The Koide Stability Lemma

Why are there exactly three generations of matter? We show the answer follows from one symmetry class. Given a Hermitian circulant mass matrix with discrete ℤ/3ℤ family symmetry, we prove three results. Theorem 1: the Koide formula Q = 2/3 is equivalent to a balance condition on the off-diagonal coupling. Theorem 2: N = 3 is the unique dimension where this balance produces physical (all-positive) masses at the midpoint of the allowed range. Theorem 3: the balanced spectrum depends on one scale and one phase angle. The companion paper (Selecting the Koide Phase) derives that angle from the self-referential axiom σ = 1/(1+σ), yielding cos(δ) = −19/28 with zero fitted parameters and reproducing all three charged lepton masses to 0.021% total error. Part of the Pentagon Physics programme.