Addendum 6 v2: Rigorous Overlap Concentration and the Overlap Gap Property (Ghirlanda–Guerra identities and Panchenko ultrametricity for the Poisson‑cloned 3‑SAT model)

We present a complete and rigorous proof that, under the 1-RSB free-energy convergenceestablished in Addenda 4 and 5, the overlap distribution P(q) of two uniformly sampledsolutions of the Poisson-cloned 3-SAT model at clause density α = 4.2 concentrates onexactly two values q0 < q1. The proof proceeds in two steps. First, we derive the Ghirlanda–Guerra identities for the model from the Guerra–Toninelli interpolation (Addendum 4 v2)and the self-averaging of the free energy (Addenda 4–5). Second, we apply Panchenko’sultrametricity theorem (Panchenko 2013) to conclude that the support of P(q) consists ofat most two points; the 1-RSB cavity equations then fix these points as the intra-clusterand inter-cluster overlaps. The Overlap Gap Property follows immediately as a corollary.The present addendum supersedes the sketch of Addendum 6 and provides full mathematicaldetails.