The Persistence Equation and the AI Black Box Problem: A Framework for Resolving Internal Model Opacity

AbstractThis paper introduces the Persistence Equation, a general framework for modeling internal coherence and systemic resilience in artificial intelligence (AI) systems. Unlike conventional evaluation methods focused on task performance or feature attribution, the Persistence Equation formalizes how systems preserve structure over time and under stress — using four interpretable variables: reversibility (η), entropy (Q), external stress (T), and sensitivity (α). These variables yield dynamic metrics of local (S(t)) and cumulative (P(n)) persistence, offering insight into how models degrade, adapt, or fail.We demonstrate how the framework applies to neural networks, large language models, and reinforcement learning systems by measuring representational reversibility, entropy growth, and input stress. Persistence metrics anticipate phenomena like catastrophic forgetting, adversarial collapse, reasoning instability, and drift in value alignment — even when task performance remains high. We also show how the framework integrates with existing interpretability, training, and monitoring tools, and how it supports architecture design, curriculum strategies, and safety interventions.Beyond AI, we map persistence theory onto human cognition, epistemology, and philosophical notions of truth, memory, and reasoning under entropy. By treating structure as something that must survive time, noise, and change, this work proposes a universal grammar of informational durability — applicable to both machines and minds.