Three Generations of Fermions as Three Levels of Self‑Similarity of the Infinium △₁ₓ₁: Resolving the Three‑Generation Problem in Δ‑Ontology

This paper examines one of the long‑standing puzzles of elementary particle physics—the existence of exactly three generations of quarks and leptons. We analyze the recent result by John Baez and his collaborators, who showed that the exceptional Lie algebra E₇ naturally accommodates the gauge group of the Standard Model and three generations of fermions, with a threefold symmetry that permutes the generations. Within Δ‑ontology, where the basic primitive is not a structureless point but the structural quantum—the right isosceles triangle △₁ₓ₁ with legs 1 and hypotenuse √2—the threefoldness ceases to be a mystery. It is shown that the three generations correspond to three successive levels of fractal self‑similarity of the infinium: the base triangle, its reduced copy, and the next copy in the hierarchy. The threefold symmetry observed in E₇ turns out to be the symmetry of self‑similarity, and the energy threshold given by the energy‑saving quantum ΔE = 2 − √2 explains why exactly three generations are observed, and no more. Thus the three‑generation problem is resolved at the level of the geometry of the first principle. ---