Layered Structural Reachability for Neural States

1. Integration point: apply this before training to an RNN or linear state-space layer with hidden dimension nd, grouped into n vector-valued nodes of width d. For a graph neural network, use graph nodes as the n blocks; for an ordinary RNN, use groups of channels or recurrent modules. Let A be the recurrent/state-transition Jacobian at initialization, or the explicit SSM transition matrix, and let B inject the external input, token embedding, residual input, or teacher-forcing signal. 2. Pseudocode: compute every block A_ij; stack their vectorizations and obtain an SVD basis M_1,...,M_r for the nonzero singular subspace; extract coefficients theta_ij^(k); build each scalar layer W^(k); run directed BFS from input blocks to obtain tau_k; form C(A,B) by repeated matrix-vector products; estimate numerical rank with threshold epsilon times the largest singular value; if rank is deficient, add input or skip edges or penalize small singular values. Use the differentiable surrogate above with s_i denoting singular values and gamma a target margin. 3. Computed from the paper's mechanism: block basis decomposition, scalar interaction layers, structural rank, and layer-specific distances. Estimated empirically: numerical rank, singular-value threshold, and the Jacobian A for nonlinear networks; evaluate A on multiple minibatches because reachability may vary with activation state. 4. First cheap experiment: compare a 2-layer GRU or 4-block linear SSM on Copy, Adding, and sequential MNIST tasks. Keep parameter count fixed; compare random input placement against placement selected by minimizing T_mix and maximizing sampled generic rank. Measure rank deficiency, smallest controllability singular value, gradient norm at horizon H, and task loss. The prediction is that models with sampled generic rank below nd will contain permanently insensitive hidden directions, whereas full-rank models will have nonzero input sensitivity. In linear systems, the measured controllability rank should match the generic-rank prediction in at least 95 percent of random coefficient trials, and the best architecture should approximately minimize the worst-layer distance T_mix.