Counting half-integral polygons with collinear interior lattice points

We count convex half-integral polygons with a prescribed number of collinear interior lattice points, up to affine unimodular equivalence. For every $i\geq3$, the count is the degree-seven polynomial conjectured by Bohnert and Springer. The proof is computer-assisted: their strip theorem reduces the polygons to two boundary chains on five integer rows, and a unique integer shear normalization gives finitely many integer linear systems. Complete rational generating functions for these systems, together with the fixed classes of the remaining reflections, give the formula by Burnside's lemma. Integral polygons are included, and translations are integral.