Question about choice of objects when calculating potential energy

If you'd physically think about such a situation, where we have A and B as two point masses at rest, being attracted by the gravitational force, nature doesn't care what you think is the relative position of A with respect to B or B with respect to A.

But even if you think about this problem mathematically, the expression for the forces on particles A and B would be -

$$vec{F_B}=frac{-G m_A m_B}{r^2} hat{r} qquad qquad vec{F_A}=frac{+G m_A m_B}{r^2} hat{r}$$

Here, $vec{r_B}=r hat{r}$ (displacement vector of B) and $vec{r_A}=-r hat{r}$ (displacement vector of A) and $hat{r}$ is the unit vector pointing from A to B.

Now, if you try to calculate the potential energy difference using either A or B as the origin, it's $$P.E.= int_{infty}^{r}vec{F_A} cdot dvec{r_A}= int_{infty}^{r}vec{F_B} cdot dvec{r_B}=frac{-Gm_A m_B}{r}$$

Therefore, this again proves that nature doesn't care what you choose as the displacement vector direction or the origin. The potential energy expression would be consistent anyway.