Constructive Emergence of Einstein–Rosen Geometry from Persistent Entanglement in Finite Holographic Codes

Constructive Emergence of Einstein–Rosen Geometry from Persistent Entanglement in Finite Holographic Codes C. J Tully Independent Researcher Engineer Australia https://orcid.org/0009-0007-5661-7332 7th November 2026 Abstract We construct an explicit, finite-dimensional dictionary that maps the mutual information of a thermofield-double-like state on a holographic quantum error-correcting code to the length of a discrete Einstein–Rosen throat. Working with perfect-tensor networks of the Pastawski– Yoshida–Harlow–Preskill (HaPPY) type and controlled interpolations thereof, we show that the minimal cut through the tensor network supplies a geodesic length whose continuum limit reproduces the spatial Kruskal metric of an eternal AdS black hole. Finite-N corrections are isolated and shown to match the structure of the Faulkner–Lewkowycz–Maldacena bulkentanglement contribution. The addition of non-stabilizer resources generates an effective positive stress tensor localized near the cut, furnishing a microscopic model of gravitational backreaction within a rigorously defined code subspace. The construction is phrased entirely in terms of observer-dependent causal diamonds, thereby providing a controlled laboratory for several of the open problems in horizon entropy first articulated by Strominger.