Residual Scaling for Language Models: An Exact Layer Score and an Error Budget for Composing It

For scalar gains on the residual contributions of a frozen language model, the signed gradient–residual product is the exact loss derivative at identity. We locate where its predicted benefit is lost when composing multiple layers. The derivative ranks single-layer amplification at Spearman $0.83$–$0.99$ on matched prompts and $0.45$–$0.98$ on held-out prompts, whereas measured ablation ranks it at $-0.19$ to $0.02$ on the same Qwen3.5-9B items. The gap between summed predictions and joint gain splits exactly into cross-item transfer, a single-coordinate finite step and cross-layer interaction. Across four checkpoints of three families, the finite step dominates on GSM8K at $\alpha =1.3$; for three-layer profiles, interaction is at most 30% of it. Transfer grows linearly and the finite step locally quadratically with strength. Coefficients frozen at $\alpha =1.1$ predict switches between these terms on ARC-Challenge: registered fp16 measurements on the original questions locate them at $1.31$–$1.35$, and on disjoint questions at $1.22$–$1.27$, within the registered $0.10$ tolerance. Choosing triples by measured single-layer gains lowers joint-loss regret from $0.041$ to $0.0014$; measuring eight derivative-ranked candidate blocks recovers this choice at an estimated 45% of the recorded single-block stage time.