RTLI Coherence Index as a Practical Proxy for Gromov–Hausdorff Convergence

This technical note introduces the RTLI coherence index η as a practical screening tool for Gromov–Hausdorff (GH) convergence in discrete metric spaces. Although exact GH distance is computationally expensive, η combines correlation length, curvature consistency, local variance, and triangle deviation into a single, dimensionless quantity that is fast to compute. Numerical tests across multiple geometric families (shrinking spheres, flat tori, random-to-circle transitions, k-NN graphs, and noisy manifolds) show a strong inverse correlation between GH distance and η, with a clear “geometry-onset” region where η stabilizes and GH distance begins to drop rapidly.

The results suggest that RTLI does not replace GH distance, but can efficiently pre-screen large datasets: when η is low or unstable, GH convergence is unlikely; once η passes the geometry-onset zone (typically near η ≈ 1.7 under current normalization), the space enters a coherent regime where GH analysis becomes meaningful. This provides a scalable way to identify promising geometric limits and reduce the computational burden of GH computations.