 Exchange square-root branch convention

Let $P_{-}$ project onto $(|01⟩-|10⟩)/\sqrt{2}$ and let $P_{+}=I-P_{-}$ project onto the symmetric sector. An isotropic exchange interaction, with a chosen common energy offset, gives the continuous family displayed above. A full exchange has $\theta =\pi$ and equals SWAP gate; the conventional half exchange has $\theta =\pi /2$ and equals $P_{+}+i{P}_{-}$. Reversing the sign of the exchange gives the same full exchange at $\theta =-\pi$ but the inverse half exchange at $\theta =-\pi /2$. Thus even continuous time-independent Hamiltonian evolution and unchanged $|00⟩,|11⟩$ do not let the full-time truth table determine a unique half-time gate. A chosen generator or spectral branch is necessary. The half-exchange eigenvalues $1,i$ are the conventional principal scalar square-root choices for $1,-1$; no analytic matrix-function branch across the negative real axis is being assumed.