Averaged Jacobian Regularity Can Misrank Few-Step Flow-Matching Schedules: A Certified Gaussian Counterexample

Few-step sampling in flow matching and stochastic interpolants requires choosing an interpolation schedule before endpoint error is observed. Averaged squared Lipschitzness of the drift, measured by the time-integrated squared $2$-norm of its spatial Jacobian, is a plausible schedule-design criterion, but it is not a solver-specific discretization-error functional. Chen, Vanden-Eijnden, and Xu propose minimizing this quantity, denoted $A_2$; they do not claim or prove that $A_2$ universally ranks equal-NFE solver error. We study that narrower surrogate question. For independent $N(0,1)$ and $N(0,4)$, the exact regularity integrals are $5\pi/8-1$ and $\pi^2/16$, so regularity prefers trigonometric VP, whereas explicit Heun, a two-stage Runge--Kutta method, with a budget of eight function evaluations (NFE $8$), prefers the linear path in Gaussian Wasserstein-2 distance. The inversion is certified by an exact rational Heun product and a nonnegative element of $\mathbb{Q}[\pi,\sqrt{2}]$. A complementary construction shows that, for every step count $N$ and every admissible endpoint log-scale, the unique integrated-regularity minimizer can have strictly larger $N$-step Euler endpoint error than a higher-regularity competitor aligned with that solver grid. A specified finite Gaussian enumeration of four candidate paths on $36$ tested blocks contains both agreement and pairwise disagreement. These results limit a universal surrogate interpretation; they do not evaluate learned velocity fields or estimate a population failure frequency.