Quadratic Completion of Admissible Spectral Pairs via Binary-Icosahedral Representation

Abstract

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.