Autonomous Emergence of Discrete Time Order via Finite Information Delay: Spectral Rigidity and Dynamic Repulsion

Abstract Note: This is the preliminary version (v1.0) focusing on the theoretical formalization of the 4n-1 selection rule. Detailed numerical datasets will be updated in subsequent versions. Finite information delay is a ubiquitous ingredient of real dissipative systems, yet its impact onmacroscopic relaxation spectra remains poorly understood. Here we show that delayed feedback can autonomously discretize relaxation into a robust hierarchy of coherent durations, even in the absence of external periodic driving. In a delayed generalized Langevin framework, the linear response develops a comb-like susceptibility whose peaks are selected by a phase-inversion condition𝜔 𝑛 𝜏 𝑑 ≈ (2𝑜 + 1)𝜋.Nonlinear saturation then stabilizes these delay-selected modes into plateau-locked temporal states characterized by quantized lifetis 𝑇 𝑛 = 𝜏 𝑑 /(2𝑜 + 1). To resolve forbidden temporal windows, we introduce a nonlinear order parameter—the Temporal Overlap Integral 𝑃(𝑇)—which reveals temporal bandgaps as a form of spectral rigidity in time. Strikingly, trajectories forced toward forbidden timescales are dynamically expelled across bandgaps and collapse into the nearest allowed plateau, exposing an intrinsic selection mechanism we term dynamic repulsion. These results identify finite information delay as a minimal route to autonomous discrete time order and establish delay-induced rigidity as a generic organizing principle in non-equilibrium relaxation.