Positive-Allocation Companion Predictors for Nonlinear Dynamics and Their Finite-Difference Diagnostics

What the math gives to ML

The paper provides a constructive stability mechanism: recurrence coefficients obtained from a nonnegative simplex allocation generate a nonnegative row-stochastic companion matrix. Consequently, the companion spectrum lies in the closed unit disk and contains the neutral eigenvalue 1, giving a directly enforceable nonexpansiveness condition for recurrent latent dynamics. The paper also shows that first and second finite differences spectrally filter modes by factors of (lambda minus 1) and (lambda minus 1) squared. The most promising neural transfer is a latent-history recurrent module whose coefficients are produced by a softmax or simplex projection, giving a stable long-memory architecture and measurable spectral and finite-difference diagnostics.