Proper-parent certificate inequality (13): $32N\sum D^\circ+32N(n-|S|)K+32kK\sum_u(|R_u|+|T_u|)\ge nNK$ (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 $A$. Let $k\ge3$ and let $V\subseteq U$ be a pool with $N=|V|\ge k+5$; write $K=\binom Nk$. Let $S\subseteq C\setminus L(\varnothing)$. For each $u\in S$ fix a deletion set $R_u\subseteq U$ and a parity certificate $w_u:U\to\mathbf F_2$ for the proper minimal witnesses of $u$ surviving the deletion: for every $A\subseteq V\setminus R_u$ with $0<|A|<k$, $u\in L(A)$ and $u\notin L(B)$ for all $B\subsetneq A$, we have $\sum_{x\in A}w_u(x)=1$. Put $T_u=\{x\in V\setminus R_u: w_u(x)=0\}$. For a $k$-set $Y$ let $J^\circ(Y)=\bigcup_{A\subsetneq Y}L(A)$, $D^\circ(Y)=n-|J^\circ(Y)|$, $J^\circ_{\mathrm{odd}}(Y)=\bigcup_{A\subsetneq Y,\,|A|\text{ odd}}L(A)$ and $Q^\circ(Y)=n-|J^\circ_{\mathrm{odd}}(Y)|$. Then (no coverage assumption is needed)

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.