Topological Dimensional Inversion: Macroscopic Spacetime as a Wilsonian Coarse-Grained Effective Field Theory of the 0D Quantum Vacuum

Standard models of dimensional topology construct macroscopic spacetime additively from the bottom up (e.g., integrating 0D points into 1D lines, 2D planes, and 3D volumes). However, when applying General Relativity to the quantum regime, this additive geometric framework inherently produces non-renormalizable ultraviolet (UV) divergences, unresolvable black hole singularities, and the unnatural fine-tuning of the Hierarchy Problem. To resolve these pathologies, this paper formalizes the Topological Dimensional Inversion model, postulating that macroscopic spacetime dimensions emerge strictly top-down. Specifically, we define emergent spatial dimensions mathematically not as additive expansions, but as thermodynamically constrained, coarse-grained effective field theories of a dimensionless quantum state. Applying the 't Hooft double-scaling limit ($N \rightarrow \infty$, $\lambda$ constant) to the Ishibashi-Kawai-Kitazawa-Tsuchiya (IKKT) Matrix Model, we formalize the 0D quantum vacuum as a divergent Hilbert space of unconstrained algebraic probability. We demonstrate that continuous geometry manifests via the eigenvalue distribution density integrated over the Haar measure of the matrix ensemble, with spontaneous symmetry breaking into a microscopic non-commutative background ($[x_\mu, x_\nu] = i\theta_{\mu\nu}$) natively generating 1D string tension. This state algorithmically structures into a 2D Holographic Tensor Network (a CFT$_3$ boundary state formulated via a Multi-scale Entanglement Renormalization Ansatz, or cMERA), where the 1D strings function as internal entanglement bonds governed by a finite bond dimension ($\chi$). Crucially, utilizing the exact isomorphism of the AdS$_4$/CFT$_3$ correspondence, we prove that the observable 3D spatial bulk (AdS$_4$) is strictly an emergent, low-energy infrared (IR) effective field theory generated by Wilsonian coarse-graining (Exact Renormalization Group flow) over the 2D boundary's UV degrees of freedom. By framing macroscopic volume, mass, and classical gravity as extreme thermodynamic informational bottlenecks rather than fundamental geometric expansions, this top-down topological formalism eliminates infinite density singularities and seamlessly preserves unitary evolution. Furthermore, this framework yields explicit, falsifiable phenomenological predictions. We resolve the Hierarchy Problem by deriving the effective Newtonian gravitational coupling strictly as a function of the boundary tensor network's bond dimension ($G_N \propto R_{AdS}^2 / \ln \chi$). Additionally, utilizing the Bekenstein-Hawking bound, we derive an exact density-dependent mass threshold for macroscopic quantum decoherence ($M_{crit} = \frac{9 \pi m_a^3}{16 \rho^2 l_p^6 (\ln \chi)^3}$), offering a direct laboratory test to differentiate algorithmic gravity from standard General Relativity.