An Exact Laplacian-Eigenvalue Degeneracy Between the Spin-½ and Spin-3/2 Representations of 2I

Abstract

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 share the same Cayley-graph Laplacian eigenvalue, $\lambda_{1/2} = \lambda_{3/2} = 18$. The spin-$\tfrac{3}{2}$ value is derived here directly: because $S$ is a union of two full conjugacy classes on which the four-dimensional character $\chi_4$ takes the constant value $1$, the corresponding group-algebra element is central and Schur's lemma gives its scalar action on $\chi_4$ exactly. Under the admissibility-window definition $A^{\max}_\rho = c_{\mathrm{BI}}/\sqrt{\lambda_\rho}$ of the spectral admissibility programme, the two sectors therefore have identical admissibility windows on $2I$. This is the paper's entire result: an exact, elementary fact about the graph Laplacian of a specific finite group and generating set. It does not by itself establish phase coherence, a singlet correlator, a Tsirelson bound, the Born rule, a renormalisation-style fixed point selecting spin-$\tfrac{1}{2}$, or any falsifiable dimension test — each of these would require additional structure this note does not supply. Read as a methodological point (interpretive, not a further result): a spectral degeneracy between two representation sectors is a fact about the finite group and generating set alone, and carries no physical selection content on its own.