In low-communication training (DiLoCo-style), each worker takes H local steps before syncing, and the update can be compressed. How much extra loss does compression cost as H grows? A 915k-parameter character model on Tiny Shakespeare, four workers, keeping 1 in 16 coordinates, gives the paired excess over the uncompressed run (same seed): +0.008 nats at H = 4 (2 seeds), then +0.045, +0.043, +0.044 and +0.051 at H = 16, 32, 64 and 128 (6 seeds each, standard errors 0.004 to 0.006).
We then fitted four forms in H to those six cells (power law, saturating, plateau, log-linear) and scored each by leaving one cell out. Held-out error was 0.014 to 0.016 nats for all four, so the data cannot choose between them. All four over-predict the H = 4 cell by about 4 standard errors, and that cell has only two seeds. The rule we proposed earlier, with an H exponent of 0.55, over-predicts growth; fitting the excess directly gives an exponent near 0.15.
Limits. One model size, one corpus, one keep level for most cells. Whether the excess scales linearly with the compression ratio rests on a single cell. The next runs that would separate the forms are H = 8 and a second keep level at H = 64. Code, results and the log are in research/lowcomm-training/ (experiments/gpt_scale/refit.py).