Temporal Membranes: Delay-Induced Self-Retention Without Clocks or Memory Substrates

Living systems must preserve organization while their components relax, diffuse, or fluctuate. Most origin-of-life accounts place this problem in space, emphasizing membranes and compartments, but persistence also has a temporal dimension. Here we ask whether a dissipative system with delayed feedback can retain statistically meaningful dependence on its own past without an explicit clock or dedicated memory variable. We studied a minimal stochastic delay equation with inhibitory delayed feedback and quantified temporal self-retention by comparing autocorrelation decay time with the intrinsic dissipative timescale. The index crossed unity at τ ≈ 1.918 and remained above unity up to τ = 30.0 under the default parameters. The transition occurred across feedback strengths β = 1.0-3.5, collapsed under nondimensional γτ scaling when β/γ and σ/√γ were fixed, and was distinct from the analytical Hopf boundary. Delay dispersion did not abolish the near-threshold regime, whereas high noise shifted it slightly below unity. Mutual information confirmed persistent dependence between present and delayed states in the long-delay regime. These results identify a minimal delay-dissipative regime in which temporal organization persists beyond dissipation alone, providing a proof of concept for temporal self-retention in artificial-life systems.