Structural Diffusion and the Correlation Scale.

This preprint defines the structural diffusion coefficient as the transport invariant of the geometric line of the Theory of Structural Articulation. The paper is synchronised with four companion works of the corpus: Projective Thermodynamic Closure, Emergence of Three-Dimensional Space and Normal Diffusion, Dirichlet Form and Reconstruction of Macroscopic Geometry, and Spectral Gap and Stability of the Geometric Phase. The central aim of the paper is to show that the diffusion coefficient is not a phenomenological fitting parameter. It is derived as a transport modulus of a projectively and thermodynamically admissible geometric phase. The same coefficient is recovered from four equivalent descriptions: the effective vacuum generator, the Gaussian heat-kernel asymptotics, the mean-square displacement law, and the Green–Kubo–Dirichlet corrector construction. The paper also formulates a rigidity result for the normal diffusion regime. Within the admissible local Dirichlet/Mosco class, anomalous transport is not treated as an alternative stable macroscopic branch. Subdiffusive regimes correspond to traps, bottlenecks, heavy waiting-time tails, or vanishing effective conductivity; superdiffusive regimes correspond to non-local jump tails or fractional-Laplacian limits. Both cases leave the local transport sector selected by the projective-thermodynamic closure. Using the protected spectral gap imported from the companion gap article, the paper defines the corresponding transport-relaxation length. This length is a phase-level correlation scale of the geometric vacuum sector. It is not identified with the Planck length and is not presented as an absolute universal constant. Physical time calibration, causal-speed closure, gravitational normalisation, and later applications are left outside the scope of this article. The result is a self-contained transport-level closure: it fixes what structural diffusion means in the TSA geometric phase, why the normal regime is selected, and how the diffusion coefficient enters the correlation scale used by subsequent sectors of the theory.