Many modern systems operate in open environments that cannot be fully observed, controlled, or reset. In such settings, reasoning about system behavior often relies on snapshots: instantaneous states, static configurations, or endpoint outcomes. While useful in closed or tightly controlled contexts, snapshot-based reasoning can miss important information about how systems actually interact with their environments over time. This paper introduces Stateful Temporal Entropy (STE), a general, domain-neutral synthesis that treats a system’s irreversible trajectory as a source of observable evidence. Rather than focusing on what a system is at a moment, STE emphasizes how a system evolves under irreversible progression, internal state, and external influence. From this perspective, entropy is understood not as noise or disorder, but as observer-side uncertainty that accumulates through lived process. Systems may appear identical at an endpoint yet remain distinguishable by the entropy of the paths they took to arrive there. STE is not proposed as a new physical law, algorithm, or inference rule. Instead, it provides a conceptual lens for identifying where evidentiary information resides in open systems and why that information is absent from snapshot-based observation. By treating trajectories as first-class evidentiary objects, STE clarifies how irreversible evolution constrains plausible explanations of system behavior. Illustrative examples from physical, cyber, and socio-technical domains are discussed to demonstrate breadth, without narrowing the synthesis to any specific application.