Addendum 3: p‑Biased Fourier Expansion for the 3‑SAT Spin System

We introduce the p-biased Fourier expansion on the infinite spin system that encodes the solution clusters of the Poisson-cloned 3-SAT model (Addendum 2). We show how the number Nk of clusters of size k can be expanded in the p-biased Fourier basis, derive the variance decomposition, and apply the hypercontractivity estimates of Keevash–Lifshitz–Long–Minzer (2024) to control the high-degree Fourier coefficients. The main result is that the variance of Nk is bounded by a polynomial in n times its expectation, which is the crucial estimate needed for the second-moment method in the 1RSB proof.