= Half-level Neveu–Schwarz vector state
{title2=$|A;p\rangle=A\cdot b_{-1/2}|0;p\rangle$}
The lowest fermionic excitation has level $1/2$. Its old-covariant physical conditions are $\alpha'p^2+1/2-a=0$ and $p\cdot A=0$. For nonzero real momentum, timelike $p$ gives a positive polarization space, null $p$ gives a nonnegative space with a longitudinal null direction, and spacelike $p$ permits a negative-norm time polarization. The limiting $a=1/2$ state is a massless gauge vector after quotienting null polarizations. This level-specific test does not prove positivity of every higher NS level.
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