Finite lattice approximation for monotone clique circuits

Take upward-closed set families of subsets of $[n]$ of size at most $\ell$, closed under the forcing rule in Razborov closure. Normalize any family containing a set of size at most one to the full family, since its clique predicate is identically true. Intersection is the meet in a lattice; closure of the union is the join in a lattice. A family represents the graphs containing a clique on at least one of its members. The approximate meet can lose positive inputs and the approximate join can acquire negative inputs. The Razborov gate-by-gate approximation lemma charges output errors to those local errors.

Independently colour each vertex with one of $m-1$ colours and join differently coloured vertices. This complete multipartite graph has no $m$-clique. If a forced set $W$ is rainbow but its $r$ witnesses are not, condition on the colours of $W$. The disjoint petals make those failure events conditionally independent. For each witness of size at most $\ell$, the probability of a colour collision is at most $\left(\genfrac{}{}{0ex}{}{\ell }{2}\right)/(m-1)$. If this is at most $1/2$, one closure step causes an error with probability at most $2^{-r}$. There are at most $n^{\ell +1}$ possible steps; apply the union bound.

Choose a uniform $m$-subset $Z$ and the graph consisting only of its clique. Acceptance by two approximating set families but rejection by their intersection requires two minimal witnesses whose union has size greater than $\ell$. At least one has size greater than $\ell /2$. The sunflower bound for minimal members of a Razborov-closed family and $\mathbb{P}(W\subseteq Z)\le (m/n{)}^{|W|}$ give $2{\sum }_{s>\ell /2}s!(r-1{)}^s(m/n{)}^s\le 4q^{\ell /2}$.