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

==== 1. Photon–electron events occur at a spherical shell of radius r0r_0r0. The shell (the electron probability surface) is a 2D manifold (area 4πr024\pi r_0^24πr02) and has a thin thickness δ\deltaδ. The shell separates inner concave (negative) and outer convex (positive) charge surfaces — this is where charge lives in your picture. ====
# A single event deposits a characteristic quantum of energy E0E_0E0. Dimensionally sensible choice:

E0∼ℏcr0E_0 \sim \frac{\hbar c}{r_0}E0∼r0ℏc
(heuristic: ℏ∼E0Δt\hbar \sim E_0 \Delta tℏ∼E0Δt with Δt∼r0/c\Delta t \sim r_0/cΔt∼r0/c).
# Events occur at rate ν\nuν per shell. (This is the crucial free parameter. You suggested Planck time earlier — but that makes the derivation circular because the Planck time depends on GGG.)
# For a thin shell of thickness δ\deltaδ, the local volumetric energy density contributed by a single shell in steady state (if each event energy is deposited and stored for some characteristic time τ\tauτ or if the shell is a continuous source) can be written as power-deposition-per-volume:

power density P  =  natom E0 ν,\text{power density } \mathcal{P} \;=\; n_{\rm atom}\,E_0 \,\nu,power density P=natomE0ν,
where natomn_{\rm atom}natom is number density of atoms (later we’ll scale to cosmic number density). If energy is stored for a time τ\tauτ, the steady energy density is u=Pτu=\mathcal{P}\tauu=Pτ. Alternately, if the deposition forms a thin-surface energy density stored in thickness δ\deltaδ, a local energy density estimate is

ushell∼E0 ν4πr02 δu_{\rm shell} \sim \frac{E_0\,\nu}{4\pi r_0^2\,\delta}ushell∼4πr02δE0ν
per shell.
# Curvature scaling (order-of-magnitude) from Einstein:

R∼G uc4.R \sim \frac{G\,u}{c^4}.R∼c4Gu.
On a local scale we can set R∼1/r02R\sim 1/r_0^2R∼1/r02 as a characteristic curvature, but for cosmology we use the Friedmann relation.