Quadratic Completion of Admissible Spectral Pairs via Binary-Icosahedral Representation

We investigate the structural nature of the pair observable $\sigma_{\mathrm{pair}}(n) = \sigma_c(n)\,\sigma_{q-c}(n)$ arising in the Weil-block analysis of the Heisenberg Cayley graphs $\mathrm{Heis}_3(\mathbb{Z}/q\mathbb{Z})$. Building on the canonical construction of conjugate pairs enforced by the parity involution and the normalisation invariance established in O17–O19, we construct an explicit dictionary between conjugate Weil blocks and rank-one matrix coefficients in a representation space. We show that, in the pre-saturation regime, $\sigma_{\mathrm{pair}}(n)$ has the same growth exponent as the Hilbert–Schmidt norm of an associated matrix trajectory, establishing a Level I identification (proved). We then formulate a hierarchy of stronger identifications: a quotient identification modulo normalisation (Level II), and a canonical representation-theoretic identification (Level III), which is a theorem conditional on a single structural hypothesis (the admissible embedding $\Phi_{q,\rho}$) in an isotypic sector of the binary icosahedral group $2I$ along the admissibility thread $Q_8 \subset 2I \subset \mathrm{SU}(2)$. We provide concrete falsifiability tests based on the effective dimension of the trajectory in $\mathrm{End}(V_\rho)$, directly computable from O25 data. A positive result would identify $\sigma_{\mathrm{pair}}$ as the restriction of a canonical Hermitian quadratic form and provide a representation-theoretic interpretation of the pair observable itself. A negative result would still determine the correct ambient representation sector, refining the admissibility hierarchy without invalidating the dictionary.