Optimizer-dependent training dynamics converge to the same one-third optimal data scaling

Neural scaling, in which loss falls as a power law with training, is central to large language models, and one recent proposal is that a 1/3 exponent emerges from learning peaked distributions. That account describes SGD, but models in practice are trained with adaptive optimizers. Here we separate two exponents the 1/3 account does not distinguish: how fast the loss falls with training steps along a single run, and how fast the optimally tuned loss falls with dataset size D. We show that the first, a dynamic exponent, is optimizer-specific while the second, an optimal data exponent, converges to 1/3 across optimizers. In an online teacher-student model we decompose the loss into norm growth (radial) and alignment toward the teacher direction (tangential), each decaying as a power law with dynamic exponents αr and αt. Under SGD, both are close to 1/3, so the data exponent is also 1/3 across different learning rates. Under Adam the two separate: αr simeq 0.48 but αt simeq 0.08. Since the total loss is minimized when these two parts are balanced, the optimal learning rate is optimizer-dependent: D-independent for SGD but falls with D for Adam. Yet tuned to that optimum, the loss returns to D-1/3 for both. A stochastic-dynamics analysis explains why: the optimizers can trade decay speed between the two channels, but they all fall on a single dynamic exponent relation, 2αr+ αt = 1, which fixes the optimal data exponent at 1/3. Across seven optimizers, including Muon, the measured exponents are consistent with this relation, and the optimal-loss envelopes agree with D-1/3 across them. The optimizer sets how fast a model learns per step; tuned optimally, it changes the prefactor but not the rate at which loss falls per sample.