The Field Potential W: A Constructive Foundation for Classical Electromagnetism

This paper presents a constructive reformulation of steady-state classical electromagnetism grounded in a single primitive vector field \mathbf{W}, termed the Field Potential. Every electric charge possesses an associated Field Potential, established outward at speed c upon the charge's creation and thereafter co-moving rigidly with it. From the defining relation \psi \equiv -\nabla \cdot \mathbf{W}, all electromagnetic fields are derived as geometric projections onto the radial propagation direction and the transverse relative velocity. Three constraints on \mathbf{W}—irrotationality, radial alignment, and transverse uniformity—are derived from a minimality principle and interaction geometry, not postulated independently. The framework reproduces Coulomb's law, derives the effective electric field \mathbf{E}_{\mathrm{eff}} = \mathbf{E}_{\mathrm{s}} / \gamma_{\perp} from a velocity triangle encoding retardation self-consistently, and obtains the Lorentz factor as the geometric ratio c / c_q without spacetime kinematics. The magnetic field emerges as a derived quantity \mathbf{B} = (\mathbf{v}_{\perp} \times \mathbf{E}_{\mathrm{s}}) / c^2, with \nabla \cdot \mathbf{B} = 0 proved as a theorem. The Helmholtz decomposition unifies scalar and vector potentials as projections of \mathbf{W}, reducing gauge freedom to a single constant shift. The steady-state Maxwell equations are derived from superposition and Stokes' theorem, not postulated. The potential energy of a moving test charge is U = \gamma_{\perp} q(V - \mathbf{A} \cdot \mathbf{v}_{\perp}), vanishing as v_{\perp} \to c. The paper argues that classical electromagnetism can be founded on a more economical and mechanistically transparent basis than the conventional formulation, with the observer playing no role in the ontology and the Lorentz factor emerging from field-encounter geometry.