Parity of zeros of a real line-bundle section (${\mathrm{PD}}_{{\mathbb{F}}_2}[s^{-1}(0)]=w_1(L)$)

For a real line bundle on a closed manifold, a transverse section has a hypersurface as its zero set. Along a closed loop transverse to this hypersurface, trivialize the pullback over the cut interval. Each simple zero reverses the sign of the component function. The total sign change is exactly the bundle's transition sign around the loop, detected by its first Stiefel–Whitney class. Thus zero-set intersection parity with every loop equals the evaluation of $w_1(L)$, proving the displayed mod-two dual class. No orientability of the manifold or bundle is required.