This paper proposes a distributed continuous-time optimization framework for multi-variable static maps that eliminates dependency on explicit gradient information. Traditional distributed methods often rely on derivative computations, limiting their applicability when only real-time objective function measurements are available. Leveraging unbiased extremum seeking and Lie bracket approximation, we develop continuous-time algorithms that utilize local measurements and neighbor-shared data to collaboratively locate static optima. The constant-frequency scheme achieves asymptotic convergence with LMI-based stability guarantees, while chirpy probing extends this to exponential and prescribed-time convergence via time-scale transformations. Key advancements include unbiased estimation of the optimal solution and customizable convergence rates (asymptotic, exponential, or prescribed-time). Numerical simulations validate the algorithms’ effectiveness.