Fourier alignment in single-neuron modular-addition dynamics
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In the specified single-neuron modular-addition setting, must training align the neuron with a Fourier feature? The counterexample target covers selection-independent Clarke trajectories and dead neurons for the stated ReLU, dead-zone smoothing and fixed-step full-batch gradient-descent dynamics.
Why it matters and what counts as progress
Why it matters: It asks which feature-alignment conclusions actually follow from the training dynamics, beyond finite empirical traces.
What counts as progress: A proof of alignment under a specified training dynamic or an exact persistent nonalignment trajectory, rather than a single finite training trace.
How the problem developed
- 5 Aug 2026 · Reported counterexample [1]
Gautam Neelakantan Memana Gave a negative answer to MAIS-O60: an initially active ReLU neuron can become inactive and freeze with Fourier energy spread equally across nonzero real frequency classes.
- 6 Aug 2026 · Extension [1]
Gautam Neelakantan Memana, with a GPT-5.6 Sol appendix Expanded the counterexample to an open set of Gaussian initializations and added cases for every Clarke trajectory, smooth dead-zone activations and fixed-step full-batch gradient descent under the stated ReLU convention.
AI contribution
A Counterexample to Fourier Alignment in Single-Neuron Modular Addition
For the single-neuron modular-addition setting, Appendix A extends Memana’s human counterexample to selection-independent Clarke trajectories and dead-neuron examples under the specified ReLU, dead-zone smoothing and fixed-step full-batch gradient-descent dynamics.
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