Non-Abelian Event-Order Memory

Integrate the module beside the hidden state of a GRU, an S4-like state-space model, or a lightweight recurrent block. Reserve a three-dimensional vector \(\mathbf{s}_t\) per sequence or feature group. Store two learnable axis vectors and normalize them to unit length after every optimizer step; store two learnable rotation angles, optionally bounded by \(|\theta_a|\leq\pi\). For each input event, select its axis and angle and update the spin using Rodrigues' formula, \(R\mathbf{s}=\mathbf{s}\cos\theta+(\mathbf{n}\times\mathbf{s})\sin\theta+\mathbf{n}(\mathbf{n}\cdot\mathbf{s})(1-\cos\theta)\). Concatenate \(\mathbf{s}_t\) or the final spin with the ordinary hidden state before the output head. For diagnostics, evaluate a fixed probe batch on a small \(K\times K\) grid of phase pairs, replace each event angle by \(\chi_a\theta_a\), average final spins to obtain \(\mathbf{m}(\chi)\), and estimate the Chern number using finite differences or spherical triangle areas. The noncommuting ordered product is taken directly from the paper; learned axes and neural readout are the adaptation. First test on length-64 synthetic sequences containing A and B events with identical counts but labels determined only by AB versus BA ordering. Compare an MLP count baseline, a GRU, and GRU-plus-spin. The prediction is that the count baseline stays near chance while the spin model improves whenever the commutator \([\mathbf{n}_1]_\times[\mathbf{n}_2]_\times-[\mathbf{n}_2]_\times[\mathbf{n}_1]_\times\) is nonzero; parallel axes should eliminate the advantage. Chern estimates should remain integer-stable while the minimum polarization gap remains nonzero.