Nonconvexity of the First Artin Cone in Six Variables

The level-one Artin cone consists of homogeneous forms that become sums of squares after multiplication by some nonzero quadratic sum of squares. We prove that this cone is not convex for sextics in six variables. The two witnesses are classical Motzkin forms in disjoint triples of variables. Each has a quadratic sum-of-squares multiplier, but their sum has none, even if inhomogeneous multipliers of degree at most two are allowed. The obstruction follows by restricting to zeros in one variable block and inspecting the lowest homogeneous part in the other. The minimum sum-of-squares multiplier degree of the sum is exactly four. Closedness then gives positive-definite witnesses by small perturbation. This answers the universal convexity question in item 9 of Oberwolfach Report 46/2015 and Open problem 2 of Ahmadi and Hall; it does not resolve the other optimization questions bundled into dataset record OWR-13941-002. This is a self-audited, AI-assisted, unrefereed preprint.