Relational Reconstruction of Spacetime Geometry from Graph Laplacians

Background:
We develop a relational and spectral framework in which metric geometry emerges from correlation structures alone, without assuming a background manifold, coordinates, or fundamental geometric degrees of freedom.

Methods:
Starting from a relational substrate equipped with a symmetric connectivity operator, we define operational distances via minimal path functionals and introduce a non-circular coarse-graining scheme separating combinatorial neighborhoods from geometry-aware weighted distances. Spectral admissibility criteria identify regimes supporting a stable continuum approximation.

Results:
In these projectable regimes, the distance matrix admits a low-dimensional embedding, yielding emergent coordinates and an effective metric structure. Proper time, spatial distance, and curvature arise as coarse-grained summaries of relational organization. In symmetric weak-field limits, the effective metric reproduces Schwarzschild geometry without postulating fundamental gravitational dynamics.

Conclusions:
Breakdown of geometry occurs when spectral gaps close or connectivity becomes non-local, providing intrinsic limits to continuum spacetime. Analytical and numerical benchmarks establish robust spectral invariants, including the $8/3$ eigenvalue ratio on $S^3$. Spacetime thus appears as an operational construct emerging from relational spectral structure rather than a primitive entity.