Action-level versus multiplied-equation regulation

The two regulated first coefficients

differ by ${\mathsf{X}}_1-{\tilde{\mathsf{X}}}_1=-{\mathsf{E}}_0({\sigma }_{\epsilon }^{\prime }{a_1}^{0\mu }[h])$, and their adjoint kernels differ by ${\sigma }_{\epsilon }^{\prime }\mathsf{B}({\mathsf{E}}_{\mathrm{A}}a)$ (Lemma lem:kernel). Both agree with the unregulated first coefficient on pure gauges supported in $\{t\ge \epsilon \}$. Only the first is in divergence form with a natural auxiliary density, which is what Lemma lem:plateaucoord uses at second order. The frozen prescription is the first.