Stabilizer Quantum Gravity II: Recoverability, Entropy Extremality, and the Einstein Bridge

This work presents the second paper of the Stabilizer Quantum Gravity (SQG) research program. In this paper, the gravitational response of emergent spacetime is derived from informational stability conditions associated with recoverable operator-algebra sectors. The central goal is to show how Einstein-like dynamics can arise not from fundamental geometry, but from deeper recoverable logical structures. The derivation is built from three main ingredients: • recoverability bounds from quantum information theory • modular Hamiltonian dynamics in local operator-algebra sectors • extremality of generalized entropy The paper shows that sectors satisfying recoverability and entropy extremality obey the linearized Einstein equation up to correction terms controlled by a stabilization deficit functional, which quantifies imperfect recoverability. Main contributions include: • formulation of a Recoverability–Einstein bridge theorem • derivation of geometric response from generalized entropy balance • connection of the bridge mechanism to null congruence focusing through the Raychaudhuri equation • introduction of a stabilization deficit as a natural source of quantum-gravitational corrections • derivation of structural constraints on the logical substrate via the Hurwitz classification of normed division algebras Taken together, these results support the interpretation of spacetime as an emergent informational phase stabilized by recoverable quantum structures. This paper is part of the broader Stabilizer Quantum Gravity (SQG) program.