ψ-Retentive Threshold: A Universal Ξ-Criterion for Self-Sustaining Structure

This work introduces a universal, domain-independent operational criterion for identifying self-sustaining (retentive) structures in systems dominated by noise, dissipation, or observational incompleteness. Within the ψ-Architecture framework, a retentive transition is defined by the joint satisfaction of three minimal conditions: a critical signal-to-noise ratio (SNR*), a minimal structural asymmetry (ε*), and a critical nucleation scale (Rc). Only when these conditions are simultaneously satisfied does a fluctuation transition from long-lived statistical persistence to a genuinely retentive configuration (Ξ-node) capable of preserving difference (Δψ) independently of continuous external driving or observation. The Ξ-threshold is formulated as a compact, machine-readable test applicable without modification across multiple domains, including cosmology, neutrino detection, laboratory time series, complex systems, and pre-biological pattern formation. The framework introduces no new forces or dynamics, but delineates a universal boundary between dissipative persistence and structural retention. This work contributes an operational bridge between ψ-Retention theory and empirical data analysis, offering a unifying criterion for detecting self-sustaining structure across physical, informational, and observational regimes.