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Looks like a few jagged edges on the rock. Could he be exploring his new home and brushing his fins on every surface? Looks like you have all live plants, but are there any plastic ones?
Tail/fin rot. This is some sort of bacterial and or fungal infection known to attack betta fish, mostly to do with bad water condition. Personally, I would not medicate but do partial water changes every other day or so and see if he improves.