Q2 — An Exact Laplacian-Eigenvalue Degeneracy Between the Spin-$\tfrac{1}{2}$ and Spin-$\tfrac{3}{2}$ Representations of $2I$

This repository contains the source of the Q2 Cosmochrony companion note An Exact Laplacian-Eigenvalue Degeneracy Between the Spin-$\tfrac{1}{2}$ and Spin-$\tfrac{3}{2}$ Representations of $2I$.

Core Result

On the binary icosahedral group $2I$ with the canonical generating set $S = 10a \cup 10b$ ($|S|=24$), the spin-$\tfrac{1}{2}$ and spin-$\tfrac{3}{2}$ irreducible representations induce the same Cayley-graph Laplacian eigenvalue, $\lambda_{1/2} = \lambda_{3/2} = 18$. The spin-$\tfrac{3}{2}$ value is derived directly here: since the generating set is a union of two full conjugacy classes on which the four-dimensional character $\chi_4$ is constant (equal to $1$), the corresponding group-algebra element is central, and Schur's lemma gives its scalar action on $\chi_4$ exactly. Under the admissibility-window definition $A^{\max}\rho = c{\mathrm{BI}}/\sqrt{\lambda_\rho}$, the two sectors therefore have identical admissibility windows on $2I$.

What this note does not claim: it does not establish phase coherence, a singlet correlator, a Tsirelson bound, the Born rule, an SU(2) fixed point selecting spin-$\tfrac{1}{2}$ in any continuum or large-prime limit, or a falsifiable dimension test. None of these follow from an eigenvalue coincidence alone. See the paper's Introduction and "What this note does not claim" section for the full list.

Keywords

Binary icosahedral group, Cayley graph, Laplacian eigenvalue, character theory, Schur's lemma, spectral admissibility, representation theory.

Repository Structure

q2/
├── tex/         # LaTeX sources (main + cosmochrony-bibliography.bib)
├── out/         # Compiled paper PDF (q2.pdf)
├── zenodo.json  # Zenodo deposition metadata
└── README.md

Links

• 🔗 DOI: 10.5281/zenodo.19616444 (concept DOI)
• 🌐 Website: https://cosmochrony.org/science/quantum-structure/q2/

Citation

J. Beau, An Exact Laplacian-Eigenvalue Degeneracy Between the Spin-$\tfrac{1}{2}$ and Spin-$\tfrac{3}{2}$ Representations of $2I$, Zenodo, 2026. DOI: 10.5281/zenodo.19616444.

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.