>When I say "singularity," I mean that the coefficients blow up (aren't continuous because they shoot toward infinity in finite time). When author of the pdf file you link to writes "singularity," he means "C^1, but not C^2." You might want to re-read my posts above with this in mind.

This is just pedantry. It doesn't matter in what space or what variable the singularity occurs if one exists the solution is undefined at time t=T. This is common knowledge. Additionally in your own words you said there wasn't a singularity and that is definitively false both technically and in terms of common mathematical parlance, there IS a singularity. So this is more of a technical correction rather than a clarification.

>You seem to have in mind some regularization procedure to get past the singularity. I don't agree this is really a valid solution

I don't see it as a valid solution EITHER. regularization is just making shit up. I don't think you understood my number theory example.

Put it this way. With number theory 1/t is undefined at t = 0. But at the exact left side and right side of t = 0 everything is defined again. There is no mathematical property preventing this from happening with numbers. Same with the ODEs. Undefined at the point of collision with no explicit mathematical rule saying it's not defined everywhere else. So really it's [0, t_0) and (t_0, infinite].

Either way there is a singularity at the top of the dome so what occurs on the dome is isomorphic to a system of n-bodies at the singularities. Simply look what happens at t > T for nortons dome. For some arbitary time T the system has an infinite but bounded amount of directions to roll off the dome. So infinty possible T's with infinite possible directions. It can at any time emerge from the singularity and roll in any direction. That is nondeterminism. Not even worth fully deriving the motion mathematically because of so many possibilities.

This is THE SAME thing for the singularity in the n body problem, which is probably why the standard procedure to "solve" this sort of problem past the singularity (and without regularization) is uncommon. At an arbitrary time T the particle may emerge from the singularity with arbitrary unknown properties. But we do know that arbitrary laws still hold like the conservation of mass and energy, we know the particle isn't going to teleport 20 light years away or anything like that. So there are an infinite amount of possible outcomes but those outcomes are bounded by the laws of newton JUST like what happens to the particle on top of Nortons Dome.

I think what's going on here is that you're bogged down by the procedural math routines to solve these types of problems. You have to think a bit outside of the box in order to deal with paradoxes and singularities, yet you must stay within the logical realm of axioms and theorems.

There is a proof that shows a contradiction that basically explains why certain things are undefined at certain points. There is no proof that says the ODE is only defined on an interval up to a singularity. The proof only demonstrates it is undefined AT the singularity.

Every possible state of the particle emerging from the singularity at every possible time t > T is a potential solution.