= Exchange square-root branch convention
{title2=$W(\theta)=P_++e^{i\theta}P_-$}
Let $P_-$ project onto $(|01\rangle-|10\rangle)/\sqrt2$ 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 $P_++iP_-$. 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 $|00\rangle,|11\rangle$ 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.
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