Theorem: Positive-scale audit identity

For every $\delta >0$ and every compact joint test,

Each entry is exactly

with ${\mathsf{I}}_j={\mathsf{I}}_{\mathrm{RR}}$ for the first three rows and ${\mathsf{I}}_j={\mathsf{I}}_{\mathrm{resp}}$ for the four response rows.

The family exists at positive scale, is strongly smooth on compact positive-scale intervals, and depends only on the actual reconstructed internal kernel.

Proof. All smoothed crossed components are smooth. The mixed matter derivative and the insertion-kernel theorem therefore give legitimate internal test pairings. The vertex transposition gives (eq:Wk)–(eq:Wmass), and linearity gives (eq:ledgeridentity). The seed theorem gives (eq:ledgerseed).

The normalization is that of Section sec:normalization: one selected mixed species pair, the prefactors ${\lambda }^2$ and $\lambda$, and two $\hslash$-weighted lines.

Null reconstructions pair to zero with every smooth test, so all reconstruction identities are respected simultaneously. On a compact positive-scale interval, the smoothing profiles and their scale derivatives have common compact support and bounded smooth seminorms. The finite-order kernel estimates then prove strong smoothness.