NEW YORK — Somehow, the U.S. Open just held even more than a stirring Thursday with exhilarating wins by an admirable African American woman (Taylor Townsend) and an admirable African American girl (Coco Gauff). It also held an inadvertent symposium on the difference between the tennis age of 15 and the tennis age of 23.

The difference is, of course, about a hundred years.

Observe age 15, which appeared by night. Absorb the sheer delight of the budding Cocomania, or just go ahead and be an unreconstructed prude. Listen to a Louis Armstrong Stadium crowd chanting, “Coco! Coco!” in honor of 15-year-old Gauff after her 6-2, 4-6, 6-4 fight through the Hungarian veteran Timea Babos, which sent Gauff into a peach of a third-round meeting with No. 1 seed Naomi Osaka. Hear the smashing reflection on Gauff’s parents and grandparents when she told the audience, “I’m really so grateful. I’m playing in front of all you guys. And you guys really believe in me.”

Swoon with enchantment when she later said of the chant, “I was thinking like maybe they feel like I’m Golden State in Game 7 or something.”

How about that exit from the court? If that wasn’t the slowest, most cordial, most accommodating exit in Grand Slam second-round history, it had to be a contender. You could lose count of the selfie count at about 10. To one man seeking an autograph, her hands full of her things: “Can you take the top off (the pen)?” To one woman who complimented her green nails: “Thank you!” To one man who asked for a selfie, whereupon she asked for one in return: “Can you take one for me on my phone?” And: “Thank you for coming.” To a woman, while posing for a photo with two other women: “Oh, where am I looking at, over here?” In the tunnel on the way out, her toss of a signed ball to a little boy failed to reach its target, so she caught it and tossed it again.

Go ahead and giggle at the whole 15-years-old of it when the Floridian said her two kid brothers motivate her and said, “My brother Cody, he’s 11. I guess he’s transitioning to be more serious in one sport … Yeah, baseball. He’s up-and-coming. He does fitness with me a lot of the times. He comes to the gym with me a lot. I see that he kind of picks up his level. He looks up to me” — and, glory, this next part — “even though he doesn’t admit it.”

Babos called Gauff “super-fast” and said, “I don’t think I have to win the point as many times against anyone as against her.” She marveled at Gauff’s “huge power, and being a 15-year-old girl.” It all felt light and airy until somebody asked Babos to peer into Gauff’s future, and Babos, 26, called the forecast sunny but provided the wise reminder that expectations can get heavy.

Gauff has played two Grand Slam tournaments. She has soared to the fourth round at Wimbledon. She has wielded her big heart to the third round at Flushing Meadows. She has reveled in being an American in New York. She has heard rave upon rave while exiting the court, one stranger saying, “We believe in you.”

Taylor Townsend, 23, merged into the U.S. Open main draw from the savage, barren road of qualifying, along which she trailed both No. 151-ranked Veronica Cepede Royg of Paraguay and No. 165-ranked Nina Stojanovic of Serbia by one set before wriggling out of both thickets. Such hardship seemed fitting.

At the end of 2012, Townsend, 16, became the first American girl in 30 years to finish a year No. 1 among the world’s juniors, setting her afloat on a raft of expectations even after a bizarre turn when the USTA tried to discourage her from coming to the 2012 U.S. Open because of her weight. On Thursday, after a long, long hunt, Townsend at last found herself “a defining moment for me to realize that I belong here.”

In between 2012 and Thursday, she waded through the wilds of lower-tier tours, scrounging for points, finding enough now and then to appear at 16 Grand Slam tournaments (nine first-round exits, six second rounds, one third round). She played in Indian Head Beach, Fla., five times; in Dothan, Ala., five times; in Charlottesville four times. She played Waco, Macon and Florence (the one in South Carolina). She played Midland (Mich.), Kirkland (Wash.), Templeton (Calif.). She went through El Paso and Daytona Beach and Bonita Springs and Wesley Chapel and Rancho Santa Fe on her way to dissipated expectations.

She played the 2019 Australian Open, lost in the first round to Sloane Stephens. She played the 2019 French Open, lost the first round to Garbine Muguruza. She played Wimbledon, lost the second round to Kiki Bertens with an unconverted match point that left her “devastated” and waking up nights haunted by a drop shot.

Finally — finally — she made a Navratilovan 106 trips to the net on Thursday, wringing 64 points out of those, including 40 out of 64 in a third set of elite boldness. “Never, never — I never played with someone coming so often to the net,” said her opponent, the reigning Wimbledon champion Simona Halep. Townsend upset Halep, 2-6, 6-3, 7-6 (7-4), and her joy looked like that of someone who knew in her bones the value of struggle.

So she sounded so much older than her mere 23. She said, “I think for some years I was lost, lost in a sea of a lot of stuff.”

She said, “Yeah, I mean, it’s been a long road. A lot of haters, a lot of people who weren’t sure. I mean, I’ve heard it for a really long time that I was never going to make it, that I wasn’t going to be able to break through or do this or do that.” She said, “So, it’s really hard to deal with that kind of stuff, like when your personal issues are publicized, because you have to make up in your mind how you deal with that. That’s always attached to you.”

She said, “I actually went on a social-media sabbatical for a year and a half. I was just tired of it. I didn’t really know why it was bothering me. I was just, like, I can’t do this anymore.” And she said, “Then I got back on. I think it was more growth, in myself saying, ‘When do you ever meet these people?’ Like, never. You know what I mean? Like realizing that the things that people say don’t matter. It’s about how you feel about yourself, about how you feel going out on the court.”

She had found herself a moment, a “huge, monumental moment” as she put it, and it resonated in her exhilarated voice, just hours before Cocomania would surge again with all its warmth and bedazzlement. Gauff, of course, became the youngest third-round female player in a U.S. Open since 1996, when Anna Kournikova reached the fourth round at age 15 years, almost three months, then ran across Steffi Graf, mandating doom.

Eventually, Kournikova reached a 1997 Wimbledon semifinal and a 2001 Australian Open quarterfinal before stopping at 21 and citing back problems. The path from 15 to the future can get jagged.

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