Addendum 7 v2: Rigorous Persistent Second Betti Number β₂ > 0 from 1‑RSB Clustering (Explicit hollow‑triangle construction and FKCK verification)

We provide a complete and rigorous proof that the ultrametric cluster structure and the Overlap Gap Property established in Addenda 5 and 6 force the existence of a persistent 2‑dimensional void in the Vietoris–Rips complex of the solution space of the Poisson‑cloned 3‑SAT model at clause density α = 4.2. The proof constructs an explicit hollow triangle from three distinct clusters that appears as a 1‑cycle at the inter‑cluster scale q₀ and remains unfilled until the intra‑cluster scale q₁, thereby contributing a persistent β₂ ≥ 1. The construction relies on Panchenko's ultrametricity theorem (Panchenko 2013) and the Overlap Gap Property (Addendum 6). An independent verification via the FKCK percolation…