Structural Origin of the Temporal Order from Set Inclusion

The direction of time is often attributed to dynamical, statistical, or entropic principles.In this work, we show that a temporal partial order can arise prior to any suchassumptions, as a purely structural consequence of set inclusion.We introduce the notion of an initial detection time associated with a set, defined as thefirst detection of any of its elements. We prove that, for two sets related by inclusion, thecorresponding initial detection times obey an observer- and path-independent inequality.This ordering follows solely from the configuration of the sets and does not rely ondynamics, probability, entropy production, or a prescribed generative rule.We further clarify the conceptual separation between this structurally induced temporalorder and history-dependent selection principles, such as entropy minimization. While thelatter determine which of several admissible developments is realized, they do not generatethe direction of time itself. This distinction is illustrated in the context of dimensionalemergence, where previous analyses relied on stepwise generative assumptions. Here, wedemonstrate that even in the absence of such assumptions, inclusion relations alone sufficeto induce a temporal order.Our results suggest that the direction of time can be understood as a minimal and robustfeature of structural inclusion, providing a foundation upon which dynamical and entropicprinciples act, rather than deriving from the