FLUSHING MEADOWS, N.Y. — Coco Gauff’s U.S. Open ended just as she had warned it might. On an error-strewn afternoon of struggle on her forehand, the first shot she and everyone else learned as a child and one of the two most important shots in tennis, Naomi Osaka beat her 6-3, 6-2.

A duel between two multiple Grand Slam winners and former U.S. Open champions never quite caught light, and for the moment they are headed in opposite directions on their missions to fix the fundamental flaws that have prevented them from being the players they want to be.

For the first time since the beginning of last year, when she returned to tennis after childbirth, Osaka appears truly ascendant. Partly because of players like Gauff and Iga Świątek, women’s tennis evolved while Osaka was sidelined. And because of previous breaks from the sport to manage her mental wellbeing and physical injuries, Osaka was mostly on the sidelines from the middle of 2021 until the start of 2024.

When she reigned over tennis, swinging away from the baseline and dictating from the hash mark at its center was a route to victory. Now, power that lacks the ability to move across the court like a grasshopper and switch direction in an instant is not enough. In the middle of her 20s, Osaka had to learn new tricks, like an open-stance backhand that would allow her to switch from defense to offense with a single stroke.

She has made massive progress on that front. When she appointed Tomasz Wiktorowski, the coach with whom Świątek won four of her six Grand Slam titles, her instant improvements in results had an air of the new manager bounce from soccer.

With each match and tournament this summer, her feet have squeaked more and more often, and her speed and surety of foot out of the corners has improved. She has also slowly begun to regain her superpower, which was to play her best tennis in the most important moment, thumping the ball at the lines without fear of the consequences.

After a run of close three-set defeats that were intermittent but went on for over a year, she last month saved two match points in an early-round match in Montreal, against Luidmila Samsonova. In the end, she got within a few points of the Canadian Open title, and since then, she has decided that anything is possible.

Osaka is in the quarterfinals of a Grand Slam for the first time since Australia in 2021. She has been there four times before. All four times, she lifted the trophy.

Gauff, like Osaka, knows better than anyone that she has plenty of work to do. That’s why why she decided to start doing it just days before arguably her most important tournament of the season, given that she is an American and one of the biggest stars in her sport.

She was double-faulting around 20 times per match this summer. That wasn’t going to work, in the U.S. Open or in any tournament. So when he suddenly became available, she hired Gavin MacMillan, the biomechanics specialist who helped fix Aryna Sabalenka’s serve and forehand, allowing her to rise to the top of the sport.

Gauff had wanted to work with MacMillan for a while. When the chance arose, she didn’t care what tournament was next on her calendar, because his approach focuses on  getting the body’s joints to the right positions, and then powering off the foundation those positions deliver.

“There’s facts and science behind it instead of someone just telling you to do this, and you just do it,” she said. “When I believe in a coach and a plan, I can play some good tennis.”

Gauff already sees progress, and it is there on the stats sheets. She double-faulted five times against Osaka — not great, but vastly better than where she was. Her double faults also went down round by round, while her serve speed went up. By Monday she was averaging 104 mph on her first serve, nearly 10 percent faster than in her opening match.

“There were some doubles, but I thought, like, that was a good performance from me serving,” she said in her news conference, using every ounce of emotional energy she had to put a positive spin on a second consecutive fourth-round exit from New York.

“After the match I was really disappointed. Kind of broke down to my team. Then hearing their perspectives and everything, it definitely is a lot of positive things. I think if I kept the way I was going in Cincinnati to here, I would have been out the first round.”

That’s one of the shots. More baffling to her are her struggles with her forehand, which goes in and out from game to game and match to match, and even from point to point, especially on her return of serve, which was far and away the worst part of her game against Osaka. Serving well and returning poorly, patching a weakness and losing a strength, had her “discombobulated.”

She recorded 24 errors on her forehand, over 40 percent of the points that Osaka won all match. Most of those go down as “unforced,” but Osaka’s strategy was so clear, and so finely executed, that it’s not that simple.

MacMillan believes a serve is just a forehand hit on a different plane. In his paradigm, fixing one should help fix the other, but that doesn’t mean the improvements will come simultaneously. They didn’t for Sabalenka three years ago, and they didn’t for Gauff during her four matches at this U.S. Open.

Coco Gauff saw immediate improvements in her service motion at the U.S. Open. (Charly Triballeau / AFP via Getty Images)

Players control every aspect of a serve, other than the wind and the position of the sun. They start the motion standing still and holding the ball. It’s essentially a closed skill. The trick is developing a repeatable motion and then practicing it 1,000 times a day, so it feels as natural as breathing.

Forehands are not like that. They are an open skill: with dependent variables out of the player’s control. It’s probably an overstatement to say that no two are alike, but given the infinite variations of velocity, spin, court position and how much movement each shot requires, they might as well be.

Gauff can restructure her forehand all she wants in practice. But then she’s going to have to execute it, sometimes standing in the middle of the court and sometimes running at full sprint off it.  She will have to master hitting it crosscourt and down the line, within six inches of the net and with six feet of clearance. She will have to hit it off slow balls and fast ones, balls that jump and spit and balls that skid and slide.

The good news is that she has the time. She won’t compete again until the China Open in late September in Beijing, giving her the opportunity for a solid training block. The more complicated news is that she did exactly this last year, firing Brad Gilbert immediately after the U.S. Open and replacing him with Matt Daly, another expert — in grips, rather than mechanics — to fix the two most important shots in tennis. Immediate results, in the slower hard courts of Asia and then the indoor WTA Tour Finals in Riyadh, Saudi Arabia, were promising. She won the China Open and those Tour Finals too.

But when the season resumed and tennis went back to being an outdoor sport, the improvements largely fell apart, barring the French Open win on a slow court in swirling conditions that grant Gauff the time she needs with the stroke as it exists now.

Gauff, as she showed with making the decision to fix her serve now in the first place, is playing the long game this time around.

“I feel like I put so much pressure on myself at my age at 21, and I realize how much the girls on tour are being successful at 25, 26, at those ages,” she said.

“If I can make that same jump of improvement, it’s a lot to be excited for the future.”

(Top photo: Sarah Stier / Getty Images)