> Let's say we added a single particle X (or, well, multiple) to our otherwise quantum world whose position and momentum we could pin down with arbitrary accuracy.
That particle X does not exist. We can't add it to the world.
It's not like I was claiming it exists. I'm asking, if you did observe one tomorrow, what would be the theory's best prediction of what would happen next?
And yes, I get the whole Fourier transform uncertainty making the math inconsistent, but that's not an answer for me here. Like if you asked Newton "if gravity didn't travel instantaneously, what would be your theory's best prediction?", he would probably be able to give you a better answer of what he expects the consequences would be than "that's impossible, the math would be inconsistent".
If you observe one tomorrow, then the theory is all wrong and people will all yell "Hey, new physics", celebrate, and try to discover why nobody has seen anything like it before. There is no "best prediction", the entire theory is invalid.
It would be like asking Newton "hey, if gravity didn't exist at all, and things traveled at path-dependent trajectories, what would your theory predict?" The answer is that it doesn't predict anything.
Your question itself assumes away quantum mechanics. There is fundamentally no fully certain particle in QM, the theory cannot meaningfully make predictions about something that's fully certain.
I've been out of school a while so might be wrong but maybe, just maybe, you could torture some mathematics into giving you some infinities if you really wanted to get a "prediction"?
But it's sort of like asking "how would linear algebra work if all nonzero matrices were invertible?" Well, all matrices aren't invertible, some definition of matrix that allows for nonzero matrices to be inverted is just different.
I think the more general question they're asking is whether it's possible to hack the math to break the uncertainty principle. As an example, if you take a Monte Carlo estimate of some quantity, you have uncertainty associated with that estimate. But there are ways to reduce that uncertainty using additional information (e.g. control variates, sampling tricks, partially solving the problem analytically, etc.). The example they're giving for a hypothetical particle with zero uncertainty might not be a great one, but I think the idea itself shouldn't be dismissed outright.
I think that misses the point of the exercise. What the OP is asking is more analogous to using negative energy in conjunction with existing models to see what would happen. It's not reflective of reality, but it tells you something about the limits of the model.
That particle X does not exist. We can't add it to the world.