Remark: Family 108. A cubic permanent–determinant lower bound

Proves an $\Omega (n^3)$ lower bound for the border determinantal complexity of the $n\times n$ permanent over $\mathbb{C}$. Even coefficientwise limits of determinants of affine-linear matrices require matrix size at least $c{n}^3$, for an absolute $c>0$ and all sufficiently large $n$; the same bound therefore holds for exact representations.

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.