The colour flux and \(\sqrt{\sigma}\)

The colour flux on the prism

The colour flux is a vector on the colour edges. On the cube every vertex meets one \(z\)-edge, so closure forces \(b_s=0\) — no confining contour. The prism adds \(\tau\), opening the \(z\tau\)-face, where the flux closes as the field strength \(F_{z\tau}\), an alternating four-cycle with \(|b_s|^{2}=4\).

Spectral decomposition over the edge operator

The flux spreads over the edge operator \(M_1=L_1+I\). Its projections give the propagator weight

\[ G_{b_s}=\frac{7/24}{1}+\frac{1/4}{5}+\frac{1/3}{7}+\frac{1/8}{9}=\frac{127}{315}=0.403175, \]

and the edge Sakharov trace, led by the seventeen zero-modes of the edge kernel, is

\[ \Sigma_E=\operatorname{Tr} M_1^{-1}=\frac{17}{1}+\frac{4}{3}+\frac{6}{5}+\frac{4}{7}+\frac{1}{9}=\frac{6368}{315}=20.21587, \]

\[ \Pi_{b_s}=\frac{G_{b_s}}{\Sigma_E}=\frac{127}{6368}=0.019943. \]

The master equation

The flux is a vector (\(p=2\)) carrying fundamental colour charge threading the octet, \(\Theta_{b_s}=3/8\). Through the master equation with the spectral weight squared,