I give a short proof that the border determinantal complexity of $F_n=x_1^n+\cdots+x_n^n$ over $\mathbb{C}$ is at least $(n-1)^2/(4e)$, and at least $(n-1)^2/(2e)$ in the symmetric model. The same argument shows that the border determinantal complexity of $x_1^d+\cdots+x_N^d$ is $Θ(Nd)$ for all $N\ge3$ and $d\ge2$. The earlier article arXiv:2606.13628, by the same author, introduced the first superlinear lower bounds and includes verification by Lean formalization. However, the AI written exposition was hard to follow. Kumar and Volk (arXiv:2609.34462) give a short proof of the exact ordinary case, but do not address the border or symmetric cases. In this article, I provide a greatly simplified exposition of the border and symmetric cases, using the implicit function theorem and an explicit section of simple solutions in place of the conormal specialization arguments of arXiv:2606.13628. I choose an explicit polynomial system with $n(n-1)^{n-2}$ simple solutions for $F_n$. I then use the implicit function theorem to ensure that each point persists under small perturbations. Finally, kernel equations and the multihomogeneous Bézout theorem bound the number of these points for a small determinant. The exposition is due to the human author. The mathematical simplification was obtained in discussion with ChatGPT Astra and refined in conversation with Claude Opus 5.5.

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