证明单神经元模加法训练中傅里叶对齐不成立，揭示模型失效机制。
A Counterexample to Fourier Alignment in Single-Neuron Modular Addition
- 构造反例：激活的ReLU神经元在有限时间内完全失活并冻结。
- 冻结状态的傅里叶能量均匀分布于所有非零实频段。
- 适用于多种训练方式与初始化，揭示对齐并非普遍规律。
我们给出了MAIS-O60问题的负解。首先构造了一个反例：一个初始激活的ReLU神经元在有限时间内完全失活，并此后保持冻结状态，其傅里叶能量在所有非零实频率类中均匀分布。该反例在初始条件的一个开集上成立，因此在高斯初始化下以正概率出现。附录由GPT-5.6撰写，进一步强化了该反例，表明在$ ext{ReLU}'(0)=0$约定下，对于ReLU的光滑死区近似以及固定步长全批量梯度下降，每个从开集出发的Clarke轨迹均可能发生相同失效。因此，单频对齐并非在单神经元模加法训练中的普遍结果。
原文摘要 · Abstract (English)
We give a negative solution to MAIS-O60. We first construct an example in which an initially active ReLU neuron becomes completely inactive in finite time and thereafter remains frozen at a limit whose Fourier energy is equally distributed among all nonzero real frequency classes. The counterexample holds on an open set of initial conditions and therefore occurs with positive probability under Gaussian initialization. An appendix prepared by GPT-5.6 Sol strengthens the counterexample by showing that the same failure can occur for every Clarke trajectory from an open set of initial conditions, under the convention $\mathrm{ReLU}'(0)=0$, for smooth dead-zone approximations of ReLU, and for fixed-step full-batch gradient descent. Thus, single-frequency alignment is not a general consequence of training a single neuron on modular addition.
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