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 $Q_8 \subset 2I \subset \mathrm{SU}(2)$. 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.