Homogeneous depth-five lower bounds for iterated matrix multiplication

Let ${\mathrm{IMM}}_{n,n}$ be the $(1,1)$ entry of a product of n independent $n\times n$ matrices of variables. Over every field of characteristic zero, every syntactically homogeneous $\Sigma \Pi \Sigma \Pi \Sigma$ circuit computing ${\mathrm{IMM}}_{n,n}$ has at least $n^{\sqrt{n}/400}$ gates for all sufficiently large n, with an absolute threshold independent of the field. Bottom linear forms may have arbitrary support, and arbitrary finite fan-in, fan-out, and gate sharing are allowed. Over every field, a block expansion gives such circuits with at most $n^{\sqrt{n}+4}$ gates for n ≥ 2. Thus the gate complexity over characteristic-zero fields is $n^{\Theta (\sqrt{n})}$.