Proposition: An infinite ambiguity at fixed normal order

On ${{\mathbb{R}}^d}_y\times {{\mathbb{R}}^{\mathbb{k}}}_n$, the distributions

have normal order zero along $\{n=0\}$, yet form an infinite-dimensional vector space. Even if their tangential support is restricted to one point, allowing arbitrary tangential distribution order remains infinite-dimensional.

Proof. Independent smooth functions $a$ give independent distributions, as follows by pairing with a test equal to one near $n=0$. For the point-supported claim, the distributions

are independent, as monomial tests with a cutoff isolate their coefficients.