Universal Spin-j Singlet from Admissibility: Completion of the SU(2) Quantum Sector

The companion paper Q2 derives the singlet correlator E(â,b̂) = -5/4(â̂) for the spin-3/2 sector of the binary icosahedral group $2I$, leaving open the identification of the unique isotropic fibre state of χ₄ with the admissibility-selected proto-state. The present paper closes this step and extends the result to all admissible sectors of the chain Q₈ 2I SU(2). We prove that for any irreducible representation χ₂ⱼ₊₁ of $2I$, the phase-coherence constraint of Q1, Q2 and the Born--Infeld (BI) neutrality of the conjugate pair (ρc, ρq-c) together force the bipartite proto-state to be the singlet |Ωⱼ = 1√2j+1∑ₘ₌₋ⱼʲ(-1)ʲ⁻ᵐ|m|-m. The singlet is uniquely selected by the diagonal $2I$-invariance of the admissible pair (Schur's lemma applied to the Clebsch--Gordan decomposition of χ₂ⱼ₊₁⊗χ₂ⱼ₊₁), and is shown to satisfy the isotropy condition M ∝ I via the singlet identity (Jₐ⁽¹⁾+Jₐ⁽²⁾)|Ωⱼ = 0, closing the identification gap of Q2. The universal correlator\[E(â,b̂) = -j(j+1)/3\,(â̂)\]is derived without new postulates for all j ∈ \1/2, 1, 3/2, 2, 5/2\ --- the complete admissible spectrum of $2I$. In the LPS limit p → ∞, the admissibility flow of~{Beau2026q2} selects j = 1/2 as the unique stable fixed point, recovering the standard singlet correlator E(â,b̂) = -â̂ of~{Beau2026q1} and completing the structural derivation of quantum mechanics for the full SU(2) sector.