Addendum 5 v2: Rigorous First‑Moment Calculation for the Number of Solution Clusters

We provide a complete and rigorous calculation of the first moment E[N_k] of the number of solution clusters of size k in the Poisson‑cloned random 3‑SAT model at clause density α = 4.2. The calculation shows that E[N_k] is exponentially large in n exactly when the free energy equals the 1RSB prediction of Mézard–Parisi–Zecchina (2002). Together with the variance estimate proved in Addendum 3, this yields the free‑energy upper bound and completes the proof of convergence to the 1RSB value. This version supersedes the heuristic sketch of Addendum 5 (v1) and provides the full mathematical details.