Stopping Criteria Below the Signal-to-Noise Floor: Window Length, Not Tolerance, Governs Convergence Detection in Architecture Comparisons

This record contains the same article in two language versions: criterio_snr.pdf (English) and criterio_snr_es.pdf (Spanish). LaTeX sources, the bibliography, and the analysis script that reproduces the reported numbers are included. Architecture comparisons in efficient sequence modeling commonly stop training with a rule of the form “improvement in validation accuracy over the last w steps is below a tolerance τ”, and then grant every architecture the resulting budget. We analyze one such rule that was pre-registered and frozen with a DOI before data collection, and show that it could not detect convergence — not because its tolerance was miscalibrated, but because the stopping decision was taken below the signal-to-noise floor of the measurement. In a multi-query associative recall benchmark, the delta-rule linear-attention condition improved at a rate of 0.31 accuracy points per evaluation in the regime treated as a plateau (positive in seven of eight seeds; t = 4.20, p = .004), while the noise of the tested difference was 1.16 points, giving a signal-to-noise ratio of 0.27. Under this ratio no tolerance separates the two errors of interest: rules that detect a plateau also stop during genuine improvement. We show that a proposed remedy — window means with a noise-calibrated tolerance — has a detection rate that is algebraically fixed at Φ(2) regardless of window length, and would have declared convergence while the model was still improving with probability .73. Decomposing the achievable gain, a factor of 7.0 comes from lengthening the comparison window and only 1.31 from replacing the two-point difference with a trend test. The failure is architecture-dependent: saturating conditions exhibit noise three orders of magnitude smaller and close immediately, so the shared budget covaries with the quantity under comparison. We give a power-based rule for choosing w that can be pre-registered without pilot data. All analyses reuse stored validation histories and require no additional training.