The edge-of-chaos hypothesis holds that recurrent systems near criticality maximize memory capacity and expressivity, motivating a natural proposal: replace the non-chaotic linear recurrence of state-space models with a learnable chaotic one. We test this with a gain-tunable chaotic cell whose expansion is set by one calibrated parameter, instrumented with a validated Lyapunov estimator, and measure delay-resolved memory capacity against the lagged mutual information each target demands. Trained end to end, the chaotic recurrence confers no benefit over a gated RNN or its own periodic-regime ablation; a gate dose–response identifies the learned gate as the suppressing mechanism. Frozen under the canonical reservoir-computing protocol, it decisively outperforms a matched linear reservoir, yet a regime-pure control ($n=20$) shows that this is not criticality: an ordered reservoir ($\lambda \approx -0.30$) matches or exceeds the critical one, and across six chaotic reservoir constructions, the ordered arm leads on 21 of 30 paired seeds. Criticality pays only where the target reaches far enough back, on Mackey–Glass and the Santa Fe laser, and delay-resolved measurement explains why: the ordered reservoir retains more at lags 1–3, the critical one retains two to three times more at lags 4–12, and across six targets, the critical margin tracks long-lag demand (Pearson $r=+0.97$), changing sign. We propose a two-factor account: bounded nonlinearity under contraction is necessary, and among nonlinear reservoirs, task-matched memory decides.