Adaptive Thresholded Signal Field (ATSF)

This paper introduces the Adaptive Thresholded Signal Field (ATSF), a dynamical framework for competitive signal selection under adaptive threshold memory. A vector-valued signal field evolves under nonlinear feedback, stochastic forcing, and competitive hypothesis channels. Each channel applies a transformation operator and aggregation functional, with activation determined by dynamic thresholds exhibiting hysteresis. The framework admits information theoretic interpretation via KL-divergence cost barriers and extends naturally to spatial continuous fields. Stability conditions are derived from the Jacobian spectrum of the coupled dynamical system. ATSF provides a unified mathematical structure for adaptive selection, memory reinforcement, and signal driven adoption dynamics across cognitive, biological, and physical systems.