Bombelli’s finite-parameter geometry distance conjecture

Prove that for any subset of Lorentzian geometries parameterized by finitely many parameters (analogous to minisuperspace models), there exists a finite n such that the Bhattacharyya-based statistical angle dn—defined from Poisson-sprinkling-induced probability distributions over n-element causal sets—is a true distance function (positive-definite) on that subset.

Background

Bombelli’s proposal defines a scale-dependent statistical distance between Lorentzian geometries via the Bhattacharyya angle between their Poisson-induced distributions on finite causal sets with n elements. For general geometries and finite n, dn is only a pseudo-distance because it depends on finitely many parameters and can vanish for non-isometric manifolds.

The conjecture asserts that restricting attention to finite-parameter families of geometries should suffice to make dn a genuine metric for some finite n, strengthening the mathematical underpinnings of comparing continuum spacetimes via causal-set sampling. The conjecture is stated as reasonable but remains unproven.