Papers

Consistent Objectives for Self-Improving Systems: Absolute, Relative and Evolutionary Assessment Over Short and Long Horizons

Peter Cotton · working draft, October 2026 · PDF · tex · certificate

A self-improving system accepts changes by a per-step objective and is judged by where it ends up. Improvement in a fixed proper score, less a constant cost per step, is the only per-step reward that selects the same trajectories as the long-term goal for every feasibility structure. It is the undiscounted case of potential-based reward shaping.

Win rates against a predecessor pay for the number of changes rather than their size. A fixed anchor makes laggards risk-seeking. Selecting a population's realised best rewards variance, which is consistent only when the deployed payoff is itself a maximum, and finite-population fitness rewards spite.

Compounded relative wealth is consistent, selecting the largest expected log return, the long-run objective of a log-score forecaster. A system that helps write its own evaluator records evaluator drift as progress. An exact identity separates the two.