We consider rigorous consequences of modular invariance for two-dimensional unitary non-rational CFTs with \(c > 1\) . Simple estimates for the torus partition function can lead to remarkably strong results. We show in particular that the spectral density of spin-J operators must grow like \(\exp \left( \pi \sqrt{\frac{2}{3}(c-1) J} \right) /\sqrt{2J}\) in any twist interval at or above \((c-1)/12\) , with a known twist-dependent prefactor. This proves that the large J spectrum becomes dense even without averaging over spins. For twists below \((c-1)/12\) we establish that the growth must be strictly slower. Finally, we estimate how fast the maximal gap between two spin-J operators must go to zero as J becomes large.
