Non-redundancy, introduced by Bessiere, Carbonnel, and Katsirelos (AAAI 2020), is a structural parameter for Constraint Satisfaction Problems ($\mathsf{CSPs}$) that governs kernelization, exact and approximate sparsification, and exact streaming complexity. It is the largest size of a $\mathsf{CSP}$ instance admitting no smaller subinstance with the same satisfying assignments. We study non-redundancy $\mathsf{NRD}_n(R)$ for Boolean symmetric $\mathsf{CSPs}$ defined by an $r$-ary relation $R$ whose value depends only on Hamming weight. An instance of $\mathsf{CSP}(R)$ has $n$ variables and constraints given by $r$-tuples; a constraint is satisfied exactly when the induced tuple lies in $R$. This class includes natural predicates such as cuts and $k$-SAT clauses. Our main result is a near-complete classification of the asymptotic growth of $\mathsf{NRD}_n(R)$ for symmetric Boolean predicates of arity at most $5$. Using computational experiments and algebraic upper- and lower-bound criteria, we resolve every predicate of arity at most $4$ and all but two predicates of arity $5$. For upper bounds, we introduce $t$-balancedness, a lifted, higher-degree version of the balancedness notion of Chen, Jansen, and Pieterse (Algorithmica 2020). We prove that $t$-balancedness is equivalent to the existence of degree-$t$ multilinear polynomials capturing $R$, and hence implies $\mathsf{NRD}_n(R)=O(n^t)$. For lower bounds, we use Carbonnel's (CP 2022) framework: predicates admitting a special reduction from $k$-ary OR inherit OR's lower bound $Ω(n^k)$. The only unresolved arity-$5$ predicates in our framework have bounds $Ω(n^2)$ and $O(n^3)$; we reduce their exact classification to natural extremal set-system questions.