Arché - A Structural Threshold Law under Temporal Compression in Complex Systems

Complex systems do not fail by accumulation alone. They transform when integration compresses the space of accessible configurations beyond structural tolerance. This paper formulates a general Threshold Law for complex relational fields, arguing that increasing integration systematically reduces effective reversibility and reshapes the topology of possibility. Rather than predicting events, the law identifies the structural condition under which a field cannot remain dynamically identical to itself. We formalize this condition through the ratio between temporally concentrated deviation and effective absorption capacity. When temporally compressed load exceeds structural tolerance, a regime shift (T-SHIFT: Temporal Stress and High-Intensity Fragility Threshold Evaluation System) becomes topologically necessary. By introducing relational field integration (Ψ), coupling intensity (κ), and second-order self-modeling (χ), we extend the framework to higher-order transitions, including reflexive configurations. The resulting formalism applies to non-linear systems endowed with memory and positive coupling, across physical, informational, biological, financial, and decision domains. Arché is introduced here as a structural measure governing regime transitions under temporal compression. The law does not describe collapse. It describes measure. This manuscript is released as a preprint and has not undergone peer review.