Spectral Split of the Assimilation Hessian Is Not a Diagnostic of Attractor Confinement: A Negative Result in Lorenz--96

We test a natural but, as we show, unfounded diagnostic criterion for attractor confinement in data assimilation: that the spectrum of the strong-constraint 4D-Var Hessian splits into a small cluster of large eigenvalues (the Lyapunov unstable directions) and a large cluster of nearly zero eigenvalues (directions along which the model is "blind" to the data) when the model lives on a false attractor. Using the 20-variable Lorenz-96 system, strong-constraint 3D-Var, and a +2.5% model-error perturbation in the forcing, we find four results. First, the number of exactly zero eigenvalues equals N minus rank(P) and is independent of model error. Second, the non-zero part of the spectrum is invariant, to within sampling noise, between the true and false models. Third, the bias of the analysis is not concentrated along the leading Hessian eigenvector but is, if anything, suppressed there. Fourth, with 50% of variables observed, the control run with the true model is itself unstable, so the +2.5% model error is invisible against a background drift. We conclude that the spectral split is an algebraic property of the observation operator, not a dynamical signature of confinement, and that the diagnostic criterion must be reformulated. We propose a corrected criterion based on the fraction of analysis bias living in the unobserved subspace, normalised against a zero-model-error control. The result is negative and, as such, more useful than a positive one: it forestalls the adoption of a plausible but non-discriminating indicator.