50A (Addendum): Stochastic Gates, Critical Precursors, and Calibration Protocols

Zero Leap Theory Series — Addendum to Paper 50 (Mathematical Foundations) Abstract: This document supplies a quantitative extension layer to the Zero Leap Theory (ZLT), aligned with the open problems of Paper 50. It models gate variables (Phi, C, eta, O) as reflected Itô diffusions on [0,1], establishing a stochastic framework for system integrity under forcing-dominant regimes. Key Contributions: Stochastic Dynamics: Formulation of gate evolution and hysteresis accumulation (H) using stochastic differential equations. First-Passage Risk: Definition of a risk functional R_T(u) that quantifies the probability of structural collapse (lock-in) over time. Critical Precursors: Identification of early-warning metrics (autocorrelation, variance inflation) that precede the critical transition G < G_crit. Optimization Protocol: Mathematical validation that periodic "Resets" (Via Negativa) combined with modulated forcing are thermodynamically superior to intensity increases for maintaining system coherence. Scope: This addendum bridges the gap between ZLT axiomatics and quantitative prediction, providing the mathematical engine for the "Intervention Admissibility System" (IAS). It is applicable to Artificial Intelligence alignment, organizational resilience, and biological entropy management. Machine Preamble: IAS-ZLT-P50A v0.3