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

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.22101631), with mechanics dissected in a companion paper (doi:10.5281/zenodo.22168300). Here we report what happens to the slogan when its confound is controlled. 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 already shown that at matched criticality saturation governs R-dim (Spearman −0.62 to −0.88; confirmed across seven activation families, two network sizes, biological E/I wiring, and two noise levels), while at matched saturation criticality's effect is bounded. 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 where the exact edge sat in a local valley). 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 negative cell decisively: 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). At controlled saturation, the edge of chaos confers no richness crown; the classic inverted-U is the signature of one-knob trajectories through a two-axis (criticality × saturation) plane. What survives — sealed separately and replicated — is a different royalty: the DURATION of the coherent response is maximal precisely at the edge (37× deep-stable, 6× deep-chaotic; relaxation law T_c ~ 1/|λ| on the stable side at +0.94), and maximal coherent time provably dissociates from maximal richness (bootstrap P = 0.99). The edge of chaos is the throne of time; richness 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 — and every number was mechanically verified against the archived results.