Proper-parent face inequality (14): $Q^\circ(Y)\le D^\circ(Y)+(n-|S|)+\#\{u\in S:Y\cap(R_u\cup T_u)\neq\varnothing\}$ (Erdős 624)

Let $U$ be a set of inputs, $C$ a finite set of colours with $n=|C|$, $g_0,g_1:\mathcal P(U)\to C$ arbitrary and $L(A)=\{g_0(A),g_1(A)\}$; let $V\subseteq U$ be finite and $k\in\mathbb N$. 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 for every $k$-set $Y\subseteq V$,

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.