Unscented Kalman Inversion

The paper’s linear-case theorem and proof are correct and complete: it derives the key information-form identity, proves exponential convergence of the precision and mean via a carefully chosen contraction mapping on a scaled precision variable, and identifies the steady-state mean as the unique minimizer of the stated Tikhonov functional. By contrast, the candidate solution reaches the same high-level conclusions but relies on two critical norm bounds that are not valid in the Euclidean spectral norm: (i) a global Lipschitz constant ≤ α^2 for T(P) in the 2-norm, and (ii) the claim that ∥I − KG∥2 ≤ 1 (strictly < 1 at the limit) in Euclidean norm. These steps are not justified in general (products such as X−1L and I − KG are non-normal), so the contraction and mean-convergence arguments as written do not hold in the stated norm. The conclusions can be repaired by switching to the paper’s symmetric similarity/scaling and spectral-radius arguments, but as written the model’s proof is flawed.