Path-dependency and emergent computing under vectorial driving

The sequential response of frustrated materials - ranging from crumpled sheets and amorphous media to metamaterials - reveals their memory effects and emergent computational potential. Despite their spatial extension, most studies rely on a single global stimulus, such as compression, effectively reducing the problem to scalar driving. Here, we introduce vectorial driving by applying multiple spatially localized stimuli to explore path-dependent, sequential responses. We uncover a wealth of phenomena absent in scalar driving, including non-Abelian responses, mixed-mode behavior, and chiral loop transients. We find that such path dependencies arise from elementary motifs linked to fold singularities, which connect triplets of states - ancestor, descendant, and sibling; and develop a general framework using pt-graphs to describe responses under any vectorial driving protocol. Leveraging binarized vectorial driving, we establish a natural connection to computation, showing that a single sample can encode multiple sequential Boolean circuits, which are selectable by driving strength and reprogrammable via additional inputs. Finally, we introduce graph-based motifs to manage the complexity of high-dimensional driving. Our work paves the way for strategies to explore, harness, and understand complex materials and memory, while advancing embodied intelligence and in-materia computing.