BCT Letter 89: Why Space Has Three Dimensions — D4 Triality, the Hopf Fibration, and BCT Compactification

Letter 89 of the BCT Superfluid Lattice Model programme. Derives from first principles why physical space has exactly three large dimensions, via three independent convergent proofs. (1) Hopf fibration: the OHC vacuum manifold is S³, whose Hopf fibration S³→S² compactifies one dimension at the Planck scale, leaving three observable dimensions. (2) D4 triality: the triality symmetry of SO(8), unique to D4, permutes three representations — 8_v (spatial dimensions), 8_s+ (quark generations), 8_s- (lepton generations) — forcing exactly three large dimensions by the same theorem that forces three quark generations (Letter 61). (3) Crystallographic: the BCT constraint c/a=√2 sets the [110] direction to Planck-scale length, compactifying it and leaving exactly three large directions. All three arguments converge. Space is three-dimensional because the vacuum is a D4-symmetric superfluid on an S³ manifold with c/a=√2. Companion to Letter 88 (10.5281/zenodo.19140621). Prediction #126.