The so-called proof was an article from academic journals JSTOR allegedly dated January 5, 2008, which talks about a person from 18th century living in Japan and named Tominaga Nakamoto. The article mentioned that “Nakamoto is the Japanese Adam Smith. Honest Ledger + Micro Cash. Satoshi is Intelligent History. Not too hard.”
Source: Modern Consensus.
Adam Smith is considered as the father of modern economics, and Craig Wright claimed that was one of the main reasons for selecting the name, Nakamoto. He went on to explain that Nakamoto's economic ideas on honest money were one of the main reasons for choosing his name for the White Paper. He explained,
“In part, yes. He wrote about money and honest money and the rational nature of things. The shogun [feudal ruler] at the time was in financial crisis and economic austerity. [...] I like the description of him, and I got into his brother, Tōka. ‘Nakamoto was upright and quiet but impatient in character’ and I thought: ‘That sounds like me.’”
Craig Wright has a history of fabricating documents and even lying under oath, which has put a lot of question marks on his credibility. In the past, he has tried passing another paper he claimed as proof for his identity as Satoshi, which later turned out to have tampered. He is currently involved in a lawsuit filed by Kleiman estate accusing Wright of stealing billions worth of Bitcoin from Dave Kleiman. Kleiman was a cyber-security expert who is believed to be among the very first set of Bitcoin developers and died in April 2013
Ira Kleinman brother of Dave Kleiman filed a lawsuit against Craig Wright accusing him of stealing $5 billion worth of Bitcoin. The court found Wright guilty of tampering with evidence and ordered him to turn over half of his Bitcoin holdings and intellectual properties to the Klieman estate.