Injectivity of the origin tangent angle mapping $\mathcal{M}: \mathcal{Z} \to [0, \pi)$ for non-trivial zeta zeros under prime log independence

In the working paper "Exact Uniqueness of the Pair $(\delta, t)$ for the Origin Tangent Angle $\alpha$ of Zeta Zeros", I am investigating the analytic properties of the geometric orientation of the tangent vector at $s=0$ for the non-trivial zero locus $\mathcal{Z} = \{s^* = 1/2 + \delta + i t \in \mathbb{C} : \zeta(s^*) = 0, 0 < \operatorname{Re}(s^*) < 1\}$. By evaluating the exact derivative $\zeta'(1-s^*)$ via Dirichlet series alongside the asymmetric functional equation factor $\chi(s) = 2^s \pi^{s-1} \sin\left(\frac{\pi s}{2}\right) \Gamma(1-s)$, one can construct an explicit closed-form expression for the origin tangent angle $\alpha(\delta, t) \in [0, \pi) \cong \mathbb{R}/\pi\mathbb{Z}$:$\alpha(\delta, t) = \frac{3\pi}{2} + \arg\left(\chi\left(\frac{1}{2} + \delta + i t\right)\right) + \arctan\left( \frac{\sum_{n=1}^\infty n^{\delta - \frac{1}{2}} \ln n \sin(t \ln n)}{\sum_{n=1}^\infty n^{\delta - \frac{1}{2}} \ln n \cos(t \ln n)} \right) \pmod{\pi}$Let $\mathcal{M}: \mathcal{Z} \to [0, \pi)$ be the evaluation map $(\delta, t) \mapsto \alpha(\delta, t)$. I am attempting to rigorously establish the global injectivity of $\mathcal{M}$ over the discrete set of non-trivial zeros $\mathcal{Z}$. My Argument for Injectivity:

1. Transcendental Phase Drift: The factor $\arg(\chi(1/2 + \delta + i t))$ depends monotonically on $t$ and varies continuously with $\delta$ via the digamma function $\psi(z) = \Gamma'(z)/\Gamma(z)$.
2. Linear Independence of Prime Logarithms: The inner arctangent quotient $Q(\delta, t)$ is driven by Dirichlet series with weights $\frac{\ln n}{n^{1/2 - \delta}}$. By Baker's theorem on linear forms in logarithms, the set $\{\ln p_k\}_{k=1}^\infty$ is linearly independent over $\mathbb{Q}$.
3. This linear independence prevents any two distinct zero coordinate pairs $(\delta_1, t_1) \neq (\delta_2, t_2)$ in the discrete spectrum $\mathcal{Z}$ from generating identical global phases mod $\pi$, asserting that $\mathcal{M}$ is strictly one-to-one ($\mathcal{M}(P_1) = \mathcal{M}(P_2) \implies P_1 = P_2$).

My Questions:

1. Is the linear independence of $\{\ln p_k\}$ over $\mathbb{Q}$ strictly sufficient to guarantee the injectivity of the quotient $Q(\delta, t)$ across the discrete zero locus $\mathcal{Z}$, or could functional relations in $\zeta'(1-s^*)$ allow two distinct isolated zeros to map to the exact same angle $\alpha \pmod{\pi}$?
2. Are there known measure-theoretic or topological obstructions when mapping a discrete set $\mathcal{Z} \subset \mathbb{R}^2$ to a 1D phase space $[0, \pi)$ via such Dirichlet series quotients that could break local injectivity? I would appreciate any insights, potential counterexamples, or references regarding the injectivity of phase maps derived from logarithmic derivatives of $\zeta(s)$.