Emergence of Three-Dimensional Space and Normal Diffusion

The paper develops the spatial-geometric sector of TSA and studies how an effective three-dimensional geometric phase arises from the projective-thermodynamic conflict front-end established in the preceding work TSA-FOUND-PTO. Starting from the imported objects of the PTO paper — the effective conflict functional, the baseline/defect decomposition, the baseline transport form, and the baseline transport metric — the paper analyzes admissible homogeneous rank slices and proves the closure of the rank-3 spatial package. The main result is the selection and stabilization of the rank-3 branch, together with the identification of the associated large-scale geometric and transport exponents: linear dimension 3, effective dimension 3, spectral dimension 3, and walk dimension 2. In this sense, the work provides the geometric-local phase, spectral dimension 3, and normal diffusion structure required by the subsequent TSA geometry papers on Dirichlet forms, spectral gap, structural diffusion, and Green–Kubo causal speed. The article also clarifies the role of rank-nondegenerate homogeneity and includes a microscopic sufficient criterion for the rank-slice cost gap through finite local grammar, local continuation types, and the projective incompatibility counter imported from TSA-FOUND-PTO.