Proposition: Regulator smoothing after construction

Let ${\rho }_{ϵ}$ be a real compact smooth approximate identity with integral one, support in a Euclidean ball of radius $ϵ$, and uniformly bounded $L^1$-norm. For a fixed compactly supported test $F$ whose support avoids the origin,

For sufficiently small $ϵ$, the contact contribution vanishes exactly, uniformly in $c_{\mathsf{P}}$. On a fixed support stage the remaining error is bounded by

where $C$ depends on a reference prescription and the uniform mollifier bound, not on $c_{\mathsf{P}}$.

Proof. Pair convolution with the reflected test convolution. The mean-value theorem bounds its difference from $F$, through four derivatives, by a constant times $ϵp_5(F)$. Apply (eq:reference-estimate) on a common enlarged compact.

The difference between any prescription and the reference is $\hslash c_{\mathsf{P}}{\delta }_0$. After convolution it is supported in the radius-$ϵ$ ball, which is disjoint from $\mathrm{supp}F$ for sufficiently small $ϵ$. Its pairing is therefore exactly zero.