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

Abstract

We prove a \(\Gamma\)-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 Carath\'eodory densities \(f_{\mathrm{in}}=\alpha_0+\alpha_1 R\) and \(f_{\mathrm{bdry}}=\beta_1 K\), 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 \(\alpha_0,\alpha_1,\beta_1\).