The Standard Model contains three generations of fermions with identicalgauge quantum numbers, differing only in mass. No accepted theory explainswhy the number is three. This note isolates a single structural argumentthat fixes the number, independent of any particular model of massgeneration. The argument has three steps. (i) By the classification ofnormed division algebras (Hurwitz) and the parallelizability of spheres(Bott-Milnor, Kervaire, Adams), a space whose unit sphere carries acoherent structure of independent rotational directions can have dimensiononly 1, 3, or 7. (ii) Dimension 1 supports no non-abelian structure.(iii) Dimension 7 corresponds to the octonions, whose non-associativitymakes the composition of a causal chain depend on its bracketing -- agrouping that carries no temporal information -- so that identicalsequences of events yield distinct outcomes with no physical quantity todistinguish them. Causal determinacy therefore excludes dimension 7,leaving dimension 3 as the unique survivor, realised by the associativequaternions. We keep a sharp line between what is proved (the number three)and what is proposed (its identification with the three observedgenerations), and we note that the argument constrains any theory claimingboth extra dimensions and causal order, not only the framework in which itarose.
Gereon Kraemer (Mon,) studied this question.