Prove the finite-volume approximation conjecture for pathology-free infinite-volume renormalization

Prove that, for an absolutely summable interaction Φ whose inverse-limit Hamiltonian H^Φ belongs to the physically admissible space and for which an absolutely summable renormalized interaction Φ′ exists under the infinite-volume renormalization-group transformation T, the finite-volume renormalized Hamiltonians satisfy R_nH_n^Φ=H_{n-1}^{Φ′}+δH_{n-1} with an error obeying lim_{n→∞}||δH_n||_∞/|Λ_n|=0.

Background

The paper constructs a pathology-free renormalization-group map on a space of inverse limits of finite-volume Hamiltonians. It proves that the map is well-defined, invariant on the physically admissible subspace, and continuous.

To establish that this inverse-limit construction reproduces the conventional infinite-volume renormalization-group transformation, the paper conjectures that finite-volume renormalizations of the Hamiltonians induced by an infinite-volume interaction converge to the renormalized infinite-volume interaction up to an error that is subextensive in the volume. The conjecture is explicitly identified as the missing result needed to connect the inverse-limit framework with infinite-volume physics.