This treatise formalizes a background-independent, self-consistent mathematicalframework unifying gauge fields and gravitation through a non-associative, hyperfinite operator network. We construct a multi-level algebraic pipeline where state regularizations over the octonionic loop ring Z[OL] map onto a subobject classifier Ωtopos within an intuitionistic topos logic. We formalize the core processing engineas an Autonomous Autoregressive Monadic Functor Engine (EAuto), operating as a self-adjoint monad triplet (E, η, µ) across an Eilenberg—Moore category ofE-algebras over a historically novel Type IV non-associative von Neumannalgebra structure (MIV). By utilizing Tomita—Takesaki modular theory and Connes modular cocycle flows, we demonstrate how a boundary operator algebra A(∂W) structured as a hyperfinite Type III1 subfactor net dynamically generates a smooth, 4D bulk Lorentzian metric gµν satisfying Einstein’s Field Equations as anecessary thermodynamic equation of state driven by non-stabilizer code contextuality ("magic"). High-dimensional lattice states are projected down to a 4D target space using an 8 × 8 coordinate-mixing rotor matrix (R45◦ ∈ SO(8)), yielding discrete Graph Laplacian spectra (∆G) whose non-stabilizer resource capacities matchfault-tolerant quantum error-correction thresholds. We explicitly resolve the nonassociative unitarity paradox via braid group 3-cocycle shifts (ω3 ∈ H3 (MIV, O)), derive the Quantum Operator Equation of Motion ([ˆK, ∆G]), and prove the Inductive Threshold Limit Theorem. Furthermore, we ground this framework in the foundational empirical baseline established by Biswas et al.and the structural milestones of Google Quantum AI, validating Faulkner—Lewkowycz—Maldacena (FLM) corrections, surface-code threshold scaling, many-body fastscrambling, and non-stabilizer ’magic’ gravitational emergence on physical quantumhardware.