The episode begins with Aelita dreaming of her long-lost mother, Anthea, and waking upcovered in sweat. After washing up, she goes to see Jeremie in his room, but finds Laura with him, both working on the virus to destroy the Cortex. Aelita, hurt from the lack of attention from Jeremie, leaves, and Jeremie, hearing the door close while listening to music, asks if someone was at the door. Laura smugly denies it.
The next day at lunch, Jeremie and Laura soon finish the virus and tells the group about their plans to implement it within a week. Aelita is clearly out of it and explains she simply had a bad dream, causing Jeremie to ask why she didn't come to see him. Aelita says she didn't want to bother him, not revealing that she did in fact go to see him. She soon asks why they don't just use the virus now to get rid of X.A.N.A. for good and, convinces the rest of the group that there is no point in waiting. They decide to implement the virus that afternoon.
During gym class, Jim is teaching the class how to use the jumping vault. Seeing Ulrich and Odd talking, Jim orders Ulrich to be the first to use it. However, while doing so, Ulrich ends up spraining his ankle. When Jim goes to look for a sprint, Ulrich complains how he won't be able to go to the Cortex with a sprained ankle. Odd blames it on "Fat Jim" for making him use the vault, which Jim overhears. As punishment for his comment, Jim gives Odd detention.
When the group (excluding Ulrich who is being checked for his ankle) discusses these abrupt complications, Jeremie is reluctant to go ahead with the mission without Ulrich and Odd,but Aelita insists they have successfully pulled off missions with only three people before, and that there was nothing to be worried about. Jeremie and the group decide to go ahead with the mission.
William, Aelita, Yumi, Jeremie and Laura head to the Factory where the first three are transferred to Lyoko to attend the mission. While in the Digital Sea travelling to the Cortex, they are attacked by several mantas. The fact the mantas can now swim stuns the Lyoko Warriors, but Jeremie realizes they can now freely swim due to X.A.N.A.'s gained power. Destroying them proves to be far more difficult that initially thought, but they soon manage to destroy all mantas. On arrival of the Cortex, William remarks how good he feels about the mission only to be taken out by a Blok. Only Aelita and Yumi remain, causing Yumi to ask if they should now abort the mission due to their decreasing chances of reaching the Core. Aelita still refuses to give up and insists they can continue, despite Jeremie's added insistence they likely will not make it to the Core without being devirtualized. Aelita requests the Megapod and he hesitantly sends it over to her. Yumi and Aelita then drive over to the Cortex.
Meanwhile, Ulrich is in his dorm resting and reading a book. Odd soon comes in and declares they are heading over the Factory, giving him some crutches, knowing Ulrich is dying to know how the mission is going. Ulrich reminds Odd of his sprained ankle and that he would have to help Ulrich over there. Odd remarks he's been "taking care of him" his whole life. The two then begin to head for the Factory.
During the ride to the Core of the Cortex, Yumi senses Aelita is troubled and asks her about her relationship with Jeremie. Aelita answers that everything is fine between them but Yumi states that she feels a considerable tension between the two and feels something is wrong. Aelita says he's just under a lot of pressure and has been working a lot lately, hence why she was never with him often any more. Yumi says she can tell her the truth and asks if she holds a grudge against him, surprising Aelita. She soon responds that she lost both her parents and that the group is the only family she has left, but that Jeremie is "special" as he was the one who turned the Supercomputer back on and materalized her back on Earth and always took care of her. This is why she believes that he saved her life and will always be the one she can trust no matter what happens. While Yumi and Aelita have this conversation, Jeremie (along with Laura and William) overhear everything in the Lab. This causes Laura and William to smile while Jeremie is unsure of how to react.
The easy tension is cut short when they see three Krabs blocking the entrance of the Cortex. Yumi knows that they don't stand a chance with just her and Aelita to fend them off. Jeremie immediately declares he is bringing the two back, but William then asks if he really needs two people to inject the virus. Laura questions his thinking, and William asks if they "ever went bowling". Instantly understanding what he is implying, Jeremie believes the move to be too dangerous, but Yumi firmly insists she try it. With Aelita leaving the Megapod, Yumi then crashes herself into the three Krabs, knocking them off the Cortex. Jeremie soon devirtualizes Yumi just as she is about to hit the Digital Void, as Aelita passes through; now successfully inside the Cortex.
She soon accesses the mainframe and Jeremie transfers the data just as Ulrich and Odd arrive to witness it. As Aelita prepares to infect the computer with the virus, she finds a webcam stream of Tyron working in his lab. She observes the stream for a minute, telling Jeremie about the webcam. Jeremie tells her to enter the codes needed to activate the virus, warning her that if Tyron realizes the webcam is on and active, he will be notified of her presence. Just as she moves to infect the Cortex with the virus, she is shocked and troubled to see her mother in the office with Tyron, also working on part of the project. Aelita is stunned as the rest of the group wonder why she was with Tyron, having seemingly vanished without a trace. Jeremie soon decides to stop the transfer as Tyron realizes the webcam is on and prepares to send Ninjas down.
Laura asks why they prevented the injection of the virus and Yumi explains that the Supercomputer is the only way to maintain contact with Anthea, thus preventing them from finding out why she was working with Tyron and why she never made contact with Aelita, her daughter after all this time. Yumi tells Jeremie things have changed now as they have another mystery to solve regarding Aelita's lost mother and that they can't afford to destroy the Supercomputer until they do. After a moment of confusion, Aelita firmly insists on destroying the Cortex and X.A.N.A. along with it, but Jeremie and the others declare that the game has changed now and bring her back to Earth.
When Aelita rejoins the others in the Lab, she is upset about not destroying X.A.N.A., but her friends insist that they have time to figure out the mystery regarding Aelita's mother. When she tries to point out X.A.N.A.'s growing amount of power, Jeremie reminds her that as long as he doesn't reach a full 100% power, he is confined to Tyron's computer. He also reminds her of how desperate she is to reunite with her mother for so long, even risking her own life to meet her, and that they just couldn't allow her to make the sacrifice of never seeing her again by destroying the Supercomputer. Jeremie then offers if X.A.N.A. reaches 95% power, they will destroy the Cortex immediately, no questions asked. Aelita agrees to the deal.
- The original French name for this episode is Obstination.
- The title,"Obstinacy," refers to Aelita's fierce determination to get rid of X.A.N.A. as quickly as possible.
- This episode possibly implies a more-than-platonic relationship between Jeremie and Laura.
- Mantas are shown to be able to survive the harsh environment of the digital sea. Also, Mantas use their mine-hatching ability for the first time in the fifth season.
- Ulrich, for the first time ever, injures himself while participating in athletic events such that he cannot participate in a mission.
- Bloks appear on the Cortex for the first time.
- Anthea is shown to be working under Tyron. However, it is currently unclear whether she is working for him by force, of her own free will or if he is blackmailing her.
- Odd mentions that he left Kiwi with his sisters, explaining the dog's disappearance.