Count-Controlled Class Difficulty Is Portable Across Learning Protocols

Neural scaling laws characterize how average test loss changes as the amount of training data grows. They do not characterize how group-wise or class-wise losses vary when a fixed data budget is reallocated. Fitting separate scaling laws for each class is also statistically unstable and computationally expensive. Motivated by this, we introduce class-count rotations, which vary class counts across training runs and jointly recover a shared response to class count and a count-controlled class-difficulty spectrum. Surprisingly, difficulty spectra estimated independently across different learning protocols are approximately affine transformations of one another. Motivated by this finding, we propose PACE, which estimates this spectrum through controlled rotations on an inexpensive source model and uses a single calibration run on a more expensive target model to forecast its per-class losses under different class allocations. We establish when rotations identify the difficulty spectrum, provide a sufficient symmetry-based mechanism for its portability, and explain why a single run on the target system can be sufficient for calibration. Our experiments across vision, language, and regression show that the difficulty spectra can be recovered reliably. Moreover, after a reusable source measurement, PACE requires only one target calibration run yet achieves substantially lower per-class loss prediction RMSE than a baseline using only class training counts, outperforming it on every held-out run of the prospectively evaluated targets. These results show that class-conditional scaling behavior can be reused across learning protocols rather than remeasured from scratch.