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

Overview

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 paper 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, an instance of the classical McKay correspondence for the binary icosahedral group. Given the singlet, the two-point correlator follows by an explicit trace computation: $E(\hat a,\hat b) = -\tfrac{j(j+1)}{3}(\hat a\cdot\hat b)$.

Central message. The diagonal-invariance hypothesis is stated as an explicit assumption throughout, not derived from admissibility or from any Born–Infeld indiscernibility argument. The theorem shows exactly how much that bare hypothesis buys — a unique state and its second moments — and no more.

Quick facts

Author

Jérôme Beau

Work type

Preprint

Focus

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

Key result

$|\psi\rangle = |\Omega_j\rangle$ given (H-inv); $E(\hat a,\hat b) = -\tfrac{j(j+1)}{3}(\hat a\cdot\hat b)$

Status

Conditional on an explicit, undemonstrated hypothesis (H-inv)

Does not establish

(H-inv) itself, phase coherence, Born rule, CHSH/Tsirelson-type bound, sector selection

Core contributions

Unique invariant singlet, all $j$: given the explicit hypothesis (H-inv) that the joint state is diagonally $2I$-invariant, the state is uniquely $|\Omega_j\rangle$ for each of the five admissible sectors — proved via $(V_j\otimes V_j)^{2I} \cong \mathrm{Hom}_{2I}(V_j^*,V_j)$, one-dimensional by Schur's lemma given irreducibility and self-duality of the restricted module.
McKay correspondence for $2I$: the five irreducible restrictions (dimensions 2, 3, 4, 5, 6) trace out the linear arm of the affine $E_8$ Dynkin diagram generated by tensoring with the defining representation, terminating because the next restriction (dimension 7) is reducible.
Casimir correlator: given the singlet, an explicit trace computation gives $E(\hat a,\hat b) = -\tfrac{j(j+1)}{3}(\hat a\cdot\hat b)$ for all five sectors.
Scope: this paper does not derive (H-inv) 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 any sector as physically preferred.

Interpretive remark

Read as a methodological point: the theorem shows exactly how much a bare diagonal-invariance hypothesis buys, on a supplied bipartite carrier — a unique state, for each of five sectors, forced by Schur's lemma alone once irreducibility and self-duality of the underlying $2I$-module are in hand. The correlator theorem then computes that state's second moments explicitly. Neither theorem derives the hypothesis it starts from, and neither by itself constitutes a physical selection principle, a probability law, or a Bell-type inequality; each of those would need its own, separately stated and separately proved bridge.

Relation to the Cosmochrony programme

This paper shares its five-sector scope with Q2's eigenvalue-degeneracy result, which fixes which sectors are in scope, but does not depend on Q2's specific eigenvalue calculation for its own argument. It does not depend on Q1, whose proved content (an unconditional Fourier-support rigidity theorem) is unrelated to this paper's conditional singlet theorem. The finite effective-composition contract typing the bipartite carrier as a supplied "K4 crossing" is discussed in the paper's own scope remark.

Open directions

Deriving (H-inv). Whether the diagonal $2I$-invariance hypothesis follows from some further, separately stated admissibility or dynamical principle — rather than being supplied as an assumption — is open.
The bipartite carrier itself. Deriving $V_j \otimes V_j$, or a general states-and-effects layer it would instantiate, from more primitive structure is a separate, open programme direction.
Multipartite extension. The five sectors raise the question of states mixing more than two sectors or more than two parties; the present single-pair construction does not address this.

References

Jérôme Beau. Diagonal 2I-Invariance Selects the Universal Spin-j Singlet and Its Casimir Correlator, 2026. doi:10.5281/zenodo.19979535