Score Addition Is Not Composition: Closed-Form Path Corrections for Training-Free Composition of Pretrained Flows

Pretrained flow-matching models are becoming a dominant class of generative models, and there is a growing need to compose two of them without retraining, as a product, a geometric mean, or a negation. Current practice relies on heuristics, such as adding velocities or annealing along summed scores, or on particle-based correctors. The heuristics do not target the composition: adding velocities overshoots the product mean of two unit-variance Gaussians by a factor of $1+\pi /2$, and summed-score annealing follows the wrong intermediate curve, potentially misallocating mass across modes even with longer equilibration. We start from an exact identity. For flows trained with independent couplings from a common source, the intermediate marginals of any tilted composition factorize into powered component marginals and a kernel-corrected overlap integral of the endpoint posteriors, and the gradient of the log-overlap term is the exact correction to summed-score annealing. Gaussian moment matching gives the term a closed form that can be evaluated from black-box flows using only velocities and velocity Jacobians, and a well-posedness certificate identifies where the correction applies. Under frozen covariances the correction is a precision-weighted force that pulls samples toward agreement between the two models' endpoint predictions. The resulting sampler, CompFM, corrects the annealing path per sample and needs no particle population. At matched score-call budgets on class-intersection tasks with independently trained audit classifiers, CompFM attains 100.0% and 96.6% audited target-class purity on MNIST and FashionMNIST, respectively, versus 79.8% and 79.3% for summed-score annealing. On CIFAR-10, it concentrates 75.3% of audited mass on vehicle groups, compared with 1.1% for summed scores at the matched score-call budget. Composing two conditional flows on colored MNIST gives 96–99% joint accuracy, against 78–87% for summed scores. Classifier-free guidance (CFG) is the equal-covariance special case of the fusion rule, which also yields a covariance-aware correction to CFG.