Emergent Time from Helical Motion on a Compact Matter Dimension: Colour, Weak Isospin and the Charges of a Family from a Double Rotation and a Knotted Line

In a framework with three-dimensional space, one compact matter dimension and no external time coordinate, built on two axioms, earlier papers derived relativistic kinematics, gravitation, electromagnetism and the Schrödinger and Dirac equations. Here we ask what the strong and weak interactions are in the same framework, and we report every test we ran, including those that failed. The helical motion of space that a line draws, locked by the axioms to turn its fibre angle and its transverse spatial plane at one rate, is an isoclinic double rotation. Of the two halves of so(4) = su(2) ⊕ su(2), the chirality clause of the axioms makes the self-dual half a gauge direction acting on one chirality only: a weak SU(2). The other half supplies the complex unit of colour. Colour is the relabelling of the three passes of a baryon’s line, read as a trefoil, on the knot’s own axes. One construction then yields SU(3) × SU(2) ×U(1)Y with the representations (3,2), (1,2), (3,1), (1,1) of one family, a right-handed neutrino included. All eight charges follow from one rule, Q = JD −s: the twist of the fibre minus the share of the line. The four anomalies cancel. Three structural facts the Standard Model takes as given also follow. The weak force acts on one chirality only. The strong force conserves parity, because colour uses the same generators for both chiralities. A junction of lines admits at most three legs, because motion at the speed of light leaves only three spatial directions there; this matches the trivalent colour junction, and lattice data exclude the variant with straight orthogonal legs at 9σ. Colour commutes with rotations and boosts. The cross-section mechanism that gave Maxwell’s equations gives Yang–Mills, and it relates the string tension to the radius of the flux tube: 0.187 fm against the lattice penetration length 0.175 ± 0.006 fm. Two predictions are sharp: a right-handed neutrino exists, and no WR boson exists. What fails is stated as plainly. The coupling values are not derived: fixed at the Planck scale, the framework’s hypercharge relation gives sin2 θW = 0.189 against 0.231. The framework has no intermediate scale of its own, and it does not yet give the size of the proton.