Proposition: An infinite derivative orbit can share one parameter

The distributions ${{\delta }_0}^{(n)}$, $n\ge 0$, on $\mathbb{R}$ are linearly independent. Nevertheless, if a parent prescription changes by $u\mapsto u+c{\delta }_0$ and derivative compatibility is imposed, then

uses the same parameter $c$ for every $n$.

Proof. In a nontrivial finite relation, let $N$ be the largest index with coefficient $b_N\ne 0$. Test with a function equal to $x^N$ near zero. Lower derivatives vanish there while the $N$th derivative is $N!$, a contradiction. The parameter statement follows by differentiating the single parent change.