Relational Reconstruction of Spacetime Geometry from Graph Laplacians

We present a relational and spectral construction of effective spacetime geometry in which metric notions arise from correlation structure alone, without assuming a background manifold, coordinates, or fundamental geometric degrees of freedom. Starting from a purely relational substrate endowed with a symmetric connectivity operator, we define operational distances through minimal path functionals and show how a stable geometric regime emerges via spectral admissibility.

A non-circular coarse-graining scheme is introduced, distinguishing pre-geometric combinatorial neighborhoods from geometry-aware weighted distances. This hierarchy allows the construction of an effective scalar descriptor whose correlations encode operational notions of time ordering and spatial separation. When relational variations become sufficiently smooth, the resulting distance matrix admits a low-dimensional embedding, enabling the reconstruction of emergent coordinates and an effective metric structure.

We demonstrate that, in this projectable regime, standard geometric observables—such as proper time, spatial distance,
and curvature—arise as descriptive summaries of relational constraints. The effective metric is shown to reproduce
general-relativistic phenomenology in appropriate limits, including the recovery of Schwarzschild geometry for isolated, approximately symmetric configurations, without postulating gravitational dynamics at the fundamental level.

The framework naturally predicts breakdowns of geometric description when spectral gaps close or relational structure
becomes non-local, providing intrinsic criteria for the limits of continuum spacetime. Numerical and analytical results supporting a universal spectral hierarchy are presented in the appendices. Overall, this work establishes a concrete pathway from relational spectral data to emergent metric geometry, positioning spacetime as an operational construct rather than a primitive entity.