Your last Ask mentioning Exector just reminded me of the recent AQURIA partnership with ASW, and hoping that this all leads to a remake of the original game (or a sequel). More on topic: What do you suppose would be done with Happy Chaos if he were ever put into custody? Since he's arguably the greatest War Criminal of all time and is immortal, do you think they would try to seal him in a Dimensional Prison since execution is (currently) impossible? Or would they try to rehabilitate him?
I hope for an Exector sequel as well (with a little less Engrish)!
Back on topic:
Ariels tried to seal the Original Sage away inside her body but that only corrupted her and drove her further in to insanity (and she's technically NOT human by any normal standard, so going insane actually did a number on her).
Assuming there was a way to seal him away through Magic, chances are he already knows about it and knows the loopholes of the Backyard to get around it.
Actually, a few paradoxes have already tried to do him in but he took advantage of them (one of them was an Alternate Closed Timeline version of Nagoriyuki).
Technically speaking though, he WAS imprisoned for nearly over 100 years already... and is as old, if not older than Asuka and Frederick (Assuming Frederick was in his 30s as of 2016, he and Asuka both would be over 200 years old).
Assuming that being imprisoned wouldn't be a reasonable way to rehabilitate him, it would probably make more sense for him to end up doing Community Service under heavy supervision, but since he's capable of creating copies of himself, decoys no less... uncovering his hidden actions would be harder to do than just simply keeping an eye on him!
At the very worst, they'd have to develop an artifact that could seal his Magic Ki pathways to prevent him from casting anything, presuming one could even put such a collar on him successfully (or even with his cooperation). Similarly, most likely, to the Slave Bomb Collar that Potemkin used to wear, but more high tech and restrictive.
That all being said... Gears "technically die" when you destroy their central core: the bundle of Gear Cells that is the core of their ability to regenerate and obtain Magic Energy. The conduit for that Magic would still be there, even if they were a dormant corpse, but at the very least it's one way to destroy an immortal.
Still, Nagoriyuki can "see" the vulnerability in Happy Chaos' hand, so exploiting that weak point could destroy his body, but the full effects on him aren't fully known, if any exist.
Furthermore, they could treat him like how some Vampires are treated: carving his body in to sealed relics and spreading each piece to the far reaches of the globe, if not jettisoned out to space.
Gruesome and Inhumane as that prospect is, if it got rid of him, that would ease peoples' concerns.
Still, catching him in the first place would have to be done, even presuming it could even be done...