Emergent Gravity via Informational Density on a Z₂-Symmetric Boundary

The standard formulation of General Relativity models gravity as the curvature of spacetime dictated by the classical stress-energy of matter. However, treating mass as an intrinsic, fundamental property rather than a derived thermodynamic state creates severe theoretical friction when attempting to unify macroscopic geometric mechanics with discrete quantum theory. In this paper, we propose a structural replacement for classical mass, defining it strictly as localized informational density computed via renormalized entanglement entropy, Sₑff. We mathematically redefine the observable spatial continuum not as an independent manifold, but as an induced Z₂-symmetric orbifold boundary embedded within a five-dimensional purification space governed by a zero-sum global entropy mandate (Sglobal = 0). By imposing universal entanglement equilibrium across all ball-shaped regions of this boundary—a structural consequence of the Z₂ topology rather than an empirical thermodynamic assumption—we demonstrate that the linearized Einstein field equations emerge as the unique local consistency condition, with Newton's constant determined directly by the area-law prefactor of the entanglement entropy. As an independent confirmation, we evaluate the extrinsic geometry of the Z₂ boundary via the Israel junction conditions and recover Poisson's equation for Newtonian gravity from the embedding data alone. Demanding consistency between the intrinsic and extrinsic derivations establishes a rigid constraint between the boundary tension and the bulk curvature scale, collapsing the framework's parameter space from three phenomenological variables to a single bulk-geometric degree of freedom. Crucially, because the geometric deformation couples to spatial entanglement rather than raw energy alone, the framework predicts that gravity acquires state-dependent corrections at subleading order in the entanglement expansion, structurally distinguishing it from General Relativity.