Hello author, first of all great fanfic I'm still reading (more specifically chapter 34) but I came to leave my thoughts: (being more suggestions) there should be a training arc more being about the physique not the powers because the Accelerator is a lot of Op,but if he doesn't have knowledge about something (magic for example since it works with different laws)he simply becomes very weak and can lose very easily as he was demonstrating in his fight with aiwass.But it is not about the powers that I speak more specifically his physique because he has a weak physique and his power does working it is useless, as it was shown in the battle against kihara and touma in which the fight against kihara he was slaughtered.More like agr in fanfic he has the imagine breaker I think he should train his physique because that is his only weakness (not counting his knowledge) plus Accelerator having a physique Strong would be nicer, he wouldn't need to abandon his powers and he's used when needed (if you decide to heal the brain that would be interesting) because it wouldn't depend so much on the of his ability, not to mention that this is toaru then that have fights like this (especially now with him in place of the protagonist)he could learn martial arts like the system,(Russian martial arts),Taekwondo,Karate etc... since one of the things present in toaru is Touma's (protagonist) fighting style, a very skilled street fighter and his imagine breaker skill,more now that the skill is with Accelerator it loses some of the essence of toaru, because Accelerator only uses it against things he has no knowledge of (magic) and esper skills when He doesn't want to calculate the vectors, so I think it would be cool to have a bow like that or something like him training physically (like seriously if he's physically strong and almost impossible to defeat He) and would come back with little of the fight we know from toaru(noises of Touma punching things) the agine breaker was going to have more prominence, and he would use it more by mixing it with his fighting style with his vector manipulation ability.
About writing: well english is not my language, i use a translator so i can't give tips or anything like that, not to mention i'm not a grammar guy but for me it's fine the way it is.
About the story: I don't have much to say, just that your story is very good, would you like to write the crossover I think it would make it cooler and open a new path for the protagonist to get stronger/Op (if it's your intention to make him stronger)Since there's going to be a crossover with fate, I recommend you read a fanfic called A Nascent Kaleidoscope.You can take inspiration to make the protagonist stronger in that style if that's the case.
Character development: I think it could be in the arc From(Othinus)the New Testament being the Accelerator who would experience the suffering of countless "worlds" there is a fanfic called The Esper and The God Magic,Very good and only has 10 chapters (I'm waiting for updates 😭) more that shows a development in the character (Accelerator) not to mention the Original othinus had a lot of contribution to Touma's character development,That's why I think you could do something in this style more precisely in the arc new testament Making Accelerator a better anti-hero than he is now.
Harem:Othinus (waifu always has to be in the present in the harem logically😎)and any other girl you find interesting, since harem is one of the things that can't be missing in toaru I remember a comment in chapter 9 wof the fanfic Who had a vote for harem that said "toaru no and toaru without a harem" (I don't remember it could have been my imagination) to say "it makes sense" to have a harem because I think it was said in the novel that one of the reasons for Touma's harem is also the imagine breaker For he nullifies the red thread of fate.
Fights As I said earlier Accelerator doesn't have to use his ability just to not be too dependent on it because in the cases mentioned before he would be useless and thus also not leaving the imagine breaker just in the background so he would also have to learn to fight normally and thus gaining experience to better use the Ib And so he would mix it with his fighting style using his power.
Skills/powers: I don't know if you plan to give him more skills/powers if it's 1 could start with The Magic of Fate Mages(not counting True magics) Magecraft Which is a type of magic that would be very Op if it were used by the espers because it involves calculations and sciences Not to mention that Accelerator could learn a lot of Magecraft and seriously it would be quite Op if combine Magecraft+Vector manipulation=Op
Since fate magic is not that different from toaru (not true magic since it would be something on the level of a magical god) I think That in the novel, Index even says that "magic and science with their own laws" (something like that).
Now if you give Accelerator new skills (not counting what he can get like black wings, white wings, platinum wings etc...)There's a character I know called Kugamawa misogi he's from a (very good) manga called Medaka box I think anyone who likes Toaru will like Medaka box, it's worth reading Medaka box I believe Despite the beginning being a bit boring (for me) it gets cool with a while.Enough of advertising and talking about Kumagawa Misogi's abilities(a different character Similar to Accelerator) those who like Accelerator I believe will like this character too.
(I'll put his skills in another comment)
I would say more things but for now and only and they are just ideas, suggestions you (author) do not You only need to follow them if you want
I admit that I would like my ideas to be put into practice despite being a bit greedy on my part, but even without that the fanfic is good and you can post it whenever you want, just don't abandon the fanfic.
Thanks for reading have a nice day (seriously a lot of patience to read this)mucho texto.