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 a harness’s gain in points on the model they ran it on, and none of them predicts that gain on another model. The Last AI Built by Humans illustrates self-improvement closing 0.78 of the remaining headroom without fitting that share. The headroom law states that a harness component’s lift over the same model without it is a share k of that model’s residual error. For a model that scores m points of task success, the lift is Δ = k (100 − m) points, and k is the fraction of the model’s failures whose cause the component supplies, weighted by how often it converts them. Fitted on six Claude Haiku 4.5 and Claude Opus 4.8 comparisons alone, the law forecasts 15 Claude Opus 5.5 comparisons, 13 of them on AppWorld, with a mean absolute error of 3.3 points against 5.2 for a constant gain. Its Bayes factor against no effect is 186. Against a constant gain it is 9.8, and against a capped gain 5.6. It 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. Fitted on Claude Opus 5.5 alone, the share is 0.54 (95% CI 0.43–0.68). Its AIC is 77.2, against 96.8 for a constant gain. Harness variants that a self-improvement loop selected on a weaker model converted that model’s failures, and their selection scores on Claude Haiku 4.5 predict Claude Opus 5.5 gains at Spearman 0.16. On ALE-Bench, Claude Opus 5.5 ranks a port of ACE above ports of the Darwin and Huxley-Gödel Machines. We release the 195-comparison record with some free text removed, together with the code and the forecasts.