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

==== You suggested treating a single photon–electron event as depositing a characteristic quantum of action/energy on a spherical shell of radius r0r_0r0. I made the minimal, physically-plausible assumptions needed to get a dimensional estimate: ====
* Use the Planck quantum of action explicitly as ℏ=h/(2π)\hbar = h/(2\pi)ℏ=h/(2π).
* Take a characteristic energy per event of order

E0∼ℏcr0,E_0 \sim \frac{\hbar c}{r_0},E0∼r0ℏc,
which is dimensionally an energy (J). This comes from the action–time idea ℏ∼E0Δt\hbar \sim E_0 \Delta tℏ∼E0Δt with Δt∼r0/c\Delta t \sim r_0/cΔt∼r0/c (the light-crossing time of the radius).
* Suppose that energy E0E_0E0 is effectively concentrated in a region of volume of order the sphere's volume V∼43πr03V \sim \tfrac{4}{3}\pi r_0^3V∼34πr03. That gives a characteristic energy (volume) density

u∼E0V∼ℏc/r043πr03=3ℏc4πr04.u \sim \frac{E_0}{V} \sim \frac{\hbar c / r_0}{\tfrac{4}{3}\pi r_0^3}
= \frac{3\hbar c}{4\pi r_0^4}.u∼VE0∼34πr03ℏc/r0=4πr043ℏc.
* Use the Einstein field-equation scaling (order of magnitude) that a curvature scale RRR is produced by an energy density uuu via

R∼G uc4.R \sim \frac{G\,u}{c^4}.R∼c4Gu.
Take a geometric curvature scale R∼1/r02R\sim 1/r_0^2R∼1/r02 (curvature from a length scale r0r_0r0). Rearranging gives

G∼Rc4u∼c4/r02u.G \sim \frac{R c^4}{u} \sim \frac{c^4 / r_0^2}{u}.G∼uRc4∼uc4/r02.
Substituting the expression for uuu gives a compact estimate:

 Gest  ≃  4π3 c3r02ℏ .\boxed{\, G_{\rm est} \;\simeq\; \frac{4\pi}{3}\,\frac{c^3 r_0^2}{\hbar}\, }.Gest≃34πℏc3r02.
(You can check units quickly: c3r02/ℏc^3 r_0^2/\hbarc3r02/ℏ has dimensions [L3M−1T−2][L^3 M^{-1} T^{-2}][L3M−1T−2] — the same as GGG.)