Maxwell from the Unified 4D Toy Model: Localized 5D Gauge Dynamics, Brane Reduction, and KK Corrections

We derive brane-effective electromagnetism from a localized 4+1 Maxwell sector embedded in the unified toy model. The gauge field is a bulk potential AM with kinetic term −Z(w)FMNFMN/(4μ0), where the transverse profile Z(w) localizes gauge dynamics near the brane at w = 0. Varying the action yields the localized Maxwell equations ∂M(ZFMN) + ξ−1∂N(∂·A) = μ0JN, the Bianchi identities, and an exact divergence consistency identity linking the gauge condition to current conservation. For brane-localized conserved sources, axial gauge, and a brane-dominant zero mode, integrating over w reduces the theory to standard 3+1 Maxwell with an effective coupling μ0eff = μ0/Zint, where Zint = ∫−∞∞Z(w) dw. For a Gaussian profile Z(w) = e−w2/λ2, μ0eff = μ0/(λ√π) and the transverse Sturm–Liouville problem yields a discrete KK tower with masses mn2 = 2n/λ2 and a brane parity selection rule. The resulting static brane potential is Coulomb plus Yukawa-suppressed corrections with a fixed coefficient pattern; the leading deviation scales as 1/2 e−2r/λ. For time-dependent sources, the brane-to-brane response depends only on the Lorentz scalar k2 (with the retarded prescription), ensuring brane Lorentz covariance; massive modes add causal tail terms inside the light cone. Finally, a minimally coupled brane matter model yields a conserved Noether current and an off-shell identity that supplies the Maxwell source consistency condition. In the updated charge ontology, a fixed defect branch carries q* = ηQe* with ηQ = ±1, while canonical normalization of the reduced zero mode gives qeff = q*/√Zint; circulation is not used to define electric charge, and the zero-mode Maxwell limit suppresses the mixed (Aw, Jw, Fμw) sector only as a controlled far-field approximation. All principal identities and reductions are backed by a compact Wolfram Language referee suite.