Nonzero-momentum condition in the massless vector norm test ($p=0,a=1/2,A=(1,0,\dots )\Rightarrow A^{\ast }\cdot A=-1$)

The usual half-level NS claim that all physical vector polarizations have nonnegative norm for $a\le 1/2$ assumes nonzero particle momentum. At zero momentum and limiting intercept, all positive-mode constraints and transversality allow a timelike polarization with negative norm, and $A\sim A+\kappa p$ removes nothing. Excluding this exceptional momentum restores the ordinary null-vector argument; retaining it makes the unrestricted elementary criterion strict $a<1/2$.