We prove that every fan-in $2$ noncommutative arithmetic circuit computing the palindrome polynomial has size $Ω(n^2)$. The proof builds on and refines a previous work of the author. The new ingredients in the proof were generated by Gemini 3.1 Pro.

翻译：我们证明，每个计算回文多项式的扇入 $2$ 非交换算术电路的大小为 $Ω(n^2)$。该证明基于并改进了作者先前的工作。证明中所引入的新成分由 Gemini 3.1 Pro 生成。