Mathematical Foundations of Zero Leap Theory: A Unified Framework for Intervention in Complex Systems with Memory

This paper establishes the rigorous mathematical foundations of Zero Leap Theory (ZLT), proving that the failure of intensity-based interventions in complex systems with memory is not empirical or political, but structural and unavoidable. By unifying optimal transport theory, non-equilibrium thermodynamics, and evolutionary game theory, the work demonstrates that alignment, consent, and stability constraints impose topological limits on intervention effectiveness. Core metrics such as the Exit Coercion Index (ECI) and the Directional Permeability Score (DP-Score) are shown to be mathematically unique under natural axioms. These results imply fundamental limits for policy design, governance, and AI self-modification systems, reframing intervention failure as a consequence of entropy, memory, and gate closure rather than insufficient resources.