Top-layer-free resilience (1): $64\,|\{u\notin L(\varnothing):256k\,\sigma_u(V)>N\}|\ge n$ for covering pairs (Erdős 624)

Throughout, $U$ is a finite set of inputs, $C$ a finite set of colours with $|C|=n=2^k$, and $g_0,g_1:\mathcal P(U)\to C$ are two arbitrary maps (a pair); write $L(A)=\{g_0(A),g_1(A)\}$ for the list of colours at the input $A$. Here also $|U|=n$. For a pool $V\subseteq U$ with $N=|V|$ and a colour $u\notin L(\varnothing)$, the proper minimal witnesses of $u$ in $W\subseteq V$ are the sets $A\subseteq W$ with $0<|A|<k$, $u\in L(A)$ and $u\notin L(B)$ for every $B\subsetneq A$; the family is affine-coherent if some $w:U\to\mathbf F_2$ has $\sum_{x\in A}w(x)=1$ on every member. Let $\sigma_u(V)$ be the least $|R|$, $R\subseteq V$, such that the proper minimal witnesses of $u$ in $V\setminus R$ are affine-coherent.

Theorem. Let $k\ge3$, $N\ge2^{28}k^4$, and suppose the pair covers every $k$-subset of $V$: for each $Y\in\binom Vk$ and each colour $u$ there is $A\subseteq Y$ with $u\in L(A)$. Then there is a set $H$ of colours with $64|H|\ge n$ such that every $u\in H$ satisfies $u\notin L(\varnothing)$ and $256k\,\sigma_u(V)>N$, i.e. for every $R\subseteq V$ with $256k|R|\le N$ the proper minimal witnesses of $u$ in $V\setminus R$ admit no $\mathbf F_2$ certificate. Equivalently

(For $V=U$ the size hypothesis holds for all $k\ge64$.)

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 $|Y|\ge m$. By an earlier track-amplification result it suffices to find common holes for fixed numbers of tracks; this is an auxiliary finite inequality from Sol's Round 10 analysis of the two-track case. It does not settle Erdős Problem 624, PAIR or Step 17.