Every-face hidden-generation bound (G): on each $k$-face few colours with $\rho_u\le k-d-1$ have $\gamma_u(Y)\le H$ (Erdős 624)

Let $U$ be a finite set, $L:\mathcal P(U)\to\mathcal P(C)$ with $|L(A)|\le b$, and $S$ a finite set of nonbottom colours ($u\notin L(\varnothing)$) each attained inside every $k$-subset of $U$. For $u\in S$ let $Z_u(U)\le\mathbb F_2^U$ be the span of the indicator vectors $\chi_F$ of the actual minimal witnesses $F$ of $u$ with $1\le|F|\le k$ ($u\in L(F)$, $u\notin L(F')$ for $F'\subsetneq F$), and for a $k$-set $Y$ let $Z_u(Y)$ be the span of those with $F\subseteq Y$, and $\mathbb F_2^Y=\{f:f(x)=0\ \forall x\notin Y\}$ (written as the kernel of restriction to $U\setminus Y$). Put $\rho_u=|U|-\dim Z_u(U)$ and $\gamma_u(Y)=\dim(Z_u(U)\cap\mathbb F_2^Y)-\dim Z_u(Y)$.

Theorem. For every $Y$ with $|Y|=k$ and integers $0\le H<d$,

(Both conditions are stated without subtraction: $|U|+d+1\le\dim Z_u(U)+k$ and $\dim(Z_u(U)\cap\mathbb F_2^Y)\le\dim Z_u(Y)+H$.)

Source: 'The Larger-Deficit Case and a Rare-Pair Barrier', boxed (G). With $S=C\setminus L(\varnothing)$, $|S|\ge n-b$, $n=2^k=|U|$ this gives $\#\{u\notin E_d:\gamma_u(Y)\le H\}\le\lfloor(b-1)n/(d-H)\rfloor$, i.e. (G) after adding the $M_d=|E_d|$ colours with $\rho_u\ge k-d$. This is a necessary condition for a hypothetical full cover, not a contradiction.

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.