Even in the highest-resolution image, examined up close, there was not so much as a discernible sliver of daylight. The margin by which the United States was eliminated from the Women’s World Cup was so microscopic that it cannot be expressed in a unit of measurement the country fully recognizes.

A millimeter, a single millimeter, is no more than 0.04 inches, yet even that most slender gap can serve as the gossamer border between two realities. Such is the unspoken truth of sports, of course: The difference between triumph and disaster, delight and dismay, can be far thinner than we choose to pretend.

For the United States, there is some comfort in that. “It is tough to have your World Cup end by a millimeter,” Alyssa Naeher, the U.S. goalkeeper, said after her team’s loss to Sweden in a penalty shootout Sunday. It does not take an especially vivid imagination to envision how the outcome might have been different.

Had Naeher intercepted Lina Hurtig’s shot at a slightly different angle, maybe the spin would have carried the ball to safety. Had Hurtig struck her penalty more softly, or more firmly, maybe Naeher would have saved it more decisively. Granted a reprieve, maybe the United States would have gone on to win that game in the round of 16, the tournament, the crown. Maybe, maybe, maybe.

That solace, though, is an illusion, and so too is the idea that the United States was eliminated by a millimeter. It was not one penalty that ended its hopes of a third straight title and, in the process, drew the veil over a whole golden, glorious generation, no matter how tempting it might be to believe. This is another unspoken truth of sports: Moments do not exist in isolation.

There is a certain irony in the fact that it was against Sweden that the United States, so limp and insipid earlier in the tournament, started to show signs of life. Naomi Girma was imperious. Lindsey Horan was dynamic. Sophia Smith, Trinity Rodman and Lynn Williams were all, at various points, electric. There were glimpses, in Melbourne, Australia, of what this team might one day be.

But that should not disguise the shortcomings of what came before. The United States was only in position to be knocked out by Sweden because it had failed to beat both the Netherlands and — more troubling — Portugal in the group stage.

The United States, the two-time reigning champion and pretournament favorite and great superpower of women’s soccer, won only one game in Australia and New Zealand, and that was against Vietnam. It was not even supposed to be in Melbourne. It was meant to be in Sydney, playing the Group G runner-up, at a time that had been specially arranged so that it was not in the middle of the long American night or early in the morning.

The spin of the ball, the single millimeter, was the culmination of a succession of failures, ones that can most immediately be traced to the last two weeks, but the roots of which stretch back not just months but years. To dismiss this disappointment as merely a cruel twist of fate is to risk failing to learn from those failures, making them endemic.

It is not enough, for example, to point the finger of blame at the coach, Vlatko Andonovski. He will, most likely, be removed from his position before his contract expires at the end of the year, and it is hard to make a case for his retention. This is the worst performance an American team has mustered at a World Cup. A price has to be paid.

But Andonovski is not the cause of the malaise. There are structural, systemic issues that have to be addressed, too. There are issues with the way the United States produces players, a fragmented system is reliant on pay-to-play youth teams in disparate leagues, unattached to elite adult teams, feeding into the college system.

That was fine when the United States effectively had a monopoly on professionalized women’s soccer, before the major men’s teams of Europe and South America decided — and let’s not cast them as the good guys here, given how long it took — that maybe women might enjoy the chance to play the sport.

In an ecosystem in which the intellectual and financial weight of global soccer can be deployed to hothouse talented young players, the American approach is not so much lacking as a guarantee of failure. So, too, is the continued emphasis on physicality, rather than cunning, that such a system favors. It is not a coincidence that the United States was eliminated from the tournament when its one player of genuine invention, Rose Lavelle, was absent. Lavelle is the one player, after all, that her country simply cannot replace.

Nurturing talent, though, is just the first problem. It is significant that Horan is the only member of Andonovski’s squad currently playing in Europe. Others, including Megan Rapinoe and Alex Morgan, have spent time there, but most have been drawn back to play in the surging National Women’s Soccer League.

That is, in many ways, good. A healthy domestic league is not only desirable but a crucial ingredient in success. But it also hints at a creeping isolationism, a disconnection from Europe’s major leagues, which are now emerging as the game’s fiscal engine and its intellectual crucible, too.

The United States needs players competing against their rivals and peers in the Champions League, not only as a finishing school but as a way to better understand their relative strength. Smith, for example, is lavishly gifted, but is she more so than Lauren James of England, Aitana Bonmatí of Spain or Linda Caicedo of Colombia? Answering that question is crucial for understanding how to set expectations.

Most immediately, though, what is required is a generational shift. It is, as Rapinoe herself put it, a “sick joke” that her last act at a World Cup will be missing a penalty. She has already confirmed she will retire at the end of the N.W.S.L. season. There are others, though, who may have to be ushered into the autumn of their careers rather less willingly.

That is never a pain-free process, and it will be all the more agonizing because of all this team has achieved. Naeher, Morgan, Julie Ertz, Kelley O’Hara and Crystal Dunn — as well as the absent Becky Sauerbrunn — have all enjoyed distinguished, glittering careers, the final, glorious ambassadors of a generation that won two World Cups.

Moving on would always be difficult in a purely sporting sense. It is made all the more charged, though, because of what this team means in a cultural one. They are, rightly, revered as players but they are also admired because of the causes — equal pay, equal rights, the struggle against racism and misogyny and homophobia — that they have willingly adopted.

They mean something to people, to fans, in a way that other athletes do not. The adoration, the loyalty, the fervor they have inspired has more in common with political or cultural idols than it does with humdrum sports fandom.

As Rapinoe has always acknowledged, though, the activism has to flow downstream from the sport. Winning, she said, is necessary because it is the precondition for people wanting to hear what you have to say. Victory has always been what allowed the U.S. players to speak their minds and to make their stands to the most people.

It follows, then, that when they are no longer almost a guarantee of winning — when they might, in some senses, make success less likely — then they cannot be protected for what they represent, for what they mean, rather than what they do. There comes a point when they have to be judged as athletes, not activists, and that means knowing when to say goodbye.

None of that would have been changed had Naeher managed to keep out Hurtig’s penalty, had the ball spun just out, had that microscopic difference worked in the Americans’ favor. This United States team was always coming to the end of its road. No matter where the ball landed, there was never any other reality than the one the United States finds itself in now, at the end of one era and the start of another.

Under an ink black Australian sky above Melbourne Rectangular Stadium, the Women’s World Cup game between the United States and Sweden on Sunday went on and on and on. For 120 minutes, it went on as the teams tried unsuccessfully to score, with nearly 28,000 fans so nervous that they could only muster a simmer of cheers. Until penalty kicks turned up the volume and decided it all.

That’s when the United States’ recent dominance in the World Cup fully ended, and the Americans were left stunned and devastated by their worst showing at the quadrennial tournament. They had arrived as the favorites after winning two consecutive championships, in 2015 and 2019. But on Sunday, in the round of 16, three missed penalty kicks and a razor-thin goal by Sweden changed their fate.

Sophia Smith, who missed an opportunity to win for the United States, had to be consoled by her teammates as she sat on the field in tears. Kelley O’Hara, in her fourth World Cup, stormed by reporters and stared straight ahead in silence after the game, moments after her penalty shot hit the right post and bounced away.

And Megan Rapinoe, the outspoken and accomplished U.S. forward who had been relegated to a reserve at this World Cup, grew teary when discussing that her international career would end with her missing a penalty kick, calling it “a sick joke.” Just a week ago, Rapinoe was asked what the team’s legacy would be if it failed to win the world title yet again. She answered, “I haven’t thought about that.”

Now she won’t forget it. Sweden won the shootout, 5-4, to eliminate the United States.

Alex Morgan, the star U.S. forward, called it “a bad dream.”

“I’m really disappointed with myself, and I wish I could have provided more with this team,” said Morgan, who was on the bench for the shootouts because she had been replaced by Rapinoe earlier. She didn’t score during the entire tournament.

Julie Ertz, who rushed back to the team after having a baby a year ago, said it was sweet to see her son in the stands after the match. “But it still hurts to lose a game like that,” she said. She walked off, wiping the wet, smeared mascara from under her eyes.

It all came apart for the United States in a flurry of 14 kicks. Here’s how they unfolded, emotions included:

Andi Sullivan, the midfielder, is up first to face Sweden’s goalkeeper, Zecira Musovic, with her teammates lined up behind her, many arm in arm. She walks over to the spot with the death stare of a gunslinger, then nails the shot into the lower left of the goal. Sullivan spins back toward her teammates and pumps a fist. The crowd finally comes alive and chants: “U-S-A! U-S-A!” U.S. 1, Sweden 0.

Fridolina Rolfo, a 5-foot-10 forward who has been on the national team for 10 years, is up first for Sweden against goalkeeper Alyssa Naeher about a month after winning the Champions League with Barcelona. She sends the ball into the right side of the net, her blond ponytail swinging behind her. She flexes her arms and opens her mouth wide to shout in celebration, and the Swedish fans, many clad in bright yellow and sitting right behind the goal, explode into cheers. U.S. 1, Sweden 1.

One of the U.S. co-captains, Lindsey Horan, has a familiar, ferocious “don’t mess with me” look on her face. It’s the look she had just before she scored the equalizer in the 1-1 tie versus the Netherlands in the group stage. It’s much tougher than the softer approach she took for much of last week with her teammates, as she encouraged the 14 World Cup rookies, one by one, to play with more confidence. The Swedish fans are booing her, competing with the U.S. cheers. But Horan is steely and delivers the ball precisely to the left side, rocketing it into the net. U.S. 2, Sweden 1.

Elin Rubensson, a midfielder who returned to soccer just two months after having a baby in 2020, evidently decides that Horan picked a wonderful place to put the ball into the net. So she sends the ball there, too — and Naeher can’t get to it. U.S. 2, Sweden 2.

Up next is Kristie Mewis, whose little sister, Sam, won the World Cup title with the United States in 2019. The elder Mewis exhales hard before she shoots with her left foot and sends the ball into the right side of the goal. The stadium starts to rumble. U.S. 3, Sweden 2.

The fans are starting to think this might never end. Nathalie Bjorn, the right back for Sweden, tries to shoot into the left corner, but the ball has other ideas. It goes flying over the goal and the Sweden fans sigh in unison. She buries her face in her hands. The momentum has changed. Peter Gerhardsson, Sweden’s coach, says after the game: “You’re just waiting. You want it to be over, and you want it to go your way.” U.S. 3, Sweden 2.

The U.S. fans go wild when Megan Rapinoe walks up. She had come in for Morgan as a substitute and was sure the ball would go straight into the back of the net, just as it had so many times before, including in the final of the 2019 World Cup. This is her final World Cup, her fourth one, after she announced in July that she would retire this year. But now, her shot isn’t even close.

She sends the ball flying over the goal. On the way back to her team, she smiles because she just can’t believe it. This is how an international career ends? She thinks she last missed a penalty shot maybe in 2018.

“That’s some dark humor, me missing,” she says after the game. “I feel like I joke too often, always in the wrong places and inappropriately, so maybe this is ha-ha at the end.” U.S. 3, Sweden 2.

Sweden’s Rebecka Blomqvist shoots and Naeher makes a superhero-like dive to knock the shot down. U.S. 3, Sweden 2.

The United States scored only four goals at this World Cup, and forward Sophia Smith scored half of them. She can win it for the U.S. team, and takes her time setting up. When she connects with the ball, it soars over the right side of the post. The win was there for the taking, and she couldn’t grab it. She buries her face in her black-gloved hands. She will not be the star today. Horan tells her later: “The best players in the world miss.” Smith explains to reporters later: “But you’ve got to remember, this is part of football. You get back up and it’s going to hurt. It’s going to hurt for forever.” U.S. 3, Sweden 2.

Hanna Bennison, a substitute for Sweden, has a chance to save her team from what had looked like disaster. She scores, sending her team into a frenzy. Gerhardsson says later: “Accept that you are nervous, so that being nervous doesn’t make you more nervous.” U.S. 3, Sweden 3.

There’s a rumble among U.S. fans when they see who is taking the next shot: It’s Alyssa Naeher, the goalkeeper. She has flipped the switch in her head and is now taking on Musovic, her counterpart. Her shot goes smack into the middle of the goal after Musovic guesses wrong. U.S. 4, Sweden 3.

Magdalena Eriksson, a seasoned center back, needs to score to keep Sweden alive. And she delivers to the upper right corner. Sweden 4, U.S. 4.

It’s up to Kelley O’Hara, in her fourth World Cup. She sprints to the spot. She wants to win this game and this tournament and has rallied her team to have confidence that it will do both. But her shot bounces off the right post and away along the baseline.

Sweden’s fans start to party, waving their blue-and-yellow flags and dancing. Naeher says she feels terrible for her teammates who missed: “They’ve trained for it. They’ve prepared for it. And, you know, unfortunately, those things happen. My heart hurts for them.” Sweden 4, U.S. 4.

Lina Hurtig, the forward who scored when Sweden humbled the United States at the Tokyo Olympics, can win it. She shoots toward the left side of the goal. Naeher leaps for it, hitting it once with both hands to make it fly upward. The ball goes up, and Naeher hits it again with her right arm while on the ground, stretched backward, to keep it out of the goal.

Did it go in, after all? Naeher insists she saved it. Hurtig raises her arms, and shadows the referee, Stéphanie Frappart, to make her case for a goal. The play is reviewed with cameras and tracking technology.

Then Frappart waves her arms: The game is over; it is ruled a goal. Hurtig takes off toward her teammates and the Swedish players run onto the field to celebrate.

The ball, indeed, had crossed entirely into the goal, according to the replay system. By the looks of it, the margin may be a millimeter. “I thought I had it. Unfortunately it must have just slipped in. But that’s tough. Ugh, we just lost the World Cup. It’s heartbreak,” Naeher says. Sweden 5, U.S. 4.