ΔΦ SPARSE THRESHOLD CONTROL v1.0 A Convergent Framework for Cognitive Correction and Cost-Constrained Control Systems

This paper investigates how systems decide when to correct themselves. Across large-scale simulations of threshold-triggered control systems (7,680 configurations), an optimal intervention regime consistently emerges at low activation density (~3–6%). Below this range, systems drift without effective correction; above it, over-intervention introduces instability and cost penalties. Independent behavioral analysis of cognitive datasets (SART, Stroop, Sternberg) identifies correction-like events occurring at ~2–3% density using an autocorrelation-based detection method. These event rates remain stable across wide parameter variation, indicating structural sparsity rather than detection artifacts. A complementary theoretical model shows that systems with memory, partial correction, and non-zero intervention cost admit a unique interior optimum at low intervention frequency (~2–8%), providing a mathematical basis for the observed regime. These three independent lines—simulation, behavioral observation, and theory—converge on the same sparse correction structure. This work does not claim that biological and engineered systems share mechanisms, but demonstrates that they independently arrive at similar solutions when optimizing correction under cost constraints. The findings suggest that sparse, threshold-triggered correction is a general structural principle governing adaptive systems.