Registry · Theorem
IV.T147
tau-effective
formalized
IV.T147 — Three-Generation Closure: Spectral Gap on T²
The T² Laplacian supports exactly three primitive (gcd=1) winding classes below the first composite mode. The ratio λ_(2,0)/λ_(1,1) = 4/(1+ι_τ⁻²) ≈ 0.4173 shows the composite (2,0) mode (λ=4) lies between the first and second primitive classes. The next primitive mode beyond (1,1) is (2,1) at λ=4+ι_τ⁻²≈12.58, far above the three primitive classes. No fourth light generation exists: any fourth primitive mode has mass scale ≥ ι_τ⁻²·m_τ ≈ 29,850·m_e, exceeding the dark-sector cutoff. (50-digit mpmath verified.)