Signed de Bruijn parity memory

Integration point: add a small exact automaton branch to an RNN or SSM before its nonlinear recurrent update. For binary or discretized inputs, maintain a one-hot context vector p_k in R^(2^M). For real inputs, begin with a straight-through threshold a_k=1{v^T x_k>0}; later replace it with a soft bit probability. At each step, compute the legal successor s'=(s_2,...,s_M,a), update p_{k+1}=P(a_k)p_k, compute the sign r_k(s)=(-1)^{q_T(s,a_k)}, and update the signed feature b_{k+1}=P(a_k)diag(r_k)b_k. Concatenate p_k or a learned low-dimensional projection of it with the ordinary hidden state h_k, then apply the standard neural update and train end-to-end. The exact transition and sign construction come from the paper; input binarization, projection, and neural readout are adaptations. Use sparse successor operations rather than dense matrices. First experiment: compare a vanilla GRU with a GRU-plus-parity-memory on synthetic length-256 sequences labeled by XORs of input pairs at offsets T, then test Copy Memory and permuted sequential-MNIST. Sweep M in {3,5,7}. The predicted signature is that parity-task accuracy remains nearly constant with sequence length up to the exact memory order, while the GRU baseline decays. The signed feature must equal the exact product of transition signs, with numerical error below 1e-6. Its empirical periodic response should also peak near phases associated with eigenvalues of B.