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

We compute the first moment E[Nk] of the number of solution clusters of size k in the Poisson-cloned 3-SAT model (Addendum 2) with the p-biased measure (Addendum 3). The calculation shows that E[Nk] is exponentially large in n precisely when the free energy equals the 1RSB prediction of Mézard–Parisi–Zecchina (2002). This completes the proof sketch of the upper bound in Addendum 4, thereby establishing the convergence of the free energy to F1RSB.