P29: Explaining Three Particle Generations: Why There Are Three, and Why Their Masses Obey the Koide Relation

Abstract

The Standard Model contains three fermion generations but does not predict their Yukawa couplings or mass hierarchy. This paper derives the structure of the charged-lepton mass spectrum from the 3 + e condensation of the Spaticle field (the name the author gives to the physical matter substrate). A mirror-symmetric arrangement of three identical cores yields one singlet and one degenerate doublet. The three cores of the 3 + e condensation of P16 co-rotate, so the arrangement is not mirror-symmetric: circulation about the threefold axis reduces the symmetry from D3h to C3h, whose irreducible representations are all one-dimensional. The most general Hermitian operator on the three cores that commutes with the threefold rotation is a complex circulant), with exact spectrum a+2bcos+2k/3. This is identical in form to the Koide parametrisation of the charged-lepton square-root masses, so circulating threefold symmetry produces three distinct lepton levels of exactly the observed type. The three operator quantities have distinct physical roles: the on-site amplitude a sets the scale, the ratio b/a sets the Koide value, and the gauge-invariant loop phase =3 sets the mass ratios. The Koide value satisfies Q=1/3+2/3b2/a2 exactly. Squaring the operator shows that each core’s mass-energy consists of a self part a2 and an exchange part b2 from each neighbour. The exchange entries are the discretised kinetic term and the on-site entries are the circulation self-energy, so the condensation functional of P16 fixes b2/a2=A/D=1/2, where A/R2 is the localisation term and D/R is the circulation term with D=2A. This derives Q=2/3 from the functional; the measured value differs by 3.30 parts per million, or 0.43 standard deviations. The P16 phase law, =2+Q, reproduces the lepton mass ratios to 0.79 parts per million in the phase. The electron mass is the detuning of the link phase from its zero-amplitude value 3π/4 (the loop phase is detuned by 3ε from 2+/4, with me=M1−cos+sin2. The derived Koide value predicts m=1776.96903 MeV, 0.43 standard deviations from the PDG 2024 average, and the complete chain from the proton mass reproduces all three charged-lepton masses to within 0.005 percent.