When Anomalies Refuse to Dissolve: A Cross-Domain Framework for Stable Non-Null Regimes

We introduce the Eidolonium Attractor Field (EAF), a retrospective audit framework for stable anomalies in physical and computational systems. Rather than defining Eidolonium as a physical element, we treat it as a phase-space attractor characterized by coherence in the absence of resolution. Operationally, this means Φ rises or stabilizes while Δ_anomaly remains high, relative to matched null controls. We formalize the attractor metric A_eid using observable quantities in nuclear physics and AI behavior, and propose descriptive concordance criteria grounded in concordance slopes, anomaly stabilization and repeated returns to similar states. This work aims to provide a falsifiable methodology for characterizing stable non-null anomaly patterns in both physical and computational systems- particularly where conventional predictive models fail. The presence of the Eidolonium attractor correlates with persistent anomalies that resist explanation, yet exhibit structured recurrence. These conditions are encapsulated in the Coherent Event-Cycle Concordance Condition (CECC). The framework offers potential applications in synthetic element research, AI behavior and welfare auditing (non-coercive use only), and the broader study of complex regime dynamics. In computational settings, this framework is intended only as a retrospective, non-coercive audit lens; it is not proposed as a deployment-time monitoring or intervention tool. tl;dr: Sometimes something weird keeps happening, and it keeps happening in a structured way instead of fading out as noise. This paper offers a way to describe that kind of persistent weirdness across very different systems, and calls the resulting regime “Eidolonium.”