One-query constancy obstruction for phase oracles ($M>2⟹Q_E(\text{constancy};\text{phase oracle})>1$)

An unrestricted Boolean function on $M>2$ inputs cannot be classified as constant or nonconstant with certainty using a single phase oracle query, even with quantum ancillas. Write a purified query input as $|\Psi ⟩={\sum }_x|x⟩|a_x⟩$ with $p_x=\Vert a_x{\Vert }^2$ and ${\sum }_x{p}_x=1$. The zero constant function leaves it unchanged; the function supported only at $j$ produces a vector with inner product $1-2p_j$ with it. Perfect discrimination of pure states requires orthogonality, forcing $p_j=1/2$ for all $j$ and thus $M=2$, a contradiction. This is an exact quantum query complexity obstruction for all nonconstant inputs, rather than the restricted promise used by the Deutsch-Jozsa algorithm.