Scretching–Planck Density and Phase in Five Dimensions I: 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 five-dimensional Kaluza–Klein geometry, taking the gravitational field equations as inputs. The covariant continuity and quantum Hamilton–Jacobi equations recover geodesic motion in the classical limit. The prescribed relativistic backgrounds give Mercury’s perihelion advance of \(42.98\) arcseconds per century, solar light deflection of \(1.751\) arcseconds, and a fractional gravitational redshift of \(2.45\times10^{-15}\) over \(22.5\) m. For neutrons above a mirror, the SP calculation gives a ground-state energy of \(1.407\) peV and transition frequencies of \(462.9\) and \(647.1\) Hz, compared with the qBounce values \(464.1(1.2)\) and \(648.8(1.6)\) Hz. The self-gravitating Schrödinger–Newton ground state has eigenvalue \(-0.1628\,G^2m^5/\hbar^2\) and satisfies the mass–radius relation \(Mr_{1/2}=3.920\,\hbar^2/(Gm^2)\). For a particle mass of \(10^{-22}\) eV/\(c^2\), this relation gives kiloparsec-scale cores at the core masses considered. In the five-dimensional model, single-valuedness around the compact circle quantizes momentum in the extra dimension. Matching one unit of that momentum to the elementary charge gives \(q=ne\). The four-dimensional gauge transformation corresponds to a shift of the compact coordinate, and the projected geodesic reproduces the Lorentz force to a reported numerical accuracy of \(9\times10^{-14}\). In the model’s normalization, \(\alpha=16\pi\ell_P^2/R^2\) gives \(R=83.0\,\ell_P\) and a first charged-mode energy of \(1.47\times10^{17}\) GeV. Eight solved problems are included.