The Prize Wheel exists; public odds and costs were not found
The official Lucky Duck badge is awarded for spinning the Prize Wheel for the first time. Use the current in-game wheel instructions, but do not rely on percentages or guaranteed rewards that are not shown by an official source.
The badge record is strong evidence for the feature because it names both the action and the first-spin milestone. It does not say where the wheel sits, what a spin costs, how often it refreshes or which outcomes are possible. Those details can change and must come from the current game interface.
A safe wheel plan is therefore short: locate the current Prize Wheel through the live UI, read every cost or cooldown before confirming, make the first spin only when the requirement is clear, and check whether Lucky Duck appears.