Explicit finite-N regularization/extension of emergent boundary algebras

Construct an explicit finite-N (type I) extension or regularization of the emergent large-N von Neumann algebras associated with boundary regions, such as the time-band algebra Y_{I_w} and the entanglement-wedge algebra X_A, in a general holographic conformal field theory. The goal is to realize these algebras at finite N as bona fide type I subalgebras whose von Neumann entropy matches the generalized entropy of the dual bulk regions, thereby making precise the finite-N version of subregion–subalgebra duality.

Background

In the large-N (G_N → 0) limit, boundary algebras associated with causal and entanglement wedges are type III_1 and capture bulk subregions. To connect these structures to finite-N physics and generalized gravitational entropy, the text argues there should exist type I finite-N extensions (e.g., Y_R^ε or X_A^ε) whose entropy reproduces the bulk generalized entropy.

However, a general, explicit construction of such finite-N type I algebras is missing. Developing it would supply a concrete bridge between algebraic large-N structures and finite-N quantum gravity observables, including generalized entropies.