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

RRSI, Harness², and other self-improvement methods report harness gains as points on the model they ran on, and The Last AI Built by Humans illustrates self-improvement closing 0.78 of the remaining headroom without a fit. None predicts a harness’s gain on a model it has not run on. We introduce the headroom law: a harness component converts only the failures whose cause it supplies, so its lift over the same model without it is a share k of that model’s residual error, Δ = k (100 − m), in points of task success. Fitted on six Claude Haiku 4.5 and Claude Opus 4.8 comparisons alone, the law forecasts 15 Claude Opus 5.5 comparisons, 13 on AppWorld, with a mean absolute error of 3.3 points, against 5.2 for a constant gain, and Bayes factors of 9.8, 5.6, and 186 against a constant gain, a capped gain, and no effect. On Claude Opus 5.5 alone the share is 0.54 (95% CI 0.43–0.68), with AIC 77.2 against a constant gain’s 96.8. Because the share depends on which failures a component matches, self-improvement that selects harness variants on a weaker model converts that model’s failures: selection scores on Claude Haiku 4.5 predict Claude Opus 5.5 gains at Spearman 0.16, and Claude Opus 5.5 ranks ported ACE above the ported Darwin and Huxley-Gödel Machines on ALE-Bench. The headroom law is the first law that predicts the lift of an agent-harness component over the same model without it before the component runs on that model. The 195-comparison record, with some free text removed, the code, and the forecasts are released.