Overview
This note records a single carrier hypothesis that would relate three otherwise separate facts of the fermionic sector: the non-observability of individual colours in the confined regime, the appearance of three massive generations, and the unit-dispersion (Koide) shape of the charged-lepton square-root masses. The proposal is one complex three-phase carrier $z_g = S + A\,e^{i(\delta + 2\pi g/3)}$, $g = 0,1,2$, read in two ways.
Its pure-phase part sums to zero over the three cube-root phases ($1 + \omega + \omega^2 = 0$), a collective and exact cancellation identified with a colour-neutral class; its real part $S + A\cos(\delta + 2\pi g/3)$ is the square-root mass profile of the unit-dispersion note.
The singlet amplitude as the master dial
Reading the carrier through the singlet (democratic) amplitude $S$ organises the three facts as three settings of one dial together with one phase.
| regime | carrier reading | phenomenon |
|---|---|---|
| $S = 0$ | pure phase, $\sum_g e^{i\phi_g} = 0$ | colour-neutral (confined) |
| $S \neq 0$ | real offset, positive values | massive sector opens |
| $S = \lVert\text{doublet}\rVert$ | singlet–doublet equipartition | Koide dispersion ($\mathrm{CV}=1$) |
| $\delta$ near a node | one real amplitude annulled | hierarchy (light electron) |
For three cosines $120^\circ$ apart, $\langle\cos^2\rangle = 1/2$, so the oscillating component has effective norm $A/\sqrt2$; equality of the differentiating and democratic norms, $A/\sqrt2 = S$, is the single condition $\mathrm{CV}(r) = 1$, the unit-dispersion form of Koide (angle $45^\circ$ to the democratic axis). The whole hypothesis compresses to one sentence: colour is the pure-phase sum-zero regime, mass is the real-offset singlet-balanced regime, hierarchy is the residual phase placement relative to a node.
A candidate admissibility filter
Decomposing the real profile into its democratic and differentiating parts gives $\lVert r_{\mathrm{dbl}}\rVert / \lVert r_{\mathrm{sing}}\rVert = \mathrm{CV}(r)$, so the Koide value is the balanced condition $\lVert r_{\mathrm{dbl}}\rVert = \lVert r_{\mathrm{sing}}\rVert$. This is not a positivity edge (the continuous envelope is positive only up to $\mathrm{CV} = 1/\sqrt2$, and the positive cone reaches $\mathrm{CV} = \sqrt2$; the Koide value lies strictly between). The candidate filter is a discrete capacity balance, $\lVert r_{\mathrm{dbl}}\rVert \le \lVert r_{\mathrm{sing}}\rVert$, saturated at $\mathrm{CV} = 1$: the differentiating generation doublet may spend at most the capacity carried by the democratic singlet. The open question is which projective capacity or entropy principle forbids the excess; until such an object is exhibited, the filter is a structured open test.
The construction charge sheet
A carrier hypothesis of this kind is a front of construction, not a result. The leptons are not to be read as hidden coloured composites: they are observed as exact colour singlets and pointlike. The safe statement is that the lepton sector retains a projective trace of the ternary carrier while its colour component is exactly neutralised in the observable. Any realisation must discharge, at least:
- produce an exact colour singlet (the sum-zero regime, not an approximate one);
- keep three discernible generations after neutralisation;
- reproduce the leptons' weak isospin and hypercharge assignments;
- preserve gauge-anomaly cancellation generation by generation;
- explain the selection rule: why the quark regime keeps an observable colour residue while the lepton regime cancels it exactly.
Items 4 and 5 are where the real work sits. In particular the colour-to-generation breaking the carrier would require along the neutralisation route is the one already shown closed in the corpus by the exact colour degeneracy and the forgetful colour projection: the breaking must come from elsewhere.
Status
The carrier's exact pure-phase sum-zero and the coincidence of its real part with the square-root mass profile are exact algebra. The organisation of colour, mass, and Koide dispersion as three settings of the singlet dial $S$ plus one phase $\delta$ is a reading, internally consistent and economical, but not derived. This note does not fix the equipartition value $A = \sqrt2\,S$, does not fix the node phase $\delta$, and does not derive the Standard-Model quantum numbers or anomaly cancellation. Whether the projection entropy of the non-injective map, or a related variational principle, forces the balanced setting of the dial is the same open test recorded for the unit-dispersion target, now phrased on the carrier. The working conclusion is only that the three facts can be written as one carrier read three ways — a candidate bridge, held at the level of a speculative outlook.
Relation to the Cosmochrony programme
This note belongs to the fermionic matter sub-programme. It combines the confined colour triplet of the non-injective colour projection with the square-root mass profile of the unit-dispersion note, proposing them as two regimes of one three-phase carrier. It is kept separate from the clean unit-dispersion result precisely because it is broader and speculative.
References
Jérôme Beau. Ternary Phase Carrier: Colour Confinement, Generation Splitting, and Koide Dispersion. Working paper, 2026. 10.5281/zenodo.21115977