An Axial Obstruction to the Uniform High-Order Matrix Damping Criterion in "Stable Singularity of the Euler Equations on R^3"

Abstract

This technical note examines the full-domain high-order matrix negativity condition (S248) in the 107-page manuscript "Stable Singularity of the Euler Equations on R^3" by Adarsh Ganeshram, Valentin Duruisseaux and Anima Anandkumar, retrieved on 8 September 2026. From the normalized axial fixed point and the diagonal formula (S243), the non-weight radial coefficient is (k+3/2)b-k*lambda, where b is the axial derivative of the reference streamfunction at the origin. For an exact critical profile with nonzero normalized amplitude, the coefficient equals 3/4 at every derivative order. A proof along the symmetry axis shows that a positive high-order weight with finite positive axial limit cannot yield the prescribed uniform negative matrix bound. For residual-corrected approximate profiles with lambda=1/2, the obstruction applies when c_u,res < 3/(2k+3), in particular c_u,res < 3/11 at k=4. The authors' spline residual correction has not been independently evaluated here. This result concerns the specified sufficient matrix certificate; it does not prove instability of the full Euler evolution or nonexistence of Euler singularities. The note contains explicit hypotheses, proofs, source equation references, a source-version SHA-256 fingerprint and an AI-assistance disclosure.