Asymptotic Inconsistency of Interpolation in Terminal-Projected Flows for PDE Inversion

Zero-shot constrained generative models have been widely adopted for solving partial differential equation (PDE) inverse problems. A prominent class of methods, such as ECI and PCFM, enforces hard physical constraints through terminal projection coupled with reverse optimal-transport (OT) displacement interpolation. The validity of these methods relies on a generally accepted asymptotic consistency assumption of the flow interpolation, that the instantaneous predicted velocity asymptotically matches the interpolation secant velocity as the integration step size vanishes. However, whether the consistency remains valid after projection has not been verified. To address this, we establish a reference framework based on Minimal Step-wise Perturbation (MSP). We prove that the asymptotic consistency fails after terminal projection: the state discrepancy does not vanish under step refinement, inducing an extraneous surplus velocity field of order $\Theta (\Delta t^{-1})$. This unrecognized surplus velocity causes trajectory distortions and distribution deformation, which cannot be interpreted as principled posterior guidance. Across parameter sweeps on PDE benchmarks, we find that model performance exhibits strong task dependence, with substantial variations and even performance reversals across different task configurations. These results demonstrate that observed performance gains cannot be directly attributed to genuine posterior-aware optimization, but instead be influenced by the surplus dynamics.