Editing Openai/6897769e-4ee4-800f-aba5-69cca34f701c (section)

=== Dimensional analysis and diagrammatic scaling suggest ===

κ~∼e4m2×C,\tilde{\kappa} \sim \frac{e^4}{m^2}\times\mathcal{C},κ~∼m2e4×C,
where mmm is the fermion mass controlling the loop suppression, and C\mathcal{C}C encodes coherence/phase factors and numerical loop integrals (order unity factors, π\piπs, etc).

Match to Einstein linearized field equation in momentum space:

⟨hh⟩∼16πGeffk2 Π(spin-2).\langle h h\rangle \sim \frac{16\pi G_{\rm eff}}{k^2}\, \Pi^{(\text{spin-2})}.⟨hh⟩∼k216πGeffΠ(spin-2).
Therefore a rough identification:

Geff∼e416π m2 C.G_{\rm eff} \sim \frac{e^4}{16\pi\,m^2}\,\mathcal{C}.Geff∼16πm2e4C.
Crucial note: this is extremely schematic. The realistic value of Newton’s GGG is tiny; obtaining it from e4/m2e^4/m^2e4/m2 for electron mass gives numbers many orders off unless C\mathcal{C}C involves huge coherence factors (large NNN effects, large-area/surface amplification, or cosmological mode counting). That’s where the QAT geometric/coherent-surface hypothesis enters: many simultaneous coherent exchanges (summing over huge numbers of atoms / boundary modes) could amplify the effective coupling.