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Rudger is from what I call the 'Family AU' - as in, certain characters of the cast are related, but there was never any overt clarification or anything on the matter in the anime, so it's stuck as a permanent headcanon. Can't be proven (although there is substantial circumstantial evidence) but also can't really be disproven, as nothing is known about Crow and Jack's family.

SO. What happened to R here? Well, he woke up.

That might not seem important, but the fact that both Rex and Rudger woke up alive is the key, seen as the last thing they were aware of, they'd been defeated as Dark Signers, and fully expected to pass on.

They did not expect the Crimson Dragon to revive them too, and for a certain white-haired ex-Dark Signer to have overheard some conversation of a sensitive nature... and then broadcast it to the entire group.

...

Rudger is from a couple of weeks or so after that. He's still having trouble talking to Jack, although it is getting easier with time and patience - a LOT of patience. Rex is in the middle of co-ordinating the construction of a new Daedalus bridge (Crow insisted they keep the name) and the reconstruction of Satellite.

Rudger's still trying to figure out where he fits in, but later on he'll go back to science and working on Momentum, and eventually will end up helping Yuusei develop Fortune.

...

Appearance: Rudger looks far more like his younger self does in flashbacks than his 'present day' Dark Signer self, having regained the colour in his hair and a paler pallor, the dark skin tone having been brought on by the poison of the conflicting marks.

Overall, he looks much like an older version of his original self. Since he amputated his arm after he'd already started becoming a Dark Signer, being revived has caused it to reappear.

Interestingly, Rex has regained his mechanical arm, and prior physique, as that was how he appeared before he allowed Rudger to defeat him.