Principal-value reference

For an unstable extension let $G_{\mathrm{PV}}$ denote the Euclidean resolvent integral with principal values at its negative transverse poles. It is a useful reference distribution, not a positive spectral ground-state covariance. The finite-rank correction to the Friedrichs kernel is locally analytic at regular diagonal pairs, because its high-momentum part is exponentially decaying and its finite poles are treated as principal-value distributions.