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 \(7.50×10¹⁸\) GeV. Under the stated normalization, fixed-radius Kaluza–Klein reduction gives \(G=α RKK^2c^3/(4)\), where \(RKK\) is the compactification radius. Separately, the paper derives the conformal curvature identity\[QSP=^2/2m\,ξ_n(Ω^2 R-R),Ω^2=(P/P_*)2/(n-2),ξ_n=n-2/4(n-1).\]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, \(N_D=α c/(Gm_em_p)=2.269×10³⁹\), gives the ratio of the electrostatic to gravitational force magnitudes between an electron and a proton.