Quantum Structure beyond Spin-1/2: Co-Admissible Sectors and the Emergence of SU(2) as a Fixed Point

The spectral admissibility programme establishes, on the binary icosahedral group 2I with canonical generating set S = 10a 10b, that the spin-12 and spin-32 sectors share the same Laplacian eigenvalue ₁/₂ = ₃/₂ = 18, hence the same admissibility window A^ = c_/18. Neither sector is preferred over the other on 2I: they are co-admissible. This paper derives the quantum-mechanical structure of the spin-32 sector from the admissibility constraints of the Foundation paper, following the same programme as the spin-12 companion paper. The central result is that the Born rule, singlet correlator, and Tsirelson bound can be derived for the spin-32 sector via the four-dimensional representation ₄ of 2I, with the Casimir operator providing the normalisation that replaces E (a, a) = -1 of the spin-12 case. The co-admissibility of the two sectors on 2I — and its lifting to strict spin-12 dominance in the LPS limit p — identifies SU (2) not as a postulate but as the unique stable sector under admissibility refinement: a fixed point of the co-admissible family under the Born--Infeld spectral filter. This replaces the rigid imposition of spin-12 by a structural selection mechanism, and provides a falsifiable prediction via the representation-theoretic dimension test r₄₅₅ = d².