A Structural Theory of Trajectory Spaces on Hypercubes: Foundational Preprints and Conceptual Framework

We introduce a structural framework for the study of minimal trajectory spaces in the discrete hypercube Qₙ. Given two vertices, we construct a canonical cubical complex encoding the space of minimal trajectories, arising naturally from independence relations between elementary transitions. These relations induce commutation structures that organize trajectories into higher-dimensional cubes. This perspective allows the trajectory space to be interpreted as an intrinsic geometric object rather than a collection of individual paths. We further define a subgroup of compatible symmetries whose action induces structural invariants on the complex. The resulting framework establishes a connection between combinatorics, discrete geometry, and group theory, providing the conceptual foundation for a broader structural theory developed across a sequence of complementary works.