Borel graphable equivalence relations

The paper proves the exact equivalence in Theorem 14—F_{ω1} is Borel graphable iff there exists a non-constructible real—and does so via (i) a backward implication in L using a hyperdegree-minimal representative (Theorem 24) and a relativized form of Friedman’s conjecture (Proposition 25), and (ii) a forward implication obtained by establishing the Borel witness coding property for F_{ω1} via Kumabe–Slaman forcing plus a Σ^1_3-absoluteness transfer (Lemma 26 and Proposition 27). The candidate solution outlines precisely this two-pronged strategy: the L-argument mirrors Proposition 25’s induction on graph distance and use of relativized Friedman + minimal hyperdegree; the forward direction matches Lemma 26/Proposition 27’s witness-coding construction and the Shoenfield absoluteness step. The link from witness coding to a diameter-2 Borel graphing is also correctly cited by the candidate and is Proposition 10 in the paper. Overall, the model’s proof sketch and the paper’s proofs agree essentially step-for-step, with only minor presentational differences (e.g., the model states a slightly stronger version of the ‘minimal hyperdegree’ fact than needed). Key touchpoints: Theorem 14, Proposition 25, Proposition 27, and Proposition 10 in the paper; plus the setup of the witness-coding framework and the use of Kumabe–Slaman forcing in the appendix.