Theorem: Infinite observation-derivative orbit

On the supported observation space $X$, the distributions

are linearly independent over $\mathbb{C}$. No finite-dimensional subspace containing $r$ is invariant under $-{\partial }_t$.

Proof. Let

On $\mathcal{R}$,

The measure $r$ is nonzero by Theorem thm:diagonal.

Suppose a finite relation ${{\sum }_{j=0}}^N{c}_j{r}_j=0$ has $c_N\ne 0$. For $N\ge 1$, pair the relation with $F^N\Phi$. When $j<N$, every term in ${{\partial }_t}^j(F^N\Phi )$ still contains a factor $F$ and vanishes on $\mathcal{R}$. At order $N$, its restriction is

The relation therefore gives

A nonnegative compact test positive near a point of $\mathcal{R}$ detects this nonzero measure, since $({\partial }_{t}F{)}^N$ has constant nonzero sign there. This is a contradiction. The $N=0$ case follows from $r\ne 0$.

An invariant finite-dimensional space containing $r$ would contain every $r_n$, contradicting independence.