Two large claims are made for Chen Tuan and both are made after his death.
The first concerns the diagram of the supreme ultimate. Zhou Dunyi's short text on that diagram stands at the head of Song Neo-Confucian metaphysics, and from the twelfth century onward writers argued that Zhou had not devised the figure but received it through a transmission line reaching back to Chen Tuan, where it had been a diagram of inner alchemical practice read from the bottom upward. The claim was pressed by critics who wanted to show that the Confucian revival had Daoist foundations, and defended against by those who did not. It has been argued about continuously since, and the evidence is a chain of assertions rather than a document.
The second concerns divination. The system of fate calculation known as Zi Wei Dou Shu, which maps a life onto a chart of notional stars, is traditionally credited to him. Its surviving textual tradition is considerably later, and the attribution follows the usual pattern of a technique acquiring a famous ancestor.
What can be said without dispute is smaller and still substantial. He was a real recluse of Mount Hua who died in 989, he was received at the Song court, he was associated in his own century with physiognomy, with breath cultivation and with the study of the Book of Changes, and he was famous enough that later traditions found it worth attaching themselves to him. The gap between those two lists is the subject.