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

Let ₐ, Vq V_ 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: involution equivariance (-), quadratic compatibility with the pair observable ₏₀₈ₑ, 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\, Ntrl. The quaternionic structure is not a choice but the unique fixed point of the three constraints. This upgrades the result of O23~Beau2026a27---which established ₑ (Im\, H) = 3---from a dimension count to a categorical universality statement: su (2) is the unique minimal admissible target through which every ₐ, must factor. Combined with the chain c_ ₏₀₈ₑ ^* of O24~Beau2026a28, this gives ^* a representation-theoretic meaning as the effective scaling exponent of norm growth in the minimal admissible non-abelian sector. The canonical construction ₐ, = is exhibited explicitly and proved unique up to unitary equivalence.