Standard models of dimensional topology construct macroscopic spacetime additively from the bottom up. However, applying General Relativity to the quantum regime within this additive geometric framework produces non-renormalizable ultraviolet (UV) divergences, unresolvable black hole singularities, and the unnatural fine-tuning of the Hierarchy Problem. This paper formalizes the Topological Dimensional Inversion (TDI) model, postulating that macroscopic spacetime dimensions emerge top-down. We define emergent spatial dimensions not as additive expansions, but as thermodynamically constrained, coarse-grained effective field theories of a dimensionless (0D) quantum state. Applying the 't Hooft planar limit to the Ishibashi-Kawai-Kitazawa-Tsuchiya (IKKT) Matrix Model, we extract a 2D continuous Multi-scale Entanglement Renormalization Ansatz (cMERA) tensor network whose finite bond dimension ($\chi$) acts as a fundamental computational bottleneck. To resolve regime conflicts, measure-theoretic fractures, and radiative loop anomalies in this topological projection, we introduce five axioms: Chronological Dithering, the Projective Coupling Horizon, Algorithmic Horizon Offloading via the Twirled Petz Map, Thermodynamic Parity Pruning, and Deterministic SVD Truncation. This framework maps to an accelerating de Sitter (dS) background via the ``bad sign'' $T\bar{T}$ deformation---requiring an imaginary-time error-correction contour---recovers General Relativity in the infrared (IR) limit without encountering the AMPS firewall, and generates six falsifiable experimental signatures. We further derive wave-function collapse as a measurable spectral hash isomorphic to the non-trivial zeros of the Riemann zeta function.