From UD Theory to Electromagnetism: The Vector Projection and the Rigorous Emergence of Maxwell's Equations

We derive classical electromagnetism as the vector projection of UD theory. The dynamical scalar fields Ud and Dᵤ form a coupled Klein–Gordon system. The vector potential is constructed as the bilinear current A^μ = κ(Ud ∂^μ Dᵤ - Dᵤ ∂^μ Ud), where κ is a constant. The electromagnetic field strength is Fμν = ∂^μ A^ν - ∂^ν A^μ. The field equations of Ud and Dᵤ imply ∂_μ Fμν = 0, giving the source-free Maxwell equations. A massive scalar mode with mass m_Ψ = m₀/√2 remains as a UV signature. Coupling to the matter current is derived from the fundamental UD interaction term, yielding the full Maxwell equations ∂_μ Fμν = e j^ν with the electromagnetic coupling constant e expressed in terms of UD parameters. The Coulomb field of a static point charge is exactly reproduced. All derivations are self-contained and rely solely on the UD axioms.