Is emission-less propulsion possible? I believe it is...

[Salviati]

52. The problem is you have to carry those anvils.

If you want to fire 1000 anvils out the back of your ship, you need to carry them, and accelerate them along with you at first. So at first, your ship is not going to have a mass of 10000 kg, it's going to be over 3x the mass - 32600 kg. This isn't the type of calculation you can do for just one cycle, and then multiply by how many cycles there are to be done. Figuring out a full system requires doing an integral and produces what's known as the rocket equation.

The rocket equation puts very hard limits on the best a rocket can do using known physics, and it just depends on 3 quantities: The Initial and final masses of your rocket, after expelling your reaction mass, and the exhaust velocity of that reaction mass.

The best change in velocity you can hope to get is: Delta V = Ve ln(mi / mf) Where mi is the initial mass of the ship, including the reaction mass you're throwing out the back, mf is the final mass, and Ve is the exhaust velocity.

Start plugging in some reasonable numbers into that equation, and it puts some pretty low speed limits on getting to different places in the solar system, let alone interstellar distances.

If you want to do better than that, you're going to have to break Newtons 3rd Law / Conservation of Momentum, which seems unlikely, but we certainly don't have everything figured out. I am fairly certain though that if there are some tricks to be had here, they're going to be subtle.