When Do Looped Transformers Need More Loops? Calibrated Measurement of External Stop Signals

Since looped transformers reveal output distributions across iterations, an observer might decide a stopping point without any retraining, halting heads, or hidden states. We test our hypothesis on canary controlled algorithmic problems to find that specific example signal is mainly due to dataset shortcuts, noise from oracle selection, and compute-frontier baselines. Ouro peaks at 2 loops on retrieval and 6 loops on one-step modular arithmetic, while within each task no simple external rule consistently or significantly improves on the validation-selected fixed loop, and none Pareto-dominates the best small fixed loop. Huginn-0125 has a smoother loop curve but remains near the capability floor, so it mainly establishes curve-shape differences rather than a strong stopping result. We provide an anonymized artifact bundle with leakage canaries, calibrated metrics, and logged trajectories for offline policy evaluation.