Learnable-order fractional recurrent networks: Structured identification of memory in non-stationary sequences

Non-stationary sequences may exhibit changing effective memory. We study the Learnable-Order Fractional Recurrent Network (LOFT-RNN), a recurrent block in which the order α_t ∈ (0, 1) of a type-I Caputo derivative is generated from the evolving state and input. The hidden trajectory follows ₀^CD_t^α(t)h(t)=f_θ(h(t),x(t)), with α(t)=α_min+(α_max-α_min)σ(g_ϕ(h(t),x(t))). The theoretical scope is deliberately limited. We give (i) a represented-class approximation proposition, conditional on a finite-dimensional fractional-state realization and uniform continuous dependence of its solution map; (ii) a uniform consistency estimate for the exact-weight L₁ formula, together with a separate interpolation-error term for the order grid; (iii) a variable-order bounded-input bounded-state estimate under dissipativity, with a Mittag-Leffler refinement only for constant order; and (iv) a regret result for an auxiliary one-dimensional reset-OGD tracker under convex surrogate losses and explicit detection assumptions. The auxiliary tracker is not identified with end-to-end neural-gate training. We also state the boundary assumptions at t=0, explain the choice of the Caputo derivative, and show that the untruncated history convolution costs O(N²d) over a length-N trajectory. The archived aggregate experiments cover a delimited-copy task, regime-switching ARFIMA sequences, Bonn EEG segment classification, and S&P 500 volatility forecasting. Because seed-level outputs, split manifests, executable code, and some requested baselines are absent from the supplied archive, the empirical comparisons are reported descriptively and no statistical-significance or state-of-the-art claim is made.