Star-Symmetric Delayed State Layer

Integrate this at the sequence-model state update or as a graph polynomial-filter block. Let h_t in R^d be the hidden state, let A in R^(d by d) be a learned linear operator, and choose delay r in {1, 2, 3}. Maintain a queue containing h_t, h_(t-1), ..., h_(t-r). Use positive coefficients by parameterizing gamma_j = softplus(u_j) + epsilon, with either one coefficient per phase j modulo r or a slowly varying schedule. The update is: (1) compute v = A h_t + B x_(t+1); (2) retrieve q = h_(t-r); (3) set h_(t+1) = v - gamma_(t-r) q; (4) apply normalization and the task nonlinearity only after this structured recurrence. In pseudocode: v = A @ h[t] + B @ x[t+1]; g = softplus(u[t % r]) + eps; h[t+1] = norm(v - g * h[t-r]). The exact mathematical structure supplied by the paper is the delayed polynomial and rotational symmetry; stability and optimization benefits must be estimated empirically because the extracted paper material does not provide a neural convergence theorem. Initialize A with spectral radius near 1 and initialize gamma values small enough that the augmented delay-state matrix has spectral radius near or below 1. First test on copy, adding, and sequential-MNIST tasks using a two-layer structured SSM with width 128. Compare against an untied r-lag residual recurrence with the same parameter count and FLOPs. Track validation loss versus sequence length, gradient norm variance, hidden-state norm, and failure rate across random seeds. Success means slower accuracy degradation on long sequences, fewer exploding or vanishing runs, or faster loss descent without increasing parameter count.