Endpoint Selection Through a Moving Fold in a Two-Neuron Group Composition Network

Which endpoint does training reach when the minimum it approaches is degenerate and a small perturbation splits it into competing branches? We study two normalized quadratic neurons trained by continuous exact cross entropy on the complete addition table of C3. With logit scale s³, the first correction to the leading feature objective has a quartic minimum in the relative phase. At unit gain, balancing this quartic against the next-order force gives a displacement of order s; relative relaxation then takes the s⁻⁸ physical clock, followed by common-phase selection on the s⁻⁹ clock. An input-gain imbalance β = 1 + ηs produces a cusp whose forcing comes from the still-moving common phases. For fixed |η| ≠ ηc and any absolutely continuous initialization law, for almost every start in the leading basin there is a start-dependent scale below which the full exact-CE flow reaches the secondary branch if and only if it enters on that side and the forcing stays below the fold threshold along the entire limiting common-phase path. The stationary landscape at the limiting phases alone does not determine the endpoint. For |η| < ηc ≈ 0.486 only the primary minimum is selected; for |η| ≥ η∗ ≈ 0.701 the entrance side alone decides; between these thresholds the path matters. Under rotation-invariant initialization, the limiting secondary probability is half the area fraction of common phases whose path keeps the forcing below the fold threshold. A first integral computes this area. We also classify the second-order unfoldings for other cyclic pairs. These are continuous-training, vanishing-scale results and give no finite-scale branch probabilities.