We present a proof of the following statement in two-dimensional conformal field theory (2d CFT). In any unitary, modular-invariant 2d CFT with central charge c>1, a normalizable vacuum, and a twist gap in the spectrum of Virasoro primary states, there exists a continuous family of twist accumulation points above the BTZ threshold. Furthermore, we derive a Cardy-like formula that counts the number of spin-J Virasoro primary states within a fixed twist window. This formula shows that, in the large-spin limit, the first three terms of the microcanonical entropy are universal.