Proposition: Jet bounds for unrooted contact

After the contact operation underlying (eq:L-deltaQ), writing ${{\partial }_w}^{\gamma }$ for the derivatives that hit ${a^{\mu \nu }}_{\lambda }$ upon pairing against $({\partial }^{\gamma }\delta )(y-w)$ (as in XVI-K, Lemma lem:amb):

1. Residual metric. Every residual-metric jet produced has order at most

   $$(2-v)+|\gamma |\le (2-v)+N_{\lambda }=(2-v)+(s+v)=2+s\le 4,$$

   since $s\le 2$ by Theorem thm:budget. The arithmetic upper bound reaches $2+s$ when all $N$ derivatives hit the residual-metric coefficient; actual attainment is not established.

2. Preparation. The preparation jet remains of order $2-s$ (equivalently $2,1,0$ by sector); ${\partial }_w$ does not raise it.

3. Switch. Derivatives ${\partial }^{{\gamma }^{\prime }}{\chi }_{\mathrm{I}}$ appear with $|{\gamma }^{\prime }|\le |\gamma |\le N_{\lambda }\le 4$.

These are statements about the pointwise jet of the coefficient in (eq:L-deltaQ), not about a distributional order after root attachment.

Proof. (1) Leibniz: ${{\partial }_w}^{\gamma }$ distributes over ${\chi }_{\mathrm{I}}(w)$ and ${L^{pq}}_{ij}(w;h)$. The portion hitting $L$ raises the residual-metric order from $2-v$ by at most $|\gamma |$. Then $|\gamma |\le N_{\lambda }=s+v$ and $s\le 2$ give (eq:L-resjet). (2) $P_A(y)$ depends on $y$ only (Definition def:La); ${\partial }_w$ does not hit it. (3) Immediate from Leibniz on ${\chi }_{\mathrm{I}}(w)$ and $N_{\lambda }\le 4$ (Proposition prop:Lsd).