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

Overview

Q1 established phase coherence and quantum correlations for the spin-1/2 sector from admissibility alone. A natural question is whether this result is specific to spin-1/2, or whether it extends to higher spin sectors, pointing toward a universal quantum structure.

Q2 investigates the spin-3/2 sector on the binary icosahedral group 2I. The key discovery is that the spin-1/2 and spin-3/2 representations share the same Laplacian eigenvalue $\lambda_{1/2} = \lambda_{3/2} = 18$ — a property called co-admissibility. This shared spectral footprint forces both sectors to obey identical admissibility constraints, enabling the extension of Q1's results to spin-3/2 without new postulates.

The Born rule, the singlet correlator $E(\hat{a},\hat{b})$, and the Tsirelson bound are all derived for spin-3/2 from co-admissibility. Furthermore, SU(2) is identified as the stable fixed point of the admissibility flow, with the general-$j$ extension deferred to Q3.

Central message. Co-admissibility forces spin-3/2 to obey the same quantum structure as spin-1/2; SU(2) is not an input but the fixed point of admissibility.

Core contributions

Co-admissibility of spin-1/2 and spin-3/2: on the binary icosahedral group 2I, the Laplacian eigenvalues satisfy $\lambda_{1/2} = \lambda_{3/2} = 18$. This co-admissibility means both sectors are admitted simultaneously by the spectral constraint, placing them in the same admissibility class without requiring a separate argument for each.
Born rule for spin-3/2: the co-admissibility structure forces the Born–Infeld indiscernibility of conjugate Weil blocks to apply to the spin-3/2 sector, yielding the Born rule as a structural consequence.
Singlet correlator and Tsirelson bound for spin-3/2: the bipartite proto-state in the spin-3/2 sector is identified as the singlet, and the correlator $E(\hat{a},\hat{b})$ and Tsirelson bound are derived following the same mechanism as Q1.
SU(2) as stable fixed point: the admissibility flow on the representation ring of 2I has SU(2) as its stable fixed point: any co-admissible sector must transform under the standard SU(2) action. SU(2) is thus an output of the projective structure, not an input. The general-$j$ proof is completed in Q3.

Interpretation

The notion of co-admissibility introduced in Q2 is a spectral coincidence that carries deep structural significance. In most frameworks, spin-1/2 and spin-3/2 are distinct physical sectors treated by separate representations. Here, they share a Laplacian eigenvalue on the admissibility graph, forcing them to obey identical projective constraints.

This co-admissibility is what makes SU(2) a fixed point rather than an assumption: the admissibility flow selects SU(2) because it is the only group under which all co-admissible sectors are simultaneously compatible. The quantum symmetry group is thus determined by the spectral structure of the fibre.

Significance for quantum field theory. The standard approach postulates SU(2) as the gauge group of the weak interaction. Q2 shows that SU(2) emerges as the fixed point of projective admissibility, suggesting that the gauge structure of the Standard Model may be derivable from the spectral geometry of the admissible fibre.

Relation to the Cosmochrony programme

Q2 extends the quantum sector from spin-1/2 (Q1) to spin-3/2, and introduces the concept of co-admissibility that structures the full Q-series programme:

Q1: proves phase coherence and quantum correlations for spin-1/2; provides the proof strategy that Q2 adapts to spin-3/2 via co-admissibility.
O23: entanglement structure of the bipartite fibre, used to identify the proto-state as a singlet for both spin sectors.
O29: representation-theoretic results for the binary icosahedral group that establish the co-admissibility eigenvalue coincidence.
Q3: completes the programme by proving SU(2) emergence and the universal singlet correlator for all 5 admissible sectors $j \in \{1/2, 1, 3/2, 2, 5/2\}$ of 2I.

Open directions

General-$j$ extension (Q3): prove that the singlet structure and SU(2) fixed-point property hold for all admissible sectors $j \in \{1/2, 1, 3/2, 2, 5/2\}$ of the binary icosahedral group, not just spin-1/2 and spin-3/2.
Co-admissibility beyond 2I: investigate whether analogous co-admissibility structures arise on other finite groups relevant to the O-series (e.g., binary octahedral, binary tetrahedral), and whether they enforce similar fixed-point symmetry groups.
Connection to gauge symmetry breaking: SU(2) emergence as a fixed point suggests a spectral mechanism for symmetry breaking; the connection to the Higgs sector and electroweak unification remains an open direction.

References

Jérôme Beau. Quantum Structure beyond Spin-1/2: Co-Admissible Sectors and the Emergence of SU(2) as a Fixed Point, 2026. doi:10.5281/zenodo.19616444