What Is the Geometry of the Alignment Tax?

Researchers present a formal geometric theory for quantifying the alignment tax - the tradeoff between AI safety and capability performance. They derive mathematical frameworks showing how safety-capability conflicts can be measured using angles between representation subspaces and provide scaling laws for how these tradeoffs evolve with model size.

• →The alignment tax rate is defined as the squared projection of safety direction onto capability subspace under linear representation assumptions.
• →Safety-capability tradeoffs follow a Pareto frontier parameterized by the principal angle between safety and capability subspaces.
• →The alignment tax decomposes into an irreducible component from data structure and a packing residual that decreases as O(m'/d) with model dimension.
• →The theory provides falsifiable predictions about per-task alignment tax rates and their scaling behavior.
• →Capability preservation can mediate or resolve conflicts between different safety objectives under certain conditions.