Data–Task Alignment as a Third Scaling Axis for Language-Model Fine-Tuning

Scaling laws for language models usually predict loss from model size and training tokens while treating the data distribution as fixed. This assumption is especially limiting in fine-tuning, where examples at the same token budget can differ substantially in their relevance to the target task. Existing laws cannot quantify that difference. We introduce data–task alignment $\alpha$, a measurable property of a training set relative to a held-out target, and study it jointly with parameter count $N$ and fine-tuning tokens $T$. Our primary experiment contains 51 controlled Qwen3 cells across four model sizes, six alignment conditions, and a $21\times$ token range. Within this grid, the first-order law $L(\alpha ,N,T)=a+b{N}^{-\beta }+c{T}^{-\gamma }-d\alpha$ predicts held-out loss with $R^2=0.993$ and RMSE $0.015$. Omitting alignment changes both fitted compute exponents, so heterogeneous data composition can be misattributed to scaling with $N$ and $T$. We then map the law's operating regime through a crossed-budget study that finds mild dependence on the token-to-parameter ratio. Selection beyond Q90 shows diminishing returns and eventual reversal as the eligible pool narrows. The alignment effect replicates on an independent math target, a code domain, Llama-3.2, and a frozen gradient selector, although the preferred functional form changes. Alignment therefore provides a useful third scaling coordinate whose interpretation remains tied to the target and measurement. These results connect task-specific data selection to finite-budget scaling by showing when better-targeted data can substitute for compute. They also identify how this return changes across regimes and eventually reverses when selection becomes too narrow.