Retention as a Physical Phase in Rare-Event Detection: A Lagrangian Architecture with Criticality, Saturation, and Two-Detector Falsification

This paper introduces a physical field model of retention in detection systems and demonstrates that memory can emerge as a bounded critical phase rather than as a statistical artifact.

We formulate a nonlinear retentive dynamics that reduces to a Hawkes self-exciting process in the linear limit and show that quadratic saturation stabilizes the system beyond the critical branching threshold. A three-phase structure (subcritical, near-critical, saturated) is derived and analyzed, together with stationary solutions, stochastic stability, and ergodicity.

A falsification protocol is provided based on five independent observables (Fano factor, correlation decay, spectral slope, cluster statistics, and cross-detector coupling), together with multiple null-models to exclude artifacts. The framework defines retention as an operationally measurable phase variable rather than as a conceptual or phenomenological construct.

This work establishes a general architecture for retention-driven detection systems and provides a theoretical backbone for future experimental and simulation studies.