Spectral Reconstruction of Spacetime Geometry

Overview

This article develops a relational and spectral approach to the emergence of spacetime geometry. Starting from abstract connectivity data encoded in Laplacian operators, it shows how effective notions of distance, dimension, curvature, and metric structure can be reconstructed without assuming a background manifold, coordinates, or fundamental geometric degrees of freedom.

The construction is intentionally kinematical. No gravitational dynamics or field equations are postulated. Geometric quantities arise only as effective descriptors in regimes where the relational substrate admits a stable and spectrally admissible continuum approximation. In that projectable regime, standard geometric observables can be treated as operational summaries of relational constraints rather than fundamental entities.

Scope statement. This page is a structured summary and entry point. The authoritative technical reference is the Zenodo preprint linked above. Code and numerical supplements are provided for reproducibility.

Core contributions

Spectral proximity and distance: operational distances derived from Laplacian eigenmodes and kernels.
Admissibility criteria: intrinsic spectral conditions delimiting when continuum geometry is meaningful.
Local metric emergence: quadratic approximation of spectral distances yielding an effective metric tensor.
Curvature as a collective descriptor: geometric quantities as summaries of relational coupling variations.
Schwarzschild-type recovery: kinematical consistency limit for static spherically symmetric regimes.

Numerical and spectral supplements

Numerical illustrations are provided as consistency and robustness checks for selected spectral claims. They do not introduce additional postulates and are not required for the logical structure of the article.

Monte-Carlo convergence of a universal spectral ratio on S³ (8/3 robustness check).
Simulation code and figure scripts

How this connects to Cosmochrony

This article isolates the geometric reconstruction problem: how metric notions can arise from relational spectral data alone. Within the broader Cosmochrony framework, it supports the claim that spacetime geometry is an effective encoding of microscopic relational connectivity, accessed through spectral reconstruction in projectable regimes.

References

Jérôme Beau. Relational Reconstruction of Spacetime Geometry from Graph Laplacians. Zenodo. DOI: 10.5281/zenodo.18356037