THE FAREY GRAPH IS UNIQUELY DETERMINED BY ITS CONNECTIVITY

The paper proves that every Π-graph contains the Farey graph as a tight (finite-branch-set) minor (Theorem 2) and deduces that, up to minor-equivalence, the Farey graph is the unique typical Π-graph (Theorem 1). The candidate solution invokes exactly this chain: it cites the tight-minor theorem, mentions the halved Farey graph and the grain-line decomposition (Theorem 6.1), and then concludes uniqueness by the immediate two-way minor relation between any Π-typical graph and F. This aligns with the paper’s strategy and results. See Theorem 1 statement and setup of typical Π-graphs , the tight-minor statement and its role , the overall proof strategy via halved Farey graph and Theorem 6.1 , the halved Farey-to-Farey contraction lemma , and the explicit note that Theorem 2 implies Theorem 1 .