  We study to what extent, and in what form, the notion of gauge-string duality
may persist at finite $N$. It is shown, in the half-BPS sector, that the states
of D3 giant graviton branes in $\mathrm{AdS}_5 \times S^5$ are holographically
dual to certain auxiliary ghosts that compensate for finite $N$ trace relations
in $U(N)$ $\mathcal{N}=4$ super Yang-Mills. The complex formed from spaces of
states of bulk D3 giants is observed to furnish a BRST-like resolution of the
half-BPS Hilbert space of $U(N)$ $\mathcal{N}=4$ SYM at finite $N$. We argue
that the identification between the states of certain bulk D-branes and the
auxiliary ghosts in the boundary holds rather generally at vanishing 't Hooft
coupling $\lambda = 0$. We propose that a complex, which furnishes a BRST-like
resolution of the finite $N$ Hilbert space of a boundary $U(N)$ gauge theory at
$\lambda = 0$, should be identified as the space of states of the dual string
theory in the $\alpha' \to \infty$ limit. The Lefschetz trace formula provides
the holographic map in this regime, where bulk observables are computed by
taking the alternating sum of the expectation values in an ensemble of states
built on each open string vacuum. The giant graviton expansion is recovered and
generalized in a limit of the resolution.
