Q3 — Diagonal $2I$-Invariance Selects the Universal Spin-$j$ Singlet and Its Casimir Correlator

This repository contains the source of the Q3 Cosmochrony companion note Diagonal $2I$-Invariance Selects the Universal Spin-$j$ Singlet and Its Casimir Correlator.

Core Result

For each of the five representations $\chi_{2j+1}$, $j \in {\tfrac12, 1, \tfrac32, 2, \tfrac52}$, obtained by restricting the spin-$j$ representation of $\mathrm{SU}(2)$ to the binary icosahedral group $2I$, this note proves a conditional theorem: given a bipartite carrier $V_{\chi_{2j+1}} \otimes V_{\chi_{2j+1}}$ (a supplied composition structure, not derived from any admissibility axiom) and the explicit hypothesis that a state on that carrier is invariant under the diagonal action of $2I$, the state is uniquely the $\mathrm{SU}(2)$ singlet $|\Omega_j\rangle = \tfrac{1}{\sqrt{2j+1}}\sum_m(-1)^{j-m}|m\rangle|-m\rangle$, up to phase. The proof uses $(V_j \otimes V_j)^{2I} \cong \mathrm{Hom}_{2I}(V_j^*, V_j)$, one-dimensional by Schur's lemma whenever the restriction of $V_j$ to $2I$ is irreducible and self-dual — which holds for all five sectors, a case of the classical McKay correspondence for the binary icosahedral group. Given the singlet, the two-point correlator follows by an explicit trace computation: $E(\hat a,\hat b) = -\tfrac{j(j+1)}{3}(\hat a\cdot\hat b)$.

What this note does not claim: it does not derive the diagonal-invariance hypothesis from admissibility or from any Born–Infeld indiscernibility argument, does not derive phase coherence, the Born rule, or a CHSH/Tsirelson-type bound, and does not select $j=\tfrac12$ as a physically preferred sector. See the paper's Introduction and "What is not claimed" section for the full list.

Keywords

Binary icosahedral group, SU(2) representation theory, Schur's lemma, Clebsch–Gordan decomposition, McKay correspondence, singlet state, Casimir identity.

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 (concept DOI)
• 🌐 Website: https://cosmochrony.org/science/quantum-structure/q3/

Citation

J. Beau, Diagonal $2I$-Invariance Selects the Universal Spin-$j$ Singlet and Its Casimir Correlator, 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.