Turing complete stationary Euler flows on standard S^3 or T^3

Determine whether there exists a Turing complete stationary solution to the incompressible Euler equations on the round three-sphere (S^3, g_round) or on the flat three-torus (T^3, g_flat).

Background

The constructions reviewed include Turing complete stationary Euler flows obtained by adapting the metric on a compact manifold and Turing complete Beltrami fields in Euclidean space with the fixed Euclidean metric. However, the existence of such flows on the standard round S^3 or flat T^3 remains unresolved.

This problem targets the canonical metrics on S^3 and T^3, asking whether Turing completeness can be achieved without metric perturbation away from these standard geometries.