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

Overview

Q2 established that the spin-3/2 sector obeys the same quantum structure as spin-1/2 via co-admissibility, and identified SU(2) as the stable fixed point of the admissibility flow on the binary icosahedral group. However, two steps remained open: the identification of the isotropic fibre state with the proto-state for spin-3/2, and the general extension to all admissible values of $j$.

Q3 closes both open steps, within a supplied bipartite tensor-product composition $V_{\chi_{2j+1}} \otimes V_{\chi_{2j+1}}$ of the conjugate pair: this composition is a construction choice (a K4 crossing), not a consequence of the admissibility axioms A1–A4. The core argument uses Born–Infeld indiscernibility and Schur's lemma: within that composition, the bipartite proto-state must be invariant under the diagonal 2I-action, and by Schur's lemma applied to the Clebsch–Gordan decomposition, the only such invariant state is the singlet $|\Omega_j\rangle$.

From the singlet identification, the universal correlator $E(\hat{a},\hat{b}) = -\tfrac{j(j+1)}{3}\,\hat{a}\cdot\hat{b}$ follows by direct computation, valid for all 5 admissible sectors $j \in \{1/2, 1, 3/2, 2, 5/2\}$ of the binary icosahedral group 2I.

Central message. Within the supplied bipartite composition, admissibility and Schur's lemma uniquely determine the proto-state as the singlet $|\Omega_j\rangle$ for all admissible $j$, completing the quantum sector of the spectral admissibility programme within that composition.

Core contributions

The bipartite composition is a K4 crossing: the ambient space $V_{\chi_{2j+1}} \otimes V_{\chi_{2j+1}}$ in which the conjugate pair's joint proto-state is sought is a supplied composition structure, not a consequence of admissibility. What Q3 proves is the state's identity within that supplied composition, not the composition itself.
Diagonal 2I-invariance from Born–Infeld indiscernibility: within that supplied composition, Born–Infeld indiscernibility forces the bipartite proto-state to be invariant under the diagonal action of 2I. This invariance condition is the structural input that makes the singlet identification unique.
Singlet identification via Schur's lemma: the Clebsch–Gordan decomposition of $V_{\chi_{2j+1}} \otimes V_{\chi_{2j+1}}$ contains exactly one trivial (singlet) subrepresentation for each admissible $j$. By Schur's lemma, the unique diagonal-2I-invariant state within that composition is the singlet $|\Omega_j\rangle$: \[ |\Omega_j\rangle = \frac{1}{\sqrt{2j+1}}\sum_{m=-j}^{j}(-1)^{j-m}|j,m\rangle\otimes|j,-m\rangle. \]
Universal correlator for all admissible $j$: from the singlet $|\Omega_j\rangle$, the bipartite correlator is computed directly: \[ E(\hat{a},\hat{b}) = \langle\Omega_j|\,(\hat{a}\cdot\mathbf{J})\otimes(\hat{b}\cdot\mathbf{J})\,|\Omega_j\rangle = -\frac{j(j+1)}{3}\,\hat{a}\cdot\hat{b}, \] holding for all 5 admissible sectors $j \in \{1/2, 1, 3/2, 2, 5/2\}$ of 2I.
Closure of the open step of Q2: Q2 left open the identification of the isotropic fibre state with the proto-state for spin-3/2, and the general-$j$ extension. Q3 closes both by the Schur's lemma argument, which applies uniformly across all admissible sectors.
Completion of the Q1–Q3 quantum programme: together with Q1 (phase coherence, spin-1/2) and Q2 (co-admissibility, spin-3/2, SU(2) fixed point), Q3 establishes that, within the supplied bipartite composition, the quantum structure of SU(2) representations follows from admissibility without any further quantum postulate.

Interpretation

The central insight of Q3 is the role of Schur's lemma as a uniqueness theorem, within a supplied bipartite composition. Admissibility does not derive that composition itself: it is a construction choice, taken as an input. Once the composition is supplied and Born–Infeld indiscernibility forces diagonal 2I-invariance of the proto-state within it, the algebraic structure of representation theory does the rest: the singlet is the only candidate. No additional physical principle beyond the supplied composition is needed.

The universality of the correlator formula $E(\hat{a},\hat{b}) = -\tfrac{j(j+1)}{3}\,\hat{a}\cdot\hat{b}$ across all admissible sectors reflects the fact that the admissibility constraints on 2I select exactly those representations for which SU(2) acts transitively on the unit sphere — the defining property of rotationally symmetric correlators.

Structural significance. The binary icosahedral group 2I has precisely 5 admissible sectors ($j = 1/2, 1, 3/2, 2, 5/2$). These are not chosen: they are the only values for which the spectral admissibility constraints are simultaneously satisfiable. The quantum structure of these sectors — singlet states, Born rule, correlators — is determined by this discrete spectral selection.

Relation to the Cosmochrony programme

Q3 closes the quantum sector of the spectral admissibility programme opened by Q1 and extended by Q2. The Q1–Q3 chain establishes:

Q1: phase coherence is a structural consequence of admissibility for spin-1/2; the singlet correlator and Tsirelson bound follow from O18 and O23.
Q2: co-admissibility on 2I extends the result to spin-3/2; SU(2) is identified as the fixed point of the admissibility flow.
Q3: Schur's lemma closes the identification for all admissible $j$; the universal correlator completes the derivation.

The result has direct implications for other parts of the programme. The universal correlator provides a falsifiable prediction: Bell-type correlations in the Cosmochrony framework should obey $E(\hat{a},\hat{b}) = -\tfrac{j(j+1)}{3}\,\hat{a}\cdot\hat{b}$ for all admissible $j$, which differs from the standard quantum mechanics result only for $j \neq 1/2$ (where $j(j+1)/3 \neq 1/4$) and could in principle be tested.

References

Jérôme Beau. Universal Spin-j Singlet from Admissibility: Completion of the SU(2) Quantum Sector, 2026. doi:10.5281/zenodo.19979535