State-Induction: A Universal Law Governing Information Flow and Behavioral Change in AI Systems

When a user tells a conversational AI “My favorite color is blue,” the system’s behavior changes—yet no training occurs, weights remain frozen, and no gradient descent takes place. We lack a formal theory explaining how information induces behavioral change in AI systems at inference time. We introduce state-induction theory, formalized as a quantitative law: B = σ∇S, where B is behavioral change (KL divergence of output distributions), σ is architectural conductivity (capacity to incorporate information), and ∇S is state gradient (informational potential). We derive this relationship from first principles and generate three falsifiable predictions: (P1) adaptation speed scales as 1/σ, (P2) behavioral change correlates linearly with information strength when σ is fixed, and (P3) the law enables quantitative bottleneck diagnosis. Across 284 data points spanning diverse architectures and tasks, we validate all predictions with correlation r > 0.94 and prediction error < 8%. The law unifies apparently disparate phenomena under one framework: in-context learning (examples provide ∇S, attention provides σ), retrieval-augmented generation (retrieval increases σ from ~0.30 to ~0.85), hallucination (occurs when σ is low despite adequate ∇S), and prompt engineering (increases ∇S through better task specification). State-induction provides quantitative principles for AI systems design, enabling architects to predict performance (B = σ∇S), diagnose failures (measure σ and ∇S independently), and optimize systematically (maximize σ subject to cost constraints). Just as Ohm’s law transformed electrical engineering from empiricism to predictive science, state-induction establishes a foundation for principled AI systems engineering.