Geometric Renormalization of the Electron Charge via Einstein-Born-Infeld Optical

The Williamson-van der Mark (WvdM) semi-classical model describes the electron as a toroidal photon soliton, successfully linking mass to electromagnetic self-energy and spin. However, under the assumption of a linear vacuum, the model predicts a bare topological charge Q0 ≈ 1.475×10−19 C, deviating by approximately 8% from the experimental elementary charge e. In this Letter, we resolve this discrepancy by embedding the soliton model within the Einstein-Born-Infeld (EBI) framework. We demonstrate that the vacuum nonlinearity in the strong-field region of the soliton induces a vacuum birefringence, creating an effective refractive index n(r) > 1. This dielectric effect reduces the local characteristic impedance of the vacuum, Zeff. Imposing the conservation of the soliton’s topological winding number, we derive a geometric rescaling factor χ = ⟨n⟩ that relates the bare charge to the observed charge via the impedance matching condition e = χQ0. Numerical analysis using the regular Bardeen metric ansatz reveals that the experimental charge e is recovered when the vacuum nonlinearity scale is β ≈ 0.196 relative to the Compton radius. This suggests the elementary charge is an emergent quantity renormalized by the non-linear optical geometry of the electron’s internal structure.