The necessary debugging distinction

The notation $L={\mathsf{E}}_{0}C$, the sharp adjoint identity, and an upper causal support bound do not, by themselves, specify the action of ${\mathsf{E}}_0^{\ast }$ on arbitrary compact sources.

In particular, the statement that advanced propagation is supported in a ball is an upper-support statement. It does not say whether propagation has a tail inside the ball, is supported on its boundary, or has a nonzero coefficient at a particular point.

We therefore distinguish:

• results following from the explicitly copied inputs;

• a sufficient boundary-trace certificate for the inherited operator;

• a fully specified flat realization for which that certificate, the support alternative, and the bridge are proved.

This separation is essential to avoid changing the fixed functionals while ostensibly keeping their names.