Proposition: Unrestricted tangential weights

If arbitrary real smooth weights were admitted, the space

would be infinite-dimensional, even among first–third exchange-symmetric weights. This proposition does not declare those weights admissible counterterms.

Proof. Consider $a_n=(t+t^{\prime }{)}^n$, which are exchange symmetric. If $P(t+t^{\prime })r=0$ for a polynomial $P$, positivity of the parameter measure implies that $P(t+t^{\prime })$ vanishes throughout every open parameter patch of $Z_U$. There is such a patch by the explicit interior witness above, and $t+t^{\prime }$ varies over an open interval on that patch. A polynomial vanishing on an open interval is identically zero. Thus the weighted distributions are linearly independent.