IV. κ-Theory: Quantitative Activation Thresholds and Model Specification

This section formalizes κ-Theory by defining the precise mathematical conditions under which history-dependent gravitational effects emerge. It introduces a nonlinear gradient-based correction to density, establishes fixed model parameters, and derives exact activation thresholds that determine when κ becomes physically relevant. By quantifying these conditions, the section transforms κ-Theory from a conceptual framework into a testable model with measurable predictions. The results demonstrate that gravitational behavior remains consistent with standard models in low-gradient regimes but exhibits controlled, measurable deviations during high-intensity structural formation, indicating a bounded form of history-dependent dynamics.