Although hippocampal grid cells are thought to be crucial for spatial navigation,their computational purpose remains disputed. Recently, theywere proposed to represent spatial transitions and convey this knowledgedownstream to place cells. However, a single scale of transitions isinsufficient to plan long goal-directed sequences in behaviorally acceptabletime.Here, a scale-space data structure is suggested to optimally accelerateretrievals from transition systems, called transition scale-space (TSS).Remaining exclusively on an algorithmic level, the scale increment isproved to be ideally√2 for biologically plausible receptive fields. It isthen argued that temporal buffering is necessary to learn the scale-spaceonline. Next, two modes for retrieval of sequences from the TSS are presented:top down and bottom up. The two modes are evaluated in symbolicsimulations (i.e., without biologically plausible spiking neurons).Additionally, a TSS is used for short-cut discovery in a simulated Morriswater maze. Finally, the results are discussed in depth with respectto biological plausibility, and several testable predictions are derived.Moreover, relations to other grid cell models, multiresolution path planning,and scale-space theory are highlighted. Summarized, reward-freetransition encoding is shown here, in a theoretical model, to be compatiblewith the observed discretization along the dorso-ventral axis of themedial entorhinal cortex. Because the theoretical model generalizes beyondnavigation, the TSS is suggested to be a general-purpose corticaldata structure for fast retrieval of sequences and relational knowledge.Source code for all simulations presented in this paper can be found athttps://github.com/rochus/transitionscalespace.