A degree-one Vassiliev invariant for planar knotoids via $ 0 $-smoothing

We introduce a smoothing-based Laurent polynomial invariant for oriented planar knotoids by combining affine index weights with endpoint winding data. For a planar knotoid diagram $ D $, we first define an auxiliary ordered winding Laurent polynomial $ H_D(x, y) $, whose summands record the affine index weight of each classical crossing together with the homology class of the associated lobe in the twice-punctured plane. We then apply $ 0 $-smoothing against the orientation at each crossing and define $ F_D(x, y) $ by combining the resulting changes in $ H_D(x, y) $ with a writhe correction. We prove that both $ H_D(x, y) $ and $ F_D(x, y) $ are invariant under planar isotopy and the classical Reidemeister moves performed away from the endpoints. We also determine their behavior under orientation reversal and mirror image and establish an additivity formula for $ F_D(x, y) $ under the planar product, subject to the specified exterior region condition. To illustrate the information supplied by smoothing, we construct an infinite family of planar knotoids for which the affine index polynomial and $ H_D(x, y) $ are independent of the parameter, whereas $ F_D(x, y) $ distinguishes all members of the family. Finally, by extending $ F_D(x, y) $ to singular planar knotoids through the Vassiliev skein relation, we prove that it is a finite-type invariant of degree exactly one. These results provide a connection between affine index methods, endpoint winding information, and smoothing operations in planar knotoid theory.