Admissible Morphisms on the Weil Pair Sector: The su(2) Action on End(Vρ) and the Open Vector Lift

Let Φq,ρ be a morphism out of the Weil pair space Vq of Heis₃(Z/qZ), subject to three constraints imposed by the admissible projection Πadm: involution equivariance, quadratic compatibility with σₚₐᵢᵣ, and factorisation through the admissible quotient. The third gives unique real-linear factorisation through Q = Vq / Πadm and nothing more; a countermodel shows that the three together impose no equivariance. Write Ntrl for the neutral traceless part of the ambient observable space. Reaching Heff = Im\,Πadm ∩ Ntrl from that quotient and transferring O23's dimension count to it are one hypothesis, Im\,Πadm = Ntrl, assumed and derived nowhere. Under it and the spinor carrier supplied conditionally by paper O23, Heff su(2) and the representation action gives a canonical equivariant map su(2) → End(V_ρ) with one-dimensional complex intertwiner space, so an equivariant map is unique only up to a scalar that the quadratic constraint, a proportionality, does not fix. No non-zero su(2)-equivariant map reaches the fundamental representation from the adjoint, Homₛᵤ₍₂₎(3,2) = 0, excluding an equivariant adjoint-factorising realisation of O26 Hypothesis 4.4 but not every quotient-factorising linear lift; that hypothesis therefore remains open, and the su(2) identification remains supplied rather than derived. A second assumption makes the parity involution descend and act by Ad(g₀). Whether the canonical map satisfies the quadratic constraint is not settled: no result here makes the quotient map an isometry. Subject to it, δₚₐᵢᵣ would acquire a representation carrier in that sector; it does not acquire a derived interpretation of β^*, which has no native carrier on the Heisenberg substrate. Interpretation, not result: the non-abelian carrier is an input to this programme's representation layer, not an output of the constraints, which supply an exact calculus on it.