(One exception came when Paul, Weiss asked Exponent if it was possible for someone to take a bag of 12 footballs into a bathroom and deflate them, at least a little, in 1 minute 40 seconds. A Patriots ball boy was seen taking balls into the bathroom on the way to the field before the game, and it became the primary theory for how the balls lost their air pressure. Pye found a small office and had several Exponent employees try. Yes, definitely, he told the lawyers in New York.)

On May 6, 2015, the N.F.L. released what instantly became known as the Wells report, named for the lead lawyer in the investigation, Theodore V. Wells Jr. It concluded that it was “more probable than not” that Patriots employees were deliberately releasing air from footballs and that Brady knew about it.

It was prominent news. And as people dug deeper into the report, past the juicy circumstantial evidence, and dipped into the science and data of Exponent’s analysis, the Ideal Gas Law had its talk-radio moment. Professors and other scientists questioned Exponent’s findings. Columnists tore into Exponent’s credentials. The “hired gun” headlines returned.

Exponent officials heard it and read it all. They remained silent as their reputation took shot after shot.

“That was difficult,” Caligiuri said. “There’s always an urge to respond to critics.”

Pye put the critics into three categories. One was the unabashed fan who was “going to make the call on feeling over questions of fact.”

“That doesn’t bother me at all,” Pye said. “That’s just the world we live in.”

Second were the armchair scientists, those who understood enough to raise reasonable questions, usually quickly dismissed. Exponent anticipated them in this case, which is one reason it conducted every experiment it could think of, even if it knew the answers. “We tried to head those people off,” Pye said.

Third were Exponent’s peers, the usual audience for Exponent’s work. They are the ones who frustrate Exponent most.