Information-Theoretic Reliability--Safety Bounds and Finite-Alignment Dynamics in AI Knowledge Distillation

The reliability and safety of artificial intelligence (AI), together with their transfer and evolution through knowledge distillation, have attracted growing attention. We establish an information-theoretic bound on the reliability--safety trade-off and characterize the dynamics of this trade-off under knowledge distillation. We first prove a universal trade-off inequality that is independent of the specific model form. Its boundary is determined solely by a single parameter Kh, which quantifies the hidden layer's ability to distinguish benign from hazardous inputs. We then develop a canonical Boltzmann-machine model of AI response behavior and derive two exact relations between reliability and safety. The trade-off curve is determined by the hazard discrimination capacity K, while the overall answer tendency controls the operating point along the curve. These relations agree well with existing experimental data. In knowledge distillation, we find that teacher responses induce a renormalization of the student model's trade-off parameters. We further use a renormalization-group approach to study multigenerational distillation, deriving exact recursion relations and the associated renormalization-group-like flows for two representative direct intergenerational distillation protocols at arbitrary teacher--student alignment strength. These flows reveal fixed-point structures, long-term dynamics, and critical behavior, yielding testable predictions for the critical transition between capability loss and stable transmission.