Oₕ-restricted operator spectrum of SYK₆ from the Weingarten calculus

This paper computes the operator spectrum of SYK₆ restricted to the octahedral symmetry group Oₕ, as required by the 4, 3, r holographic tensor network. Three ingredients are presented with explicit verification: the character table of S₆ (11 irreducible representations, verified by all 242 orthogonality relations), the homomorphism Oₕ → S₆ defined by the action of the octahedral group on the six faces of the cube (48 elements classified by cycle type), and the decomposition of the bilinear operator space into Oₕ irreducible representations. The six-face representation decomposes as A₁₆ ⊕ Eg ⊕ T₁ₔ. The antisymmetric bilinears give 15 dimensions across six irreps; the symmetric bilinears give 21 dimensions across six irreps. Three Oₕ irreps (A₁ₔ, A₂ₔ, Eᵤ) are absent from the bilinear space entirely. The Weingarten scaling at each order in χ is classified, and the SYK₆ conformal block spectrum is computed numerically for both singlet and non-singlet channels. All results are exact and computationally verified.