Two Charged Cross Edges Minimally Close One Messenger Cycle
A complex weak-singlet scalar of hypercharge one can couple the up and down messenger sectors at both stages. Both cross vertices are renormalizable and hypercharge neutral.
One cross-edge leaves the interaction incidence acyclic. Two cross-edges give twelve rows of rank eleven and exactly one continuous and modulo-two cycle. The corresponding coupling ratio survives every field rephasing.
The charged scalar must retain zero vacuum expectation value to preserve electromagnetism. Its cycle invariant remains a free source coordinate, and the new field changes thresholds, loops, decays, and instrument requirements. This is minimal internal relative-probe capacity, not numerical selection.
Evidence: research/flavor/flavor-charged-two-edge-cycle-carrier.md and
research/flavor/results/wp635_charged_two_edge_cycle_carrier.json.