L.4 Coherent extension preserving XVI-K

In particular, the XVI-K defects of XVI-K, Lemma lem:defect, are recovered on ${\mathcal{V}}_{\mathrm{K}}$ by Proposition prop:RL(3): $D_{\alpha }=D_{\alpha ,S^2}$ and $D_{{\gamma }^{\prime }\alpha }$ of that lemma equal the corresponding $D_{\alpha ,u}$ for $u=S^2$ and $u=S{\partial }_{\alpha }S$.

ledger (finite). The differential relations inside ${\mathcal{V}}_{\mathrm{adm}}$ that can produce a nonzero $D_{\alpha ,u}$ are the Leibniz identities among products ${\partial }^{A}S\cdot {\partial }^{B}S$ whose total numerical order $n=|A|+|B|$ satisfies $n\le 3$ (so that the image of numerical order $n+1$ still lies in the declared span with $N\le 4$). Equivalently, in scaling-degree notation, $u\in {\mathcal{H}}_k$ with $k=4+n$ and $4\le k\le 7$. The typed sources are

(factor orders), a finite list; each yields finitely many multi-index directions $\alpha$ and finitely many contact coefficients with defect order $|\gamma |\le n+1\le 4$ (numerical orders $1$–$4$ of the image, since $\mathrm{sd}(D_{\alpha ,u})\le k+1=5+n$ forces $|\gamma |\le (k+1)-4=n+1$). No value of any $c_{\alpha ,u,\gamma }$ is asserted; the ledger records existence and finiteness only. Category (4) of Section sec:l5 treats these defects as fixed differential-normalization data, not as free counterterms.

-properties. As for ${\mathfrak{R}}_{\mathrm{K}}$: ${\mathfrak{R}}_{\mathrm{L}}$ is not claimed Lorentz covariant; no Leibniz rule is claimed beyond the controlled defects of Lemma lem:Ldefect; ${\mathfrak{R}}_{\mathrm{L}}$ is not claimed to commute with $\mathcal{P}$; no wavefront set of ${\mathfrak{R}}_{\mathrm{L}}({q^{\circ }}_{\lambda })$ at $z=0$ is asserted beyond point support.

extended kernel. With $\Phi (y,w)=y-w$, set

On $y\ne w$ this restricts to (eq:QL2). Root attachment is deferred to L.7; no claim of L-A is made here.