Smallest Magic Cube is of Order 3

Theorem
Apart from the trivial order $1$ magic cube:

the smallest magic cube is the order $3$ magic cube:

Proof
Suppose there were an order $2$ magic cube.

Take one row of this magic cube.

From Magic Constant of Magic Cube, the row and column total is $9$.

Any row or column with a $1$ in it must therefore also have an $8$ in it.

But there are:
 * one row
 * one column

both of which have a $1$ in them.

Therefore the $8$ would need to go in $2$ distinct cells.

But $8$ appears in a magic cube exactly once.

Hence there can be no order $2$ magic cube.