But Dr. Naismith wasn’t first, said the volunteer, who directed Mr. Flansburg to a little-known book from 1952 called “I Grew Up With Basketball” by Frank J. Basloe, a Herkimer notable who had managed barnstorming basketball teams in the early 1900s. Mr. Basloe told the story of Lambert Will, the teenage director of the town’s new Y.M.C.A., who was said to have thought up basketball while throwing cabbages into a bushel basket at Ausman’s General Store on Main Street. This was in late 1890, a full year before Naismith was said to have posted the 13 rules.

The first-ever basketball game, Mr. Basloe wrote, was played between the Herkimer Y.M.C.A. 9 and the Herkimer Businessmen 9 on Feb. 7, 1891. A second match, against a team from nearby Little Falls, was played two weeks later. His book includes a photograph of the original Herkimer 9 team, including Mr. Will. One player is holding a ball on which someone has written “91-92” — as in 1891-92 — though it is unclear when that date was added.

Mr. Flansburg’s initial response: Nonsense. “It seemed like an easy thing to negate,” he said.

After his father’s death, Mr. Flansburg went back to his Arizona paradise, only to return to Herkimer in 2020 when Covid restrictions canceled his mathematics roadshow. He decided to dig in and disprove.

Burrowing into the historical society’s dusty corners, among copies of long-gone newspapers (The Mohawk Courier, The Ilion Citizen) and evocative artifacts (the trombone used by Mr. and Mrs. Harvey Shaul in the 1920s, the high-wheeled bicycle that George Nellis rode to San Francisco in 1887), he found some corroboration in Mr. Basloe’s book, including the locations of the market, still standing, and the Y.M.C.A. building, now a parking lot.

Intrigued, Mr. Flansburg put on a mask and visited the archives of libraries and historical societies across the region. He came away convinced that, at the very least, Central New York — and Herkimer in particular — had been pivotal in basketball’s early evolution (a claim also made by Holyoke, Mass.).

“Something was up,” he said

He tracked down two historians, the brothers George and Darril Fosty, who had written a very brief article about the Herkimer angle in 2010. “We just got ridiculed,” George Fosty recalled. “People just thought we were nuts.”

Or worse.

A few months after the Fostys published their squib online, a wealthy alumnus of the University of Kansas, where Dr. Naismith coached and taught for most of his career, paid $4.3 million for the 13 original rules after the document was put up for auction by the doctor’s grandson, Ian Naismith. (The document, regarded as basketball’s sacred scripture, is now enshrined at the university.)

The Kansas City Star described the grandson at the time as being sick of hearing about Mr. Will. “I want Herkimer out of my life,” he said, before getting really mad.

“I’m ready to kick some ass,” Ian Naismith, who died two years later at 72, was quoted, precisely, as shouting. “I will hurt somebody. Whoever says it can face me, and I’m 260 pounds.”

George Fosty, who weighed about 300 pounds at the time, was not concerned. Still, he and his brother set aside the project until being contacted a dozen years later by a man calling himself the Human Calculator. Soon they were delving deeper into the past with Mr. Flansburg and another Herkimer native, Brion Carroll.

In 2022, the Fosty brothers, with Mr. Carroll, published a book with a flagrant foul of a title: “Nais-MYTH: Basketball’s Stolen Legacy.” It combines circumstantial evidence and what Mr. Carroll called “if this, then that” logic to upend the game’s origin story.

The book recounts Lambert Will’s late-life recollection that in 1890, the Springfield Y asked the greater Y.M.C.A. network for suggestions of new indoor athletic endeavors. He sent in his thoughts, which included the rudimentary rules for what became basketball, but he never heard back. And so, the story goes, Mr. Will arranged for his new game to be played in Herkimer, with two teams of nine men.

The book points to a Utica Daily Press article from 1898 that said Herkimer had a basketball team in 1891. It refers to a Herkimer Evening Times article from 1940, about a 50th anniversary basketball reunion, that said basketball began in the Mohawk Valley “at the Herkimer Y.M.C.A. in the fall of 1890.” It names a man who, in later years, had said he witnessed the first game at the Herkimer Y, and had produced his 1891 Y.M.C.A. membership card as proof.

It raises questions about Dr. Naismith’s shifting accounts of the game’s development, as well as his other odd claims, like eating rabbit could make someone run faster, or that he had invented a stretching machine to make people taller, or that he had invented the hair dryer.

The book also notes a curious detail about the 13 original rules, now displayed at the University of Kansas. A handwritten date at the bottom of the document, “Feb. 1892,” has clearly been erased and replaced with “Dec. 1891” — a date that would better align with the official Naismith narrative.

Dr. Naismith is believed to have changed the date when he signed the document at the behest of one of his sons, in 1931. But a University of Kansas website and the 1996 reprint of Dr. Naismith’s book, “Basketball: Its Origin and Development,” include an image of the document bearing the original February 1892 date.

But, but, but. As with Mr. Basloe long ago, Mr. Flansburg and his team have found no smoking-gun document. No news clippings of games played in 1891, no Y.M.C.A. documents, no correspondence. Some records, in fact, are missing, leading to mutters of a conspiracy.

“It’s like a special ops preceded us,” Mr. Carroll said.

What they do have is the recollection of Mr. Will, who served in the Spanish-American War, volunteered as a firefighter for more than 60 years and ran a printing business. He joined Mr. Basloe in the 1950s in a protracted campaign to have the Basketball Hall of Fame located in Herkimer (1950 population: 9,400) — only to lose out to Springfield (1950 population: 162,000) — and died the day before his 88th birthday in 1964.

“All I know is that I played my first basketball with a head of cabbage in 1890 and played the first game on my mother’s birthday, Feb. 7, 1891, after I had made up enough rules for a game,” he once said. “I am not looking for any glory for what I did for basketball, which I am thankful to think a head of cabbage gave so many people and myself a great deal of pleasure.”

Another day at Crazy Otto’s Empire Diner, Mr. Flansburg sat with one of Mr. Will’s grandsons, Dennis Will. He is 66, with a long gray beard.

Though his memories of his grandfather are faint, he clearly remembers joining two dozen Will relatives on a 1994 trip to Springfield, where the Basketball Hall of Fame held a small celebration of Mr. Will centered on a display of a few mementos borrowed from the family. It was a private videotaped affair, no press, that lasted less than a half-hour.

The display quickly disappeared, and Mr. Will said that the mementos have never been returned to the family, despite repeated requests. The moment seemed to define the decades-long struggle, and Mr. Will choked up in a booth at Crazy Otto’s.

“Credit where credit’s due,” he said, with reddened eyes.

Mr. Flansburg escorted Mr. Will and his wife, Laurie Will, along Main Street, its sun-baked emptiness interrupted by an old bar here, a new restaurant there. They paused in front of the once-grand but long-shuttered building where Lambert Will threw cabbages into a barrel.

The Human Calculator talked of Herkimer: what was, what is, and what, maybe, could be. He talked like every sports fan who ever was, knowing the odds but rooting still for the long shot.