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

==== Assumptions (QAT style) ====
* Characteristic shell radius: r0r_0r0 (electron probability shell; e.g. Bohr radius or some other QAT radius).
* Characteristic event energy: choose the dimensionally sensible form you suggested earlier, e.g.

E0∼ℏcr0E_0 \sim \frac{\hbar c}{r_0}E0∼r0ℏc
(the light-crossing / action estimate).
* Event rate (events per second per shell): ν\nuν. You suggested the Planck time tPt_PtP as a natural short timescale, so ν∼1/tP\nu \sim 1/t_Pν∼1/tP. (Important: tPt_PtP depends on GGG. I discuss that circularity below.)
* Energy deposition is concentrated on a thin shell of thickness δ\deltaδ (e.g. δ∼ℓP\delta\sim\ell_Pδ∼ℓP, the Planck length, if you want the extreme thinness).

Surface power flux (W/m²) — energy per unit time per unit area:

Φ  =  E0 ν4πr02.\Phi \;=\; \frac{E_0\,\nu}{4\pi r_0^2}.Φ=4πr02E0ν.
If this is an electromagnetic-like flux, the associated local energy density just outside the shell (for radiation) can be estimated by dividing flux by the speed of propagation, ccc:

urad  ≈  Φc  =  E0 ν4πr02 c.u_{\text{rad}} \;\approx\; \frac{\Phi}{c}
\;=\; \frac{E_0\,\nu}{4\pi r_0^2\,c}.urad≈cΦ=4πr02cE0ν.
If instead you want a true volumetric energy density inside the shell (if the energy is lodged into a shell of small thickness δ\deltaδ), then

ushell  ≈  E0 ν4πr02 δ.u_{\text{shell}} \;\approx\; \frac{E_0\,\nu}{4\pi r_0^2 \,\delta}.ushell≈4πr02δE0ν.
(Units: J/s divided by m²·m gives J/m³ — energy per unit volume. This is appropriate if the shell stores energy in a small thickness δ\deltaδ.)

Either estimate is physically plausible; pick which picture best matches the QAT idea (radiation flux vs. localized surface energy). For linking to curvature (Einstein eqns) you want a bulk energy density uuu (J/m³), so the shell thickness formula is convenient.