From Geometry to Clocks to Scales Rotor Circulation and the Fermion Mass Spectrum

We define mass as the modulus-squared of a complex rotor —a quantised circulation in a source/sink medium— 𝑚 = (𝑘/2𝑐2) 𝑧𝑧‾, the elastic energy of a trapped vibration, in which the rotor phase cancels so that the Lorentz invariance of mass is manifest. From this single definition the fermion mass spectrum acquires a unified structure with four principal results. First, the Koide constant is a scale-invariant angle, 𝑄 = 1/(3cos2𝜙), so its constancy is geometry and its value the single condition 𝜙 = 45∘; each of the four sectors is then generated by one clock, one phase, and one amplitude, and every fermion mass is reproduced from its sector triple (charged leptons to 0.01%, up quarks to 0.5%, down quarks to 0.1%, neutrinos exactly in their inverse form). Second, the four clocks descend from a single clock at the grand-unification scale: read in its proper inverse form the neutrino clock is that scale, 𝑀 ≃ 2 × 1016 GeV — emergent, not inserted, and coinciding with the supersymmetric unification scale — and the charged-lepton clock descends from it through one chain, dimensional transmutation to 𝛬QCD, the rotor soliton to the nucleon, and the forced 𝑁𝑐 = 3 to the per-rotor share 𝑚𝑁/𝑁𝑐 = 313 MeV, matched to better than half a percent. Third, the mechanism that carries a scale across the colour boundary is proven, not assumed: the stiffness multiplies a modulus that colour transformations leave exactly invariant (colour-blindness), and in the vortex ontology the free lepton and the confined quark share that scale by Kelvin’s circulation theorem — circulation is a topological invariant, the same free or bound. Assembling these, the charged-lepton sector reduces to a closed form, 𝑚𝑘 = (𝑚𝑁/𝑁𝑐)[1 + √2cos(2/𝑁2 + 2𝜋𝑘/𝑁𝑐)]2: three masses from the nucleon mass, the forced colour count, and one balance condition (𝑏/𝑎 = √2, shown to be the equal-energy point of the two circulation modes). Fourth, the sector phase is not an input but is fixed by gauge quantum numbers: it has the closed form 𝜃 = 𝑢(1 − 𝑢) (the same balance law as √2), with 1/𝑢 = 𝑁 1+𝑇3+1/2 (𝐶 (𝟑))𝑞 built from the colour count 𝑁 = 3, 𝑐 2 𝑐 the weak isospin 𝑇3, and the 𝑆𝑈(3) Casimir 𝐶2(𝟑) = 4/3, giving the clean rationals 𝜃 = 2⁄9 , 3⁄16 , 8⁄81 , 11⁄144 for the lepton, down, neutrino, and up sectors to 0–3%. A consequence reaches inter-sector mixing: because the phase formula forces different sectors to different phases, the up and down (and the charged-lepton and neutrino) generation triangles are necessarily misaligned, so CP violation is forced to exist in both the CKM and PMNS sectors — nonzero precisely because the sectors carry different quantum numbers, with the quark misalignment exactly 𝜃𝑑 − 𝜃𝑢 = 1/9. The mixing angle magnitudes, by contrast, are not geometric: the three-fold symmetry that gives the clean spectrum forbids differential mixing, so the observed hierarchy is a mass-ratio (Fritzsch) effect. We are equally precise about the boundary. The overall scale remains an input, with the four clocks sharing no universal geometric base. The up sector, riding the electroweak vacuum value through the order-unity top Yukawa, is not an isolated exception but the 𝑇3 = + 1⁄2 branch of a weak-isospin selection rule: the up-type sectors (up, neutrino) ride the high electroweak-vev scale and the down-type sectors (down, charged lepton) the low QCD/rotor scale, so the same isospin flag that sets the phase also selects which condensate the clock couples to, with only the two condensate scales remaining inputs. Behind the phase formula the one residual step is to prove the family-holonomy multiplier is the colour Casimir rather than a coincidence of equal value. The underlying phenomenology —the circulant, the sector pattern, the seesaw extension, the inverse relation— is due to Koide, Foot, Brannen, Żenczykowski, and Rivero, whom we credit throughout; ours are the rotor definition of mass, the hydrodynamic reading of its scale through the unification-scale chain, the proofs above, and a boundary map that has reduced every remaining question to a specific calculation. The honest one-line summary: the geometry of circulation forces the form of the mass spectrum and, through the gauge quantum numbers it carries, the sector phase; the overall scale alone remains medium dynamics and input.