Dimensional Emergence in QIW–EC: Cayley–Dickson Degeneration and Hausdorff/Spectral Dimensions (v1.0)

We develop a computable framework for dimensional emergence in the Quantum Intrinsic Wormhole–Einstein–Cartan (QIW–EC) program. Spacetime is modeled as a discrete wormhole network at Planck scale; smooth geometry and continuum QFT are IR effective images. Each directed edge carries a Cayley–Dickson (CD) algebra-valued variable. We introduce an action that penalizes nonassociativity, norm non-multiplicativity, and zero divisors. As the CD level nnn increases (dimension 2n2^n2n), algebraic pathologies suppress coherent propagation. Under RG flow this induces an algebraic natural selection toward a stable island n≤3n\le 3n≤3 (at most octonions), yielding a macroscopic world with one time plus at most four spatial dimensions.
We quantify dimensional flow via spectral dimension dsd_sds and Hausdorff/ball-growth dimension dHd_HdH, discuss dimensional collapse (ds→0d_s\to 0ds→0) at extreme densities (GUT window, black-hole cores), and provide a minimal, reproducible prototype. We outline a motivated route from an octonionic phase (and exceptional group G2G_2G2) to the Standard Model gauge structure, and propose a “dimension detector” based on ds(E)d_s(E)ds(E), quasi–Kaluza–Klein spectra, causal-stability tests, and the presence of an O\mathbb{O}O-window. The paper states falsifiable predictions and a concrete plan for numerical scans, without invoking extra geometric dimensions as fundamental.