Debiased perturbational complexity peaks at the edge of chaos: reproducible response dimensionality in recurrent networks

The perturbational complexity index (PCIst) is the most clinically validated readout of consciousness-related brain complexity, yet what dynamical property it tracks has remained unclear. Using a fully out-of-sample variant of the index (PCIst-XV) that removes its selection biases (companion paper), paired sham controls, and conditional Lyapunov exponents λ, we characterize perturbational complexity across 366 recurrent networks spanning ordered, edge-of-chaos and chaotic regimes, in two connectivity families (Gaussian echo-state and Dale's-law E/I). Debiased complexity Δ follows an inverted-U in λ: near-floor in deeply ordered networks (median 11), maximal at |λ| < 0.01 (median 94), and collapsing in chaos (median 5; edge vs order p = 1.8×10⁻³, edge vs chaos p = 5.9×10⁻³; trend on the ordered side Spearman ρ = 0.56, p = 5.5×10⁻⁶, n = 92 networks). The decay into chaos is gradual, not step-like (preregistered step test rejected), consistent with a divergence-horizon account: reproducible response structure survives for ~1/λ steps within the measurement window. Three controls sharpen the interpretation. (i) Within nominally identical networks, Δ tracks each network's measured λ, not its construction parameter — tanh networks with spectral radius 1.0–1.5 self-organize to λ ≈ 0, an effective-criticality regime. (ii) Response energy does not explain the effect (Spearman(Δ, energy) = 0.06, p = 0.72; the edge peak survives energy residualization at p = 7.9×10⁻⁶): deeply ordered networks respond strongly but simply. (iii) The quantity that does explain Δ is the response's reproducible dimensionality (R-dim): the number of principal response directions that replicate across independent trial halves (Spearman ρ = 0.74, p = 2.1×10⁻¹¹), which itself traces the same inverted-U (median R-dim 1.4 → 6.2 → 0.4 across order → edge → chaos). A stereotyped excitable network reproducing the slow-wave response of the unconscious brain sits at the floor (Δ = 8.5) despite maximal response energy and reliability, because its response is two-dimensional. We propose that perturbational complexity measures reproducible causal dimensionality — a property requiring simultaneous sensitivity (lost in order) and stability (lost in chaos) — and is therefore maximal only at the edge of chaos approached from the stable side. The ordered-to-edge rise and the R-dim relationship replicate in Dale's-law E/I networks (edge vs order p = 8.6×10⁻⁴; ρ = 0.54), where E/I balance itself prevents deep chaos — an incidental observation consistent with cortical dynamics living near, and stabilized short of, the edge.