Scretching–Planck Density and Phase in Five Dimensions II: Kaluza–Klein Electromagnetism, Charge from Phase Circulation, and Gravity from the Laboratory to Dark-Matter Cores

This paper applies the Scretching–Planck (SP) density–phase representation, \(\psi=\sqrt{P}\,e^{iS/\hbar}\), to curved spacetime and a five-dimensional Kaluza–Klein model, taking the gravitational field equations as inputs. The polar form of the Klein–Gordon equation yields a continuity equation and a relativistic Hamilton–Jacobi equation. Its eikonal limit gives geodesic motion. On the prescribed relativistic backgrounds, the paper recovers the standard predictions for Mercury’s perihelion advance, solar light deflection, and gravitational redshift. For neutrons above a mirror, the calculated ground-state energy is \(1.407\) peV. The \(1\to3\) and \(1\to4\) transition frequencies are \(462.9\) and \(647.1\) Hz, compared with qBounce measurements of \(464.1(1.2)\) and \(648.8(1.6)\) Hz, respectively. The normalized one-particle Schrödinger–Newton ground state has eigenvalue \(-0.16277\,G^2m^5/\hbar^2\) and obeys the half-mass relation \(Mr_{1/2}=3.925\,\hbar^2/(Gm^2)\). In the Kaluza–Klein model, single-valuedness around the compact circle quantizes momentum in the extra dimension. Matching the first momentum mode to the elementary charge gives \(q_n=ne\). With rationalized electromagnetic units, \(\alpha=4\ell_P^2/R^2\) gives \(R=23.41\,\ell_P\) and a first charged-mode energy scale of \(5.215\times10^{17}\) GeV. Projecting the five-dimensional geodesic onto four-dimensional spacetime recovers the Lorentz force. Eight solved problems are included.