Theorem: Complete diagnostic expansion

The difference of complete families satisfies

with coefficients (eq:diagnosticcoeff) and their evaluated moments (eq:diagnosticmoments). For every $L⋐U$,

at sufficiently small scales.

Every coefficient is supported in $S_{\kappa }$. In particular,

This leading coefficient is nonzero on every chart containing a nonempty portion of $S_{\kappa }$.

Proof. The profiles $P_j$ have a common compact $S,T$-support: $S$ lies in $\mathrm{supp}p$, and $S-T$ lies in the support of a derivative of $p$. Thus $T$ is bounded.

Apply Lemma lem:diagnosticTaylor to the $j$-term in (eq:diagnosticrescale), with $M=3-j$. Before multiplication, the error is

Multiplication by ${\delta }^{j-3}$ gives $O(\delta )$. Since $F_j$ contains $j$ ambient derivatives, the total test order is at most $8-j$, hence at most eight. For small scales the tangential support stays in a fixed compact set; this follows from compact test support, bounded profile support, and the positive bounded $B$-range. The estimate is therefore uniform for all tests supported in $L$.

Collecting $e=j-3+n$ gives (eq:diagnosticcoeff). All coefficient functionals use finite jets along $S_{\kappa }$, proving support there.

For $e=-3$, only $j=n=0$ occurs. Its moment is

This proves (eq:diagnosticleading). A compact normalized smooth profile cannot have derivative identically zero. A test with positive nonzero restriction to the attachment detects the coefficient.