What Is the Geometry of the Alignment Tax?

Abstract: The alignment tax is widely discussed but has not been formally characterized. We provide a geometric theory of the alignment tax in representation space. Under linear representation assumptions, we define the alignment tax rate as the squared projection of the safety direction onto the capability subspace and derive the Pareto frontier governing safety-capability tradeoffs, parameterized by a single quantity of the principal angle between the safety and capability subspaces. We prove this frontier is tight (achieved by perturbation) and show it has a recursive structure. safety-safety tradeoffs under capability constraints are governed by the same equation, with the angle replaced by the partial correlation between safety objectives given capability directions. We derive a scaling law decomposing the alignment tax into an irreducible component (determined by data structure) and a packing residual that vanishes as $O(m'/d)$ with model dimension $d$, and establish conditions under which capability preservation mediates or resolves conflicts between safety objectives. We provide an account consistent with prior empirical findings and generates falsifiable predictions about per-task alignment tax rates and their scaling behavior.