Exact-$k$ witness localization for a covering colour with valid $\mathbb F_2$-labels (Erdős 624)

Setting: $U$ is a finite set of inputs, $L:\mathcal P(U)\to\mathcal P(C)$ an arbitrary list map (the actual colour lists), $k$ a natural number. A colour $u$ is nonbottom if $u\notin L(\varnothing)$ and covering if every $k$-subset $Y\subseteq U$ contains some $A$ with $u\in L(A)$. An actual minimal witness of $u$ is $F\subseteq U$ with $u\in L(F)$ and $u\notin L(F')$ for every $F'\subsetneq F$. A valid label for $u$ is a map $p:U\to V$ into an $\mathbb F_2$-vector space with $\sum_{x\in F}p(x)=0$ for every actual minimal witness $F$ with $1\le|F|\le k$. For $T\subseteq U$, $\operatorname{rk}(T)=\dim\operatorname{span}p(T)$ and $\operatorname{rk}(U)=\dim\operatorname{span}p(U)$ (the dimension of the valid parity space). Theorem. Let $u$ be nonbottom and covering, $p$ a valid label for $u$, and $T\subseteq U$ with $|T|\le k$ and $\operatorname{rk}(T)+k\le|T|+\operatorname{rk}(U)$ (i.e. at least $k-|T|$ further coordinate labels are independent modulo $\operatorname{span}p(T)$). Then there is a nonempty actual minimal witness $F\subseteq T$ of $u$ with $\sum_{x\in F}p(x)=0$.

This is the 'exact-$k$ extension / restriction' argument used repeatedly in Sol's Round 13 notes (e.g. 'Individual Spectra and Common-Input Degree', 'Actual Witness Spans').

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 13 notes on the two-track (PAIR) case. It does not settle Erdős Problem 624, PAIR or Step 17; the initial near-critical population those notes start from is not established.