Unpredictable but Confined: Thin Chaos and the Scope of Kellert's Impossibility Argument

This paper argues that Stephen Kellert’s well-known claim in In the Wake of Chaos (1993)—that chaotic systems require impossible accuracy for useful prediction—applies only to one kind of task. Kellert focuses on predicting the precise value of a variable within a stated tolerance at a future time. Sensitive dependence on initial conditions, measured by a positive Lyapunov exponent, forces the required initial accuracy to grow exponentially, rendering many such point-value tasks impossible for any finite observer. The paper contends that a second, equally common task falls outside this impossibility: predicting whether the system will remain confined within a specified region over a given interval. This distinction is framed through the ideas of thin and thick chaos introduced in the author’s earlier work. Thin chaos involves local instability that never carries the system across the functional boundaries that matter for its purpose; thick chaos does. The thinness ratio quantifies this by comparing the typical displacement produced by chaotic motion with the displacement that would count as failure. Four elementary results establish the claim. The first constructs an example in which the Lyapunov exponent can be fixed while the confinement outcome is varied at will, showing the two questions are logically independent. The second shows that the thinness ratio is controlled by Kellert’s own measure of predictive hopelessness scaled by the ratio of the system’s accessible span to the failure threshold; Kellert’s criterion is recovered exactly when the failure scale is half that span. The third supplies a distribution-free upper bound on the probability that the system has left the safe region by the end of the interval, proves the bound cannot be improved without further assumptions, and extends it to the entire interval when successive displacements behave like a martingale. The fourth replaces the root-mean-square displacement with a quantile of the running maximum, making the ratio equal to one precisely when confinement holds at a stated confidence level. The results are limited to logical relations and inequalities. They do not establish that any particular physical system remains confined; every confinement claim remains conditional on the adequacy of the model that supplies the confining geometry. A positive Lyapunov exponent alone therefore settles neither risk nor safety. Practical reporting of chaos should state the relevant variable, failure scale, horizon, and confidence level, and reliability claims based on the root-mean-square form of the ratio are correspondingly weak unless stronger distributional assumptions are justified. The paper’s contribution is to separate the prediction of location from the prediction of continued confinement and to supply a measure for the latter.