Abstract On any stationary, shear-free, non-rotating null three-surface Σ with compact spacelike cross-sections S ∼= S2, we consider the spatial SU(2) connection in time gauge and its curvature. The oriented tangent plane T S picks out an su(2) axis τˆ generating rotations of TS. We show that ∫∫ () λ(S):=−4 TrFτˆ = dΓ=2πχ(S), SS where F is the SU(2) curvature, Γ is the SO(2) Levi–Civita connection of the induced metric on S, and χ(S) is the Euler characteristic. Thus for S ∼= S2 we obtain λ(S) = 4π. The definition is gauge-invariant and does not depend on the particular time gauge once expressed as 2πχ(S). On a stationary, shear-free, non-rotating null surface, the intrinsic metric on Su is invari- ant along the generators, and the associated Γu changes at most by a gauge transformation. As a consequence, λ(Su) is independent of the cut Su. If in addition one chooses a spin structure along the null generators that is either periodic (t = 1) or anti-periodic (t = 2), and identifies this with the weight-t representation of the reduced U(1) bundle, the corresponding flux is λt(S) = tλ(S) = 4πt, and in particular λ2(S) = 8π. We briefly discuss how this classical normalization echoes the ubiquitous 8π factors in loop quantum gravity treatments of isolated-horizon entropy, without claiming a derivation of the quantum result.