An Exact Saturation Law for Pairwise-Agreement Symbolic Entropy and Why Entropy-Saturation Criteria Cannot Recover the Decorrelation Delay

Source abstract

A recent family of methods selects the phase-space reconstruction delay $\tau$ as the point where a symbolic entropy $H(m,\tau)$ stops growing and saturates, on the premise that this onset marks the decorrelation delay. We prove the premise structurally unsound \emph{within the pairwise-agreement class of symbolic schemes that these methods use}. The agreement symbol is invariant under complementation of the thresholded sign vector, so the effective alphabet has exactly $2^{m-1}$ symbols and $H(m,\tau)\in[0,m-1]$, with $H\to m-1$ for any mixing process. Saturation is therefore an artefact of a ceiling imposed by the symbolisation rather than a property of the dynamics, and the delay read off it is the argument of the approach to a self-imposed asymptote; the law is verified to three decimal places on i.i.d.\ noise. The conclusion is specific to saturation criteria built on pairwise agreement and carries no claim about symbolic entropies of other design, such as permutation entropy, whose alphabet is not complement-degenerate. Testing the family, which comprises SEM, multiscale sample entropy, symbolic and multiscale fuzzy entropy, the C-C method, and low-pass filtered symbolic entropy (LPF-SEM), against average mutual information and false nearest neighbours on five chaotic systems at six noise levels, all with independently verified positive Lyapunov exponents, we find that on clean data LPF-SEM returns delays $1.17$--$4.69$ times the reference and carries the largest bias of the six estimators at every noise level; that the low-pass filter, \emph{doubles} the delay bias ($28.8\to56.4$ samples) rather than reducing it; that the returned delay spans a fortyfold range across the filter's own parameter grid; that the threshold which recovers the reference varies fivefold between systems and does not exist for two of five; and that the dimension output is the method's own fallback constant in $100\%$ of runs for four of five systems. The engineering outcome is counterintuitive and cuts against the theorem's practical weight: on real MFPT bearing data the comparison inverts, the most biased estimator classifying best ($0.930$) and the most accurate one worst ($0.599$). Neither the delay's accuracy nor its length explains it: a sweep at fixed dimension is flat or non-monotone, and the calibrated pair alone reproduces $88\%$ of the winner's score. Saturation-based \emph{pairwise-agreement} entropy criteria should not be used as delay estimators; where only classification accuracy is wanted, LPF-SEM may serve as a black-box feature heuristic, but its $\tau$ is not the decorrelation delay.