Product Manifold Flow Matching: Low-NFE Generative Efficiency and Domain-Dependent Finite-Step Quality Optima

Generative flow quality need not improve monotonically as numerical integration is refined when the learned velocity field is itself approximate. We investigate this behavior in Product Manifold Flow Matching (PMFM), which represents image patches on Products of hyperbolic factors and samples with exponential-map Euler updates. Across 32 M and 129 M CIFAR-100 models, Product at NFE = 4 achieves lower FID than the original Euclidean baseline at NFE = 32 in all six architecture–training-seed configurations, with an approximately eightfold reduction in measured sampling time. A targeted 32 M ablation shows, however, that this crossover is not specific to Product geometry: a factor-wise Euclidean analog of the Product target-speed reweighting improves the Euclidean baseline across the evaluated NFE range and reproduces the cross-budget advantage in all three training seeds. Removing the same reweighting from the Product target degrades FID from NFE ≥ 4. Product FID reaches an early optimum at NFE = 8 on CIFAR-100 and at NFE = 4 on a balanced synthetic pile-driver corpus, while same-initial-condition step-doubling discrepancies continue to decrease. These results separate numerical refinement from generative fidelity and identify target-speed reweighting as a major contributor to low-NFE quality, without supporting a curvature-only causal interpretation.