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

==== Einstein scaling (order of magnitude): ====

R∼G uc4,R \sim \frac{G\,u}{c^{4}},R∼c4Gu,
where RRR is a curvature scale. For a process with characteristic length r0r_0r0, a natural curvature estimate is R∼1/r02R\sim 1/r_0^2R∼1/r02. Rearranging:

G∼R c4u∼c4/(r02)u.G \sim \frac{R\,c^4}{u} \sim \frac{c^4/(r_0^2)}{u}.G∼uRc4∼uc4/(r02).
Use the shell volumetric density ushell=(E0 ν)/(4πr02δ)u_{\text{shell}}=(E_0\,\nu)/(4\pi r_0^2 \delta)ushell=(E0ν)/(4πr02δ). Substitute:

G  ∼  c4/r02(E0 ν)/(4πr02δ)  =  4πc4δE0 ν.G \;\sim\; \frac{c^4/r_0^2}{(E_0\,\nu)/(4\pi r_0^2 \delta)}
\;=\; \frac{4\pi c^4 \delta}{E_0\,\nu}.G∼(E0ν)/(4πr02δ)c4/r02=E0ν4πc4δ.
Now substitute E0 ⁣∼ ⁣ℏc/r0E_0 \!\sim\! \hbar c / r_0E0∼ℏc/r0:

  G  ∼  4πc3 δ r0ℏ ν  (surface-thin-shell model).\boxed{\; G \;\sim\; \frac{4\pi c^3 \, \delta \, r_0}{\hbar\,\nu}\; }
\qquad\text{(surface-thin-shell model).}G∼ℏν4πc3δr0(surface-thin-shell model).
This is a clear, compact formula you can use to explore consequences.

===== If you assume the thin-shell thickness δ\deltaδ equals the Planck length ℓP\ell_PℓP and the event rate is one per Planck time tPt_PtP, then δ\deltaδ and ν\nuν are expressed in terms of ℏ,c,G\hbar, c, Gℏ,c,G as: =====

ℓP=ℏGc3,tP=ℏGc5.\ell_P=\sqrt{\frac{\hbar G}{c^3}},\qquad t_P=\sqrt{\frac{\hbar G}{c^5}}.ℓP=c3ℏG,tP=c5ℏG.
Plug these into the expression for GGG. You’ll get an equation in which GGG appears on both sides (because tP,ℓPt_P,\ell_PtP,ℓP depend on GGG) — i.e. the relation becomes circular. That means: assuming Planck scales for thickness and rate presupposes GGG, so you cannot derive GGG from them — you only check self-consistency.

Algebraic sketch of the circularity (quick):
* With δ=ℓP\delta=\ell_Pδ=ℓP and ν=1/tP\nu=1/t_Pν=1/tP,

G∼4πc3 ℓP r0ℏ⋅tP=4πc3r0ℏ(ℓPtP).G \sim \frac{4\pi c^3 \, \ell_P \, r_0}{\hbar} \cdot t_P
= \frac{4\pi c^3 r_0}{\hbar} (\ell_P t_P).G∼ℏ4πc3ℓPr0⋅tP=ℏ4πc3r0(ℓPtP).
* But ℓPtP=ℏGc3⋅ℏGc5=ℏGc4\ell_P t_P = \sqrt{\frac{\hbar G}{c^3}} \cdot \sqrt{\frac{\hbar G}{c^5}} = \frac{\hbar G}{c^4}ℓPtP=c3ℏG⋅c5ℏG=c4ℏG.
* Substitute back:

G∼4πc3r0ℏ⋅ℏGc4  ⇒  G∼4πr0c⋅G,G \sim \frac{4\pi c^3 r_0}{\hbar} \cdot \frac{\hbar G}{c^4}
\;\Rightarrow\; G \sim \frac{4\pi r_0}{c} \cdot G,G∼ℏ4πc3r0⋅c4ℏG⇒G∼c4πr0⋅G,
which cancels GGG and reduces to a geometric consistency condition giving a particular ratio for r0r_0r0. (You can rearrange to find the value of r0r_0r0 that makes the identity hold — it will be Planck-scale. This is why previous simple estimates forced r0r_0r0 down to Planck lengths.)

Bottom line: assuming Planck thickness and Planck rate forces the model either to be trivially consistent (only if r0r_0r0 is Planck-scale) or circular.