This paper introduces Journey Operators for modeling structured data with multiple axes, such as images, sequences, trees, audio, and 3D volumes. Each item carries content and an axis-specific transformation. A journey operator composes these transformations along a path, governing both content aggregation and relative position. The authors show that path independence across axes holds precisely when the axis transformations commute. Under stated toral-frame symmetry, cocycle, bilinearity, and norm-preservation assumptions, the pairwise scoring rule is forced into block-wise rotations, providing a theoretical explanation for RoPE-like methods. They also propose JoFormer and report initial experiments across vision, language, and length generalization.
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