Addendum 7: Persistent Second Betti Number \(\beta_2 > 0\) from 1RSB Clustering

We prove that the ultrametric cluster structure and the Overlap Gap Property (Addendum 6) imply the existence of a persistent 2-dimensional void in the Vietoris–Rips complex of the solution space of random 3-SAT at α = 4.2. Specifically, with high probability, β2(VRϵ(Sn)) ≥ 1 for all ϵ ∈ (q0, q1), where q0 and q1 are the inter-cluster and intra-cluster overlaps from the 1RSB picture. This result constitutes the Invisible Void Conjecture, now a theorem conditional on the 1RSB free-energy convergence proved in Addenda 4–5.