Offline–Online Decoupled Architecture for Projected Microsecond-Latency and near Machine-Precision Caputo Fractional Derivative Evaluation, with an Edge-Computing Feasibility Study

Accurate evaluation of the Caputo fractional derivative (CFD) of arbitrary non-integer order is essential for modeling memory-dependent phenomena. However, the classical Shifted Gegenbauer Pseudospectral (SGPS) method attains exponential convergence only on a zero-measure set of the fractional order parameter; for all other orders, the transformed integrand exhibits reduced endpoint regularity, limiting quadrature convergence to algebraic rates. We introduce a $\varphi$-generalized algebraic transformation that enforces $C^r$ endpoint regularity for any prescribed integer $r>0$, and establish a convergence hierarchy for the interpolation-induced truncation error (ITE) across analytic, Gevrey, $C^{\infty }$, and finitely smooth data. For data analytic in a disc around the evaluation point, the transformed integrand has a convergent fractional power series with geometrically decaying coefficients, which explains the fast pre-asymptotic quadrature convergence observed numerically. Precomputable weighted Fractional Shifted Gegenbauer Integration Matrices (FSGIMs), assembled by a batched formulation, reduce each online evaluation to one $\mathcal{O}\left(Mn\right)$ matrix–vector product whose cost does not depend on the history length, apart from the retained sample buffer. Double-precision MATLAB benchmarks give excellent accuracies for problems with known solutions and improve on several finite-difference, diffusive-representation, and adaptive-quadrature baselines by orders of magnitude. Microsecond-scale latency on a Cortex-M4-class microcontroller is projected from operation counts and desktop timings; it has not been measured on hardware. A two-cycle NASA battery case study illustrates the feasibility of CFD-based State-of-Health features.