= Three-axis obstruction to unitary qubit orthogonalization
No fixed <unitary operator> maps both <eigenstate> rays of each of the three <Pauli matrices> to their orthogonal partners. Such a map would give $U^\dagger XU=-X$, $U^\dagger YU=-Y$ and $U^\dagger ZU=-Z$. But <unitary conjugation> preserves multiplication. The relation $XY=iZ$ would imply both $U^\dagger(XY)U=iZ$ and $U^\dagger(iZ)U=-iZ$, a contradiction.
Two axes alone do not suffice: $U=Y$ flips the X- and Z-basis labels while preserving the Y-basis labels. Thus six specified input rays already obstruct a unitary implementation, without requiring tests on every point of the <Bloch sphere>.
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