Finite-Window Practical Identifiability Under Hidden Forcing in Mixed Integer- and Fractional-Order Systems

Inference from a finite observation window in fractional dynamics depends on what is known about the forcing. We study whether fractional orders can be distinguished under uncertain excitation and whether calibrated interventions can exclude a competing model family. For passive observations, the known-forcing case provides a conditional reference. Covariance-weighted projection quantifies the order information that remains when the forcing and physical coefficients are uncertain. With 33 forcing coefficients and memory-only observations, the retained fraction of the squared order-sensitivity norm decreases from 2.01×10−3 after forcing elimination to 9.40×10−7 after two physical coefficients are also profiled. Different optimizers attain matching objective values even though the fully profiled order remains poorly constrained. In the active cascade, causal convolution contrasts eliminate a common unmeasured input. A positive annihilator removes unknown decay and ordinary dynamics, and a quartic deterministic error certificate tests the single-order family. Assuming zero common history, bounded gains, repeatable inputs, and the stated waveform regularity, the fine-resolution acquisition excludes that family for both blocks throughout the prescribed continuous order interval. The coarse-resolution acquisition remains inconclusive. These diagnostics inform forcing restrictions, the choice of complementary measurements and acquisition precision; they do not establish global recovery or field validation.