Hall Obstructions in Regular Tournaments and the Strong Seymour Conjecture Through Order 13

This preprint studies inclusion-minimal Hall obstructions at vertices of regular tournaments. For a regular tournament of order 2k+1, it derives exact identities for the directed cuts surrounding such an obstruction. At order 13 these identities leave only obstruction sizes three and four, and each case yields an explicit matching that produces a strong Seymour vertex. Combined with the theorem of Bai, Li, and Park for oriented graphs of minimum outdegree at most five, this proves the strong Seymour conjecture for all tournaments of order at most 13. The paper also derives structural consequences at orders 11 and 13, proves an equivalence with the positive-integer weighted formulation, and includes a reproducible catalogue audit that is logically separate from the direct proof.