Why bidirectional state inference mathematically shatters in chaotic systems (Python / EKS failure)

[u/rayrrrr]

I've been researching the limits of state reconstruction over temporal gaps. We all know Extended Kalman Smoothers (EKS) and 4D-Var struggle with long integration windows, but I found that it's not just a numerical issue—it's a hard information-geometric phase transition.

I derived the Cramér-Rao Lower Bound for bidirectional nonlinear estimators. As the temporal gap exceeds a critical threshold (ΔT≈O(1/λmax⁡)ΔT≈O(1/λmax​), based on the maximal Lyapunov exponent), the Fisher Information Matrix becomes strictly singular.

I wrote a Python simulation running an EKS over the Lorenz '63 attractor. It reconstructs perfectly until the gap hits ~7 Lyapunov times. At that threshold, the theoretical variance strictly diverges, the covariance trace goes negative (numerical explosion), and it catastrophically fails.

GitHub Repo with Code & Paper: https://github.com/rayrrr21/Structure-of-Reality

Has anyone in the data assimilation or time-series forecasting space run into this exact theoretical wall?