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

Let $\Phi_{q,\rho}$ be a morphism out of the Weil pair space $V_q$ of $\mathrm{Heis}_3(\mathbb{Z}/q\mathbb{Z})$, subject to three constraints imposed by the admissible projection $\Pi_{\mathrm{adm}}$: involution equivariance, quadratic compatibility with $\sigma_{\mathrm{pair}}$, and factorisation through the admissible quotient. The third gives unique real-linear factorisation through $\mathcal{Q} = V_q / \ker \Pi_{\mathrm{adm}}$ and nothing more; a countermodel shows that the three together impose no equivariance. Write $\mathrm{Ntrl}$ for the neutral traceless part of the ambient observable space. Reaching $H_{\mathrm{eff}} = \mathrm{Im}\,\Pi_{\mathrm{adm}} \cap \mathrm{Ntrl}$ from that quotient and transferring O23's dimension count to it are one hypothesis, $\mathrm{Im}\,\Pi_{\mathrm{adm}} = \mathrm{Ntrl}$, assumed and derived nowhere. Under it and the spinor carrier supplied conditionally by paper O23, $H_{\mathrm{eff}} \simeq \mathfrak{su}(2)$ and the representation action gives a canonical equivariant map $\mathfrak{su}(2) \to \mathrm{End}(V_\rho)$ 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 $\mathfrak{su}(2)$-equivariant map reaches the fundamental representation from the adjoint, $\dim \mathrm{Hom}_{\mathfrak{su}(2)}(\mathbf{3},\mathbf{2}) = 0$, 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 $\mathfrak{su}(2)$ identification remains supplied rather than derived. A second assumption makes the parity involution descend and act by $\mathrm{Ad}(g_0)$. Whether the canonical map satisfies the quadratic constraint is not settled: no result here makes the quotient map an isometry. Subject to it, $\delta_{\mathrm{pair}}$ would acquire a representation carrier in that sector; it does not acquire a derived interpretation of $\beta^*$, 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.