The lifetime of the coherent response: sealed laws of its rise, its peak, and its fall

When a recurrent network is perturbed, how long does its stimulus-locked response live? We define T_c as the post-stimulus time by which the coherent (cross-trial-reproducible) response accumulates 80% of its energy, and report its life-cycle as sealed results (public preregistrations before any blind quantity, single runs, failures included), each given both as it fell and under a pedestal-exclusion criterion that separates the response's lifetime from the noise floor of the estimator. Rise: on the stable side, coherent lifetime is set by the relaxation time — Spearman(T_c, 1/|λ|) = +0.943 (p = 3.5 × 10⁻²¹, 43 virgin networks), and +0.937 under a post-hoc quality gate: the paper's most robust result. Peak: lifetime is maximal precisely at the edge of chaos — median 110.5 samples versus 3.0 in deep order and 18.0 in deep chaos (37× and 6×), replicating wholesale in a second sealed protocol's fresh networks; under the quality gate the peak persists but roughly halves (19.5× and 3.2×), because near the edge the response is long-lived and weak per trial, exactly where a clipped estimator's noise pedestal accumulates. Fall: on the chaotic side the sealed protocol returned Spearman −0.567 with an ln–ln slope of −0.559 (95% CI [−0.79, −0.33]), excluding the −1 exponent of our own naive divergence derivation. That direction is robust — it survives every deletion we tried, and partialling out the generator parameter leaves −0.511 — but the magnitude is not: gated, the correlation falls to −0.438 (below the sealed bar) and the slope to −0.342, so the honest interval for the exponent is −0.34 to −0.56 and "λ^(−0.56)" should be read as its upper end. We also report a methodological result of independent interest: the split-half reliability of T_c is +0.976 across an order-to-chaos population but only +0.103 within the chaotic band where the decay claim lives — a target's reliability must be measured on the domain of the claim, not on a wider population where ranking is easy. Maximal coherent time is nonetheless not maximal complexity: the sealed protocol that replicated the edge peak established that the λ stratum maximizing T_c is not the one maximizing reproducible dimensionality, and a companion paper (doi:10.5281/zenodo.22168818) shows that at controlled saturation the edge confers no richness advantage at all. We close with the measurement boundary, mapped by three known-truth benches: at realistic recording geometries every T_c estimator we built fails validation — one would have fabricated an inverted result — so the lifetime laws remain laws of the simulated world, awaiting high-trial-count recordings.