7 The n=3 Fermion: Quarks and Confinement

7.1 Fractional occupancy

For a three-site equal superposition, each site carries probability |ψk|2 = 13. This is the per-site occupancy of a quark in a colour triplet. The value 13 requires no postulate: it is the Born-rule probability of finding the fermion on any one colour site.

7.2 Colour structure (m = 1)

For winding m=1, the compression residual has off-diagonal entries with equal magnitude:

|B  | = 1  for all colour pairs i ⁄= j.
  ij   3

This is exact colour symmetry, emerging from the equal-superposition constraint and Theorem 1. No SU(3) symmetry is imposed.

What Standard Model language calls a “gluon” is the name given to the correlation between quark colour changes. The mixed symmetry of B for n=3, m=1 means this correlation has no self-contained symmetry class — see Section 8.

7.3 Scalar sector (m = 0)

For winding m=0, the residual is purely symmetric: a spin-0, colour-neutral scalar with three-fold internal structure. Whether this corresponds to the physical Higgs boson requires derivation of the mass generation mechanism, which is open.