Proposition: The g=0 prescription

If $g=0$, then ${\mathsf{V}}_{1,g}={\mathsf{V}}_{2,g}=0$ and ${\rho }_g[q;{\varphi }_0]={\varphi }_0$, while ${\mathsf{X}}_q$ is unchanged. Hence

which differs from $\int f{\varphi }_0$ in general: in the setting of Proposition prop:sourced the first coefficient is ${\mathsf{E}}_0{\partial }_t({\sigma }_{\epsilon }\psi )\ne 0$ for suitable $\psi$. The dressing survives at $g=0$; $\Phi \ne \varphi$. The declared free prescription is: harmonic-type dressing with respect to the edge-regulated density, no scalar correction. At $g=0$ the set $P_g$ is empty, so no plateau $U$ satisfies (eq:U) and the on-shell statement (v) is vacuous; the off-shell statements (iii)–(iv) remain valid for fields supported in any $U^{\prime }$ with $\overline{U^{\prime }}\subset \{\epsilon <t<T\}$.

Proof. Immediate from Lemma lem:Eexp, Definition def:rho, Proposition prop:sourced, and the proof of Theorem thm:main(iv).