Three distinct test certificates

A verification record must distinguish:

1. Fixed-test certificate: one ${\Phi }_{\ast }\in \mathcal{T}$, independent of scale, with a proved dominant asymptotic such as (eq:dominant).

2. Bounded-set certificate: a specified bounded set $B\subset \mathcal{T}$ and uniform control of ${\sup }_{\Phi \in B}|(t_{\delta }-T)(\Phi )|$, or a proved failure of that control.

3. Moving-witness calculation: a family ${\Phi }_{\lambda }$, with its support and derivative growth recorded, and the behavior of its pairings.

For example, if

then its derivatives have polynomial frequency growth. For nonzero $b$, differentiation in a direction pairing nontrivially with $\xi$ shows that this is not a bounded test family.

The distinction is not cosmetic. On $\mathbb{R}$, take $A={\partial }_x{\delta }_{\mathrm{D}}$, and $\psi =1$ near zero. The constant family $t_{\delta }=A$ converges strongly, but

Thus growth on an unbounded moving test family can coexist with strong convergence of the distributional family.

Witness identities can check signs and selected coefficients. A specified partial contraction is not automatically injective on kernels: already a single nonzero test functional on ${C_c}^{\infty }$ has a nontrivial kernel. Adding a tensor product with a function in that kernel preserves the partial evaluation while changing the full kernel.