Harness Gains Are Linear in Residual Error: A Weak-to-Strong Headroom Law Forecasts Claude Opus 5.5 from Weaker Claude Models

Papers on self-improving agent harnesses, such as RRSI, report the gain on the model they evaluate and explain only in words why stronger models gain less. Yet a gain in points transfers poorly to a stronger model. We show that a component’s gain is a fixed share of the model’s residual error, the points the model misses without the component. We call this relation the headroom law. When a component leaves successes intact, a model that scores m points gains Δ = k (100 − m) in expectation, where k is the expected share of failures the component fixes. A single share k forecasts the gains on two models when the component fixes the same fraction of each model’s failures. Fitted on weaker Claude models, the law misses fifteen held-out Claude Opus 5.5 gains by 3.3 points on average, against 5.2 for a forecast that gives every model the same gain. It is the first law that predicts an agent-harness component’s gain on a model before the component runs on that model. A self-improvement loop that scores candidates with Claude Haiku 4.5 keeps variants that fix Haiku’s failures, yet their Haiku scores barely predict their Claude Opus 5.5 gains (Spearman correlation 0.16). Improving a frontier model’s harness therefore requires scoring candidates with that model or one whose failures have the same causes.