The translation gcd property for rational polygons: Classification and finite witnesses

We characterize the rational convex polygons for which every rational translate has an Ehrhart quasi-polynomial with the gcd property, defined using its minimum period. They are exactly the rational translates of integral zonotopes. The proof reduces to centered half-integral polygons and computes the net jump from a nonintegral edge and its opposite edge. A preimage of this boundary under multiplication by three produces distinct constituents with indices 1 and 3 for a dyadic translation. We also construct a finite set of rational translations that detects every failure of the property. If qP is integral and the primitive edge normals have maximum infinity norm M >= 1, every candidate translation in the constructed finite set has denominator at most 124416 q^7 M^7.