Erasure-cost lower bound: a parity certificate must delete a point of each zero-sum triple block (Erdős 624 cross-review)

Let $P_1,\dots,P_t$ be pairwise disjoint finite sets of coordinates and, for each $j$, let $E^1_j,E^2_j,E^3_j\subseteq P_j$ satisfy $\chi_{E^1_j}+\chi_{E^2_j}+\chi_{E^3_j}=0$ over $\mathbb F_2$ (relation (Z)). Let $R$ be a set of deleted coordinates and $w$ an $\mathbb F_2$-valued form such that every $E^i_j$ disjoint from $R$ has $\sum_{x\in E^i_j}w(x)=1$. Then $|R|\ge t$.

In the cross-review's construction these are the three proper minimal witnesses of a colour $u_z$ in each block, so every successful deletion meets all $t$ blocks: $\sigma_{u_z}(U)\ge t$ (the lower half of (EC)).

Context: Erdős Problem 624 asks whether $H(n)-\log_2 n\to\infty$, where $H(n)$ is the least $m$ such that some $f:\mathcal P(X)\to X$ with $|X|=n$ satisfies $f[\mathcal P(Y)]=X$ for all $|Y|\ge m$. This is an auxiliary finite statement from Sol's Round 12 Step 17 cross-review / resilience analysis of the two-track (PAIR) case. It does not settle Erdős Problem 624, PAIR or Step 17.