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

The companion paper Q2 derives the singlet correlator E (a, b) = -54 (a) for the spin-32 sector of the binary icosahedral group 2I, leaving open the identification of the unique isotropic fibre state of ₄ 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 ₂₉+₁ of 2I, the phase-coherence constraint of Q1, Q2 and the Born--Infeld (BI) neutrality of the conjugate pair (c, ₐ-₂) together force the bipartite proto-state to be the singlet |ⱼ = 12j+1₌=-₉^j (-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 ₂₉+₁₂₉+₁), and is shown to satisfy the isotropy condition M I via the singlet identity (Jₐ^ (1) +Jₐ^ (2) ) |ⱼ = 0, closing the identification gap of Q2. The universal correlator (a, b) = -j (j+1) 3\, (a) derived without new postulates for all j \1{2, 1, 32, 2, 52\} --- the complete admissible spectrum of 2I. In the LPS limit p, the admissibility flow of~Beau2026q2 selects j = 12 as the unique stable fixed point, recovering the standard singlet correlator E (a, b) = -a of~Beau2026q1 and completing the structural derivation of quantum mechanics for the full SU (2) sector.