A Uniqueness Result for the Vacuum Edge Representation in Loop Quantum Gravity

We prove that the spin-1 (adjoint) representation is the unique SU(2) representation j for which the curvature representation space V₁ ⊗ Vⱼ at a 4-valent vertex in loop quantum gravity (i) contains the spin-2 representation, and (ii) contains no representation of spin greater than 2. The proof follows directly from the Clebsch–Gordan decomposition. This result recovers the adjoint representation carried by the classical Ashtekar–Barbero connection and provides a kinematic rationale for the preference for low-spin boundary data in spin foam models, complementing evidence from EPRL vertex asymptotics and graviton propagator calculations.