COOPERSTOWN, N.Y. -- Among the more than 40,000 three-dimensional artifacts in the National Baseball Hall of Fame and Museum, one stands out as profoundly poignant on this induction weekend.

It’s a baseball with two fingers and a thumb traced in blue ink over the curved red seams and white horsehide, demonstrating the grip for the legendary cutter Mariano Rivera used to rack up a record 652 saves and a unanimous selection to the Hall.

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As the placard at the Hall notes, the late Roy Halladay carried that ball with him from the All-Star Game in 2008 through the rest of his Cooperstown-worthy career.

“When you have All-Star Games, you always chit-chat with the guys,” Rivera said here Saturday. “So I was talking to Halladay, and he asked me how I throw a cutter and would I teach him?”

We all know what happened next.

“He did good,” said Rivera, putting it lightly. “And my guys [on the Yankees] got mad at me.”

Halladay was still with the American League East rival Blue Jays at the time of this cutter counseling session, and his cutter usage increased dramatically from 2008 forward. The pitch helped him put together the finishing touches on a Hall of Fame resume that, sadly, Halladay is not here to see recognized at these festivities, at which Rivera, Edgar Martinez, Mike Mussina, Lee Smith and Harold Baines will give their acceptance speeches. Halladay passed away in a private plane crash in November 2017.

Halladay’s widow Brandy, who will speak on his behalf, donated the cutter grip ball to the Hall.

“I wish he could be here,” Rivera said of Halladay. “He earned it, and he deserved it, and he gave us a lot of great memories. He’s a guy that was always giving his best.”

Of course, the same can be said of Rivera, and that extends to the expertise he offered, even to opponents.

“If I would have charged the guys I gave advice, I would have made a lot of money,” Rivera joked. “It’s part of me. I always wanted people to do good and have success.”

That didn’t always sit well. Rivera had to pay a fine in the kangaroo court in his Yankees clubhouse for helping Halladay in 2008. But perhaps distancing himself from the situation slightly, Rivera did assert that, contrary to the Hall placard’s assertions, he did not trace his own fingers on Halladay’s ball. He dispensed the advice, and Halladay took it from there.

In the culture of baseball, the influence of excellence -- even if it’s opponents comparing notes -- is always in the air. It’s how many players tap into their potential and compile the kinds of careers that get them to the dais where 59 living legends will convene Sunday.

For Rivera, there is one especially big influence he wishes could be on that stage with him. When asked which Hall of Famer he was most looking forward to meeting this weekend, Rivera thought for a moment before offering his answer.

“He’s already passed,” Rivera said solemnly. “That was Mr. Jackie [Robinson]. That was the guy I always wanted to meet.”

How does Rivera summarize the impact Robinson had on him?

“No words for that,” Rivera said. “As a minority, he gave us everything that he had, for us to come in. And he did it with class. So for me to be a minority and wear his number [42] is something special.”

That brings up an interesting question: Could Robinson have handled the cutter that Rivera mastered on his own and then taught to Halladay, among many others?

“He would have, he would have,” Rivera said. “I would have put it in there for him to hit.”

Hall of Famers helping Hall of Famers. The concept exists in Rivera’s imagination, and in the case of that ball now sitting in the Hall’s New Inductees exhibit, it is on display in Cooperstown.