Addendum 3 v2: Rigorous Variance Control for the Number of Solution Clusters (Hypercontractivity on the p‑biased Cube)

We provide a complete and rigorous proof of the variance control Var[N_k] ≤ n^O(1) E[N_k] for the number of solution clusters of size k in the Poisson‑cloned 3‑SAT model at clause density α = 4.2. The proof combines the p‑biased Fourier expansion (Addendum 3 v1) with the sharp global hypercontractivity estimates of Keevash–Lifshitz–Long–Minzer (2024). This result is the central ingredient for the second‑moment method that establishes the free‑energy upper bound (Addendum 5 v2) and completes the 1RSB proof. The present addendum supersedes the heuristic sketch of Addendum 3 v1 and provides the full mathematical details.