Theorem: Sharp-width convergence in one fixed test space

For each fixed input satisfying (eq:separation),

For every $N\ge 0$, there is a constant $A_N$ and a sufficiently small positive width bound such that

Consequently,

The limiting tests are independent of the choice of fixed ramp profile with the stated endpoints.

Proof. Proposition prop:propagation controls every coordinate derivative on a fixed neighbourhood of $K$. The first bare test is a finite derivative of $f{w}_{\epsilon }$. For the second, write

Uniform boundedness of the coordinate derivatives and their $O(\epsilon )$ differences gives the same rate for every derivative of $B_{\epsilon }-B_0$. The Leibniz rule applied to (eq:bare) proves the bare-test estimates, with support in $K$.

Lemma lem:reduction proves the completed-test estimates and their support in $C$. The inclusion ${\mathcal{D}}_C\to \mathcal{D}$ and quotient map $\mathcal{D}\to Q_m$ are continuous.

The limiting data (eq:velocities) do not depend on interior profile moments. Cauchy uniqueness in (eq:limit-cauchy), followed by (eq:limiting-tests), proves profile independence.