Actual functionals, not an undecorated symbol algebra

The notation $\mathsf{B}(f)$ may be used for the admitted element $B_f$, but it is not a new independent field. Its classical functional is exactly (eq:B).

Likewise $\mathsf{J}(a)$ is the actual coordinate functional (eq:Jcoord). It can have nontrivial relations among different source tests. Those relations are determined by equality of the compact kernels $q_a$, not by an arbitrarily chosen quotient.