Amplitude from Trace

Description: The character table of the binary icosahedral group 2I encodes coupling constants through the trace of its five-fold conjugacy class: χ(2, C₅) = 2cos(π/5) = φ. This paper identifies the complementary face. The off-diagonal entry of the same SU(2) rotation matrix gives the transition amplitude sin(π/5), and the contraction rate σ² = 1/φ² of the axiom σ = 1/(1+σ) filters it into the observable sector. Two CKM parameters follow as direct formulas: the Cabibbo mixing element |V_us| = sin(π/5)/φ² (0.3σ from PDG) and the CP-violating phase δ = π/φ² (0.1σ from CKMfitter). Combined with the empirical ansatz β = π/(2φ³) and the identity φ² = φ + 1, unitarity implies α = π/2 exactly. The right angle is conditional on the β ansatz, which is the weakest link at 1.4σ. Belle II will test the prediction at decisive precision by approximately 2028. Each claim is classified by epistemic status: direct formula, controlled approximation, empirical conjecture, or conditional consequence. Paper 44 of the Pentagon Physics programme. Keywords: CKM matrix, quark mixing, golden ratio, binary icosahedral group, character table, Cabibbo angle, CP violation, unitarity triangle, five-fold symmetry, transition amplitude, contraction rate, self-referential equation, Pentagon Physics, Belle II, flavour symmetry