does not supply those estimates.