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

==== 1. Geometry & fields: define the spacetime manifold and the thin spherical boundary (S² shell). Choose Euclidean 4D manifold with a static spherical shell of radius RRR and small thickness ε\varepsilonε. ====
# Action & boundary conditions: write the full classical action (EM + Dirac + gauge fixing + ghosts) in SI units with minimal coupling Dμ=∇μ+iqAμ/ℏD_\mu = \nabla_\mu + i q A_\mu/\hbarDμ=∇μ+iqAμ/ℏ. Specify physical boundary conditions (conducting shell for EM; MIT bag or antiperiodic conditions for Dirac).
# One-loop effective action via heat kernel: reduce the problem to functional determinants of second-order elliptic operators. Use the heat-kernel expansion to extract the coefficient of the curvature scalar RRR in the effective action. That coefficient determines the induced 1/G1/G1/G.
# Evaluate coefficients & numerics: compute (or use known tables for) the heat-kernel coefficients (bulk + boundary) for the chosen fields and boundary conditions. Combine bosonic and fermionic contributions to produce κind\kappa_{\rm ind}κind and evaluate numerically for candidate cutoffs Λ∼1/L⋆\Lambda\sim 1/L_\starΛ∼1/L⋆ (Planck length vs atomic length).