Diffusion models primarily rely on general-purpose ODE-based numerical methods for training-free accelerated sampling, but performance degrades with coarse time discretization. In this paper, we propose optimal linear leaps (LEAP), a specialized method grounded in theoretical and empirical investigations of sampling trajectory geometry. Built upon low-dimensionality and cross-trajectory consistency, LEAP formulates a supervised least-squares optimization problem within the local trajectory subspace, achieving a precise approximation of future states without model evaluations. Extensive experiments demonstrate that LEAP outperforms previous methods across diverse modalities, including text-to-image (Stable Diffusion 3.5, FLUX.1-dev) and image-to-3D (TRELLIS.2) generation. Furthermore, LEAP can be seamlessly integrated into existing solver-based methods, offering a plug-and-play enhancement for modern diffusion models. Code is available at https://github.com/zju-pi/LEAP.

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