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

Let Φq,ρ Vq → V_ρ be a morphism from the Weil representation space of Heis₃(Z/qZ) to a representation space of 2I ⊂ SU(2), subject to three structural constraints imposed by the admissible projection Π: involution equivariance (χ ↦ -χ), quadratic compatibility with the pair observable σₚₐᵢᵣ, 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 su(2), identified with the admissible neutral traceless sector Im\,H = Im\,Π ∩ Ntrl. The quaternionic structure is not a choice but the unique fixed point of the three constraints. This upgrades the result of O23 —which proves R (Im\,H) = 3 for the traceless sector of a supplied spinor carrier (O23 Theorem 3.1)—from a dimension count to a categorical universality statement: su(2) is the unique minimal admissible target through which every Φq,ρ must factor. The universality statement is conditional on that supplied carrier; the carrier selection and the observable identification remain open bridges of O23. 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 Φq,ρ = ρ ∘ ι ∘ π is exhibited explicitly and proved unique up to unitary equivalence.