A cubic lower bound for border determinantal complexity of the permanent

We prove that the complex border determinantal complexity of the $m\times m$ permanent is $\Omega (m^3)$, allowing arbitrary affine-linear determinant representations and coefficientwise limits. It also gives cubic lower bounds for exact determinantal complexity and for the numbers of vertices and edges in affine-linear algebraic branching programs, including coefficientwise limits with a fixed vertex or edge budget.