Proper odd-palette inequality (11): $32\sum_{Y\in\binom Vk}Q^\circ(Y)\ge n\binom Nk$ (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 $A$. For a $k$-set $Y$ define the proper odd palette $J^\circ_{\mathrm{odd}}(Y)=\bigcup_{A\subsetneq Y,\ |A|\text{ odd}}L(A)$ and $Q^\circ(Y)=n-|J^\circ_{\mathrm{odd}}(Y)|$. Let $k\ge3$ and $V\subseteq U$ with $N=|V|\ge k+5$. Then

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.