Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Marko Lela

TL;DR

The paper proves a Γ-convergence result showing that a diffeomorphism-natural discrete MDL functional converges to the Einstein–Hilbert action with the Gibbons–Hawking–York boundary term on boundary-fitted, shape-regular meshes. It identifies interior and boundary Carathéodory densities as $f_{in}=\alpha_0+\alpha_1 R$ and $f_{bdry}=\beta_1 K$, and proves the liminf/limsup inequalities via a recovery sequence based on reflected Fermi smoothing, yielding $F(g)=c_0\int_M dV_g + c_1\int_M R_g dV_g + c_2\int_{\partial M} K_g dS_g$ with $c_0=\alpha_0$, $c_1=\alpha_1$, and $c_2=\beta_1$. A boundary first-layer analysis shows leading $O(h^{d-1})$ contributions per boundary cell and a global $O(h)$ boundary remainder, while the interior remainder is $O(h^2)$. The results are established under equi-coercivity, bounded geometry, and boundary regularity, and Appendix F provides a reproducible protocol for rate checks and calibration of the constants. This work situates EH+GHY as the natural continuum limit of a data-efficient, diffeomorphism-natural discretization, advancing the MIS program's aim to derive continuum physics from information-theoretic discretizations.

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 \(f_{\mathrm{in}}=α_0+α_1 R\) 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 \(α_0,α_1,β_1\).

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

TL;DR

The paper proves a Γ-convergence result showing that a diffeomorphism-natural discrete MDL functional converges to the Einstein–Hilbert action with the Gibbons–Hawking–York boundary term on boundary-fitted, shape-regular meshes. It identifies interior and boundary Carathéodory densities as and , and proves the liminf/limsup inequalities via a recovery sequence based on reflected Fermi smoothing, yielding with , , and . A boundary first-layer analysis shows leading contributions per boundary cell and a global boundary remainder, while the interior remainder is . The results are established under equi-coercivity, bounded geometry, and boundary regularity, and Appendix F provides a reproducible protocol for rate checks and calibration of the constants. This work situates EH+GHY as the natural continuum limit of a data-efficient, diffeomorphism-natural discretization, advancing the MIS program's aim to derive continuum physics from information-theoretic discretizations.

Abstract

We prove a -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 and , and obtain the bounds via a recovery sequence based on reflected Fermi smoothing. A boundary first-layer asymptotics shows that boundary cells contribute at order , 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 .

Paper Structure

This paper contains 99 sections, 25 theorems, 119 equations.

Table of Contents

  1. Related work and positioning
  2. Regge calculus and measure convergence.
  3. Boundary terms in discrete gravity.
  4. Graph and metric-measure formulations.
  5. Continuum structure and boundary terms.
  6. Finite elements and boundary-fitted meshes.
  7. MDL and model selection.
  8. Positioning and contribution.
  9. Numerical scope.
  10. Proof roadmap
  11. Scales and windows.
  12. TP1 (interior blow-up).
  13. TP2 (boundary blow-up, first layer).
  14. TP3 (Carathéodory and quasi-additivity).
  15. TP4/TP5 (liminf/limsup).
  16. ...and 84 more sections

Key Result

Proposition 5.1

Let $c_n\subset M^\circ$ be interior cells with $\mathrm{dist}(c_n,\partial M)\ge C\,R_n$ and shape-regularity uniform in $n$. Assume the normalized, isotropic moment conditions up to order 2 for the interior window, and $\ell\in C^2$ on the compact feature set $K_{\mathrm{feat}}$. Then there exist uniformly on compact subsets of $M^\circ$. In particular,

Theorems & Definitions (62)

  • Remark 3.1: Motivation for (BA2)
  • Proposition 5.1: TP1: interior cell asymptotics
  • proof
  • Remark 5.2: Uniformity and mesh independence
  • Proposition 6.1: TP2: boundary cell asymptotics, first layer
  • proof
  • Remark 6.2: Origin and propagation of the $O(h)$ remainder
  • Corollary 6.3: Riemann–sum convergence and global remainder
  • Remark 6.4: Window-shape invariance at fixed $\mu_1$
  • Remark 6.5: Boundary regularity
  • ...and 52 more