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In episode 10 (Misaka Mikoto) Misaka 10032 finds a black cat. Touma tells her to choose a name for the cat, since she was the one who found it. She says that the cat will be called dog, which Touma dismisses, proposing that she chooses a "more majestic" name. So she says Tokugawa Ieyasu, which Touma dismisses as "too majestic". She than chooses Schrödinger, which, once again, Touma dismisses as being "a taboo for cats".

It was not clear to me which became the definitive name for the cat, although I have only seen up to episode 14.
Is it ever said, either in the anime or in the novels, which name ended up being chosen? Or did she end up not naming it?

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It does not appear as though the viewer (or reader) is ever told the true name of Misaka 10032's cat in either the anime or the novels (though I didn't check the manga).

From a translation of the novel (volume 3):

"That's right! A name! This is your cat, so take responsibility and give it a name!"

(Some irrelevant stuff omitted)

"Dog."

"Hah?"

"Misaka is naming this black cat Dog. ...Dog even though it is a cat. Heh heh."

(Some irrelevant stuff omitted)

"...No, um...Please give it a more serious and dignified name that fits the type of animal it is."

"Then Tokugawa Ieyasu, says Misaka after reconsidering."

"That's too dignified! Wait, are you the kind of character that pretends to think about things but doesn't think at all!"

"Then what about Schrödinger?"

"Hell no! Even if it was just an example, some professor who would happily come up with a story about sticking a cat in a box and spraying poison gas inside couldn't have liked cats!"

In the end, they decided to name the cat later. However, Kamijou had a bad feeling that they would be unable to agree on a name later either and she would end up literally nicknaming the cat "Later".

It does not appear to be mentioned later whether or not they did agree to name the cat, and it is always referred to as "black cat" or simply "the cat". From the Toaru Kagaku no Index Wiki:

It is not known if Misaka 10032 has decided on a name for the cat. Furthermore, since all the tentative names given by Misaka 10032 can be used on males, it can be assumed that the cat is also male.

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