//This is where Norton gets tricky with us. He posits another solution to the equation ...
Norton didn't posit this as a solution, it is a solution.
//This is baffling. It’s bizarre to stitch two different solutions to an equation
Norton never stitches together two solutions. I don't know if Dr. Davies is unfamiliar with a function defined using a case statement, but there's nothing Frankensteinish about it.
//but in fact the top equation is not Newtonian at the apex (clearly since it moves despite the absence of a force there).
Alas, Dr. Davies has missed the biggest delight of Norton's Dome! The ball can nondeterministically roll off of Norton's dome and yet still obey all of Newton's laws :-). It sure looks like it wouldn't be--after all, how can the ball start to move in absense of a force?
But at time t=T, the ball isn't moving yet. Its not even accelerating yet. So at time t=T its obeying Newton's laws. An at every time before and after that it also obeys Newton's laws.
I wouldn't say "that's all." If it actually does admit nondeterministic solutions (I am not convinced that there is no property or axiom buried in the formalism that forbids this case somehow), then the next question is, does quantum mechanics, which implies classical mechanics, admit nondeterministic solutions? If not (wave mechanics has a way of ignoring weird stuff when it only happens at a point), that would make quantum mechanics a consequence of determinism, a fairly shocking possibility.
The fact that the author quibbles about units, when they've failed to notice the Norton took `g=1`, makes it seem as if they're not familiar with conventions in mathematical physics. The addition of a constant `k` adds no value other than converting between units, which aren't of interest here other than as a bookkeeping mechanism, because we're not dealing with any numerical quantities.