Hyperbolic Rapidity Parametrization of the Coulomb Field: Gaussian Envelope, Two-Regime Electrostatics, and the Fine-Structure Constant as a Rest-Frame Ratio

We examine whether the hyperbolic rapidity parametrization developed in earlier work for the interior metric of a Schwarzschild black hole can also parametrize the electrostatic field of a charged particle described by a Gaussian envelope. The construction is motivated by the observation that the Lorentz-contracted width of a Gaussian wavepacket, σ(φ) = σ₀ sech φ, has the same functional form as the interior metric of a Schwarzschild black hole, r/rₛ = sech² β. Treating the Compton width σ₀ as a boundary between an exterior Coulomb regime and an interior regime, we obtain a two-regime electrostatic model whose interior profile is algebraically equivalent to that of a uniformly charged sphere. The electrostatic self-energy of this configuration is (3/5) α mₑ c², where α is the fine-structure constant. We also observe that the ratio of the classical electron radius rₑ = α σ₀ to the dynamical Compton horizon σ(φ) equals α cosh φ in a frame of rapidity φ, a purely kinematic identity that reduces to α at rest. Finally, we record a differential identity satisfied by ψ(β) = sech β, whose second-order form is algebraically identical to the Ginzburg–Landau field equation. The mathematical identities are exact; the physical interpretations are offered as conjectures.