Diffusion Transformer Text-to-Video models have achieved remarkable synthesis quality, yet fine-grained spatial controllability remains a significant challenge. While existing training-free methods produce solid overall results in spatially grounded generation, \ie, placing a specific object in a designated location, they rely on gradient-based optimization techniques that incur prohibitive computational overhead, a bottleneck amplified in modern large-scale architectures.

To address this limitation, we present Gradient-free Analytical Trajectory Optimization Video Generation (GATO-Vid), a novel training-free and gradient-free approach for precise spatial guidance. Rather than relying on costly backward passes, we introduce an alternative cross-attention score and solve it analytically to obtain an exact, closed-form solution.

To use our analytical solution, we propose an on-the-fly injection mechanism tailored to the topological manifold of the transformer's latent space. Our experiments demonstrate that GATO-Vid significantly outperforms existing baselines in localization accuracy while introducing minimal computational overhead.