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 — 8ᵥ (spatial dimensions), 8ₛ+ (quark generations), 8ₛ- (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.
Michel Robert Cabrié (Sun,) studied this question.