  Large $N$ factorization ensures that, for low-dimension gauge-invariant
operators in the half-BPS sector of ${\cal N}=4$ SYM, products of holomorphic
traces have vanishing correlators with single anti-holomorphic traces. This
vanishing is necessary to consistently map trace operators in the CFT$_4$ to a
Fock space of graviton oscillations in the dual AdS$_5$. We investigate the
regimes at which the CFT correlators do not vanish but become of order one in
the large $N$ limit, which we call a factorization threshold. Quite generally,
we find the threshold to be when the product of the two holomorphic operator
dimensions is of order $N\log N$. Our analysis considers extremal and
non-extremal correlators and correlators in states dual to LLM backgrounds, and
we observe intriguing similarities between the the energy-dependent running
coupling of non-abelian gauge theories and our threshold equations. Finally, we
discuss some interpretations of the threshold within the bulk AdS spacetime.
