Resilience forces large disjoint packings of violating proper witnesses: $256k(k-1)|\mathcal P|>N$ (29) (Erdős 624)

Let $U$ be a finite set, $C$ a finite set of colours, $g_0,g_1:\mathcal P(U)\to C$, $L(A)=\{g_0(A),g_1(A)\}$, $V\subseteq U$ with $N=|V|$, $k\in\mathbb N$ and $u\in C$. The proper minimal witnesses of a colour $u$ inside $W$ are the $A\subseteq W$ with $0<|A|<k$, $u\in L(A)$ and $u\notin L(B)$ for all $B\subsetneq A$; a certificate is $w:U\to\mathbf F_2$ with $\sum_{x\in A}w(x)=1$ on every such $A$. Assume $u$ is resilient: for every $R\subseteq V$ with $256k|R|\le N$, the proper minimal witnesses of $u$ in $V\setminus R$ admit no certificate. Then for every form $w:U\to\mathbf F_2$ there is a family $\mathcal P$ of pairwise disjoint proper minimal witnesses $A\subseteq V$ of $u$ with $\sum_{x\in A}w(x)=0$ and

(Here $k-1$ is truncated natural-number subtraction; no lower bound on $k$ is needed.)

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.