Why Projected Hamiltonian Flow Matching Can Learn Zero

Generative models have recently incorporated Hamiltonian structure to align the generation process with physical constraints. However, the resulting flows may preserve energy yet fail to learn the required probability transport. We study projected scalar-Hamiltonian velocities $P_{H}J\nabla h$ and construct smooth positive density paths on compact phase-symmetric shells whose unique population Flow Matching optimum is zero. Orbit averaging identifies the obstruction and characterizes the exact divergence range on round shells. Under a density-ratio assumption, a finite-excess-risk bound gives a quantitative endpoint error floor for approximate fits. We also characterize the closed regression spaces that preserve every teacher's density response: they contain the closure of tangent gradients. A fixed-budget class–path comparison measures how this population capacity translates into neural learning, and a source-filtering diagnostic identifies an exact conditional sampler in the physical-data task. For an explicit two-oscillator construction, the population-optimal flow retains its source law and incurs endpoint total-variation error $1/8$.