Even bipartite non-local magic need not cause response

For the two-qubit syndrome, the indicator can be evaluated:

Indeed, the largest overlap with a product state is ${\cos }^2\kappa$, whereas the largest overlap with a maximally entangled two-qubit state is $(\cos \kappa +\sin \kappa {)}^2/2$. These are the two possible Schmidt spectra of a two-qubit pure stabilizer state. Both overlaps are strictly below one for $0<\kappa <\pi /4$.

This counterexample concerns bipartite non-local magic, including in a complete Choi state. It does not identify that quantity with the specialized perturbative tripartite quantity of CaoEtAl2026. The fixed syndrome carries no logical-input response.