Existence of two distinct Hamilton-path extensions

Prove that, for n ≥ 5, opposite-parity vertices u and v, and a matching M of Q_n, two distinct Hamilton paths between u and v extending M exist if and only if none of the C-conditions holds for M.

Background

The two-path formulation is a strengthened auxiliary conjecture introduced to support the paper’s inductive construction across canonical subcubes. The paper verifies it computationally for dimension 5 and uses it to prove the Hamilton-path characterization for matchings spanning at most five directions. Its validity for general dimensions remains unresolved; the analogous statement fails in dimension 4 because of explicit obstructions.