Proposition: The retained continuity obstruction

The construction does not remove the VII-C incompatibility under its original simultaneous requirements: unchanged old-product agreement, sharp admission with specified normal admission continuity, bounded-set hypocontinuous quantum multiplication, and continuous scalar extraction.

In particular, it does not imply $B_{f,\epsilon }\to B_f$ in ${\tau }_{\Sigma }$.

Proof. The unchanged regular triangle has a positive logarithmic coefficient on a separated detecting test by the inherited VII-C record VIIC. Under those simultaneous normal-continuity requirements, the converging regular input kernels and hypocontinuous multiplication would force convergence of that extracted scalar sequence, contradicting its logarithm.

The present product asserts no such continuity. Its old coefficients are unchanged, while its sharp scalar assignment is $U_c$. The carrier separator already excludes the asserted unchanged convergence in ${\tau }_{\Sigma }$. No proof above alters either fact.