Quadratic and boost alternatives

For a proper cone, $\mathsf{Q}$ implies the Lorentz-cone conclusion by Theorem thm:signature. Its boundary is then contained in $\{Q=0\}$. If sharp operational displacements are to mean that boundary, the robust-endpoint interpretation in Section sec:kp must also be supplied. Quadraticity remains a representation input in this route.

A boost route is also possible, but it adds explicit continuous symmetries.