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

==== If we invert the relation to find the radius r0r_0r0 that would give the observed GGG, we get ====

r0,required≃3ℏGobs4πc3.r_{0,\text{required}} \simeq \sqrt{\frac{3\hbar G_{\rm obs}}{4\pi c^3}}.r0,required≃4πc33ℏGobs.
Plugging in GobsG_{\rm obs}Gobs gives

r0,required≈7.9×10−36 m.r_{0,\text{required}} \approx 7.9\times10^{-36}\ \text{m}.r0,required≈7.9×10−36 m.
For comparison, the Planck length is
ℓP=ℏG/c3≈1.62×10−35 m.\ell_P = \sqrt{\hbar G / c^3} \approx 1.62\times10^{-35}\ \mathrm{m}.ℓP=ℏG/c3≈1.62×10−35 m.

So the required radius is of order 10−3510^{-35}10−35 m — i.e. on the order of the Planck length (about 0.5 ℓP\ell_PℓP in the crude algebra above). In other words: to get the correct measured GGG with this simple QAT estimate you need the characteristic shell radius to be a Planck-scale length, not an atomic scale.