Proposition: Integrated-vertex asymptotics

On the compact outer null cone,

For the uses below, the remainders and their direct scale derivatives are controlled by $\Vert \Phi {\Vert }_{17,H}$. The $D^{2}M$ estimate remains valid after two $r$ derivatives.

In particular, $D^{3}M$ has no inverse-quartic term.

Proof. The first three statements follow from Lemma lem:master. Total $S$ differentiation splits into propagation and weight differentiation. The only inverse-cubic terms in $D^{2}M$ are

They give $P_2$. Parity removes the nominal second-order master coefficient, so $DM=O({\delta }^{-1})$.

For $D^{3}M$, use Theorem thm:first-correction. At zero,

Thus, for frozen weights,

For a spatial propagation derivative, define

In layer coordinates, at fixed unscaled time,

Therefore

For frozen $w$,

Summing (eq:fourth-time) and (eq:mixed-fourth) with weights ${\partial }_{x^{\rho }}\varphi$ gives

The kinetic vertex contributes minus one half of this.

When one of the three total time derivatives falls on the weight, the coefficient is $-3\kappa \mathcal{G}[D_{\mathrm{w}}{g}_{\varphi }]/2$. Terms with two or three weight derivatives are lower order. The mass vertex contributes $\kappa \mathcal{G}[\varphi ]/2$ at inverse-cubic order. These are exactly (eq:P3).

The first-correction estimate uses fourteen weight derivatives; the output derivative raises this to fifteen. Allowing two incidence spectators gives seventeen. The remaining leading-master operations require no more: the interior estimates below use at most eleven. Repeating the calculation with ${\rho }^{[1]}$ gives the scale-leading coefficients $-3P_2$ and $-3P_3$, and the same remainder bounds.