Counterexample to Single-Frequency Alignment in Modular Addition
Gautam Neelakantan Memana constructs an open set of initial conditions for which a single ReLU neuron trained on modular addition freezes without aligning to one Fourier frequency; an appendix developed with GPT-5.6 Sol strengthens the counterexample across Clarke trajectories, activation conventions, smooth approximations, and gradient descent.
In a preprint first submitted on August 5, 2026, Gautam Neelakantan Memana gives a negative answer to MAIS-O60, which asks whether training a single ReLU neuron on modular addition forces its limiting direction to align with one Fourier frequency. The paper constructs an open set of initially active states whose neuron becomes completely inactive in finite time and then remains frozen with its Fourier energy distributed across all nonzero real frequency classes. Because the set is open, the behavior has positive probability under Gaussian initialization.
Memana states that he worked out and wrote the main results, with GPT-5.6 Sol assisting with literature review and clarification. An appendix initially drafted by GPT-5.6 Sol and then reviewed and edited by Memana strengthens the construction: failure can occur for every Clarke trajectory from an open set, under the convention , for smooth dead-zone approximations, and for fixed-step full-batch gradient descent. The result therefore supplies a concrete, robust counterexample rather than merely evidence against the proposed alignment behavior.