Top-layer-free packing (29): $n/64$ colours with $256k(k-1)|\mathcal P_u(w)|>N$ for every form $w$ (Erdős 624)

Throughout, $U$ is a finite set of inputs, $C$ a finite set of colours with $|C|=n=2^k$, and $g_0,g_1:\mathcal P(U)\to C$ two arbitrary maps; $L(A)=\{g_0(A),g_1(A)\}$. Here $|U|=n$. 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 $A$. Theorem. Let $k\ge3$, $V\subseteq U$ with $N=|V|\ge2^{28}k^4$, and suppose the pair covers every $k$-subset of $V$. Then there is a set $H$ of colours with $64|H|\ge n$, all outside $L(\varnothing)$, such that for every $u\in H$ and every form $w:U\to\mathbf F_2$ (chosen independently per colour) there is a family $\mathcal P$ of pairwise disjoint proper minimal witnesses $A\subseteq V$ of $u$, each with $\sum_{x\in A}w(x)=0$, and $N<256k(k-1)|\mathcal P|$.

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.