Theorem: Sharpened non-tip expansions

For a fixed compact non-tip localization of either $47$ or $5+$, there are distributions such that

with

The singular coefficients are supported on the corresponding localized seed incidence intersected with $\Sigma$. Both inverse-cubic coefficients vanish:

Inverse-quartic and inverse-quintic terms are absent already by derivative placement.

Proof. Localize the complete joint test before the seed operator acts; all localization derivatives are retained. Expand by the finite Leibniz rule, transfer crossed derivatives to homogeneous derivative profiles, and use the exact seed measures. The residual variables are compact smooth spectators. Lemma lem:placement gives $q\le 3$. Theorem thm:scalar with $N=q$ yields possible inverse powers through three and the stated logarithmic and finite terms.

At inverse-cubic order, every crossed derivative contributes a longitudinal $\ell$-factor in the log-profile moment. In $47$, all three printed operator terms contain the input contraction ${\mathsf{A}}^{\sigma \nu ,I}{\ell }_{\sigma }(e^I{)}^{\kappa \gamma }{\ell }_{\kappa }$. In $5+$, it is ${\mathsf{A}}^{\alpha \beta ,I}{\ell }_{\alpha }(e^I{)}^{\kappa \nu }{\ell }_{\kappa }$. Both vanish by (eq:Bkin-null). The mass vertex has crossed order at most two and cannot contribute at inverse-cubic order.

The scalar bound costs at most $36$ amplitude derivatives. At most five pre-existing test derivatives occur. Order $44$ is conservative. Locality sets $v=\xi =0$, while the seed sets $u=0$. No condition $r=0$ follows.