Selecting the Koide Phase

The companion paper (The Koide Stability Lemma) reduces three charged lepton masses to one unknown phase angle. This paper closes that angle. We impose two independent algebraic conditions on the circulant eigenvalue hierarchy. First, the golden-ratio balance Q_n = 1/φ² at the fourth-root rung, derived from the self-referential axiom σ = 1/(1+σ) with no mass data. Second, a rational phase constraint cos(δ) = −p/q with denominator bounded by the conductor of the cyclotomic field ℚ(ζ₅). An exhaustive scan over all rationals with q ≤ 50 under positivity and the balance condition yields exactly one survivor: cos(δ) = −19/28. No other numerator with this denominator, and no other denominator in the conductor-bounded range, produces physical masses. The two conditions form a closed system determining both the phase and the rung with zero fitted parameters. Predicted masses: electron 0.5111 MeV (0.020%), muon 105.651 MeV (0.007%), tau 1776.884 MeV (0.001%), total error 0.021%. Section 10 provides the full independence chain showing neither condition uses mass data; agreement with PDG values is therefore a prediction, not a fit. Appendix A gives the complete computational protocol for reproduction. Part of the Pentagon Physics programme.