AdS$_3$ Quantum Gravity and Finite $N$ Chiral Primaries

String theory on AdS$_3$ $\times$ S$^3$ $\times$ $\mathcal{M}_4$ provides a well-studied realization of AdS$_3$/CFT$_2$ holography, but its non-perturbative structure at finite $N \sim 1/G_N^{(3)}$ is largely unknown. A long-standing puzzle concerns the stringy exclusion principle: what bulk mechanism can reproduce the boundary expectation that the chiral primary Hilbert space of the symmetric orbifold contains only a finite number of states at finite $N$? In this work, we present a bulk prescription for computing the finite $N$ spectrum of chiral primary states in symmetric orbifolds of $\mathbb{T}^4$ or K3. We show that the integer spectrum at any $N$ is reproduced exactly by summing over one-loop supersymmetric partition functions of the IIB theory on (AdS$_3$ $\times$ S$^3$)/$\mathbb{Z}_k$ $\times$ $\mathcal{M}_4$ orbifolds and their spectral flows. Using the worldsheet in the tensionless limit, we verify that the terms appearing in our proposal coincide with the partition functions of these orbifold geometries and their asymmetric generalizations. These partition functions contribute with alternating signs due to BPS modes with negative conformal dimensions and charges in twisted sectors. The resulting alternating sum collapses via large cancellations to the finite $N$ polynomials observed in symmetric orbifold CFTs, providing a bulk explanation of the stringy exclusion principle. We identify different Stokes sectors where different infinite subsets of these geometries contribute to the path integral, and propose a classification as functions of the chemical potentials.