On Approximability of Satisfiable $k$-CSPs: VII

Let ${\mathrm{\Sigma }}_1,\dots ,{\mathrm{\Sigma }}_k$ be finite alphabets, and let $\mu$ be a distribution over ${\mathrm{\Sigma }}_1\times \cdots \times {\mathrm{\Sigma }}_k$ in which the probability of each atom is at least $\alpha$. We prove that if $\mu$ does not admit Abelian embeddings, and $f_i:{\mathrm{\Sigma }}_i\to \mathbb{C}$ are $1$-bounded functions (for $i=1,\dots ,k$) such that $$\mid {\mathbb{E}}_{(x_1,\dots ,x_k)\sim {\mu }^{\otimes n}}[f_1(x_1)\dots f_k(x_k)]\mid \ge \epsilon ,$$ then there exists $L\text{\hspace{0.17em}⁣}:{\mathrm{\Sigma }}_1^n\to \mathbb{C}$ of degree at most $d$ and $\mathrm{\parallel }L{\mathrm{\parallel }}_2\le 1$ such that $\mathrm{\mid }⟨f_1,L⟩\mathrm{\mid }\ge \delta$, where $d$ and $\delta >0$ depend only on $k,\alpha$ and $\epsilon$. This answers the analytic question posed by Bhangale, Khot, and Minzer (STOC 2022). We also prove several extensions of this result that are useful in subsequent applications.