Arbitrary precision, not arbitrary length. Even "from [a] mathematical viewpoint, given an exact initial condition, we can gain mathematically reliable trajectories of chaotic dynamic systems" to only a "finite...interval" [1]. (This is due to "numerical noises, i.e. truncation and round-off error, where truncation error is determined by numerical algorithms and round-off error is due to the limited precision of numerical data, respectively.")
For a physical system like the weather, uncertainty "mainly comes from limited precision of measurement," though there is also the "inherently uncertain/random property of nature, caused by such as thermal fluctuation, wave-particle duality of de Broglie’s wave, and so on."
[1] https://www.sciencedirect.com/science/article/abs/pii/S10075...
For example, I can calculate the Fibonacci sequence to an arbitrary length but not infinite.
Skim the paper. Numerical noise means you cannot calculate the 3-body problem to an arbitrary length. There is a finite, mathematical limit even with perfect knowledge of initial conditions.
There isn’t any claim that mathematically exact starting values can’t be propagated with arbitrary precision to arbitrary length, and I would claim that this is possible (but not practical due to compute being limited, of course).
But there’s no hard limit of precision and length where a simulation can’t be made if the starting conditions are exact. The point of the paper is that starting conditions are never exact which limits the length you can propagate.
It talks about that. Which is relevant when we're talking about the weather. But it opens by discussing the hard mathematical limits to numerical methods.
> there’s no hard limit of precision and length where a simulation can’t be made if the starting conditions are exact
Wrong.
Read. The. Paper. Numerical methods for chaotic systems are inherently, mathematically uncertain.
Beyond a certain number of steps, adding precision doesn't yield a more precise answer, it just produces a different one. At a certain point, the difference between the different answers you get with more precision covers the entire solution space.
Even with perfect knowledge of initial conditions, numerical noise limits the forecast interval.