At a nonzero temperature T, a constant field $overline{A}_0 sim T/g$ generates nontrivial eigenvalues of the thermal Wilson line. We discuss contributions to the free energy of such a holonomous plasma when the coupling constant, $g$, is weak. We review the computation to $sim g^2$ by several alternate methods, and show that gauge invariant sources, which are nonlinear in the gauge potential $A_0$, generate novel contributions to the gluon self energy at $sim g^2$. These ensure the gluon self energy remains transverse to $sim g^2$, and are essential in computing contributions to the free energy at $sim g^3$ for small holonomy, $overline{A}_0 sim T$. We show that the contribution $sim g^3$ from off-diagonal gluons is discontinuous as the holonomy vanishes. The contribution from diagonal gluons is continuous as the holonomy vanishes, but sharply constrains the possible sources which generate nonzero holonomy, and must involve an infinite number of Polyakov loops.
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