How Should LLMs Consume High-Quality Data? Optimal Data Scheduling via Quality-Aware Functional Scaling Laws

High-quality data is scarce in large language model (LLM) training, yet how to schedule its use with optimization dynamics lacks theoretical guidance. We extend functional scaling laws with time-varying data quality and derive asymptotically optimal joint data-quality and batch-size schedules within a feature-space regression model. The solution reveals two regimes and dual uses of high-quality data: in the noise-limited regime, a smaller batch converts cleaner data into more signal at comparable noise; in the signal-limited regime, late placement suppresses terminal noise without sacrificing signal accumulation. This explains why conventional decay schedules can conflict with curriculum-style pipelines. Motivated by the theoretical structure, we propose Drop-Stable-Rampup for LLM midtraining: drop the batch size at the quality transition, keep it low to accumulate signal, then ramp up to suppress noise. On a 15B MoE model midtrained on 108B tokens of general-domain proprietary data, Drop-Stable-Rampup improves average accuracy over Warmup-Stable-Decay by +1.70 and Cosine-decay by +2.98, including +4.23 on GSM8K and +2.80 on MATH. On a public math-and-code mixture, it leads all reported STEM, mathematics, and code benchmarks, improving the overall mean over the strongest baseline by +3.27 on a 600M dense model and +5.25 on the same MoE architecture.