Q3 — Universal Spin-$j$ Singlet from Admissibility

This repository contains the source of the Q3 Cosmochrony paper Universal Spin-$j$ Singlet from Admissibility: Completion of the $\mathrm{SU}(2)$ Quantum Sector.

The companion paper Q2 derives the singlet correlator $E(\hat a,\hat b) = -\tfrac54(\hat a\cdot\hat b)$ for the spin-$\tfrac32$ sector of the binary icosahedral group $2I$, leaving open the identification of the unique isotropic fibre state with the admissibility-selected proto-state. This paper closes that step and extends the result to all admissible sectors of the chain $Q_8 \hookrightarrow 2I \hookrightarrow \mathrm{SU}(2)$.

Core Result

For any irreducible representation $\chi_{2j+1}$ of $2I$, the phase-coherence constraint (Q1, Q2) and the Born–Infeld neutrality of the conjugate pair $(\rho_c, \rho_{q-c})$ together force the bipartite proto-state to be the singlet $|\Omega_j\rangle = \frac{1}{\sqrt{2j+1}}\sum_{m=-j}^{j}(-1)^{j-m}|m\rangle|-m\rangle$. The singlet is uniquely selected by the diagonal $2I$-invariance of the admissible pair (Schur's lemma on the Clebsch–Gordan decomposition) and satisfies the isotropy condition $M \propto I$ via $(J_a^{(1)}+J_a^{(2)})|\Omega_j\rangle = 0$.

The universal correlator $$E(\hat a,\hat b) = -\frac{j(j+1)}{3},(\hat a\cdot\hat b)$$ follows without new postulates for all $j \in {\tfrac12, 1, \tfrac32, 2, \tfrac52}$ — the complete admissible spectrum of $2I$.

Keywords

Spin-$j$ singlet, binary icosahedral group, admissibility, Bell correlator, Schur's lemma, Clebsch–Gordan, isotropy, SU(2) quantum sector.

Repository Contents

q3/
├── tex/         # LaTeX sources (main + cosmochrony-bibliography.bib)
├── out/         # Compiled paper PDF (q3.pdf)
├── zenodo.json  # Zenodo deposition metadata
└── README.md

Links

• 🔗 DOI: 10.5281/zenodo.19979535
• 🌐 Website: https://cosmochrony.org/science/quantum-structure/q3/

Citation

J. Beau, Universal Spin-$j$ Singlet from Admissibility: Completion of the $\mathrm{SU}(2)$ Quantum Sector, Zenodo, 2026. DOI: 10.5281/zenodo.19979535.

Acknowledgements

Portions of the editorial refinement benefited from iterative interactions with large language models, used as analytical assistants. All claims and final formulations remain the sole responsibility of the author.