Thermodynamic Selection of Macroscopic Spacetime: Topological Phase Transitions and Stability Basins in Stabilizing Quantum Gravity

This document presents a major conceptual and computational milestone within the Stabilizing Quantum Gravity (SQG) research program. It directly addresses one of the most profound challenges in fundamental physics: explaining the specific macroscopic features of our universe (such as its dimensionality and geometric stability) without resorting to an unconstrained anthropic "multiverse" landscape. In this manuscript, the SQG framework reformulates the landscape of possible emergent universes as a rigorous thermodynamic ensemble of finite-depth stabilizer codes. We demonstrate that the emergence of macroscopic spacetime is not a random draw from an infinite landscape, but a mathematically strict Topological Phase Transition. Core Mechanisms & Computational Benchmarks: The Topological Free Energy Functional: We introduce a dynamical selection principle governed by $\Phi = \alpha C - \beta D_S$, where the system balances the drive for logical capacity ($C$) against the energetic penalty of the stabilizer deficit ($D_S$). Mapping to Lattice Gauge Theory: We provide a formal theorem establishing that the SQG stabilizer deficit measure is mathematically isomorphic to the partition function of a 4D generalized Lattice Gauge Theory (LGT) or spin-glass model, grounding the theory in established statistical mechanics. MCMC Simulation & The Stability Basin: Accompanying this paper is a Markov Chain Monte Carlo (MCMC) numerical simulation. The simulation proves that random, non-geometric informational configurations strictly collapse into a narrow Stability Basin characterized by an effective spectral dimension of $d_s \to 4$. Elimination of Fine-Tuning: The framework algorithmically predicts that high-dimensional spacetimes are destroyed by uncorrectable stabilizer deficits, while low-dimensional spacetimes suffer from suboptimal logical capacity. Thus, our 3D/4D macroscopic geometry emerges naturally as the ultimate thermodynamic attractor. Significance for Fundamental Physics: By replacing the ad hoc anthropic landscape with the exact thermodynamic relaxation equations of statistical mechanics, SQG provides a deterministic, computable, and falsifiable mechanism for the selection of the universe's geometry. Furthermore, the residual unsaturated deficit in the early universe yields falsifiable signatures, including a natural source for Early Dark Energy and deviations in primordial spectral tilts.