Earlier quoted context omitted.
This game isn't writing the numbers (complex-valued probability amplitudes of states, classical EM vectors even) anywhere so it's going to be pretty hard to relate what you're seeing here to the math that you'll find if you look up interference elsewhere. I'll make an attempt here, though. I think the effect on level 4 is totally classical so it shouldn't take too much background. 1. The solution to a problem involvi…
Thanks for the great explanation! If you don't mind, I had a question about '3.' I get how it's used in making your larger argument, but as a thing in itself I'm kinda lost. Part of what I'm wondering is how it's okay to say 'L(W) = 0' when earlier you described L(W) like 'L(W: wavefunction) => X: wavefunction'; so is the zero in 'L(W) = 0' just shorthand for something like 'L(W) = (f(x,t) = 0)'? (I'm also curious wh…
There is a deeper point to be made here that I'm glad you brought up. Functions form a vector space (because they satisfy the axioms of vector behavior, basically because they can be added to each other and scaled by constant multiples). In linear algebra the symbol 0 often does double-duty as the zero vector, which is defined as the vector that doesn't change other vectors when it's added to them. So, here, when I write L(W) = 0 I'm implicitly invoking 0 = f_zero(x,t) = 0.
As for why "mapping to zero" has a physical basis, well, it's really more of a thing we're always guaranteed to be able to do. You can always subtract everything from the right-hand side of an equation! For example, Wikipedia introduces the one-dimensional wave equation as D_t^2 u = a^2 D_x^2 u. I can also write that as L[u] = D_t^2 u - q^2 * D_x^2 u = 0, so L[u] = 0. (In my notation, D_x is the derivative with respect to x, and D_x^2 is the second derivative with respect to x.)
The real question is why the addition thing works; if I had to try explaining it I would just say it's just fundamental that Maxwell's equations are linear, and when dealing with things that aren't, we usually approximate them with linear functions anyways[0]. That's how gravitational waves emerge from GR, by the way: at low energies the nonlinear equations behave nearly linear, and in that approximation the familiar wave equation falls out.
[0] If you zoom in to a small enough range in the graph of all but the most esoteric functions, the thing on your screen will look like a line. Try it, it's a good intuition to have.