A Geometric Closure of the Kaluza Unification

We construct a fully consistent five–dimensional model in which four–dimensionalEinstein–Maxwell theory and quantum charge fields arise as the low–energy effectivetheory of pure gravity plus matter on a circle bundle. Starting from a five–dimensionalEinstein–Hilbert action with a compact S1 fiber, we perform a complete Kaluza–Kleinreduction in which the four–dimensional gauge field is encoded in the rotational componentsof the higher–dimensional metric. We retain the radion field describing thebundle radius and show how a generic stabilizing potential leads to a fixed vacuumexpectation value, thereby justifying the truncation to Einstein–Maxwell at low energies.We then introduce five–dimensional fermions and scalars, derive the Kaluza–Kleinspectrum, and show that four–dimensional electric charge is identified with quantizedmomentum along the compact direction. Einstein’s equations arise on the projectedbase once the Kaluza fiber is identified as the quantum phase:A_K = −i(ψ, δψ)The four–dimensional Newton constant, gauge coupling, and Kaluza–Klein massscale are expressed in terms of the underlying five–dimensional parameters and thestabilized radius. The resulting theory is predictive and falsifiable.