Emergent Time from Helical Motion on a Compact Matter Dimension: The Koide Relation and the Number of Generations

The Koide relation, (me + mµ + mτ )/( √ me + √mµ + √ mτ ) 2 = 2/3, holds to a few parts in 106 and has had no accepted explanation for four decades. In a framework with three-dimensional space, one compact matter dimension and no external time coordinate, we show that it follows from a 45◦ helix. Along each strand of a particle the motion of space advances by one circumference 2πλ ¯ along the matter dimension while it turns once in space at radius λ ¯ ; both facts were established in the companion papers, so the motion is shared equally between the matter dimension and space. If the three charged-lepton generations are the eigenmodes of a threefold-symmetric operator about the matter dimension, whose eigenvalues are the radii of the bundles, the equal share makes the diagonal and off-diagonal parts of that operator equal, which is Q = 2/3 exactly, whatever the phase, provided the three radii are positive. The same geometry says where the relation should not hold: neutrinos are untwisted single-layer bundles and give Q = 0.527 in the framework, and quarks, which are knotted, give 0.888 and 0.746 at a common scale; the quark values themselves are not derived. Separately, the number of generations is the number of stationary untwisted bundles, three by the planar kissing number, and anomaly cancellation carries it to quarks: no fourth generation exists at any mass, and no light sterile neutrino exists beyond the three right-handed partners, as LEP, the LHC, MicroBooNE and KATRIN currently find. The Koide phase is one third of a gauge-invariant loop phase. Its empirical value, close to 2/9 rad, corresponds to a loop phase of 2/3 rad, which can be written as the framework’s ΩΛ = 2/3 times one radian; this reading was found after the fact and carries no weight. Whether 2/3 and 2/9 can both be exact turns on bound-state quantum electrodynamics in muonium, a computation we leave to others. Failed attempts are reported, among them three mechanisms for lepton mixing and quark–lepton complementarity