The Coulomb singularity as a real blow-up: the source of a point charge carried by a projective plane

We replace the point O carrying a point charge q by its real blow-up: O becomes the surface Σ of lines through O, a real projective plane. The ball B minus O becomes a compact manifold M, orientable, with boundary the sphere S; Σ is an interior surface of M, non-orientable and one-sided. The Coulomb field is unchanged off Σ. The flux 2-form, unbounded near O for the Euclidean metric, lifts to a form that is bounded for every smooth metric on M, with no continuous extension to Σ; its exterior derivative in the sense of currents is carried by Σ: this is the source, a uniform density of total mass q on Σ. Gauss's law becomes Stokes's theorem on M. Since Σ is non-orientable, the charge is the class of a density, H²(Σ ; o) ≅ ℝ, where o is the orientation local system of Σ, which the Thom isomorphism and the connecting morphism of the pair (M, ∂M) relate to the flux through S. We compare with the spherical blow-up, which replaces O by the sphere of oriented directions. The construction is a geometric reformulation of the point charge: it resolves neither the divergence of the self-energy nor the dynamics of the charge, and Maxwell's equations do not select it.