Top-layer-free resilience on the full domain, $k\ge51$ (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 $k\ge51$ and suppose the pair covers every $k$-subset of $U$. Then there is a set $H$ of colours with $64|H|\ge n$ such that each $u\in H$ lies outside $L(\varnothing)$ and, for every $R\subseteq U$ with $256k|R|\le n$, the proper minimal witnesses of $u$ disjoint from $R$ admit no certificate (i.e. $256k\,\sigma_u(U)>n$).

Sol states the threshold $k\ge64$; the exact threshold for the needed inequality $2^k\ge2^{28}k^4$ is $k\ge51$ (it fails at $k=50$), which is what is proved here.

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.