I'll try to reword it then...
The cube isn't moving while it's on the platform, true, but it IS moving when it exits the stationary blue portal. The state of the cube as it entered the portal doesn't matter. When it EXITED the portal, it is moving and would have no reason whatsoever to STOP moving at the same speed (not counting wind resistance and gravity, of course).
As the item exits the blue portal, it is now in motion and the laws of inertia apply to it accordingly. These laws are based on the objects CURRENT state, not it's previous state. It doesn't matter how it started moving (in this case, it skips acceleration completely), it will continue moving.
If you're staring at the blue portal as the cube exits it, you'll see it pop up as though shot. It's coordinates have changed in relation to the surface of the portal. This is the very definition of velocity, so why would it stop moving at the end of the cube's length? The cube doesn't have any memory of being at rest, it will only continue to do what it's current state tells it to do, which is move upwards at the same rate that it was already going (minus gravity and wind resistance, blah blah blah).
I don't have any better way to explain this... Since it's an impossible scenario in the first place, I don't think infinite energy arguments are sufficient to disprove what I'm saying.
And lets be clear, I'm continuing this discussion because I think it's an interesting and fun topic. I don't care how pointless it is, so you needn't bother pointing it out.