Topological and Combinatorial Invariants of Trajectory Spaces on Hypercubes

We introduce a canonical cubical structure on the space of minimal trajectories between two configurations of a discrete hypercube. From commutation relations between independent elementary transitions, we construct a cubical complex X(x, y) whose vertices correspond to minimal trajectories and whose higher-dimensional cells encode families of trajectories related by commutation. This construction is fully determined by the combinatorial structure of the hypercube and does not depend on auxiliary choices. As a consequence, the trajectory space is interpreted as an intrinsic geometric object rather than a representation dependent on a chosen ordering of transitions. The framework naturally yields combinatorial and topological invariants and establishes the structural principle that the geometry of the trajectory space emerges from independence relations between transitions.