Exact Non-Renormalisation of the Coulomb Charge Under Refinement: Why Refinement Invariance Cannot Fix a Non-Gaussian Coupling

Any programme that extracts a gauge coupling from a discrete substrate must define that coupling byan estimator acting on a truncated, refined geometry. We prove that the discrete Gauss law reproduces the continuum Coulomb potential exactly — on every shell, for every mesh ratio, and for arbitrary refinement sequences — if and only if the flux is evaluated on the sphere whose radius is fixed by Eq. (4). The criterion is per-shell and refers to no symmetry group, no lattice spacing and no dimensionful scale. In three dimensions the required radius is the geometric mean of the two shell radii; in \( d \) dimensions it is given by Eq. (3), and the three-dimensional statement is not the general one. Two consequences follow. (i) The enclosed static charge is exactly marginal: the refinement \( \beta \)-function of the Coulomb sector vanishes identically, and a measured drift is the closed-form bias of the estimator. Left-endpoint sampling returns \( \alpha_{\rm eff}/\alpha=q \) exactly, arithmetic-mean sampling returns \( 4q/(1+q)^2 \), and a finite-refinement extrapolation of the former is biased by 10% at refinement level \( k=4 \), two to three orders of magnitude above the level at which refinement-based agreements are usually quoted. (ii) A geometry-only estimator is exact only for the marginal power \( p=d-2 \) of a trial potential \( V\propto r^{-p} \): exactness and a vanishing anomalous dimension are the same condition, so exact refinement invariance of a static flux makes the flux directionexactly marginal, and a marginal direction does not select the coupling value. This is a statement about the static flux sector, not about the theory: a conserved current has zero anomalous dimension in an interacting theory, so nothing here forces the theory to be free. The no-go is stated with explicit hypotheses: it applies to static flux observables and to estimators that depend on shell geometry alone, and it does not apply when the estimator embeds dynamical fit information. We separate throughout the refinement flow of the discretisation from the renormalisation-group flow of the theory, since the two are routinely conflated, and give the operational null test, the Cartan-sector extension to non-Abelian groups with its conditions and proof, and the procedure for lattices without spherical shell decomposition.