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

We prove a $\mathrm{\Gamma }$-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 Carathéodory densities $f_{\mathrm{i}\mathrm{n}}={\alpha }_0+{\alpha }_{1}R$ and $f_{\mathrm{b}\mathrm{d}\mathrm{r}\mathrm{y}}={\beta }_{1}K$, and obtain the $\mathrm{lim\hspace{0.17em}inf}\mathrm{/}\mathrm{lim\hspace{0.17em}sup}$ 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 ${\alpha }_0,{\alpha }_1,{\beta }_1$.