Mr. Laffer's Curve, which illustrates the theory that cutting tax rates can increase tax revenues, is enjoying a revival. The idea played a starring role in Republican campaigns for tax cuts in 1981 under President Ronald Reagan and in 2001 under President George W. Bush. Now Republicans are once again leaning on the Laffer Curve to argue that cutting tax rates would not increase the federal debt.
Speaker Paul D. Ryan, Republican of Wisconsin, who has invoked Mr. Laffer's theory during the current debate, keeps on the wall of his personal office a framed napkin illustrated and signed by Mr. Laffer and inscribed "to my friend, Paul Ryan."
The Smithsonian presents its napkin as the tablecloth that started this tax revolt.
"Economist Art Laffer sketched a new direction for the Republican Party on this napkin," the display says, "illustrating his theory that lowering taxes increased economic activity."
Brian Domitrovic, a historian at Sam Houston State University who has studied and written about the creation of the Laffer Curve, said the sum of the available evidence shows that the Smithsonian's napkin is "most certainly an ex post facto creation."
But Peter Liebhold, the curator for the Smithsonian's division of work and industry, who acquired the napkin in 2013, said that the museum was confident of its authenticity.
"As you know, oral history is wonderful but subject to problems," he wrote in an email. "The Two Continents meeting happened many years ago, and some of the participants' memories of details may be slightly off. In this case, we have a piece of primary material that clearly documents the occasion. I look forward to your story, but strongly stand by the authenticity of the Smithsonian artifact."
The Laffer napkin has long been shrouded in mystery, partly because none of those present at the original meeting thought that it was particularly historic at the time.
Mr. Wanniski, the hype man of the tax cut movement, was the first to publish an account in 1978, which he embroidered in multiple retellings. He was working for The Wall Street Journal's editorial page in 1974, and he said that after the Republicans suffered sweeping losses in the 1974 midterm elections, he arranged a meeting between Mr. Laffer, whose ideas he considered important, and Mr. Cheney.
The economy was mired in stagflation, the unfortunate combination of high inflation and high unemployment. Mr. Laffer, then a professor at the University of Chicago, was a leading spokesman for a new school of economic thought, supply-side economics, that saw tax cuts as the best way to increase economic growth while reducing inflation.
The two men met for drinks with Mr. Wanniski in early December at a restaurant across the street from the Treasury Department. Mr. Wanniski also invited Grace-Marie Arnett, a Republican aide who was interested in supply-side economics.
In Mr. Wanniski's account, Mr. Laffer grew frustrated as he tried to explain the value of tax cuts to Mr. Cheney, finally grabbing a paper napkin so he could draw a visual aid. The Laffer Curve, which looks like the nose of an airplane, is meant to show that higher tax rates can reduce tax revenues — and lower tax rates can bring in more money.
Ms. Arnett, now known as Grace-Marie Turner, said that she recalled the moment vividly because she had never seen the Laffer Curve before, and she found it illuminating.
She said she was certain that the meeting happened after the elections because until then, she was working for a Republican House candidate in Texas. And, she added, "I absolutely know that it was a paper cocktail napkin."
Mr. Cheney also wrote in his memoirs that the meeting happened after the midterms.
Mr. Laffer has always said he does not have a clear recollection of the original meeting, but he said that he does not regard the Smithsonian's napkin as the authentic original.
"There was a napkin done and I don't know where it ended; I guess it was probably in the trash," he said. "And Jude Wanniski wanted me to do another example of it two years later, and I believe that this is the napkin that Jude Wanniski asked me to do for him."
Mr. Wanniski never mentioned the cloth napkin in his public writings. In June 2005, shortly before his death, he published an account of the original meeting on his website. He wrote again that Mr. Laffer drew the curve on a cocktail napkin, in December 1974, and he specifically denied that Mr. Rumsfeld was there.
Patricia Koyce Wanniski, his widow, said in an interview that she found the napkin at the back of a drawer of Mr. Wanniski's clothing after his death.
Why did Mr. Laffer dedicate the napkin to Mr. Rumsfeld and write the date "9/13/74"?
One possible explanation is suggested by Mr. Rumsfeld's 2011 memoir. He wrote that his datebook showed he had dined with Mr. Laffer and Mr. Cheney on Sept. 16, 1975, and that he had made a note at the time that Mr. Laffer drew a curve on a napkin.
Is it possible Mr. Laffer reprised that napkin for Mr. Wanniski, creating a reproduction of a reproduction? Mr. Laffer said he simply was not sure.
"When you get old," he said, "it's hard to say which napkin was which and when and where."