arXiv:2611.00414v1 Announce Type: new
Abstract: We study the local differential geometry of a Fourier-based modular-addition circuit in a one-layer transformer whose algorithm has previously been reverse-engineered. For each of the five key Fourier frequencies, we restrict the model's multilayer perceptron (MLP) contribution to a corresponding Fourier input plane and readout, and measure its directional Beltrami coefficient under continuous input perturbations. Whereas nondegenerate restrictions of ideal complex multiplication are conformal, the trained maps exhibit nonconformal distortion (median $|\mu|=0.304$–$0.446$), spatial organization in the Beltrami argument, and regions of local orientation reversal. The median across frequencies decreases from $0.939$ at initialization to $0.390$ after training. The key-frequency planes also show lower median distortion than most random-plane controls, although their distributions overlap and the coordinate frames differ in conditioning. Across 500 checkpoints in one training run, most of the aggregate reduction in Beltrami magnitude occurs before the principal increase in the measured Fourier-circuit formation. At the individual-input level, Beltrami magnitude shows negligible association with the correct-logit margin ($|\rho|\leq0.026$). Late-checkpoint measurements across six runs show comparable distortion for most selected frequencies. These results provide a quantitative characterization of local deformation in a frequency-projected component of an identified neural circuit, complementary to its algorithmic reconstruction.
Abstract: We study the local differential geometry of a Fourier-based modular-addition circuit in a one-layer transformer whose algorithm has previously been reverse-engineered. For each of the five key Fourier frequencies, we restrict the model's multilayer perceptron (MLP) contribution to a corresponding Fourier input plane and readout, and measure its directional Beltrami coefficient under continuous input perturbations. Whereas nondegenerate restrictions of ideal complex multiplication are conformal, the trained maps exhibit nonconformal distortion (median $|\mu|=0.304$–$0.446$), spatial organization in the Beltrami argument, and regions of local orientation reversal. The median across frequencies decreases from $0.939$ at initialization to $0.390$ after training. The key-frequency planes also show lower median distortion than most random-plane controls, although their distributions overlap and the coordinate frames differ in conditioning. Across 500 checkpoints in one training run, most of the aggregate reduction in Beltrami magnitude occurs before the principal increase in the measured Fourier-circuit formation. At the individual-input level, Beltrami magnitude shows negligible association with the correct-logit margin ($|\rho|\leq0.026$). Late-checkpoint measurements across six runs show comparable distortion for most selected frequencies. These results provide a quantitative characterization of local deformation in a frequency-projected component of an identified neural circuit, complementary to its algorithmic reconstruction.
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