Theorem: Explicit complete smooth-data expansion

Status: Proved mathematical consistency benchmark.

For every $N\ge 0$,

and on compact joint test supports

In particular the complete family has limit ${A_0}^{\zeta }$ with an order-two rate.

Proof. At small scale, $c_{\delta }(x)=\int \rho (s)C(x-\delta s)\mathrm{d}s$. Taylor-expand through degree $2N+1$, uniformly with crossed derivatives through order five. Evenness removes odd terms and gives the raw expansion with a $C^5$ remainder of order ${\delta }^{2N+2}$. The moments of $\kappa$ give the differentiated expansion and the same remainder order independently. Insert both into the finite exact adjoints. Their localized seed measures have finite mass, proving the test estimates.

Proof. Near the translation origin,

This follows by testing the convolution representation on profiles supported in that neighborhood. The smooth germ has derivatives

uniformly bounded by $\Vert \Phi {\Vert }_{5,H}$. Taylor expansion against $\rho$ and separately against $\kappa$ gives the same coefficients and errors.