Möbius-compressed circular latent states

Integrate this at a circular embedding or latent-state layer that currently stores a tensor of N trainable angles per example, token group, graph node set, or attention head. Fix reference coordinates \(\xi_i=i/N\). Store only three unconstrained parameters per population: \(u_\alpha\), \(u_\beta\), and \(u_A\), with \(\alpha=\pi\tanh(u_\alpha)\), \(\beta=(1-\epsilon)\sigma(u_\beta)\), and \(A=\pi\tanh(u_A)\). At each forward pass compute \(z_i=\pi\xi_i+A\) and generate \(\theta_i=\phi(\alpha,\beta,A,\xi_i)\). For numerical stability, do not directly evaluate a discontinuous \(\arctan(\tan z_i)\) implementation; instead use a wrapped angle or an equivalent complex-number circle map and represent the output as \([\cos\theta_i,\sin\theta_i]\). These features can be concatenated to node embeddings or used as attention biases \(b_{ij}=\tau\cos(\theta_i-\theta_j)\). Pseudocode: initialize \(u\); convert to \(\alpha,\beta,A\); generate all \(\theta_i\); form circular features or pairwise biases; run the ordinary network; backpropagate through the analytic map. The paper supplies the parameterization and constraint; estimate \(\lambda\), \(\tau\), and initialization empirically. First test a small graph neural network or synthetic circular-mixture classifier with N=64 phases. Compare against N independent learned angles and a Fourier positional embedding. Record parameter count, accuracy, loss curves, gradient norms, and discontinuity or NaN events. Success is at least 10x fewer phase parameters with no more than 1% absolute accuracy loss, or improved training stability at comparable accuracy.