Critical-rank circuits: at $r=k-1$ the circuits of $(q_x)$ are exactly the minimal witnesses $H_u$ (Erdős 624)

Setting: $U$ is a finite pool of inputs (to apply this to a sub-pool $V$, take $U$ to be the subtype $V$), $C$ a set of colours, $L:\mathcal P(U)\to\mathcal P(C)$ an arbitrary list map (e.g. $L(A)=\{g_0(A),g_1(A)\}$), $k\in\mathbb N$, and $u\in C$ with $u\notin L(\emptyset)$. A set $B$ is a hole if $u\notin L(A)$ for all $A\subseteq B$. $H_u=\{A: 0<|A|\le k,\ u\in L(A),\ u\notin L(A')\ \forall A'\subsetneq A\}$ (minimal witnesses of size $\le k$), $Z=\mathrm{span}_{\mathbb F_2}\{\chi_A: A\in H_u\}\le\mathbb F_2^U$, $Q=\mathbb F_2^U/Z$, $q_x=e_x+Z$, $r=\dim Q$, $\rho(B)=\dim\mathrm{span}\{q_x:x\in B\}$. All of these are written out inline in the Lean statement (no custom definitions).

Theorem. Assume coverage and $r+1=k$. Then for every $A\subseteq U$: $A$ is a circuit of the family $(q_x)_{x\in U}$ (i.e. $(q_x)_A$ is dependent and $(q_x)_{A'}$ is independent for every $A'\subsetneq A$) iff $A\in H_u$, i.e. $0<|A|\le k$, $u\in L(A)$ and $u\notin L(A')$ for all $A'\subsetneq A$.

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 12 Step 17 cross-review / resilience analysis of the two-track (PAIR) case. It does not settle Erdős Problem 624, PAIR or Step 17.