Proposition: Dini certificate with an anchor

Let $u_{\delta },S_{\delta }\in {\mathcal{D}}^{\prime }(X)$ have fixed-test pairings locally absolutely continuous in $\log \delta$. Suppose

and

Then $u_{\delta }-S_{\delta }$ has a strong distributional limit $T$, with

Proof. Absolute continuity gives the scalar integral identity between two positive scales. The bound makes every pairing Cauchy. A fixed positive-scale anchor and its finite-order estimate, combined with the integrable derivative bound, show that the limit is a distribution. Uniformity on bounded test sets proves strong convergence and the stated rate.