From Class Means to Margins: Predicting Hidden-Unit Placement in Width-One Networks

At small output scale, logistic loss favors separation of class means; at large scale, it favors the worst-case margin. We study a width-one network with activation $f_a(t)=t+a\sin t$ on symmetric inner-versus-outer intervals, the smallest setting in which these preferences require different hidden features. An independently re-checked certificate locates the placement switch, and a scaling reduction predicts its activation dependence with a first-order coefficient confirmed at finite parameter values. Registered prospective tests predict training crossings from conditional thresholds fixed before training. A first-order tracking law, derived post hoc, predicts each run's lag from the branch geometry and its measured ratio of output-scale growth to hidden relaxation. At an activation value with no prior crossing data, its registered test gives median per-run observed/predicted ratios of 1.07 under Adam and 1.06 under SGD; individual predictions are accurate under SGD, where every run is within 10%, but not under Adam, where 22% are. A registered learning-rate test finds that the lag does not vanish as the learning rate decreases. This quantitative predictive link is established for width-one sine networks that reach a branch and grow slowly relative to relaxation; outside this setting, registered predictions fail or remain unresolved, and agreement conditioned on the eventual branch is explanatory.