L.7 Root attachment, exact regressions, and propagated counterterms

The root kernel is the imported $B_{f,g}(y,x)$ of Hypothesis ${\mathrm{H}}_{\mathrm{root}}$ (XVI-K, Section sec:hyp), used only through the unchanged covector-shape hypotheses (R1)–(R3): existence of the composition; compact support of $B_{f,g}$; $\mathrm{WF}(B_{f,g})\subset Z_2\setminus 0$. No property of $B_{f,g}$ beyond (R1)–(R3) is used.

-attached distributions. With $\pi (y,x,w)=(y,x)$ and ${\Phi }_3(y,x,w)=y-w$, and with ${Q^{\mu \nu }}_{\mathrm{L}}$ the unrooted extended kernel (eq:QLext), define

and the smeared coefficient

(The same formulae with ${\mathfrak{R}}_{\mathrm{L}}$ replaced by ${\mathfrak{R}}_{\mathrm{L},\mu }$ define ${U^{{\mathfrak{R}}_{\mathrm{L},\mu }}}_{\mathrm{L}}$ and ${{\mathcal{C}}^{{\mathfrak{R}}_{\mathrm{L},\mu }}}_{\mathrm{L}}$.)