Pairwise Character Geometry and the Classification of Neutral Generators in SU(2)

We study the geometry of \emph{neutral generating sets}: symmetric generating sets $S = S^{-1}$ of a finite group $G$ equipped with an irreducible representation $\rho\colon G \to \mathrm{SU}(2)$ satisfying $\chi_\rho(s) = 0$ for all $s \in S$. The central result is that the geometry of the associated unit axes $\{\vec{u}_s\}_{s \in S} \subset S^2$ is entirely encoded in the pairwise character values: $G_{st} := \vec{u}_s \cdot \vec{u}_t = -\tfrac{1}{2}\chi_\rho(st)$.Using this identity together with the second-moment tensor $M = \sum_{s \in S^+} \vec{u}_s \vec{u}_s^T$, we give a complete character-theoretic classification of all admissible groups: $M \not\propto I$ $\Longrightarrow$ $\rho(G) \cong \mathrm{Dic}_n$, $n \ge 3$; $M \propto I$, $G_{st} = 0$ for all $s, t$ with $t \neq s^{\pm 1}$ $\Longrightarrow$ $\rho(G) \cong Q_8$; $M \propto I$, $G_{st} \notin \{0, \pm 1\}$ for some $s, t$ $\Longrightarrow$ $\rho(G) \in \{2O, 2I\}$. In particular, the binary tetrahedral group $2T$ (projective image $A_4$) admits no neutral generating set, a structural exclusion we prove directly. The finite subgroups of $\mathrm{SU}(2)$ compatible with spectral neutrality are exactly $Q_8$, $\mathrm{Dic}_n$ ($n \ge 3$), $2O$, and $2I$. This completes the Spectral Admissibility Sub-Programme of Cosmochrony.