Drift-Assisted Hyperbolic Recurrent Layer

Use this as a recurrent or state-space block rather than as a generic optimizer. Split the hidden state into a scalar clock c, a stable bank h_s, and optionally an unstable or bidirectional bank h_u. Set c_(k+1) = c_k + delta, with delta parameterized as a signed nonzero value, and define h_(k+1) = G_theta(h_k, c_k, x_k). Represent the metric implicitly using diagonal weights P_ii(c) = exp(2 beta_i c), so no dense matrix is needed. At each sampled timestep, compute the next state, draw a few random stable probe vectors u, and use automatic differentiation to obtain Jacobian-vector products J_k u. Compute the metric ratio r(u) = norm_P(z_(k+1))(J_k u) divided by norm_P(z_k)(u), then add the penalty lambda times max(0, log(r(u)) + alpha) squared. The paper supplies the affine fixed-point obstruction and drift-enabled metric mechanism; the implementation estimates local Jacobian gains and finite-time metric exponents empirically. First experiment: compare a vanilla linear RNN, an orthogonal RNN, and this drift-assisted layer on Copy Memory and sequential MNIST at lengths 100, 500, and 1000. Sweep beta times delta and the residual gain. The quantitative prediction is that the measured long-horizon gradient slope changes approximately linearly with beta times delta, crosses zero near lambda_Euclidean plus beta delta equal to zero, and becomes unstable when the estimated stable-channel metric gain exceeds exp(-alpha). Agreement within 20 percent is the required initial test.