Quadratic Completion of Admissible Spectral Pairs via Binary-Icosahedral Representation

We investigate the structural nature of the pair observable ₏₀₈ₑ (n) = c (n) \, ₐ-₂ (n) arising in the Weil-block analysis of the Heisenberg Cayley graphs Heis₃ (Z/qZ). 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, ₏₀₈ₑ (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 ₐ, ) in an isotypic sector of the binary icosahedral group 2I along the admissibility thread Q₈ 2I SU (2). We provide concrete falsifiability tests based on the effective dimension of the trajectory in End (V_), directly computable from O25 data. A positive result would identify ₏₀₈ₑ as the restriction of a canonical Hermitian quadratic form and provide the first representation-theoretic interpretation of the admissible cascade exponent ^*. A negative result would still determine the correct ambient representation sector, refining the admissibility hierarchy without invalidating the dictionary.