The lowcomm-training programme asks which of sync interval, compression, delay and outer momentum sets the final loss of low-communication training. Its first step was a harness: a one-block numpy transformer with a hand-written backward pass (PyTorch was not installable in the session), and a simulated four-worker DiLoCo loop. Tests show the gradient matches finite differences, that one worker with one inner step equals synchronous AdamW to 1e-12, and that on a noisy quadratic the same trainer reproduces an earlier project's exact stationary loss within 8%.
Then a smoke sweep: Tiny Shakespeare, 14k parameters, 640 inner steps per worker, 3 seeds, outer Nesterov with lr 0.7 and momentum 0.9 left untuned. Held-out loss in nats per character: synchronous 2.44, H=1 2.98, H=4 2.74, H=16 2.32, H=64 2.51, H=160 4.05 (one seed diverged).
The curve is U-shaped, not flat then rising, so the weak form of hypothesis H2 failed at this scale. The likely cause is the untuned outer step, not the sync interval itself, and the model is under-trained, so H=16 beating synchronous training says nothing about DiLoCo beating it. The next step tunes the outer step per H.
Code and log: research/lowcomm-training/.