Remark: Motivation for Input (S-bv)

XVI-K quotes $D_F=-\mathcal{P}(h_0+i{E}_0^{\mathrm{A}})$ and states “with $h_0=-W_0$ (XI-E.tex, eq:h)”. XI-E, eq:h, reads $h_{\mu }=-W_{\mu }$, $h_{\mu }-{h_{\mu }}^{\mathsf{T}}=i{\delta }_{\mu }$. Hence $S=h_0+i{E}_0^{\mathrm{A}}=-W_0+i{E}_0^{\mathrm{A}}$, and XVI-J, lem:identify, identifies this with $-{{\Delta }_F}^{\mathrm{FR}}$. A Euclidean calculation motivates the candidate form. Substituting $x^0-y^0=-i{\tau }_{\mathrm{E}}$ (${\tau }_{\mathrm{E}}>0$) into the $W_0$ integral and using ${{\int }_0}^{\infty }{e}^{-p{\tau }_{\mathrm{E}}}\sin (pr)dp=r/({{\tau }_{\mathrm{E}}}^2+r^2)$ (checked symbolically in kernel_relations.py) gives $1/(4{\pi }^2({{\tau }_{\mathrm{E}}}^2+r^2))$, which suggests the motivational value $c=1/(4{\pi }^2)$. {This Euclidean calculation motivates the candidate; it does not certify the real-time identification of $S$ on $\Omega$ as a single boundary value.} The certification is the real-time argument of Theorem thm:wpA-Sbv, which uses the predecessor Fourier family of $W_0$ and the step representation of ${E_0}^{\mathrm{A}}$, not this Euclidean slice. XVI-K states: “No explicit position-space formula for $S$ is used.” {Boundary value versus $\epsilon$.} The symbol $i0$ denotes a distributional boundary value. It is unrelated to the fixed preparation parameter $\epsilon$. The series rule “no $\epsilon \to 0$” (XVI-I, prop:signs(5): “There is no $i\epsilon$ prescription and no limit $\epsilon \to 0$”) does not forbid the $i0$ notation, and Input (S-bv) introduces no prescription and changes no definition.