> There is no time when the particle is accelerating, or even moving, while not experiencing a net force in this setup.

I agree! But at the same time, there is no moment at which a net force compels the particle to leave a state of rest. The criteria you are putting forwards with that sentence isn't that of the first law, it's just a consequence of the second law. The first law makes a statement that force is required to cause a body to leave a state of rest, but that's not what we have here.

We just postulate that it must stop being at rest immediately after T, and then we find a force as a result of it not being at rest anymore. This inverts the causality that NFL requires, and because it doesn't have give a cause for the particle to leave the state of rest, which the 1st law requires, it's not a valid physical solution in Newtonian mechanics.

Here is Wikipedia's translation of the first law, which is as good as any:

> Every body continues in its state of rest, or of uniform motion in a straight line, unless it is compelled to change that state by forces impressed upon it.

In our trajectory, the body is not compelled to change its state by a force impressed on it. It's in a stable state, then we arbitrarily decide that it will change it's state immediately after T, and then we show that this leads to a trajectory which satisfies the second law. But since it changes state without being compelled by a force, it's no longer a trajectory which satisfies the first law.

> The argument you'd have to make would be to change Newtonian mechanics so the first law is no longer a special case of the second law

I'm not changing Newtonian mechanics - Newton's first law is not a special case of the second law. It makes a statement about cause and effect. The second law doesn't - it's just a differential equation, which does not have such a content. The article redefines the first law to strip it of its causal content and instead make it a trivial statement, but that's not what the first law plainly states. In Newtonian physics, force is necessary to cause a change in velocity, it's not simply that force and acceleration co-occur.

> but actually says something nontrivial about all of the time-derivatives of position rather than just the first two

No, this isn't necessary. If you want, you can instead understand it as something about the nth derivative of position, and that way you can recover perfect reversibility, but you don't have to. You can just keep Newtonian mechanics as is, and recognize that they describe a causal system, and not just a system of differential equations (which make no statements about causation in and of themselves), but then you lose perfect reversibility for some pathological trajectories like this one.

> Even then I'm not sure that would save you in general, since it should be possible to cook up examples where the motion is some non-analytic thing like a portion of a "bump function".

I'm not arguing for the infinite derivative modification either, but I don't see how that's the case - if you have a non-analytic function then some derivatives don't exist, and if they don't exist then they wouldn't be able to satisfy the infinite derivative formulation. Seems to me that such a law would directly eliminate all non-analytic trajectories.