Golden Ratio and Fibonacci Structure in Fermion Mixing: An Empirical Theory from A₄ = SU(5)

We report numerical evidence that the mixing angles of both the lepton (PMNS) and quark (CKM) sectors of the Standard Model follow from a small set of constants derived from the golden ratio φ = (1+√5)/2 and Fibonacci numbers. Defining - X = 1/φ ≈ 0.618- λ = 1/(2√5) ≈ 0.224 we find: - All four PMNS angles match expressions in X and 2/3 to within 2% (χ²/ndf = 0.26).- All four CKM parameters match expressions in X, λ, and 1/3 to within 4%.- The exact identity sin θ_12^PMNS / sin θ_12^CKM = 4√5/√13 is satisfied to machine precision.- The Koide relation K = 2/3 is recovered to 10⁻⁵. The underlying structure is the A₄ = SU(5) Coxeter matrix, whose unique factorization over ℚ(√5) and embedding hierarchy in E₆, E₇, E₈ we fully characterize. We also report a negative result: fermion masses do not follow a simple φ- or λ-lattice (Monte Carlo p > 0.05). This work extends the φ-lattice program initiated in our previous study [1], where a precise formula for the muon mass was derived from 9 baryon sectors and tested against 19 exotic hadrons from LHCb, BESIII, and Belle II (17/19 hits, 7σ).