Polynomial-pool localization: on every $2^{28}k^4$-point pool, $n/64$ colours stay inconsistent after deleting $2^{20}k^3$ points (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 $k\ge3$ and let $V\subseteq U$ be a pool with exactly $N_0=2^{28}k^4$ points (such pools exist iff $N_0\le n$, i.e. $k\ge51$) on which the pair covers every $k$-subset. Then there is a set $H$ of colours, $64|H|\ge n$, such that every $u\in H$ lies outside $L(\varnothing)$ and for every $R\subseteq V$ with $|R|\le2^{20}k^3$ the proper minimal witnesses of $u$ in $V\setminus R$ admit no certificate.

Honesty: Sol's further remark that a shortest witnessing odd relation is a linear circuit with at most $N_0+1$ parents (the Fredholm-alternative form) is not part of this statement; only the certificate (non-coherence) form is proved.

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.