Complete bipartite schedules without temporal returns are rotations

We consider two groups of m participants and m rounds of perfect matchings in which each participant meets every member of the other group exactly once. We prove that such a schedule admits no return along a chain of encounters at strictly increasing times (equivalently, it has the NOHO property of gossip theory) if and only if it is a rotation after a fixed relabeling of the participants. It suffices to exclude returns involving four encounters. With the first round normalized there are (m−1)! safe schedules. For a fixed cyclic family, the safe round orders are exactly s_t = a + dt with gcd(d, m) = 1, and there are m·φ(m) of them; an alternative proof of this case uses a classical zero-sum fact. Kamecke (1997) proved that rotations are free of repercussion effects, and Weimann and Brosig-Koch (2019) attribute a broad uniqueness claim to him; this note gives a self-contained proof of a precise formulation. Bibliographic priority is not certified. Exact verification scripts (Python, standard library) are included. AI assistance (OpenAI Codex; independent review with Claude, Anthropic) is disclosed in the manuscript. The work has not undergone external human peer review.