arXiv:2610.40683v1 Announce Type: new Abstract: We consider an undirected edge-traversal game on a finite bouquet of simple cycles. Each of two players controls a token, both tokens begin at the common root, and the players alternate traversing edges that have not been used by either token. A player with no legal move loses. We prove that the first player wins precisely when the number of cycles of odd edge-length is odd, without assuming equal cycle lengths. The proof uses a simultaneous induction on root and defense configurations, with an exact analysis of the forced race between two opened cycles. We also explain how the classification corresponds to a completed Lean 4 formalization of the underlying game on finite edge sets.
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