Quaternionic Rigidity of Admissible Morphisms: Every Admissible Φq,ρ Necessarily Factors through su(2)

Let $\Phi_{q,\rho} \colon V_q \to V_\rho$ be a morphism from the Weil representation space of $\mathrm{Heis}_3(\mathbb{Z}/q\mathbb{Z})$ to a representation space of $2I \subset \mathrm{SU}(2)$, subject to three structural constraints imposed by the admissible projection $\Pi$: involution equivariance ($\chi \mapsto -\chi$), quadratic compatibility with the pair observable $\sigma_{\mathrm{pair}}$, and naturality with respect to admissibility. The main result of this paper is a rigidity theorem: any morphism satisfying these three constraints is forced to factor through the three-dimensional real Lie algebra $\mathfrak{su}(2)$, identified with the admissible neutral traceless sector $\mathrm{Im}\,\mathbb{H} = \mathrm{Im}\,\Pi \cap \mathrm{Ntrl}$. The quaternionic structure is not a choice but the unique fixed point of the three constraints. This upgrades the result of O23 —which established $\dim_{\mathbb{R}} (\mathrm{Im}\,\mathbb{H}) = 3$—from a dimension count to a categorical universality statement: $\mathfrak{su}(2)$ is the unique minimal admissible target through which every $\Phi_{q,\rho}$ must factor. This gives the pair-capacity exponent a representation-theoretic carrier in the minimal admissible non-abelian sector. It does not derive the separate capacity-to-rate map: that prescription combines a changing-degree LPS equation with a fixed-degree Heisenberg observable and has no native carrier on the latter substrate. The canonical construction $\Phi_{q,\rho} = \rho \circ \iota \circ \pi$ is exhibited explicitly and proved unique up to unitary equivalence.