Computational modeling demonstrates matrix-valued coupling in vector synchronization on sparse graphs, indicating that topological changes are detectable as internal-time regime shifts.
The D-Operator generalizes Kuramoto synchronization to vector states, but its coupling term uses a scalar weight per edge. FieldMap, however, is defined around d x d coupling matrices -- anisotropic, direction-dependent influence between nodes, which a scalar weight cannot express. We introduce FieldDOperator, a drop-in generalization of the D-Operator's coupling term from a scalar weight to a matrix, implemented without modifying the original D-Operator file. We verify it reduces exactly to the scalar case, actually exhibits anisotropic behavior a scalar coupling cannot, respects FieldMap's sparse (non-all-to-all) edge structure, and ports to PyTorch with differentiable coupling parameters. We then connect it to StructuralTime and show that a change in the coupling topology -- not just a change in target, as in the companion paper -- is also detectable as an internal-time regime change, with a real interpretive error made and corrected along the way.
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ARIMOTO et al. (2026) studied this question.
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