ZLT-P50: Mathematical Foundations of Zero Leap Theory — Rigorous Formalization of Gates, Resistance, and Intervention Impossibility (v2.0)

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. v2.0 contribution: Introduces a formal ontological partition of the five gate operators {C, η, Φ, H, O} into two functional layers — the Transmission Gate G_T = Φ·C·η (intervention efficiency) and the System Aperture Function G_A = G_T·ψ(O)·χ(H) (structural admissibility). This resolves a terminological tension present since Paper 1 and canonizes the Ĝb diagnostic introduced in Paper 50A and the five-gate aperture used in Paper 71. All v1.0 axioms, theorems, and proofs are preserved verbatim. No downstream results are invalidated. Principal results: (i) Gate family-invariance and generalized impossibility as topological obstructions; (ii) correspondence between admissible interventions and constrained geodesics in Wasserstein space; (iii) entropy production bounds; (iv) ESS characterization of gate configurations; (v) uniqueness of the Exit Coercion Index; (vi) [NEW v2.0] System Aperture Ordering (Prop. 5.10) and Structural Admissibility Condition (Cor. 5.11). Part of the Zero Leap Theory series (50+ papers). Series entry points: IAS-ZLT-CORE (doi:10.5281/zenodo.18408846), Paper 1 (doi:10.5281/zenodo.18264279), Paper 52 (doi:10.5281/zenodo.18344445).