Proposition: Positive scale and intrinsicness

For every $\delta >0$, $t_{\delta }$ is defined on all of $\mathcal{T}$. It is strongly smooth on compact positive-scale intervals.

It depends on the actual internal distribution, not on a presentation. In particular, reconstruction-null presentations give the zero positive-scale family.

Proof. The convolution of the compact distribution $\chi C$ with a compact smooth profile is smooth. Formula (eq:vertextest) therefore pairs each internal distribution with a smooth compact test.

On a compact positive-scale interval, the profiles and all scale derivatives have common compact support and bounded test seminorms. Their convolutions have locally bounded smooth seminorms. Local finite-order estimates for $K$ justify differentiation, uniformly on bounded joint test sets.

All operations start from the actual $K$. A zero reconstructed distribution has zero pairing with every resulting test.