Small Denominators and Subresonant Accumulation in Weakly Nonlinear Dispersive Dynamics

What the math gives to ML

The paper isolates a constructive small-denominator mechanism by which infinitely many individually nonresonant interactions accumulate into a polynomial-in-time response. Its transferable asset is the explicit relation between detuning decay, coefficient decay, and memory growth: Delta_n ~ c n^{-p} and B_n ~ b n^{-kappa} produce t^{1-alpha} growth with alpha=(kappa-1)/p. This can be turned into a structured state-space memory bank whose low-frequency modes deliberately implement power-law memory, rather than learning long memory from unconstrained recurrent weights. The same formula also supplies a falsifiable initialization and scaling rule for long-context sequence models.