This study addresses a current research gap in Mathematics concerning Partial Differential Equations for traffic-flow optimization in Tanzania: finite-element discretization and error bounds in Tanzania. The objective is to formulate a rigorous model, state verifiable assumptions, and derive results with direct analytical or practical implications. A theorem-driven mathematical framework was developed under explicit regularity assumptions, with stability and convergence analysis of the proposed estimator. The main results show stability of the proposed functional under bounded perturbations and convergence of the estimator to a well-defined limit, characterised by $R(x)=argmin_theta L(theta;x)$. The findings provide a reproducible analytical basis for subsequent theoretical and applied extensions. Recommendations are not applicable for this abstract type. Partial Differential Equations for traffic-flow optimization in Tanzania: finite-element discretization and error bounds, Tanzania, Africa, Mathematics, commentary on article This work contributes a formal specification, transparent assumptions, and mathematically interpretable claims.