Superpolynomial monotone clique lower bound ($\mathrm{size}({\mathrm{CLIQUE}}_{n,\lfloor n^{1/4}\rfloor })\ge \exp (c(\log n{)}^2)$)

In the finite lattice approximation for monotone clique circuits, set $m=r=\lfloor n^{1/4}\rfloor$ and $\ell =\lfloor {\log }_{2}n\rfloor$. A nonuniversal output accepts at most ${\sum }_{s=2}^{\ell }s!(r-1{)}^s(m/n{)}^s\le 2(\ell rm/n{)}^2$ of uniform positive cliques. Its missing positives require the circuit size times the positive clique error of a truncated lattice meet to be bounded below. A universal output accepts every negative colouring input; those errors require the circuit size times the negative colouring error of a forced-set closure to be at least one. Both alternatives imply the displayed superpolynomial bound by the Razborov gate-by-gate approximation lemma.