IX-A-ST and its separator

The IX-A-ST record preserves a specific unchanged sharp insertion in a topology that detects its initial boundary charge IXAST,IXA. Its continuous separator distinguishes that insertion from the regular approximants whose corresponding boundary coordinates vanish.

The logical implication is elementary: if a continuous linear map $b_{\Sigma }$ satisfies

then $F_{\lambda }$ cannot converge to $F_{\mathrm{sh}}$ in that topology.

The present family is not identified with that pair. Its sharp linear background kernel is $\theta {\mathcal{C}}^{\ast }b$, while the unchanged multiplied-equation kernel carries the additional surface term in (eq:multiplied-kernel). Moreover, the theorem freezes the classical backgrounds and concerns smooth scalar output tests, not convergence of the full mixed metric–scalar functional kernels. No map identifying the IX-A-ST topology with ${\mathcal{D}}_C$ or $Q_m$ is asserted.