The Threshold of Change: A Concentration Theory for Complex Systems in Mathematics, Finance, and Economics

This paper develops a unified framework for analyzing complex systems through the concept of a concentration threshold, defined as the point at which a distribution ceases to behave as an extended object and begins to act as a point mass. Inspired by mathematical models based on the Laplace distribution and its collapse toward the Dirac delta function, the framework is extended to finance—through the volatility threshold in diversified portfolios—and to economics—through wage compression associated with the Kuznets curve. This extended version incorporates conceptual figures, empirical simulations, and a formal discussion of limitations. The result is a general theory of the Threshold of Change, applicable to any system in which the marginal derivative of a critical variable approaches zero.