Modern deep learning involves training costly, highly overparameterized networks, thus motivating the search for sparser networks that require less compute and memory but can still be trained to the same accuracy as the full network (i.e. matching). Iterative magnitude pruning (IMP) is a state of the art algorithm that can find such highly sparse matching subnetworks, known as winning tickets, that can be retrained from initialization or an early training stage. IMP operates by iterative cycles of training, masking a fraction of smallest magnitude weights, rewinding unmasked weights back to an early training point, and repeating. Despite its simplicity, the underlying principles for when and how IMP finds winning tickets remain elusive. In particular, what useful information does an IMP mask found at the end of training convey to a rewound network near the beginning of training? We find that—at higher sparsities—pairs of pruned networks at successive pruning iterations are connected by a linear path with zero error barrier if and only if they are matching. This indicates that masks found at the end of training encodes information about the identity of an axial subspace that intersects a desired linearly connected mode of a matching sublevel set. We leverage this observation to design a simple adaptive pruning heuristic for speeding up the discovery of winning tickets and achieve a 30% reduction in computation time on CIFAR-100. These results make progress toward demystifying the existence of winning tickets with an eye towards enabling the development of more efficient pruning algorithms.