Emotion first bubbled up at 1-1 in the second set, when Williams exchanged words with chair umpire Carlos Ramos. Ramos had assessed Williams a coaching violation after her longtime coach, Patrick Mouratoglou, motioned from the player’s box that Williams should go to the net more. Williams, 36, disputed the violation and then told Ramos that she is not a cheater.

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Williams was under the impression that Ramos had taken back the violation, but he had not. Mouratoglou later admitted on ESPN’s broadcast that he had been coaching.

The six-time U.S. Open champion received another code violation four games later, leading 3-2 in the second set. Osaka had just broken her serve, and Williams smashed her racket onto the court so hard that it broke. Ramos assessed her a point penalty under WTA and Grand Slam rules, the second code violation of the match.

Williams and Osaka played two more games, then during the changeover with Osaka leading 4-3, Williams spoke with Ramos again, demanding an apology for “stealing” a point from her.

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“You will never, ever be on a court of mine as long as you live. You owe me an apology,” Williams said to Ramos. “Say it. Say you’re sorry. . . . I have never cheated in my life.”

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Williams called Ramos a “thief,” and he assessed her a third code violation for verbal abuse, resulting in a game penalty that put Osaka up 5-3 and one game from the championship.

Williams only grew more irritated and said she was being treated differently from male players who, she argued, get away with much harsher language and behavior on court. Williams called the tournament referee and the Grand Slam’s supervisor, who chatted with Williams but did nothing.

Osaka won the match soon thereafter as boos cascaded from the crowd at Arthur Ashe Stadium, where fans packed into the tournament’s main stage with hopes of witnessing history. Throughout the match they shouted encouragement at Williams.

Osaka, who is of Haitian-Japanese descent and was raised in the United States but plays for Japan, acknowledged the crowd once, offering a wave and a warm smile as she walked out. From then on, she kept her eyes lowered when she wasn’t playing. A few times she shrugged her shoulders as if to relieve tension.

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It worked; her poise was impressive, only melting at the post-match ceremony when the crowd began to boo loudly.

Williams held her face in a forced, closed-lip smile as tears ran down Osaka’s cheeks while U.S. Tennis Association President Katrina Adams spoke. Williams hugged Osaka multiple times, first at net after the match and again when they stood side by side in front of a stage while the crowd railed.

“I’m sorry it had to end like this,” Osaka said when it was her turn to speak.

Williams ordered the crowd to stop booing.

“I know you guys were here rooting, and I was rooting, too, but let’s make this the best moment we can,” Williams said as the crowd quieted. “We’ll get through it. . . . Let’s not boo anymore. Let’s get through this.”

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The 20-year-old Osaka, who is the first Japanese woman to reach a Grand Slam final, hit harder and bigger against Williams, especially in the first set.

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She showed off stellar defense in returning even the most smartly placed of Williams’s shots and defended her serve well, an aspect of her game she made her forte during this U.S. Open. Osaka has saved 24 of 28 break points she has faced over the fortnight. Against Williams, she faced six and saved five.

But perhaps most impressive was her fierce reserve in the face of the theatrics that closed the match and a star-studded crowd at Arthur Ashe Stadium that was almost wholly against her.

“I think I was able to do that because it was my first Grand Slam final,” Osaka said. “I felt like I shouldn’t let myself be overcome by nerves or anything and I should just really focus on playing tennis because that’s what’s gotten me to this point. I just thought, like, no matter what happens outside of the court, for me, when I step on the court, it’s just about tennis.”

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For Williams, there was plenty to mull over.

The American was attempting for the second time this summer to tie Margaret Court’s record of 24 Grand Slam titles. The Wimbledon finalist, who gave birth to her first child a year ago, also was trying to break Chris Evert’s record of six U.S. Open singles titles.

Williams complimented Ramos and said the two don’t have history, although Saturday’s incident did bring her back to controversies in 2004 and 2009 at the U.S. Open.

In 2004, Williams received an apology from the USTA after a poorly called third set by chair umpire Mariana Alves played a part in Williams’s quarterfinal loss to Jennifer Capriati. In 2009, after an outburst in which Williams threatened a line judge, she was assessed a penalty that concluded a semifinal loss against Kim Clijsters.

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Williams explained she was flashing back to those instances Saturday. Asked in the immediate aftermath, she said she wouldn’t change her behavior during the match and that she was fighting for women’s rights.

“I can’t sit here and say I wouldn’t say he’s a thief, because I thought he took a game from me,” Williams said. “I’ve seen other men call other umpires several things. I’m here fighting for women’s rights and for women’s equality and for all kinds of stuff. For me to say ‘thief’ and for him to take a game, it made me feel like it was a sexist remark. He’s never taken a game from a man because they said ‘thief.’

“I just feel like the fact that I have to go through this is just an example for the next person that has emotions and that want to express themselves and want to be a strong woman. They’re going to be allowed to do that because of today. Maybe it didn’t work out for me, but it’s going to work out for the next person.”