Monotonicity and bounds for the zero count before maximal volume

For each parameter $a>0$ or $b>0$, let $\mathcal{C}(a)$ or $\mathcal{C}(b)$ denote the number of zeroes of $v_0$ before the maximal-volume orbit of $\Psi_a$ or $\Psi_b$, respectively.

Zero-count conjecture. The count satisfies:

1. $\mathcal{C}(a)$ and $\mathcal{C}(b)$ are decreasing in $a$ and $b$, respectively.
2. For $a>0$ sufficiently small, $\mathcal{C}(a)=1$; for $b>0$ sufficiently small, $\mathcal{C}(b)=2$.
3. $\mathcal{C}(a)=0$ for all $a\geq\sqrt{3}$, and $\mathcal{C}(b)=1$ for all $b\geq1$.

These parameter-counting assertions are numerical conjectures intended to control when the Doubling Lemma applies; their analytic verification remains open.