Corollary: All-profile criterion

Consider all real normalized compact smooth profiles. Under a common certificate:

1. raw convergence for every such profile implies $H_{\lambda ,J_{\lambda }}[\Phi ]=0$ for every $\lambda <0$ and every $\Phi$;

2. it also implies $H_{0,j}[\Phi ]=0$ for $j<J_0$;

3. given these conditions, profile independence of the raw limit holds exactly when $H_{0,J_0}[\Phi ]$ is a constant distribution in the translation variable, for every $\Phi$.

The statements about degree zero are omitted when that degree is absent.

Proof. A distribution vanishing on all normalized compact tests is zero: differences of normalized tests span the zero-integral test subspace, and adjoining one normalized test spans the full test space. Apply this observation to the coefficients whose vanishing is required by raw convergence.

For profile independence, a distribution has equal values on all normalized compact tests exactly when it annihilates all zero-integral tests. It is then a constant multiple of the integration functional.