From Simplicial Minimality to Spectral Dimension: Why 4 × 2 = 8 Does Not Yield ds = ln8

Abstract

A non-degenerate simplex in three dimensions has four vertices. Separately, the SU(2)- invariant intertwiner space of a four-valent j = 1/2 node is two-dimensional. Their product is eight. It is tempting to set beff = 8 and ds = ln 8.That identification is not a theorem. This note separates three quantities—geometric coordination z, internal multiplicity M , and dynamical branching bdyn—and shows that finite internal multiplicity together with local finite-range dynamics does not change the leadingspectral exponent. On a three-dimensional Hausdorff geometry one still has dw = 2 and ds = 3. If one assumes dH = 3 and requires ds = ln 8, the walk dimension must be dw = 6/ ln 8 ≃ 2.885. That is an anomalous-transport target, not a derivation. An alternative programme (reduced Hausdorff dimension dH ≃ ln 8 with ordinary diffusion) is a different hypothesis and is not treated as proved here.The missing object is an explicit tetrahedral coarse-graining that couples the intertwinersector to spatial transport. Until that object exists, 4×2=8⇏ds=ln⁡8.