Simplex-Stable Companion Memory

1. Exact integration point: use this module as the linear memory core of an RNN, SSM, or latent world model. An encoder maps observation x_t to a latent vector u_t. Maintain a history of latent states h_t and predict the next state with the simplex-constrained companion recurrence. Inputs can be incorporated through a bounded additive term B u_t, or through a residual nonlinear branch whose output is clipped or norm-normalized: h_{t+1} equals the weighted history plus g_theta(x_t). For the cleanest stability test, first use the autonomous recurrence without the residual branch. 2. Pseudocode: initialize logits a in R^m and set alpha to softmax(a). Maintain a buffer containing h_t through h_{t-m+1}. Compute h_next as the sum over j of alpha_j times buffer[j], shift the buffer, decode x_hat_next from h_next, and optimize prediction or reconstruction loss. If coefficients adapt to context, compute logits a_t from a small network and apply softmax at every step; each instantaneous transition remains row-stochastic, although a common fixed spectral decomposition no longer applies. 3. Computed versus estimated: the simplex constraint and infinity norm bound are exact. Estimate spectral radius, Euclidean transient amplification, and modal decay numerically from learned coefficients. 4. First cheap experiment: train one-step and 50-step predictors on FitzHugh-Nagumo or SIR trajectories, comparing a GRU, an unconstrained companion recurrence, and this simplex companion with memory lengths 8, 16, and 32. The quantitative prediction is that the constrained model has spectral radius no greater than 1 plus 10^-6 and contains an eigenvalue within 10^-6 of 1. Its infinity norm power bound should satisfy ||C^k||_infinity <= 1 for every k. For each eigenmode, first-difference amplitude should scale as the absolute value of lambda minus 1, and second-difference amplitude as its square. Regress log difference amplitude against log absolute lambda minus 1 and expect slopes near 1 and 2.