The Hlawka inequality compares the norms of three vectors with those of their pairwise sums and total sum. The question is the smallest comparison constant that works in every finite dimension.

This mission records the proved extension of the sharp diagonal Hlawka constant to every real $p\ge12$. It uses the unchanged coordinate norm and cyclic constant from the foundation mission, for arbitrary complex vectors, including unequal and zero norms.

The cutoff-12 goal theorem is already proved on Prove2Me. Sixteen accepted interval proofs cover $[12,16]$ and join the proved cutoff-16 result. The conclusion includes both the inequality and optimality of its constant, uniformly over all finite complex coordinate dimensions.

This is an entry in the sharp diagonal Hlawka campaign, whose conjectured endpoint is $p\ge2$. The cutoff-2 mission remains open; 12 is a proved sufficient cutoff, with no claim that it is optimal. The broader Schatten-norm question for general matrices is described by Audenaert and Kittaneh, §8.2.