A Topological Origin of Colour SU(3)

We give a topological/geometric origin for the gauge structure SU(3) of quantum chromodynamics within the Harvey complex-time manifold—a five-dimensional framework whose compactified imaginary-time axis τ ∈ S¹ = U(1) serves as a fibre. In this framework a particle’s electric charge is encoded by the phase fraction p_Q advanced per circuit of the τ fibre (Q = 2p_Q e), gauge return requiring 1/p_Q circuits. We first establish that this charge winding is a conserved topological invariant—mass-independent and strictly conserved under Lorentz boosts and gravitational potentials, with conserved charge equal to the Bohr–Sommerfeld loop action J = ∮ P_τ d(cτ) = 2πℏ—a prerequisite for binding fractional charge to a stable colour structure. For a quark p_Q = 1/3, gauge return needs three circuits passing through three equally spaced phase sites, which we argue are the three colours. On this basis we establish six claims: Colour = a Z₃ winding-site structure; A commensurability criterion = colour arises from the incommensurability of the charge-winding period (3 for quarks) with the universal spinor double-cover period (2), explaining why quarks carry colour while leptons (period 2, commensurate with spin) do not, with colour number = charge-winding period = 3; Z₃ → su(3) via the closure of the Weyl–Heisenberg (clock–shift) operator algebra under a complex structure; The geometric identity of the eight generators (two Cartan = relative site azimuths, six ladder = inter-site hops, non-Abelian because triangular hops do not commute); Gluons = the su(3) connection compensating the winding mismatch, with a Yang–Mills action obtained as the continuum limit of node-network plaquette holonomies and masslessness forced by gauge invariance; and Confinement = a winding-closure criterion—only net-integer (triality-0) colour windings close, have finite energy, and propagate freely. We show this topological criterion agrees term by term with the colour-singlet content of SU(3) representation theory (Haar projection), and that the leading group factor C_A = N = 3 and the antiscreening sign of asymptotic freedom are geometrically forced. As a conditional corollary we note that if a universal ceiling "1/|p_Q| < 4" holds, then k = 3 is the unique colour-bearing winding and the gauge group is forced to be exactly SU(3); we report honestly that the origin of this ceiling remains open (it cannot be inherited from a gravitational horizon, a definite negative result owing to a scale obstruction). As an energy-side corollary we identify the rest mass of colour-neutral bound states (baryons, glueballs) with the τ-precession of confined massless colour fields (M = E_conf / c², a geometrisation of “mass without mass”), the absolute scale (Λ_QCD) being non-derivable while the mechanism and the ratio M_glueball / M_p ≈ 1.5–1.8 are. This paper reproduces rather than modifies QCD and predicts no deviation from it.