An Exact Laplacian-Eigenvalue Degeneracy Between the Spin-1/2 and Spin-3/2 Representations of 2I

Overview

The spectral admissibility programme assigns, to each irreducible representation of a finite group and a fixed generating set, an admissibility window built from the induced Cayley-graph Laplacian eigenvalue. Two sectors with equal eigenvalue are co-admissible in this precise sense.

This note proves one such degeneracy: on the binary icosahedral group $2I$, with the canonical generating set $S = 10a \cup 10b$ (a union of two conjugacy classes, $|S|=24$), the spin-1/2 and spin-3/2 representations induce the same eigenvalue, $\lambda_{1/2} = \lambda_{3/2} = 18$. Because $S$ is a union of full conjugacy classes on which the four-dimensional spin-3/2 character is constant, the corresponding group-algebra element is central, and Schur's lemma gives its scalar action exactly.

Central message. This is a fact about the graph Laplacian of $2I$ and one specific generating set — nothing more. It does not identify a physical spin observable, a joint quantum state, or a probability law, and by itself it selects no physical dynamics or preferred sector.

Core contributions

Exact eigenvalue degeneracy: $\lambda_{1/2} = \lambda_{3/2} = 18$ on $2I$ for the canonical generating set, proved directly for the spin-3/2 case via Schur's lemma (the spin-1/2 value is cited from the spectral admissibility programme).
Co-admissibility: under the admissibility-window definition $A^{\max}_\rho = c_{\mathrm{BI}}/\sqrt{\lambda_\rho}$, the two sectors have identical admissibility windows on $2I$ — an immediate corollary of the eigenvalue equality.
Scope: this paper does not derive phase coherence, a singlet correlator, a Tsirelson-type bound, the Born rule, or an SU(2) fixed-point/sector-selection argument in any continuum or large-prime limit.

Interpretive remark

Read as a methodological point for the wider spectral admissibility programme: an eigenvalue degeneracy between two representation sectors of a finite group is exactly that — a combinatorial fact about the group, the generating set, and the sectors' dimensions. On its own it selects no physical state, no dynamics, and no preferred sector. Any claim that such a degeneracy drives a physical selection mechanism needs its own, separately stated and separately proved bridge; none is supplied here.

Relation to the Cosmochrony programme

This is a self-contained finite-group spectral result. It shares its generating-set data with the spectral admissibility programme's Laplacian-eigenvalue table for $2I$, and its eigenvalue equality is used, as an explicit hypothesis-independent fact, by Q3 to identify which five sectors are in scope there. It does not itself feed a phase-coherence, correlator, or Born-rule result anywhere in the programme.

References

Jérôme Beau. An Exact Laplacian-Eigenvalue Degeneracy Between the Spin-1/2 and Spin-3/2 Representations of 2I, 2026. doi:10.5281/zenodo.19616444