State-dependent splitting errors for anisotropic transport and diffusion: Source-verified counterexamples and an auxiliary estimate
We audit the source statements in arXiv:2603.08799v1, identified by a versioned PDF and its hash. An explicit smooth periodic shear contradicts the derivative-norm conservation assertion, which is also verified in the publisher's public Note 10. A positive-coefficient diffusion example exposes omitted mixed third derivatives in the preprint's displayed splitting prefactor. We quote the targeted preprint passages, distinguish a changed publisher note from the inaccessible publisher body, and make no claim that the unverified published diffusion formula is identical. For fixed trigonometric data we give the exact centered-stencil commutator and its order of convergence. Cosine-only numerical checks report initial and evolved-supremum comparators, finite-time limiting coefficients, and step refinement from pinned code. A standard auxiliary autonomous matrix estimate records the intermediate-state hypothesis needed for a corrected analysis; no novelty is claimed for that estimate. The results are specific counterexamples and proof obligations, not an exhaustive audit, an unconditional time-dependent PDE theorem, or a demonstrated quantum speedup.