2. Let $m$ be a real number, vector $\boldsymbol{a}=\left(1+2^{m}, 1-2^{m}\right), \boldsymbol{b}=\left(4^{m}-3,4^{m}+5\right)$, then the minimum value of the dot product $\boldsymbol{a} \cdot \boldsymbol{b}$ is $\qquad$
$$
\boldsymbol{a} \cdot \boldsymbol{b}=\left(2^{m}+1\right)\left(4^{m}-3\right)-\left(2^{m}-1\right)\left(4^{m}+5\right)=2 \cdot 4^{m}-8 \cdot 2^{m}+2=2\left(2^{m}-2\right)^{2}-6 \text {, }
$$

So when $m=1$, the minimum value of $\boldsymbol{a} \cdot \boldsymbol{b}$ is -6.