Fixed-axis spin obstruction for positive-energy Dirac spinors ($p_y^2+p_z^2=0$)

A positive-energy standard-representation Dirac spinor has upper component $\xi$ and lower component $(\mathbf{\sigma }\cdot \mathbf{p})\xi /(E+m)$. To be an eigenstate of $\mathrm{diag}({\sigma }_x,{\sigma }_x)/2$, both components must have the same ${\sigma }_x$ eigenvalue. With $\xi ={\chi }_{\pm }$, the commutator obstruction $2i(p_y{\sigma }_z-p_z{\sigma }_y){\chi }_{\pm }$ has squared norm $4(p_y^2+p_z^2)$. It vanishes precisely for momentum parallel to x, including zero. Exact transverse-spin eigenstates therefore fail at nonzero transverse momentum; the obstruction is not a universal failure at every relativistic energy.