Proposition: Persistence of the normal-topology obstruction

There is no sequentially continuous extension of the unchanged regular scalar coefficient to the sharp pair in the normal kernel stages of Assumption ass:tensor, agreeing with that coefficient on regular kernels.

Consequently the VII-C algebra-extension requirements cannot all hold simultaneously: unchanged old-product agreement, exact sharp admission with the specified normal admission continuity, bounded-set hypocontinuous quantum multiplication, and continuous scalar coefficient extraction.

Proof. Choose the separated witness in Theorem thm:diagonal. Normal convergence of the two insertion kernels is part of the inherited VII-C input. Sequential continuity of a scalar extension would make its values on that convergent pair converge. Equation (eq:log) contradicts this.

For the algebra consequence, hypocontinuity gives separate continuity and continuity along convergent pairs of sequences: split a product difference into two terms and use uniformity over the bounded convergent family in the other factor. Apply this first with the fixed middle factor, then with the other convergent insertion. Continuous scalar extraction would force convergence of the same divergent coefficient.