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After being damaged in battle (by a bullet IIRC) , Accelerator had to start to use a necklace for being able to use his powers. What does this necklace does? Why does Accelerator need a necklace to use his powers?

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So, Accelerator's ability requires a ton of calculation and mental power in order to function. Seeing as how Accelerator himself possesses extremely high intellect, he was able to perform these calculations by himself no problem.

However, once he got shot, his frontal lobe was greatly damaged. He lost many of his motor skills, including the ability to walk. He also lost a huge amount of his mental capacity, hindering his ability to speak, read, and perform normal mental functions. Naturally, this affected his ability to make the calculations required for his power.

The device that he wears on his neck was developed by Heaven Canceller. It's connected to the Misaka Network, and the Misaka Network performs the calculations for him. This allowed him to regain enough mental capacity to think, speak, and read normally, as well as granting him the ability to walk again. It also helps performs the calculations that are needed for his powers to work, although it's stated that the calculation ability provided by the device is less than half of what he once had.

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