Differentially Positive Recurrent Core

1. Exact integration point: replace the recurrent matrix in a small GRU, vanilla RNN, or diagonal-plus-low-rank SSM by a nonnegative transition matrix. For a first implementation use $h_{t+1}=\phi(W_hh_t+W_xx_t+b)$ with $W_h=\operatorname{softplus}(V_h)$ and a monotone activation such as softplus; this guarantees positive-orthant cone preservation when no sign-changing skip terms are added. A less restrictive version keeps unconstrained weights and adds the Jacobian penalty. 2. Pseudocode: initialize $V_h$; at each sequence step compute $h_{t+1}$, retain $J_t=\partial h_{t+1}/\partial h_t$ using forward-mode Jacobian-vector products or exact Jacobians for a small model; set $L=L_{task}+\lambda L_{cone}$; backpropagate and update $\theta$. For larger models estimate violations with $m$ random vectors $v_k\geq0$: penalize $\operatorname{ReLU}(-J_tv_k)$ instead of materializing $J_t$. 3. Computed from the paper: cone preservation and the order-based recurrent interpretation. Estimated empirically: Jacobian entries, cone-violation rate, Lyapunov exponents, and whether recurrent hidden states form ordered or unordered clusters. Define violation rate as the fraction of sampled $(t,k)$ for which any component of $J_tv_k$ is negative. 4. First cheap experiment: train a one-layer RNN on adding, copy-memory, and sequential-MNIST tasks, comparing an unconstrained RNN, a positive-transition RNN, and the soft-penalty model at matched parameter counts. Sweep transition gain $\alpha$ in $W_h=\alpha\operatorname{softplus}(V_h)$ and sequence length. The quantitative prediction is that the hard-positive model has zero cone violations, while the soft model's violation rate decreases monotonically with $\lambda$; ordered perturbation pairs should preserve componentwise order over time. A falsifiable long-horizon signature is that positive models avoid the abrupt error increase seen when the unconstrained model's largest estimated Jacobian-product Lyapunov exponent becomes positive.