Stabilizer Quantum Gravity (SQG): Emergent Spacetime, Gauge Symmetry, and Matter from Recoverable Quantum Codes

This comprehensive manuscript introduces Stabilizer Quantum Gravity (SQG), a foundational quantum-information and operator-algebraic framework in which spacetime geometry, the Standard Model gauge structure, and cosmological dynamics emerge from a finite-depth, recoverable logical substrate. Rather than quantizing a pre-existing metric or inserting gauge symmetries axiomatically, SQG posits that physical reality operates as a macroscopic quantum error-correcting code. Gravitational and gauge dynamics arise strictly as the effective responses of dynamically stabilized sectors within a net of von Neumann algebras. Key Framework Contributions & Results: Operator-Algebraic Foundations: Establishes the core principles of SQG using approximate recoverability maps, finite logical depth, and stability under environmental deformations. Emergent Gauge Structure (The Standard Model): Introduces the Logical Composition Ceiling theorem. It proves that multiplicative norm preservation restricts local admissible sectors to a Hurwitz bound of $d \le 8$ (the octonionic limit). Continuous Yang-Mills gauge symmetries (and the $SU(3) \times SU(2) \times U(1)$ factorization) are formally derived as the inherent stabilizer redundancies of this maximal sector. Matter from Code: Identifies chiral fermions and matter multiplets as topological defect sectors of the stabilizer architecture. Mass hierarchies and generation multiplicities are linked to categorical associators (F-symbols) and modular commutant constraints. Geometric Emergence & Gravity: Demonstrates how Lorentzian causal structure and intrinsic time emerge from Tomita-Takesaki modular flow. The semiclassical Einstein equations are recovered in high-saturation regimes via an entanglement extremality principle (generalizing Ryu-Takayanagi/QES limits). Cosmology & Black Holes: Reinterprets the Big Bang as a pre-geometric percolation transition and models late-time accelerated expansion (Dark Energy) through the macroscopic relaxation flow of the stabilization network. Black hole islands and the Page curve are modeled as strict stabilization phase transitions. Rigid Phenomenology & Falsifiability: Derives a constrained modified-gravity response where departures from General Relativity are strictly confined to a scalar/environmental channel. The tensor sector remains symmetry-protected ($c_T = c$), yielding falsifiable relations between structure-growth ($\mu$) and weak-lensing ($\Sigma$) kernels. Mechanized Verification: Core structural theorems (Composition Ceiling, Stability Phase Transition, and Screening-Tensor Compatibility) are accompanied by formal blueprints for mechanized verification in Lean 4. This document serves as the primary, unified exposition of the SQG framework, detailing its mathematical spine, theorem targets, and experimental/observational program.