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.
Jérôme Beau (Sun,) studied this question.