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

Abstract

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 — frozen in a later protocol of the program, hence post hoc with respect to every seal reported here — 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 the post-hoc quality gate: the paper's most robust result. Peak: lifetime is maximal 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 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. Corrected in this version, the peak is not a law independent of the rise: the relaxation law alone, continued through λ = 0 and truncated at the 600-sample window, predicts an edge median of 344 and passes the sealed 3× bar by itself, so the sealed criterion could not separate "the edge is special" from "relaxation slows down near zero". What is new in the data is the shortfall — the edge lives about three times shorter than that naive prediction (110.5 sealed, 58.5 gated, against 344). 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 the deletions and the partialling reported in the archived review (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. A sealed follow-up that factorized the exponent into a rate clause and a quantile clause returned 3/3 and is withdrawn here: its rate clause excluded the null by 0.002 of bootstrap interval and does so in only 60% of bootstrap seeds, and the quantile clause was tested against a null that is wrong for the pipeline. We also report a methodological result of independent interest, corrected in this version: two split-half reliability estimates of T_c, each on ten networks inside the chaotic band, gave +0.952 (λ from 0.007 to 0.029, no pedestal-dominated network) and +0.103 (λ from 0.010 to 0.262, two pedestal-dominated networks); under the pedestal gate the second becomes +0.964. v1 described the first population as spanning order to chaos, which is false. The lesson survives in sharper form: a ten-network reliability estimate without a quality gate bounds nothing, and the population on which a target's reliability is measured must be declared and must contain the domain of the claim. Maximal coherent time is nonetheless not maximal complexity: the sealed protocol that replicated the edge peak carried a secondary clause whose sealed statistic — the bootstrap probability that the λ stratum maximizing T_c lies at least as close to the edge as the stratum maximizing reproducible dimensionality — was 0.987; the strict reading, that the two strata differ, is 0.92 (0.912 under the gate). A companion paper (doi:10.5281/zenodo.22168818) reports that at controlled saturation the edge stratum's median R-dim was the lowest of its three cells (bootstrap probability that the edge is the maximum: 0.20). 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. That boundary is a limit of our estimators, not evidence about the laws in recordings.