Hurwitz's Theorem and the 3+1 Dimensional Brane (Brane-Bulk Series, Paper XXIX)

The dimensionality of spacetime 3+1 is not a free parameter — it is fixed by Hurwitz's theorem: the only normed division algebras are R (1D), C (2D), H (4D), and O (8D). The universe must have a brane of dimension 2^n - 1 (spatial) where 2^n is the division algebra dimension: R gives 0+1D (trivial), C gives 1+1D (strings), H gives 3+1D (our universe), O gives 7+1D (parent spacetime). The 3+1D brane we inhabit is forced by the quaternionic structure of matter H+ = {1,e1,e2,e3}. The 7+1D bulk is forced by the octonionic parent. Spacetime dimensionality is algebraically inevitable. Part of the One Octonion brane-bulk series (Papers I-XL), anchor DOI:10.5281/zenodo.19120873.