Boolean polynomial obstructions to binary expansion of balanced Cayley complexes

The proposed coboundary expansion of random balanced Cayley complexes with binary coefficients and logarithmically many group-product layers fails in every fixed dimension $k\ge2$. For $G=(\F)^d$ and every $A\subseteq G$, we prove \[ _{}H^{k-1}(Y_{A,k};) _{j=0}^{(k,d)} dj-|A|. \] The proof maps a Boolean polynomial vanishing on $A$ to a cohomology class by writing the polynomial of a total group sum as a simplicial coboundary on the complete join. Consequently, for any fixed positive constant $C$, every subset of size $\lceil C\log D(G)\rceil$ has zero binary coboundary expansion for all sufficiently large $d$, where $D(G)$ is the sum of the degrees of its irreducible complex representations.