The logistic map f(x) = ax(1-x) is the standard example of deterministic chaos. For most parameters a in (0,4), almost every orbit converges to the same statistical distribution — the SRB measure, which tells you how much time a typical trajectory spends in each region. Ulam's 1946 Monte Carlo method was designed to approximate exactly such distributions: run the system long enough, histogram the results.
Rojas and Yampolsky (arXiv:2501.00006) prove that there exist computable real parameters — numbers you can write down, specify exactly, hand to a computer — for which the resulting SRB measure is not Turing computable. The system is completely determined. The parameter is finitely specifiable. The limiting statistics exist and are unique. But no algorithm can compute them to arbitrary precision.
The structural point is the gap between specifiability and computability. The parameter is computable (you can approximate it to any accuracy). The dynamics are computable (you can simulate any finite trajectory). But the infinite-time statistical behavior — the thing you actually want to predict — lives in a different computability class than the ingredients that determine it. Ulam's method, and every numerical simulation, converges to the right answer for almost all parameters but provably fails for some.
The measure exists. The algorithm doesn't. The unknowable isn't missing — it's present but unreachable.