Clebsch–Gordan Coefficients: Orthogonality, Recursion, and Lattice QCD Applications — E8 Intelligence Research

FINDING: Search results are pedagogical resources on Clebsch–Gordan (CG) coefficients, with one lattice-QCD application paper; no explicit proof of golden-ratio absence is found in the returned material. | MATH: CG coefficients ⟨j₁ m₁; j₂ m₂ | J M⟩ are the overlap matrix elements between uncoupled and coupled angular-momentum bases; they satisfy orthogonality ∑_{m₁,m₂} ⟨j₁m₁;j₂m₂|JM⟩⟨j₁m₁;j₂m₂|J′M′⟩ = δ_{JJ′}δ_{MM′}, and recursion relations from ladder operators. The lattice paper uses double-valued irreps of the octahedral group (O_h^D), whose dimensions are {1,2,3,4} — no φ appears. | CONNECTION: CG coefficients for SU(2) are algebraic numbers from square roots of rationals (e.g., √(2/3), √(1/2)); the golden ratio φ = (1+√5)/2 is quadratic irrational but not of the form √(p/q) with p,q integers unless p/q = φ² — which is impossible since φ² is irrational. Thus φ cannot appear as a CG coefficient magnitude. However, φ *can* appear in SU(3) or higher-rank tensor products via ratios of

Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com