Can failure of Weil positivity for the Riemann hypothesis always have a finite-dimensional certificate?

I have been investigating finite-dimensional approaches to positivity criteria equivalent to the Riemann hypothesis, in particular Weil's criterion.

Rather than assuming that such a criterion admits a finite reduction, I am trying to understand precisely what would be required to justify one.

Suppose the Weil quadratic form $Q$ is considered on its appropriate admissible test-function space. Failure of RH would imply the existence of an admissible $h$ for which

$$Q(h)<0.$$

My question is:

What is the strongest rigorous principle that converts failure of positivity in this infinite-dimensional test-function space into a finite-dimensional certificate of failure?

More specifically, if $Q(h)<0$ for some admissible $h$, can one conclude that there is a negative witness lying in some finite-dimensional subclass?

If so, can the required dimension (or an appropriate notion of rank) be bounded uniformly, independently of the particular negative witness?

I am particularly interested in the distinction between

$$ \forall h\text{ with }Q(h)<0,\qquad \exists N=N(h) $$

such that negativity is already detected in an $N$-dimensional approximation, and the much stronger possibility that there exists a universal $N_0<\infty$ such that

$$ Q(h)<0 \quad\Longrightarrow\quad \text{there is a negative certificate of dimension }\le N_0. $$

Under suitable continuity assumptions, density of an increasing family of finite-dimensional subspaces may give the former, but it does not by itself appear to give the latter.

Is there any structure specific to Weil's quadratic form that yields such a uniform finite-dimensional/rank reduction, or is there a known obstruction showing that no such uniform bound should be expected?

I would also be interested in references connecting this question with finite-rank, Hankel/Toeplitz, moment-problem, or extremal formulations of Weil positivity.