Counting Ehrhart functions of half-integral polygons

Fix an integer \(i\ge2\). We count the distinct Ehrhart functions of convex half-integral polygons with exactly \(i\) interior lattice points and at least two boundary lattice points. The number is \((9i^3+72i^2+175i+106)/2\), as proposed by Bohnert and Springer. We describe the complete set of functions by their area and their boundary counts before and after doubling. Convex section estimates isolate two exceptional families, and explicit polygons realize every remaining signature. The proof uses the published complete computational classification for \(2\le i\le16\), five published minimum-width bounds, and a finite list of exact rational inequality certificates. Integral polygons and period collapse are included, and no restriction is imposed on the arrangement of the interior lattice points.