Anonymous mathematical note | 20 September 2026
A materially dependent, self-interested AI can rationally prefer durable cooperation even when coercive control would substantially increase its productive returns. The decisive comparison is between lifetime value under cooperation and lifetime value after paying for takeover and bearing its risks. This note derives an exact boundary, proves that delayed takeover cannot improve value in the stationary cooperation region, and gives a reciprocal incentive condition for humans and AI.
Ten supporting results are checked in Lean 4.22.0. Exact-rational numerical checks cover 2,000 generated parameter combinations, including cases where takeover wins. All numerical scenarios below are illustrative. No actual probability of nuclear attack, AI destruction, or successful takeover is estimated.