If you had a point mass of amplitude, an infinitely sharp spike in the quantum arena, the amplitude distribution would not be twice differentiable and the future evolution of the system would be undefined.

This just caught my eye, and it's not clear to me what the actual mathematics behind it is. An "infinitely sharp spike" is an intuitive description of something that can be formalised, but not as a function mapping points of configuration space to amplitudes (because "infinity" in this context is not a number). The concept of an infinitely sharp spike can be formalised, but in that setting, I believe it is infinitely differentiable. So either you rule out infinitely sharp spikes from the start for some other reason, or you have to allow them in all the way. Am I missing something?