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

=== Propose a 4D bulk + 3D boundary action (SI units): ===

Bulk (unchanged physics preserved):

Sbulk  =  ∫d4x −g[c316πGR  −  14μ0FμνFμν  +  ψˉ(iℏc γμDμ−mc2)ψ].S_{\rm bulk} \;=\; \int d^4x\,\sqrt{-g}\left[
\frac{c^3}{16\pi G} R \;-\; \frac{1}{4\mu_0} F_{\mu\nu}F^{\mu\nu}
\;+\; \bar{\psi}\left(i\hbar c\,\gamma^\mu D_\mu - mc^2\right)\psi
\right].Sbulk=∫d4x−g[16πGc3R−4μ01FμνFμν+ψˉ(iℏcγμDμ−mc2)ψ].
Boundary (each interaction shell Σ\SigmaΣ):

Sbdy  =  ∑Σ∫Σd3ξ ∣h∣[c38πGK    +    α AμJμ    +    β ϕ21R2    +    γ εabcAa∂bAc].S_{\rm bdy} \;=\; \sum_{\Sigma}\int_{\Sigma} d^3\xi\,\sqrt{|h|}
\left[
\frac{c^3}{8\pi G} K \;\;+\;\; \alpha\, A_\mu J^\mu
\;\;+\;\; \beta\, \phi^2 \frac{1}{R^2}
\;\;+\;\; \gamma\, \varepsilon^{abc} A_a \partial_b A_c
\right].Sbdy=Σ∑∫Σd3ξ∣h∣[8πGc3K+αAμJμ+βϕ2R21+γεabcAa∂bAc].
* KKK: Gibbons–Hawking–York term (ensures well-posed metric variation; gives correct GR boundary dynamics).
* AμJμA_\mu J^\muAμJμ: surface EM coupling to the boundary current (your absorption/emission channel).
* ϕ\phiϕ: scalar density of cumulative exchange; ϕ2/R2\phi^2/R^2ϕ2/R2 encodes the 1/r21/r^21/r2 geometric weighting.
* Chern–Simons–like A∧dAA\wedge dAA∧dA term (topological helicity/phase memory; optional but useful for spin/handedness bookkeeping).

What the variations give:
* δS/δAμ⇒\delta S/\delta A_\mu \RightarrowδS/δAμ⇒ Maxwell in bulk and correct jump conditions on Σ\SigmaΣ: n^ ⁣⋅ ⁣(D+−D−)=σbdy\hat{n}\!\cdot\!(\mathbf{D}^+ - \mathbf{D}^-) = \sigma_{\rm bdy}n^⋅(D+−D−)=σbdy, n^ ⁣× ⁣(H+−H−)=Kbdy\hat{n}\!\times\!(\mathbf{H}^+ - \mathbf{H}^-) = \mathbf{K}_{\rm bdy}n^×(H+−H−)=Kbdy.
* δS/δgμν⇒\delta S/\delta g_{\mu\nu} \RightarrowδS/δgμν⇒ Einstein eqs in bulk and Israel conditions on Σ\SigmaΣ with Sab∼S_{ab} \simSab∼ (EM surface energy–momentum + ϕ\phiϕ term + topological stress). This is your direct bridge from spherical EM exchange to curvature.