Quadratic Completion of Admissible Spectral Pairs via Binary-Icosahedral Representation

We investigate the structural nature of the pair observable σₚₐᵢᵣ(n) = σc(n)\,σq-c(n) arising in the Weil-block analysis of the Heisenberg Cayley graphs Heis₃(Z/qZ). Building on the canonical construction of conjugate pairs under the parity involution c ↔ 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, σₚₐᵢᵣ(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 Φq,ρ) in an isotypic sector of the binary icosahedral group $2I$ along the admissibility thread Q₈ ⊂ 2I ⊂ SU(2). Under the involution-equivariance condition of that embedding, every Level III outer product is a symmetric square uuT, so the trajectory spans at most Sym²(V_ρ), of dimension $3$ when C V_ρ = 2: the rank-four target in End(V_ρ) 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 Heff rather than in End(V_ρ); it yields finite-data diagnostics compatible with a supplied adjoint carrier and does not identify a representation sector.