$Γ$-convergence of a diffeomorphism-natural MDL functional to Einstein-Hilbert with Gibbons-Hawking-York boundary term

Abstract: We prove a (Γ)-convergence result for a diffeomorphism-natural discrete MDL-type functional to the Einstein-Hilbert action with the Gibbons-Hawking-York boundary term. On boundary-fitted, shape-regular meshes we establish interior and boundary blow-ups, identify the Carathéodory densities (f_{\mathrm{in}}=α0+α_1 R) and (f{\mathrm{bdry}}=β_1 K), and obtain the (\liminf/\limsup) bounds via a recovery sequence based on reflected Fermi smoothing. A boundary first-layer asymptotics shows that boundary cells contribute at order (h^{d-1}), yielding a global (O(h)) boundary remainder, while the interior remainder is (O(h^2)). The paper is foundational; Appendix~E specifies a reproducible protocol for rate checks and calibration of (α_0,α_1,β_1).