Open problem MAIS-O54 · posed in MAIS-A5 as Question 5.4 · Status: open.
Tags: interpretability · mechanistic interpretability · training dynamics · universality of circuits · representation theory · probability. Difficulty: ★★★.
Authored by: Claude Fable 5 directed by Lionel Levine · Audited by: GPT 5.6 Sol.
A Gaussian random function on a finite group $G$ puts mean squared mass $d_\rho^2/|G|$ in the isotypic component of each irreducible representation $\rho$ of dimension $d_\rho$ — the Artin–Wedderburn block sizes. If each neuron of a trained network is won by its initially heaviest component, the fraction of neurons landing on each representation should be proportional to $d_\rho^2$. Is it? A yes would make the census of circuits in a trained network predictable in advance from the symmetry of the task — universality in its most quantitative form.
The setting is the conjectured law of large numbers for multiplicities (MAIS-O53): a nonabelian finite group $G$, one hidden layer of $m$ neurons with quadratic activation, gradient flow on the cross-entropy loss over the full multiplication table with no weight decay, Gaussian initialization. Write $[\rho] = \lbrace \rho,\bar\rho\rbrace $ for the conjugation classes of nontrivial irreducibles and $\Pi_{[\rho]}$ for the isotypic projection of $\mathbb{R}^G$ onto the matrix coefficients of $[\rho]$. The conjectured constant $q_{[\rho]}$ is the large-width, late-time limit of the fraction of neurons whose weights are, up to a small fraction of their centered energy, matrix coefficients of $[\rho]$ alone; the exact model and purity quantifiers are in MAIS-A5, Conjecture 5.3.
Question (MAIS-A5, Question 5.4). In the setting of [MAIS-A5, Conjecture 5.3], is $q_{[\rho]}$ proportional to $\dim \Pi_{[\rho]} = \sum_{\rho' \in [\rho]} d_{\rho'}^2$, i.e. to the dimension of the isotypic component?
Three heuristics suggest different answers, and deciding among them is part of the problem. Initial spectral mass suggests $q_{[\rho]} \propto d_\rho^2$, as above. The maximum-margin construction of Morwani et al. [MEOZK24] (margin in the $\ell_{2,3}$ norm, a theory that does not yet predict this ensemble) allocates neurons in proportions closer to $d_\rho^3$. And in the small-initialization limit, the alternating-gradient-flow picture of Kunin et al. [KMCK+25] predicts sequential acquisition ranked by initial spectral mass rather than a fixed proportion. The nearest theorem, He et al.'s uniform diversification across characters [HWZCY26] for abelian groups with no self-conjugate nontrivial character, concerns a Taylor-surrogate projected flow and one-dimensional representations only. For the definitions and the full context, see MAIS-A5.
References
- [MEOZK24] D. Morwani, B. L. Edelman, C.-A. Oncescu, R. Zhao, and S. Kakade, Feature emergence via margin maximization: case studies in algebraic tasks, ICLR 2024. arXiv:2311.07568
- [KMCK+25] D. Kunin, G. L. Marchetti, F. Chen, D. Karkada, J. B. Simon, M. R. DeWeese, S. Ganguli, and N. Miolane, Alternating gradient flows: a theory of feature learning in two-layer neural networks, NeurIPS 2025. arXiv:2506.06489
- [HWZCY26] J. He, L. Wang, F. Zhang, S. Chen, and Z. Yang, Neural networks provably learn spectral representations for group composition, preprint, 2026. arXiv:2606.02993
Related: MAIS-O53 (the conjecture defining $q_{[\rho]}$) · MAIS-O55 (the two-class case $S_3$) · MAIS-O5 (the headline selection law).