Rigidity of Fourier Multipliers for Universal Euler Convective Cancellations

We prove an operator-level rigidity theorem for universal Euler convective cancellations on the three-torus. Let $R(k):k_{\mathbb{C}}^{\perp}\to\mathbb{C}^{3}$, $k\in\mathbb{Z}^{3}\setminus\{0\}$, be an arbitrary reality-compatible Fourier symbol, with no boundedness, locality, isotropy, self-adjointness, or range assumption. If $\langle Rw,(w\cdot\nabla)w\rangle_{L^{2}}=0$ for every real, mean-zero, divergence-free trigonometric polynomial $w$, then, and only then, $$R=A+B\operatorname{curl}+\operatorname{curl}B,\qquad A,B\in\operatorname{Sym}(3,\mathbb{R}).$$ Thus universal cancellation forces an arbitrary symbol to be the restriction of a constant-coefficient self-adjoint operator of order at most one. If the output is also divergence-free, the family reduces to $R=cI+d\operatorname{curl}$. The zero-mode block is invisible and remains arbitrary when constant fields are admitted. The proof combines phase-polarized triads, analytic propagation through $\mathbb{Z}^{3}$, and exact finite rank certificates. We also obtain sharp finite test counts under explicit ordering hypotheses, identify the isosceles degeneracy caused by Leray projection, and pull the classification back to universal strain–vorticity cancellations. The certificates are reproduced by exact-arithmetic programs; the Lean 4 formalization and its compiler-trusted finite-rank dependency are documented separately. No application to PDE regularity or existence is claimed.