Is Einstein’s equation the thermodynamic limit of quantum local consistency conditions?

Question

Question.

In holographic bulk reconstruction and quantum error-correcting codes (QECC), the global bulk geometry relies on the compatibility of overlapping boundary subregions.

Let us formalize a generalized framework: define the bulk not by an a priori continuous manifold, but as an emergent space defined by a dense network of overlapping local von Neumann sub-algebras.

For these sub-algebras to coexist, their respective local density matrices \(\rho _{i}\) must satisfy global compatibility constraints (the quantum marginal problem).

Ⅰ. Can the smooth macroscopic metric \(g_{\mu \nu }\) be mathematically derived as the coarse-grained constraint manifold of these local quantum consistency conditions?

Ⅱ. In this view, is the classical gravitational potential simply a thermodynamic consequence of the Monogamy of Entanglement across overlapping observer patches?

Ⅲ. Are there any existing toy models where the dynamical equations of spacetime (like Einstein's equations) directly emerge from the compatibility constraints of localized quantum states?