It's rather complex for something so simple but here is all you'll need. The vector space analysis is the most useful, I grant you that but the Hamiltonian and the Lagrangian are vital to understanding how the energy is dissipated. Of course you're only going to want to consider the Lagrangian but you're not accounting for the total energy in the system and that's crucial when you change the direction of the momentum of the TWO systems.
Don't forget, if you have an equal and opposite reaction, you now have TWO momenta to account for until both system rejoin their momenta. What you do to one can't affect the other until the system rejoin which is of course the reasons rockets and also combustion engines work in the first place. So this has to work or your car cannot... LOL
https://qmichaellewis.blogspot.com/2025/02/a-full-exploration-of-dynamics-of-lewis.html
This is all the math and physics you need to disprove this idea. I'm not quite sure how you can disprove it as it would mean that rockets can't fly and a V8 can't drive a wheel. The best course of action is just to not believe any of it and not waste your time trying to understand the math and physics involved. If you look at the video, you see how impossibly complex I made this contraption. You need a platform with a ramp, a ball and linear accelerator of some sort. Clearly, a piece of foam, a spring and a pool ball can't be used to imagine the physics of this system as any shifting of the platform, although expected, wouldn't prove anything. Even all the math, proved and validated cannot change perception. In that regard, you are correct. The laws of physics are sadly very immutable.