HISTORY
Accelerator's background is dropped sparingly across a few novels, some of it being made explicit and other parts being strongly insinuated.
Like all other espers, Accelerator was presumably born to normal Japanese parents who registered him into the Power Curriculum Program at Academy City around the age of five. He doesn't remember his life before Academy City, nor anything of his biological family. At most, he remembers his real name is very Japanese and common, and that his surname consists of two characters and his given name three.
After going through the program and registering with the school, it's strongly implied that his parents abandoned him and he became a Child Error (a term used for children whose families use Academy City as a dumping ground for unwanted kids). After undergoing the Curriculum he was placed in a "special class," presumably referring to the Special Esper Dual Modification Technology Research Institute, a combination school and research facility that studied the possibility of espers developing more than one ability. It was a notoriously cruel and inhumane place that experimented on the brains of espers. Early on it became clear espers could only develop one ability, but out of stubbornness and a desire for fame the Special Esper Institute continued its research, mainly using Child Errors as they had a tendency to slip through the cracks in the system. There were rumours that bodies were disposed of on the grounds, though Accelerator states that it was more like a garbage dump to dispose of living humans. Narration and Accelerator himself states that he saw and experienced horrible things while living there.
Eventually, the city's law enforcement Anti-Skill caught wind of what was going on in the facility, and shut it down. Accelerator, having been there until he was nine, had been moved to another research facility prior to this, so he never learned of its shutdown until years later.
Around the age of ten he presumably began working with a researcher named Kihara Amata, of the infamous and ethically questionable Kihara family. It's stated that Amata was the one to develop Accelerator's ability, though he hated Accelerator even when he was a small child and wanted to kill him (only holding off because of Accelerator's young age). Accelerator, meanwhile, was apparently attached enough to Amata that he began to mimic his personality and sense of style.
Also around the age of ten an incident occurred in which some classmates of his tried to attack him. (He never mentions classmates before or after this singular incident, and never mentions having friends at all, so the insinuation is that he's never fit in with his peers.) Said classmates hit his reflection and had their bones broken. When a teacher tried to intervene, they were injured or killed as well. Another group of adults followed, and things continued to escalate, with Anti-Skill and Judgement outright attacking him with weapons and powers. The entire time he was terrified.
The incident ended when, facing down unmanned helicopters and Anti-Skill in robotic suits, Accelerator realized this was only going to continue until the entire world got involved. He was a kaiju, a monster, and if he didn't stop this he was going to get the world destroyed.
So he surrendered, and Academy City shoved him into a classroom on his own in a school with two thousand students, where he sat all day, alone, not even taking part in things like school events or extracurriculars. Meanwhile, as the Special Esper Institute could no longer handle him, he floated from research institute to institute (Imaginary Number Institute, the Wisdom Institute, a Kirigaoka affiliate and so on) every two months, each time being hated and feared by the researchers and moving on when the institute realized it couldn't handle his power.
During this time, he reached his official rank, the #1 Level 5, and began to get attacked at all hours by other espers wanting to prove their strength, and assassins from organizations and countries external to Academy City.
Eventually he landed on the Level 6 research institute Yoshikawa Kikyou was part of. He was well received, though the researchers apart from Yoshikawa continued to fear him, and he stayed longer than a couple of months. Likely a few years, given timelines.
A large number of institutes were involved in developing the Level 6 Shift Experiment. Espers are categorized from 0 to 5 based on their powers, with Level 6 only being theoretical in nature and considered to be on par with God. According to the supercomputer the Tree Diagram's calculations, Accelerator is the only esper who can reach Level 6 and remain stable (AKA not go supernova), with it taking 250 years at his current rate of development. In order to speed things up, researchers calculated using the Tree Diagram that if Accelerator were to fight the #3 Level 5, Misaka Mikoto, 128 times, he would achieve Level 6. However, as neither of these options were possible the Experiment utilized a previously defunct project called Radio Noise, in which attempts were made to clone Mikoto and create more Level 5 electromagnetic espers. Radio Noise had been shut down when it became clear that cloning technology was limited to the creation of espers at a maximum of Level 3.
Radio Noise was restarted, and these Level 2 and 3 clones of Mikoto were instead used for the Level 6 Shift. Instead of fighting #5 128 times, Accelerator would fight 20 000 Level 2s and 3s in predetermined combat scenarios. Researchers claimed to Accelerator that he would not be fighting actual people, but dolls, the equivalent of fleshy robots. The clones, the Sisters, were connected in a hivemind situation using their electromagnetic ability, and share memories and combat experience with each other. Over time they would grow stronger, pushing Accelerator until he reached Level 6.
Accelerator agreed to this out of a desperate desire to no longer be just the strongest, but to reach an insane level of power that just the thought of challenging him would be utterly ridiculous to everyone. He hoped that by becoming invincible, he wouldn't have to fight anymore and for the first time in his life, have friends and a family.
So he began killing the Sisters. Early on, even with reassurance from the researchers, he tried talking to the clones. This was in fact going against the experiment's rules, as once it began none of the researchers or Sisters made an active attempt to talk to him. But he did it anyway, because he wanted to see whether they were real, sentient beings or dolls like he had been told. Specifically, he wanted to hear even one of them say that she was scared and didn't want to die, as he would have stopped if that had been the case. But the Sisters performed to their specifications, and gradually Accelerator sunk deeper and deeper into the violence of the experiment. Eventually his attempt at conversation turned into verbal abuse, trying to scare the Sisters into stopping. However, by the time the experiment is shut down (with #10 032) he wouldn't have been able to stop, even if a Sister told him she didn't want to die.
In May the experiments move from indoor locations to outdoors. By this point they are on trial 9803. A week or two before August 23rd (as depicted in the Railgun anime), #9982 is killed by Accelerator in a train switchyard, and he has an encounter with Mikoto, meeting her for the first time. He attempts to connect with her as a fellow Level 5 but fails when Mikoto attempts to kill him several times for having killed a Sister. However, she is no match for him. Before he can kill her a clean-up crew of dozens of Sisters arrive and dissuade him, explaining that killing Mikoto would throw the entire experiment off. He shrugs and leaves, and Mikoto has a bit of a breakdown.
The light novel Toaru Majutsu no Index Old Testament 3 picks up from here.
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