Even rosters and rotating byes have exact cycles
With n even participants, a full cycle has n − 1 rounds and includes every unordered pair once. With n odd participants and one bye, a cycle has n rounds, includes every real pair once, and gives each person one bye. The initial arrangement is shuffled with secure browser randomness, then the circle method constructs the rounds. Longer plans repeat the same cycle. Across a partial cycle, bye counts differ by at most one. The page still limits a generated plan to fifty rounds, even when the mathematical cycle would be longer.
Actual repeated contacts and a mathematical lower bound
A contact is one unordered pair of people working together in a round. A pair contributes one contact, a trio contributes three, and a bye contributes zero. A contact counts as repeated when those two IDs already met in an earlier planned round. If a pair appears three times, it contributes two repeated contacts. There are n × (n − 1) / 2 possible distinct pairs. The reported lower bound is total planned contacts minus this capacity, or zero when that difference is negative.
A zero lower bound is not a trio guarantee
The capacity calculation ignores how contacts must fit into simultaneous rounds, so its bound can be too optimistic for trios. Five people in two rounds of one pair plus one trio cannot avoid every repeated contact, even though eight contacts fit below ten possible pairs. The tool reports the actual repeats it found and labels trio results as heuristic. A positive bound proves that at least that many repeats are unavoidable. A larger observed number does not prove that no better schedule exists.