**Abstract.** This work proves a Γ-convergence result for a diffeomorphism-natural discrete “MDL” (minimal description length–type) functional to the full Einstein–Hilbert action with the Gibbons–Hawking–York (GHY) boundary term. On boundary-fitted, shape-regular meshes we establish interior and boundary blow-up analyses (TP1–TP5), identify the Carathéodory densities \(f_{\mathrm{in}}=\alpha_0+\alpha_1 R\) and \(f_{\mathrm{bdry}}=\beta_1 K\), and prove the \(\liminf/\limsup\) inequalities via a recovery sequence based on reflected Fermi smoothing. As a result, the discrete energies Γ-converge to \[c_0\!\int_M dV_g + c_1\!\int_M R_g\,dV_g + c_2\!\int_{\partial M} K_g\,dS_g,\]and stability of minimizers holds under equi-coercivity. **Version 1.1 notes (2025-11-13).** - Added a concise **Conclusion** with MIS (Meta Information Symmetry) context and outlook. - **Unified boundary sign convention**: outer unit normal; \(K=\mathrm{tr}_h II>0\) on spheres. Visible reminder near the main theorem; consistent references in TP2/TP4/TP5/TP6/TP8 and “Scope”. - **Citations expanded** across TP3–TP6 and the appendices (e.g., Federer, Lovelock, Kolář–Michor–Slovák, Ciarlet/Brenner–Scott, Regge; Hartle–Sorkin). Text-only additions; **no changes to statements or proofs**. - **Typesetting fixes**: cleaned abstract/math display of \(F(g)\); resolved minor math-mode issues. - **Appendices**: Smoothing (App. A) extended with \(L^1\) stability for \(R\) and \(K\) and a complete **Lemma U**; Rate protocol (App. E) added; Scan indifference/BA3 (App. F) added. **Boundary-first analysis.** A first-layer asymptotics shows that boundary cells contribute at order \(h^{d-1}\) (not \(o(h^{d-1})\)), yielding the GHY term in the limit and a global \(O(h)\) boundary remainder, while the interior remainder scales as \(O(h^2)\). **Foundational scope and reproducibility.** No numerical results are claimed. Appendix E provides a reproducible “sanity-check protocol” (test geometries, expected rates, calibration of \(\alpha_0,\alpha_1,\beta_1\), and suggested diagnostic plots) to independently confirm the predicted \(O(h)\)/\(O(h^2)\) global rates. **Files in this record.** - PDF of the paper (v1.1). - LaTeX source bundle (v1.1) with .tex/.bib, figures, and build scripts, for full reproducibility. **Keywords:** Γ-convergence; Einstein–Hilbert action; Gibbons–Hawking–York boundary term; Regge calculus (discrete gravity); diffeomorphism-natural energies; Carathéodory densities; reflected Fermi coordinates; boundary-fitted meshes; variational limits; stability of minimizers. **Primary subjects:** Mathematical Physics; Differential Geometry; Analysis of PDEs; General Relativity and Quantum Cosmology. **Notes for readers:** - This is a theory paper; numerical sanity checks are deliberately deferred. - The appendix includes all assumptions (“constants bracket”), window/mesh setups, and calibration steps to enable independent replication of the predicted rates. If you cite this deposit, please use the Zenodo DOI and the title: “Γ-convergence of a diffeomorphism-natural MDL functional to Einstein–Hilbert with Gibbons–Hawking–York boundary term.”