Why Exactly Three Generations: Geometric Necessity from Planck-Scale Spacetime and Atemporal Hyperspace Projection

The Standard Model accommodates three generations of fermions without explaining why this number is precisely three. We demonstrate that the existence of exactly three generations is not a free parameter but a geometric necessity arising from the dimensional structure of spacetime. Within the framework of the Teoría Unificada Floriano (TUF), we show that fermions emerge as projections from an atemporal hyperspace Ξ onto the observable spacetime M⁴×S¹, where wave configurations in Ξ can sustain at most d distinct harmonic modes for d spatial dimensions. Since our universe has d=3 spatial dimensions, precisely three fermion generations exist. We derive fermion masses from topological quantization mₙ = n·mₑ where quantum numbers n arise from discrete symmetry groups {A₄, A₅, PSL(2,p)}, predicting the nine charged fermion masses with mean accuracy 0.90% and zero free parameters (PDG 2025). The reduced χ² = 0.02 confirms agreement well within experimental uncertainties. A topological invariant R_L + R_U + R_D = 5/2 is derived from the fibration structure of M⁵.