Addendum 4 v2: Rigorous Guerra–Toninelli Lower Bound for the Poisson‑Cloned 3‑SAT Model

We provide a complete and rigorous proof of the free-energy lower bound for the Poisson-cloned 3-SAT model at clause density α = 4.2. The proof adapts the Guerra–Toninelli interpolation to the sparse hypergraph setting, building on the contiguity established in Addendum 2, the stability of the 1RSB solution (Montanari–Parisi–Ricci-Tersenghi 2004), and the interpolation techniques developed for NAE-SAT (Nam–Sly–Sohn 2024). Together with the upper bound proved in Addendum 5 v2, this establishes the convergence of the free energy to the 1RSB prediction F1RSB. The present addendum supersedes the heuristic sketch of Addendum 4 and provides the full mathematical details.