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The edge of chaos is the throne of time, not of richness: a sealed dissection of the field's favorite slogan

Year: 2026
Type: preprint
Abstract: "Complexity is maximal at the edge of chaos" is among the most repeated claims in the dynamics of neural systems, and our own program confirmed its standard form: debiased perturbational complexity (R-dim, the reproducible dimensionality of the evoked response) traces an inverted-U as a network's spectral radius is dialed from order to chaos (doi:10.5281/zenodo.22100826), with mechanics dissected in a companion paper (doi:10.5281/zenodo.22168299). Here we report what happens to the slogan when its confound is controlled, in one model family (echo-state reservoirs of tanh units; §8). In one-knob designs, criticality (measured maximal Lyapunov exponent λ) and saturation depth (how far activity is driven into the nonlinearity's ceiling) rise together; a sealed two-axis campaign had shown that at matched criticality saturation governs R-dim (Spearman −0.62 to −0.88 in every λ stratum, 95% intervals excluding zero; replicated across seven activation families, two network sizes, biological E/I wiring, and two noise levels), while at matched saturation criticality's effect is not resolved at this n (point estimates +0.04 / +0.05 / −0.19; widest 95% interval [−0.45, +0.11]; the sealed "bound" criteria those estimates passed are satisfied by pure noise 89% of the time and are not counted as evidence). The remaining question — does an R-dim peak survive AT the edge once saturation is held fixed? — resisted three sealed attempts (indeterminate in ρ; a "just-inside-stability" hypothesis killed on virgin ground; indeterminate again on a fine λ-grid). The pre-declared, properly powered last word (protocol sealed publicly before any network was generated, run once; 180 virgin networks in three λ-cells of 60 at fixed saturation) returned its no-crown cell: the edge cell's median R-dim was the lowest of the three (6.41 vs 6.65 near-stable and 6.52 near-chaotic; bootstrap probability that the edge is the maximum: 0.20; edge-minus-flank differences −0.23 and −0.11 with 95% intervals [−0.92, +0.71] and [−0.85, +0.86]). At controlled saturation, no richness crown of the edge is detected at this power: a crown of ≥ 0.5 dimensions has bootstrap probability 0.06–0.10, while the one-knob sweep on the same family spans 3.0 to 6.7 dimensions. The classic inverted-U is, on this evidence, the signature of one-knob trajectories through a two-axis (criticality × saturation) plane. A subsequent sealed test of the opposite reading — an edge penalty, sighted twice — also failed (§7). What the edge does own, sealed separately and replicated, is duration: the coherent response lives longest in the edge stratum (median 110.5 samples against 3.0 deep-stable and 18.0 deep-chaotic as sealed; 58.5, i.e. 19.5× and 3.2×, under the pedestal-exclusion criterion of §5), with the relaxation law T_c ~ 1/|λ| on the stable side at +0.94 either way. The edge-maximum criterion is, however, also passed by a pure relaxation model with no chaos physics; the informative fact is that the edge lives about 3× shorter than that model predicts. The sealed dissociation criterion passed (the T_c-maximizing stratum is at least as close to the edge as the R-dim-maximizing one, bootstrap P = 0.987), but the strict statement — T_c peaks at the edge while R-dim does not — has P = 0.83, below the sealed 0.90 bar; that time and richness are maximized in different places is what these data suggest, not what they seal. The edge of chaos is the throne of time; richness, wherever we have measured it, belongs to freedom from saturation. Every claim above followed the program's standing discipline — public preregistration before blind quantities, single runs, failures reported with the successes, including five sealed deaths reported here for the first time — and every number was mechanically verified against the archived results.
DOI: 10.5281/zenodo.22168818
Language: English
Open Access status: green
OpenAlex ID: W7204730816
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