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

=== Setup (SI units): ===
Treat each atomic “shell” as a fast boundary system (EM + matter) living on a 2-D surface with induced metric γab\gamma_{ab}γab. Integrating out these fast boundary modes yields an effective geometric action. By standard heat-kernel / EFT reasoning (Sakharov-style induced gravity), the coarse-grained bulk action contains an Einstein–Hilbert term

Seff[g]  =  ∫d4x −g c316πGQAT R  +  ⋯ .S_{\rm eff}[g]\;=\;\int d^4x\,\sqrt{-g}\,\frac{c^3}{16\pi G_{\rm QAT}}\,R\;+\;\cdots .Seff[g]=∫d4x−g16πGQATc3R+⋯.
The parametric size of GQATG_{\rm QAT}GQAT is set by:
'' a micro thickness/curvature scale L∗L_''L∗ of the boundary layer (your geometric half-radius scale),
* an effective count of fast boundary degrees of freedom per physical 3-volume, nbdyn_{\rm bdy}nbdy (how many active shells/modes per m³),
* a dimensionless spectral coefficient κind∼10−2 ⁣− ⁣1\kappa_{\rm ind}\sim 10^{-2}\!-\!1κind∼10−2−1 (depends on the spectrum/statistics of the boundary fields).

A robust dimensional estimate is

  1GQAT  ∼  κind ℏc3 nbdy 1L∗2  ⇒GQAT  ∼  c3κind ℏ L∗2nbdy.\boxed{\;
\frac{1}{G_{\rm QAT}}
\;\sim\;
\kappa_{\rm ind}\,\frac{\hbar}{c^3}\,n_{\rm bdy}\,\frac{1}{L_*^{2}}
\;}
\qquad\Rightarrow\qquad
G_{\rm QAT}\;\sim\;\frac{c^3}{\kappa_{\rm ind}\,\hbar}\,\frac{L_*^{2}}{n_{\rm bdy}}.GQAT1∼κindc3ℏnbdyL∗21⇒GQAT∼κindℏc3nbdyL∗2.
How to use this formula (plug-and-play):
# Pick L∗L_''L∗ from your boundary geometry (e.g., half-radius of the active shell thickness; for an atomic shell, L∗∼12a0≃2.6×10−11 mL_''\sim \tfrac{1}{2}a_0\simeq2.6\times10^{-11}\,\mathrm{m}L∗∼21a0≃2.6×10−11m; for condensed/plasma environments, choose the relevant EM skin/penetration scale).
# Estimate nbdyn_{\rm bdy}nbdy as the number of active boundary modes per m³ (e.g., atoms/m3×\text{atoms/m}^3 \timesatoms/m3× modes per atom at the relevant temperature/ionization; or an EM mode density for plasma cells).
# Take κind\kappa_{\rm ind}κind in the conservative range 10−2 ⁣− ⁣110^{-2}\!-\!110−2−1.

If the resulting GQATG_{\rm QAT}GQAT comes out within an order or two of 6.67×10−11 m3 kg−1 s−26.67\times10^{-11}\,\mathrm{m^3\,kg^{-1}\,s^{-2}}6.67×10−11m3kg−1s−2 for realistic L∗L_''L∗ and nbdyn_{\rm bdy}nbdy (without fine-tuning), that’s strong evidence the boundary-induction route is viable. If not, it tells us precisely where the micro picture must be sharpened (e.g., which modes dominate, what L∗L_''L∗ truly is in matter vs plasma, etc.).

: