Remark: Family 135. Sharp homogeneous depth-five complexity of matrix products

Over every characteristic-zero field, the $(1,1)$ entry of a product of $n$ independent $n\times n$ variable matrices requires $n^{\Theta (\sqrt{n})}$ gates in homogeneous depth-five sum–product circuits. This sharp bound allows shared gates and bottom linear forms involving all variables.

The authors are OpenAI. This companion does not verify the proof. The claim is not a theorem of the Enter series and not a premise of any registered sheet. It does not close O.1. O.1 stays NOT_CLOSED. It is not a de Sitter state and not the QdS external-leg scheme. No quantum coefficient is admitted. No compilation is asserted.