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

Abstract

For each of the five representations $\chi_{2j+1}$, $j \in \{\tfrac{1}{2}, 1, \tfrac{3}{2}, 2, \tfrac{5}{2}\}$, obtained by restricting the spin-$j$ representation of $\mathrm{SU}(2)$ to the binary icosahedral group $2I$, we prove 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: the chain of irreducible restrictions of dimension $2,3,4,5,6$ is exactly the linear arm of the affine $E_8$ diagram generated by tensoring with the defining representation, terminating because the next restriction (dimension $7$) is reducible. Given the singlet, the two-point correlator follows by an explicit trace computation: $E(\hat{a},\hat{b}) = -\tfrac{j(j+1)}{3}(\hat{a}\cdot\hat{b})$. This paper 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 $j=\tfrac{1}{2}$ as a physically preferred sector. Read as a methodological point (interpretive, not a further result): the theorem shows precisely how much a bare group-invariance hypothesis buys on a supplied composition — a unique state and its second moments — and how much more a full quantum derivation would still require.