Bayesian Multi-Layer Adaptive Network Dynamics (B-MAND) Framework for Modeling, Mitigation, and Optimal Control of Misinformation Propagation in Multi-Layer Socio-Technical Systems: Mathematical Foundations, Stochastic Simulations, Global Sensitivity Analyses, and Quantitative Policy Implications

Misinformation and disinformation constitute a profound, systemic challenge in contemporary digital socio-technical ecosystems. This paper introduces the Bayesian Multi-Layer Adaptive Network Dynamics (B-MAND) framework---a mathematically rigorous synthesis of network science, Bayesian inference, stochastic optimal control, and Popperian falsifiability. B-MAND models the information ecosystem as a time-evolving multi-layer graph with per-node continuous Bayesian belief states and layer-specific adaptive controls derived via Pontryagin's minimum principle.We derive the complete Itô stochastic differential equations for node-level and mean-field dynamics, prove local and global stability (Jacobian, Lyapunov, Routh-Hurwitz), identify transcritical and Hopf bifurcations, and obtain explicit optimal control Hamiltonians. All derivations are supported by reproducible Python simulations on Barabási--Albert scale-free networks (N=500, m=5) calibrated to peer-reviewed empirical parameters. Comprehensive analyses include global Sobol indices (first-order β_f=0.682), PRCC, eFAST, 10,000-run Monte-Carlo uncertainty quantification (95% CI for misinformation peak [0.71,0.89]), conjugate Bayesian posteriors, and explicit falsifiability criteria.B-MAND demonstrates superior fidelity (R²>0.972) over baselines and provides concrete examples of control reducing misinformation peaks by more than 40%. The work extends to critical real-world applications, including public-health infodemics and digital election integrity, culminating in a quantitative policy roadmap. All code, data, and figures are fully self-contained, ensuring exact reproducibility.