The finite N BPS spectrum of the symmetric orbifold CFT of K3 contains "polar" states below the black hole threshold whose holographic origin has remained unclear. We propose a bulk prescription for computing these degeneracies exactly at finite N, given by a sum over one-loop partition functions of the IIB theory on spectrally-flowed (AdS3 x S3)/Z_K x K3 orbifold backgrounds. For half-BPS chiral primaries, the infinite sum over orbifold saddles collapses to a finite polynomial via large cancellations due to bosonic negative modes, providing a bulk explanation of the stringy exclusion principle. We then extend these results to the 1/4-BPS polar states of the finite N elliptic genus. The contributing saddles and their Stokes phenomena are found to be governed by the analytic structure of the grand-canonical partition function.