Bombelli’s positivity and convergence conjecture for dn

Establish that for any two distinguishing, finite-volume, non-isometric Lorentzian manifolds (M,g) and (M′,g′), there exists a finite n such that dn((M,g),(M′,g′)) > 0, and moreover dn((M,g),(M′,g′)) → 1 as n → ∞, where dn is the Bhattacharyya-based statistical angle between the manifolds’ Poisson-sprinkling-induced distributions on n-element causal sets.

Background

The dn distance compares Lorentzian geometries via finite causal-set samples drawn by Poisson sprinkling. Although dn is generally a pseudo-distance for finite n, the conjecture aims to show that for any pair of non-isometric geometries, finite samples suffice to separate them (dn>0), and that increasing sample size drives them to maximal separation (dn→1).

Confirming this would strongly support the probabilistic discrimination of continuum spacetimes using causal-set statistics and provide a pathway toward rigorous formulations of the Hauptvermutung in probabilistic terms. The conjecture is presented as reasonable but explicitly noted as unproven.