As Simone Biles closed out her final transcendent floor routine and acknowledged the judges, she held her arms above her head in a salute for as long as she could. It was partly a cheeky, sarcastic gesture, a reference to the deduction she seemingly received in the preceding final for not properly saluting to all judges, but it was also perfect. The Bercy Arena, filled again for one last glimpse of her this year, responded by saluting her in turn with a long, heartfelt ovation.

Although the American did not close out her Paris Olympics with a golden picturesque finish in a chaotic last day of artistic gymnastics, her final day of competition here was rather an exhibition of the sportsmanship and humanity that has accompanied her greatness. After a fall on the balance beam led to a fifth-place finish, Biles won a silver medal on the floor exercise.

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With gold medals in the all-around, team and vault competitions, plus the silver medal on the floor exercise, the 27-year-old leaves Paris with four more Olympic medals. She is now the joint-second most decorated female gymnast at the Games with 11 medals in total and she has also extended her own record as the most decorated gymnast of all time, male or female, with 41 Olympic and world championship medals.

Much of the final day was a demonstration of how difficult and fraught elite gymnastics is. In the balance beam final, four gymnasts fell and another, the silver medallist Zhou Yaqin, lost her balance on a leap and put her hands down on the beam. As so many of her peers faltered, Italy’s Alice D’Amato produced a cool beam routine that would seal her first Olympic gold medal.

Having been drawn seventh in the lineup, Biles stepped up to the beam knowing she could win gold with just a solid performance, but she instead fell off on her backwards layout stepout somersault. Despite her otherworldly talents, she is still human. She is still susceptible to the nerves and stresses of every gymnast. After competing in 17 routines across a frantic nine days, the physical and mental fatigue has long set in. Her humanity is what makes her success so special.

The afternoon ended on the floor exercise and, with a medal on the line, Rebeca Andrade produced her very best floor routine of the Olympics. Having incurred significant landing deductions all week with her complex opening pass, a full-twisting front layout somersault to a full twisting back tumbling pass, this time she stuck it cold. Her landings on her subsequent tumbling passes were excellent and she set the bar high with a score of 14.166.

Although Biles’s first and third passes – her triple twisting double-back somersault and the Biles – were both spectacular, Biles landed directly out of bounds on her second and fourth passes, which each incur a 0.3 deduction in addition to the landing deductions. She finished with a score of 14.133, moving into second place behind Andrade, who clinched her second Olympic gold medal after winning vault in Tokyo. As soon as Biles received her score, she walked over to Andrade and congratulated her.

Throughout the past week, Biles has been pushed to the limit by the Brazilian, who has offered her less room for error than in any competition over the past 10 years, and Biles has responded to that challenge brilliantly. Finally, Andrade has beaten her. While Biles has further cemented her status as the greatest gymnast of all time, Andrade is building a special career as one of the greats in her own right.

The floor exercise final ended with more drama as Jordan Chiles, Biles’s USA teammate and training partner in Houston, was initially in fifth place. Chiles had been awarded a lower difficulty score than the routine she had attempted so USA Gymnastics submitted an inquiry about Chiles’s difficulty score.

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As Ana Barbosu of Romania celebrated what she thought was her first Olympic bronze medal, a Romanian flag wrapped around her, Chiles’s inquiry was accepted, her difficulty score was raised and she leapfrogged her in the final standings to become the bronze medallist. Biles reacted to her close friend’s first individual Olympic medal with more glee than in any of her own medals combined.

Despite being beaten in a floor final for the first time in her career, Biles immediately took the defeat in her stride. As Andrade was presented to accept her gold medal in the first Olympic gymnastics podium with three black medallists, both Biles and Chiles turned to face Andrade and they bowed to the new champion. “She’s so amazing,” Biles said. “She’s queen. First, it was an all-black podium so it’s super-exciting for us. But then Jordan was like: ‘Should we bow to her?’ And I was like: ‘Absolutely.’”

Three years after suffering from the twisties, being forced to watch her rivals from her stands and being subjected to waves of personal attacks just for recognising that she had a mental injury, Biles’s redemption tour is complete. Even after an imperfect climax, she will leave Paris having re-established herself as the best gymnast in the world after one of the most legendary Olympic comebacks of all.

“I’ve accomplished way more than my wildest dreams, not just at this Olympics, but in the sport,” she said. “So I can’t be mad at my performances. A couple years ago, I didn’t think I’d be back here at an Olympic Games. So competing and then walking away with four medals, I’m not mad about it. I’m pretty proud of myself.”