Formulate and prove the causal set Hauptvermutung

Establish a precise formulation and proof of the causal set Hauptvermutung: for any causal set (C,≺) that admits faithful embeddings—i.e., injective maps that preserve causal relations and realize uniform Poisson density at the Planck scale with characteristic length much greater than the mean spacing—into Lorentzian manifolds (M1,g1) and (M2,g2), show that there exists a C-preserving diffeomorphism θ: M1→M2 that is an approximate isometry, thereby demonstrating the essential uniqueness of the continuum approximation determined by (C,≺) above the Planck scale.

Background

The Hauptvermutung (‘main conjecture’) is central to causal set theory’s program of recovering Lorentzian geometry from discrete causal sets. Faithful embeddings require preserving causal order and matching spacetime volume by Poisson density at the Planck scale, together with slowly varying geometry at scales larger than the mean spacing. The conjecture asserts the essential uniqueness of the continuum approximation: any two faithful embeddings of the same causal set should yield Lorentzian manifolds related by a diffeomorphism that approximately preserves metric structure above the Planck scale.

A precise mathematical formulation of “approximate isometry” and the associated uniqueness criteria is currently lacking, and proving the conjecture would rigorously secure the recovery of continuum spacetime from discrete causal structure in the physically relevant pre-limit regime.