The A-Philosophy dynamic Riemann Hypothesis

For $N\in\mathbb{N}$, let $\mathcal{Z}_N$ be the parameter space of functions

$$Z_N(t;\overline{a})=Z_0(t)+\sum_{k=1}^{N}\frac{a_k}{\sqrt{k+1}}\cos\bigl(\theta(t)-\ln(k+1)t\bigr),$$

with $\overline{a}\in\mathbb{R}^N$. A path $\gamma(r)$ in this space starts at the core function $Z_0(t)$ when $r=0$ and ends at $Z_N(t)$ when $r=1$; the consecutive zeros $t_n(r)$ and $t_{n+1}(r)$ are non-colliding if they do not collide along the path. A-Philosophy dynamic RH. For every $n\in\mathbb{N}$, there exists a path $\gamma(r)$ in $\mathcal{Z}_{\left[\frac{t_n}{2}\right]}$ with $\gamma(0)=Z_0(t)$ and $\gamma(1)=Z_N(t)$ that is non-colliding for the consecutive pair of zeros $t_n(r)$ and $t_{n+1}(r)$ for all $r\in[0,1]$. This reframes the reality of the zeros as a path-connectedness and non-collision problem in the parameter space, but the source gives no resolution.