Nielsen Quaternionic Hyperbolic Latent Layer

1. Integration point: use this map as the transition operator of a recurrent state-space model. Store each latent slot as two normalized 4-vectors representing unit quaternions \(q_1,q_2\); use an encoder to initialize them from an input window and a decoder to predict observations. Start with fixed \(A=\begin{pmatrix}2&1\\1&1\end{pmatrix}\), and learn only the encoder, decoder, and optionally a small residual observation-dependent perturbation. 2. Pseudocode: factor \(A\) offline into Nielsen generators. At every recurrent step, apply each factor \((q_1,q_2)\leftarrow N_{i_j}^{\varepsilon_j}(q_1,q_2)\), using quaternion multiplication and normalization after each product. For inverse generators use quaternion inverses, \(q^{-1}=\bar q\). Decode the resulting pair. 3. Computed exactly: group multiplication, normalization, generator composition, the commuting residual \(r_c=\|q_1q_2-q_2q_1\|\), and \(\lambda_\pm\). Estimated empirically: whether the encoder places data near a common maximal torus, prediction error, and tangent growth rates. If residual forcing is added, constrain its norm so the analytical signature remains measurable. 4. First cheap experiment: train this model and same-size GRU, linear SSM, and unconstrained 8-dimensional MLP recurrence on synthetic sequences generated by the quaternionic map, using one-step training and 100- to 1000-step rollout evaluation. Initialize perturbations inside a common torus and project them onto the eigenvectors of \(A\). The falsifiable prediction is that log-distance versus time has slopes \(\log\lambda_+\) and \(\log\lambda_-\) within 10 percent before saturation, while the unconstrained baseline has no fixed slope and larger long-horizon drift. Also verify that commuting pairs remain commuting to numerical precision and quaternion norms remain within \(10^{-6}\) of one.