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

==== 1. Set up background coordinates (spherical coordinates) and express metric and induced boundary geometry; compute extrinsic curvature KKK and boundary intrinsic curvature R\mathcal{R}R symbolic in terms of RRR. ====
# Linearize and identify Laplacians: - Vector: gauge fixed operator Δμν\Delta_{\mu\nu}Δμν. - Ghost: scalar Laplacian. - Fermion: squared Dirac operator D2D^2D2.
# Pick boundary conditions (Robin with parameter β\betaβ for scalars, impedance for vectors, MIT bag for fermions) and write them in the standard table form. Map your physical σ,η\sigma,\etaσ,η to those table parameters.
# Pull A2A_2A2 formulas from Vassilevich for each operator and boundary type. Substitute geometry (for small curvature you can treat RRR as symbolic). The A2A_2A2 expression will contain combinations like ∫R\int R∫R, ∫∂MK\int_{\partial M} K∫∂MK, ∫∂MR\int_{\partial M}\mathcal{R}∫∂MR, and boundary coupling polynomials in σ,η\sigma,\etaσ,η.
# Assemble A2totA_2^{\rm tot}A2tot adding bosonic contributions and subtracting fermionic ones with multiplicity. Include ghost subtraction.
# Perform the ttt-integral up to the cutoff Λ\LambdaΛ and isolate the coefficient of ∫R\int R∫R. If using a sharp cut-off: ∫Λ−2∞dtt t−1∼12Λ2+⋯\int_{\Lambda^{-2}}^{\infty}\frac{dt}{t}\,t^{-1} \sim \tfrac{1}{2}\Lambda^2 + \cdots∫Λ−2∞tdtt−1∼21Λ2+⋯ (explicit factors depend on your precise expansion normalization — track (4π)−2(4\pi)^{-2}(4π)−2 normalization from the heat kernel!).
# Extract κind\kappa_{\rm ind}κind by matching to 1/Gind=(c3/ℏ) κind Λ21/G_{\rm ind} = (c^3/\hbar)\,\kappa_{\rm ind}\,\Lambda^21/Gind=(c3/ℏ)κindΛ2.
# Numerics: evaluate for: - Λ=1/ℓP\Lambda = 1/\ell_PΛ=1/ℓP (Planck cutoff), - Λ=1/(a0/2)\Lambda = 1/(a_0/2)Λ=1/(a0/2) (half Bohr radius), - Some intermediate values. Vary σ,η\sigma,\etaσ,η across physically plausible ranges.