But none of that mattered, because the mere presence of Kerr changes everything. Her absence had hung over Australia from the moment it was confirmed, a couple of hours before the team’s opening game, that she had sustained a calf injury and would need a week or so before she could even begin to think about playing in this tournament.

A player of her quality, of course, would always be a loss, tactically and technically and in a sporting sense, but the blow was more than anything psychological. There is no shortage of ability in Tony Gustavsson’s squad: Foord, Raso and Mary Fowler — imperious against Denmark, the instigator of both goals — all stand among the game’s top rank.

But Kerr is Australia’s totem, its talisman. That status had been accentuated in the months building up to the tournament. Kerr was the face of the World Cup, the player depicted in murals and on magazine covers. It felt, at times, as if this was her tournament, her chance to cement her legacy, as much as it was Australia’s and New Zealand’s.

The best measure of her significance was the extent to which, without her, Australia had seemed uncertain, diminished. It labored against Ireland, lost against Nigeria. Only when her teammates seemed to grasp the nettle, to interpret their task as staying in the competition for long enough that she might get to play in it, did things change.

They powered past Canada. After a nervy start at Stadium Australia, they swatted Denmark aside, too. Their job was, in effect, that of a cycling sprint train: They had to deliver Kerr as close to the line as they could.

The nerves had spread from the team to the country. The challenge, for the fans, was to continue to believe, even though the player in whom they had ultimate faith was missing. Australia has embraced this tournament in a way that has, if anything, exceeded its own expectations. On Monday, the Sydney Opera House was illuminated in gold and green. The fan parks were packed. It just needed to stay the course until she returned.

The Australia team is, after all, a markedly different proposition with Kerr than it is without her. The long, slow process of coaxing her back to something resembling fitness — and, to be clear, she will still be some way short of that now — has been a cautious, tortuous one.

She was back in the squad against Canada, though she did not warm up. That was step one. She has gradually increased her training load. She was on the bench here, too, as everyone expected, but then she appeared, after little more than an hour, and started — with no great intensity, it has to be said — to go through a gentle warm-up.

That might have been enough for the fans, for the players, the mere sight of her a tonic, a glimpse of a brighter tomorrow. But then, with 73 minutes played, she walked back to the bench, tugged at her socks, tightened her laces, and stood with her hands on her hips. She was ready, or as ready as she could ever be.

The stadium, when it realized what was happening, stood and cheered. Flags waved. Fans bounced on their heels, the excitement almost too much for them. Australia was already through. But now Kerr was back, and everything was different.

There was a moment, a few minutes after her introduction, when she fell awkwardly as she contested a high ball. The fans, delirious by that stage, fell silent, waiting to see what had happened. All eyes fixed on her. Slowly, she got to her feet, wincing slightly, trying to walk it off. She picked up her pace, stretched out her leg. She put out a hand, letting her teammates and the coaching staff know she was fine. No lasting harm done.

The exhalation almost had a physical form. Kerr has arrived. Now the tournament, the one she has been waiting for, can begin.