Parity-Conserving Reaction-Diffusion Memory

Integrate this as a parallel recurrent memory branch in a small sequence model, preferably a GRU or linear state-space baseline. For a sequence of length $L$, maintain hidden features $h_t\in\mathbb{R}^{L\times d}$ and one scalar charge probability $p_t\in[0,1]^L$. At each step, first compute local input-driven event logits $u_t=W_xx_t+W_hh_t$ and map them to bounded rates $D_t=\operatorname{softplus}(a_D)$, $\lambda_t=\operatorname{softplus}(a_\lambda+u_t)$, and $\tau_t=\operatorname{softplus}(a_\tau+u_t)$, with a small fixed step size $\Delta t$ chosen so all event probabilities remain below $0.1$. Use periodic or reflective boundary conditions. Compute diffusion with the nearest-neighbor Laplacian, then compute pair annihilation and pair nucleation on edges $(i,i+1)$ using $r^{\rm ann}_{t,i}=\Delta t\lambda_t\,p_{t,i}p_{t,i+1}$ and $r^{\rm nuc}_{t,i}=\Delta t\tau_t(1-p_{t,i})(1-p_{t,i+1})$. In the simplest differentiable MVP, add the net edge reaction contribution to both endpoints and clip $p$ to $[0,1]$; in the exact stochastic version, sample mutually exclusive edge events and flip both binary occupancies, which preserves total parity exactly. Then update the neural state with a charge gate: $h_{t+1}=\operatorname{GRU}(x_t,h_t;\,g_t)$ where $g_{t,i}=\sigma(w_g p_{t+1,i}+b_g)$, or equivalently use $h_{t+1,i}=g_{t,i}\tilde h_{t+1,i}+(1-g_{t,i})h_{t,i}$. The mathematical quantities computed directly are the Laplacian transport and pair rates; learn $a_D,a_\lambda,a_\tau,w_g$ by backpropagation, and optionally estimate the empirical charge lifetime and parity violation rate from trajectories. First test on adding-machine, delayed-copy, and synthetic parity/XOR sequence tasks with length 256 or 1024, comparing equal-parameter GRU, S4-style diagonal SSM, and a standard gated memory. Report accuracy versus sequence length, gradient norm stability, and wall-clock cost. A positive signal is substantially better accuracy on delayed parity and cancellation tasks at long horizons, with no increase in exploding/vanishing gradients; for stochastic binary mode, parity violations should be exactly zero and the learned defect lifetime should increase when the task requires longer memory.