Proposition: Continuous faithful bulk inclusion

The inclusion

is injective. On each fixed-degree, fixed-support, fixed-wavefront bulk stage with its normal topology, it is continuous.

All boundary-jet coordinates of a bulk polynomial vanish. The topology induced on the bulk is its strong coefficient topology; it is not asserted to coincide with the finer normal topology.

Proof. Injectivity follows because both spaces consist of the same actual functionals with unique kernels.

Lemma lem:bulkderivative puts every relevant first metric derivative in ${\mathcal{M}}_1$. The map $\mathsf{b}$ vanishes there. Thus every boundary-jet coordinate of a bulk polynomial is zero.

The remaining topology on the bulk is the strong coefficient topology. The normal topology contains the strong distribution seminorms, so its inclusion into this topology is continuous. No converse continuity is needed or claimed.