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 under the parity involution $c \\leftrightarrow q-c$, the candidate fibre structure of the pair sector (a structurally motivated hypothesis; O18 states the fibre identification as an open problem), 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 $Q_8 \\subset 2I \\subset \\mathrm{SU}(2)$. Under the involution-equivariance condition of that embedding, every Level III outer product is a symmetric square $uu^{\\mathsf T}$, so the trajectory spans at most $\\mathrm{Sym}^2(V_\\rho)$, of dimension $3$ when $\\dim_{\\mathbb{C}} V_\\rho = 2$: the rank-four target in $\\mathrm{End}(V_\\rho)$ is excluded under that contract, without refuting other embeddings or constructions. The conjugate-pair covariance computable from O25 data is formed in the measured carrier $H_{\\mathrm{eff}}$ rather than in $\\mathrm{End}(V_\\rho)$; it yields finite-data diagnostics compatible with a supplied adjoint carrier and does not identify a representation sector.