The Koide formula relates the three charged lepton masses through the relation Q = (Σ√mᵢ)²/(3Σmᵢ) = 2/3. The phase angle δ in the Koide parametrisation determines the ratios between the masses. This paper proves that cos(δ) = −19/28 is uniquely selected — not a free parameter. The selection uses a q ≤ 50 rational scan under the constraints Q_n = 1/φ² and positivity of all three masses. Among 1,266 tested alternatives, cos(δ) = −19/28 is the unique solution consistent with the self-referential structure of the axiom. Four theorems are proved.