New lower bounds on tensor rank of $(n,2,m)$ matrix multiplication with GPT-6

The tensor rank of $n\times 2$-with-$2\times m$ matrix multiplication is at least $(n+2)m$ if $n\ge 4$ and at least $\left\lceil \frac{24m+2}{5} \right\rceil$ if $n=3$, over arbitrary fields. This result matches the Hopcroft-Kerr upper bound for $n=3$ and $m\le 6$.