Causal Set Theory Hauptvermutung (approximate isometry from a shared discrete approximation)

Establish or refute the causal set theory Hauptvermutung by proving that if a single causal set is approximated at a fixed discreteness scale V_c by two distinct spacetimes, then those spacetimes are approximately isometric (i.e., identical up to the discreteness scale V_c).

Background

In causal set theory, a spacetime continuum is an emergent approximation of a fundamentally discrete, locally finite partially ordered set produced by a Poisson sprinkling at density ρ=1/V_c. To ensure that a causal set has a unique continuum interpretation at a given scale, the field posits its Hauptvermutung.

Confirming the Hauptvermutung would formalize the uniqueness (up to approximate isometry) of the continuum spacetime corresponding to a given causal set at scale V_c, thereby underpinning the continuum limit and strengthening the link between discrete dynamics and continuum general relativity.