Homogeneous depth-five lower bounds for iterated matrix multiplication with unrestricted bottom fan-in (Bera-Chakrabarti question, balanced regime)

Claims that 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 large $n$, with gate sharing, arbitrary fan-in and bottom linear forms of arbitrary support allowed, and a block construction gives $n^{\sqrt n+4}$ over every field, so the gate complexity is $n^{\Theta(\sqrt n)}$. The measure is the rank of a partial-derivative-times-multiplication operator. It does NOT treat the Bera-Chakrabarti regime $\mathrm{IMM}_{n^q,n}$ directly, positive characteristic, non-homogeneous circuits, or depths beyond five.

Claims that 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 large $n$, with gate sharing, arbitrary fan-in and bottom linear forms of arbitrary support allowed, and a block construction gives $n^{\sqrt n+4}$ over every field, so the gate complexity is $n^{\Theta(\sqrt n)}$. The measure is the rank of a partial-derivative-times-multiplication operator. It does NOT treat the Bera-Chakrabarti regime $\mathrm{IMM}_{n^q,n}$ directly, positive characteristic, non-homogeneous circuits, or depths beyond five.