I think you're asking questions that some are afraid to ask.
It appears to me that some people have become accustomed to working with approximations, and have accepted the map for the terrain.
Fundamentally, I don't see how you can use continuous math to explain a discrete system.
No, here we are discussing the formalism without approximations associated with an instance of its approximate application.
And QM says "The map is the terrain".
You might want to be a little more specific, and rely less on approximations.
I am aware of what the Copenhagen interpretation states, thanks
Here we discard Copenhagen and move forward.
Schrodinger/Dirac/Feynman.
A wave is a product, trigonometric functions do not exist.
Gerard hooft was on Curt Jaimungal's youtube channel a while back, I generally agree with him, discrete systems cannot be explained by real numbers, only integers