  Working within the framework of both the $A_\infty$ and the Berkovits open
superstring field theory, we derive a necessary and sufficient condition for a
Neveu-Schwarz marginal deformation to be exact up to third order in the
deformation parameter. For a specific class of backgrounds, we find that this
condition localizes on the boundary of the worldsheet moduli space, thus
providing a very simple computational prescription for recovering algebraic
constraints (generalized ADHM equations) which need to be satisfied by the
moduli. Applying our results to the $\mathrm{D(-1)/D3}$ system, we confirm up
to third order that blowing up the size of the D-instanton inside the D3 brane
worldvolume is an exact modulus of the full string theory. We also discuss
examples of more complicated backgrounds, such as instantons on unresolved ALE
spaces, as well as the spiked instantons.
