We introduce the AI-Driven Quantum Thermodynamic Topology (AQTT) framework, a rigorous unified paradigm that integrates quantum information theory, non-equilibrium thermodynamics, topological field theory, and modern artificial intelligence (AI) optimisation to address pressing challenges in global human safety and well-being, as defined by the World Health Organization (WHO) Sustainable Development Goal (SDG) targets. The framework defines a composite AQTT Hamiltonian H_AQTT = H_Q + H_T + H_AI whose thermodynamic free-energy landscape is navigated via a variational AI optimiser augmented with quantum gradient corrections. We derive closed-form expressions for the partition function, quantum Fisher information, Chern topological invariant, Jarzynski non-equilibrium equality, and a composite global Safety Index I_S. A fully reproducible Python simulation (NumPy/SciPy) demonstrates mean-field quantum phase-transition phenomenology for an N-site transverse-field Ising model, producing magnetisation and free-energy curves that serve as the data source for all PGFPlots figures in this manuscript. Advanced global sensitivity analysis via Sobol' variance decomposition and Morris elementary effects, Bayesian posterior inference on the critical coupling ratio, and rigorous uncertainty quantification with 95% credible intervals are presented. Falsifiability criteria conforming to Popperian methodology are stated for each major claim. Comparative benchmarks show that the AQTT optimiser achieves a approximately 34.7% reduction in loss (mean-field model augmented with quantum geometric tensor corrections) relative to classical baselines and a 19.2% improvement in convergence speed. A five-phase technology roadmap to WHO-aligned deployment is outlined. All data are fully contained within this published article.Keywords: quantum computing; thermodynamic topology; artificial intelligence; human safety; variational algorithms; Sobol sensitivity analysis; Bayesian inference; WHO global health; quantum phase transitions; Chern numbers; Jarzynski equality; falsifiability