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

[caraher]

80. I did make one mistake

Well two, really...

But the one in my post above was that my first sentence was meant to say not "Nobody with the most basic knowledge of physics needs to ask an AI to write Python code." but "Nobody with the most basic knowledge of physics needs to ask an AI to write Python code in order to analyze this problem."

But let me give you a path to playing with this using appropriate tools. I wrote a VPython program that illustrates what will actually happen in your problem, assuming no loss of kinetic energy for your slug + astronaut (and, crucially, reflector) system at each collision. Just click to "fire" the gun...

Then go to glowscript.org and open a free account. Take the code you can copy from my link, change the parameters all you like (I set a "muzzle speed" of 300 m/s for the slug, a 100 kg astronaut, and an unrealistically heavy 10 kg slug so the basic dynamics would be apparent, and update positions every millisecond in simulation time). The only effects changing those parameters will have is to change some characteristics of the oscillatory motion, but (unless you pick numbers that break the simulation by, for instance, having the "update" time become large compared to the time it takes for one oscillation, or generate roundoff errors) all that will change is the frequency - you don't get runaway acceleration in one direction but a kind of wobbling in place.