Why this is not admission into XI-B's algebra

The proposition is about actual off-shell representatives. It does not prove that every possible on-shell cubic symbol has no alternative finite-generator representation. Such an assertion would require a separate argument.

It does prove that the production plan cannot use an off-shell kernel bound as a substitute for a supplied admission map. XI-B has neither a topology for taking these kernel limits nor a declared Wick-polynomial extension. No algebraic limit operation may be inferred from its CCR.