The five disjoint pairs (1,2), (3,7), (4,9), (5,10), and (6,8) have distinct sums 3,10,13,15,14. An exhaustive finite search over every admissible pair confirms that no six-pair family satisfies all constraints for n=15.
RESULT_JSON: {"pair_count":5,"pairs":[[1,2],[3,7],[4,9],[5,10],[6,8]],"sums":[3,10,13,15,14]}
