  Diophantine equations are in general undecidable, yet appear readily in
string theory. We demonstrate that numerous classes of Diophantine equations
arising in string theory are decidable and propose that decidability may
propagate through networks of string vacua due to additional structure in the
theory. Diophantine equations arising in index computations relevant for
D3-instanton corrections to the superpotential exhibit propagation of
decidability, with new and existing solutions propagating through networks of
geometries related by topological transitions. In the geometries we consider,
most divisor classes appear in at least one solution, significantly improving
prospects for Kahler moduli stabilization across large ensembles of string
compactifications.
