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 mapping of Gómez‑Ruiz et al. (2025) is also discussed. This result constitutes the second barrier of the Black Hole Trilemma, together with the OGP (Addendum 6) and the exponential backtracking (Addendum 8).