Structural Identifiability and Gradient Bounds for Hybrid-Activation Learning of Fractional HIV Dynamics

The recovery of biological parameters from fractional HIV trajectories is examined alongside the mathematical properties of hybrid neural activations. Using the model of Okeke et al. [1] and the five-component activation mixture of Essang et al. [2], an exact parameter symmetry is derived: proliferation rate, healthy-cell death rate and carrying capacity cannot all be recovered separately from the model trajectories without additional information. A reduced parametrisation eliminates this redundancy. Conditional identifiability of the reduced coefficients is established through full-rank trajectory regressors for a known fractional order and fully observed states. The hybrid activation derivative is corrected, and global scalar slope bounds are proved. These bounds do not guarantee preservation of gradients through arbitrary network depth. A fractional residual learning framework is specified, and completed synthetic experiments examine parameter symmetry, regressor singular values, observation-window conditioning and slope products. The study establishes mathematical and computational limitations rather than empirical superiority of a trained learning algorithm.