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We combine the large-$c$ ST modular bootstrap equations with the Cardyformula for the asymptotic growth of the density of states to prove that any$2d$ unitary, compact, conformal field theory (CFT) with no higher spinconserved currents leads to conflicting inequalities whenever the entirespectrum of non-trivial primaries lies above the BTZ threshold. As aconsequence, the holographic dual of $3d$ pure gravity, if it exists, cannot bea $2d$ CFT. Consistent solutions of ST bootstrap equations require additionalprimaries lying below the BTZ threshold. The lowest non-trivial primarynecessarily has odd spin

Three-dimensional quantum gravity according to ST modular bootstrap