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

=== I recommend the following sequence (I can do each step and produce a LaTeX file you can share): ===
# Boundary action → surface stress-energy Start with your surface Lagrangian Sbdy=∫d3y Lbdy(ψ,A,h)S_{\rm bdy}=\int d^3y\,\mathcal{L}_{\rm bdy}(\psi,A,h)Sbdy=∫d3yLbdy(ψ,A,h). Derive Sab  =  −2−hδSbdyδhab,S_{ab} \;=\; -\frac{2}{\sqrt{-h}} \frac{\delta S_{\rm bdy}}{\delta h^{ab}},Sab=−−h2δhabδSbdy, and compute S00S_{00}S00 for a minimal model: EM energy density + mechanical tension term + fermion zero-mode contribution.
# Compute EM self-energy for shell of finite thickness (repeat the thin-shell calculation but allow a finite thickness δ\deltaδ to see regularization issues). Produce numerical estimates (use electron values and a few plausible δ\deltaδ) and show where the factor 1/2 comes from.
# Apply Israel junction conditions to get the jump in extrinsic curvature and find the weak-field Newtonian potential (Poisson equation). Compute the effective GMGMGM and compare to the mass from the integrated surface energy.
# Include Maxwell TμνT_{\mu\nu}Tμν and show explicitly how Poynting flux depositing onto the shell appears as a time-changing S00S_{00}S00 consistent with the continuity equation (analysis of photon-by-photon deposition -> incremental mass growth).
# Address stability: include a simple Poincaré stress term in the surface Lagrangian, calculate its contribution to SabS_{ab}Sab, and show the shell equilibrium condition (force balance).
# Discuss quantum/field issues: what parts need second-quantization (discrete photons on the boundary), topological issues for spinors on S2S^2S2 (spin structures), and where Planck’s constant h/2πh/2\pih/2π enters (quantization conditions and minimal action).
# Numerical checks & predictions: compute example numbers, e.g. thin-shell electron gives R=re/2R=r_e/2R=re/2; compute energy per photon and how many photons at given flux would be needed to generate comparable mass, check orders of magnitude. Suggest observational checks (e.g., small corrections to classical electron radius, electromagnetic contributions to inertia under extreme fields, or astrophysical plasma effects).