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Bibliographic Details
Main Author: MAILLOT, François
Format: Recurso digital
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Published: Zenodo 2026
Subjects:
finite-scale geometry, curvature reconstruction, sectional curvature, Riemann tensor, Regge calculus, numerical relativity, discrete geometry, general relativity, finite-resolution observables
géométrie à échelle finie, reconstruction de courbure, courbure sectionnelle, tenseur de Riemann, calcul de Regge, relativité numérique, géométrie discrète, relativité générale, observables à résolution finie
Online Access:https://doi.org/10.5281/zenodo.19739460
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Table of Contents:
  • <div> <h3><strong>Finite-Scale Reconstruction of Curvature Fields from Polygonal Observables</strong></h3> <p><strong>Subtitle:</strong> <em>A finite-resolution framework for curvature-field reconstruction, sectional curvature, and effective geometric dynamics.</em></p> <h3><strong>Abstract</strong></h3> <p>This work develops a finite-scale framework for reconstructing curvature fields from polygonal geometric observables. Starting from the observable <span class="math-inline">$\mathcal{I}_n(K,r)$</span>, defined as the ratio between inscribed and circumscribed polygonal areas, the paper:</p> <ul> <li> <p><strong>Derives local inversion formulas</strong> to extract curvature at a specific point.</p> </li> <li> <p><strong>Extends reconstruction to curvature fields</strong>, allowing for the mapping of spatially varying manifolds.</p> </li> <li> <p><strong>Constructs finite-scale analogues of differential operators</strong> (gradient, Laplacian).</p> </li> <li> <p><strong>Outlines a route toward tensorial curvature reconstruction</strong> from sectional curvature measurements.</p> </li> </ul> <p>Numerical tests on synthetic curvature fields (including Schwarzschild-like profiles) confirm <strong>algebraic consistency</strong> and <strong>controlled noise amplification</strong>.</p> <blockquote> <p><strong>Core Philosophy:</strong> This framework is not proposed as a replacement for General Relativity, but as an <strong>operational finite-resolution reconstruction layer</strong> compatible with continuum geometry in the infinitesimal limit.</p> </blockquote> <h3><strong>Key Features</strong></h3> <ul> <li> <p><strong>Resolution Parameters:</strong> Introduces a <span class="math-inline">$(n, r)$</span> parameter space (angular discretization and probing scale).</p> </li> <li> <p><strong>Noise Stability:</strong> Demonstrates an optimal reconstruction scale <span class="math-inline">$r^* \sim \sigma^{1/3}$</span> to balance truncation and measurement errors.</p> </li> <li> <p><strong>Reproducibility:</strong> All results and figures are fully reproducible via the provided Python scripts.</p> </li> </ul> </div>

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