Polynomial-size commutative aggregates cannot be half of a covering pair (Erdős 624)

Theorem. Fix $a\ge0$. There is $k_0$ such that for all $k\ge k_0$ the following holds. Let $|U|=n=2^k$, let $M$ be any finite commutative monoid with $|M|\le n^a$, and let $v:U\to M$, $\psi:M\to U$, $g:\mathcal P(U)\to U$ be arbitrary. Put $h(A)=\psi\bigl(\prod_{x\in A}v(x)\bigr)$. Then there are a $k$-set $Y\subseteq U$ and a point $u\in U$ missed by both $h$ and $g$ on all subsets of $Y$.

This covers, for example, any fixed number of sums in $\mathbb F_2^k$, $\mathbb F_{2^k}$ or $\mathbb Z/2^k\mathbb Z$, products in a ring of size $n$, mixtures of these, and boundedly many elementary symmetric functions, each followed by arbitrary postprocessing. Consequently, a pair of maps covering every $k$-set for all large $k$ (which would be needed to refute the two-map common-hole route to Erdős Problem 624) cannot have either map of this polynomial-state commutative form. Context: Erdős Problem 624 asks whether $H(n)-\log_2 n\to\infty$, where $H(n)$ is the least $m$ such that some $f:\mathcal P(X)\to X$ ($|X|=n$) satisfies $f[\mathcal P(Y)]=X$ for all $|Y|\ge m$. By an earlier track-amplification result it would suffice to show that, for large $k$, any two maps $\mathcal P(U)\to U$ with $|U|=2^k$ have a common hole: a $k$-set $Y$ and a point $u$ missed by both maps on $\mathcal P(Y)$. These results do not settle Problem 624.

Formalization Note. $U$ and $M$ range over arbitrary finite types in Type.