Accountability-Aware Fractional Control for Embodied Intelligent Systems: Mittag-Leffler Stability and Conditional Proxemic Safety

This paper develops an accountability-aware fractional control framework for embodied intelligent systems in shared human environments. The approach combines a Caputo fractional-order stabilizing law, an intent-evidence realization with softmax belief reconstruction, and a conditional proxemic safety layer. Sufficient conditions are established for local Mittag-Leffler stability of the augmented error dynamics and forward invariance of the safe set. Numerical results are presented as a theorem-validation benchmark. For the base case with $\alpha =0.9$, the augmented error norm decays from $1.2359$ to $9.90\times {10}^{-3}$ while the safety margin remains strictly positive, and the robustness condition is satisfied with a margin of $1.8641$. An $\alpha$-sweep and a step-size convergence study further show that the fractional order induces a systematic safety–performance trade-off and that the reported behaviors are numerically stable. Additional simulations with four intent classes, bounded observation noise, and Monte Carlo uncertainty stress tests are included to strengthen the numerical evidence beyond the two-intent theorem-validation case. The manuscript also clarifies the quantitative interpretation of the accountability index, the conditional nature of the safety theorem, and an implementable sampled safety-filter realization for concrete robotic platforms. The results support the proposed framework as a mathematically consistent tool for shaping the balance between regulation and proxemic safety.