General Relativity and Newton's Constant with Scretching–Planck Matter: Four Routes and a Geometric Identity II

This paper writes the stress tensor of a minimally coupled scalar field in Scretching–Planck (SP) density and phase variables and uses it to compare four approaches to gravity. MICROSCOPE constrains composition-dependent free fall. A bound on anomalous coupling to the SP quantum stress also requires a model of how that stress contributes to the test bodies. The four approaches use massless spin-2 self-coupling, local horizon thermodynamics, a regulated one-loop matter action, or five-dimensional gravity. In the induced-gravity example, 100 minimally coupled real scalars reproduce the measured \(G\) when the proper-time cutoff scale is set to \(7.50\times10^{18}\) GeV. Under the stated normalization, fixed-radius Kaluza–Klein reduction gives \(G=\alpha R_{\mathrm{KK}}^2c^3/(4\hbar)\), where \(R_{\mathrm{KK}}\) is the compactification radius. Separately, the paper derives the conformal curvature identity\[Q_{\mathrm{SP}}=\frac{\hbar^2}{2m}\,\xi_n\bigl(\Omega^2\widetilde R-R\bigr),\qquad\Omega^2=(P/P_*)^{2/(n-2)},\qquad\xi_n=\frac{n-2}{4(n-1)}.\]The hydrogen \(1s\) state provides a flat-space check. Each route requires input beyond the SP matter sector to fix \(G\). Dirac’s dimensionless number, \(N_D=\alpha\hbar c/(Gm_em_p)=2.269\times10^{39}\), gives the ratio of the electrostatic to gravitational force magnitudes between an electron and a proton.