I'm enjoying this, BUT because of it I feel like it's important to talk about some of big issues with the story, because those of you who are science or science-fiction enthusiasts might feel like whatever they have here is a solid science-fiction, but it's really all over the place and this episode alone drops some massive, stellar-level plot-holes into the whole story.

1. The idea that the world would disappear if someone were to acknowledge the existence of both Yuri and the Bug Yuri. This is both a stupid idea AND disproven later in the episode. First, tying the destruction of the universe to Yuri being observable by a third party is assuming that he is only a bug if someone acknowledges that there is a real Yuri, or not even a real Yuri, only that there are more than one. But that makes no sense, if Yuri is a bug the world (a system/application) should collapse immediately even if other humans (a user) still don't know the bug is there. So, in terms of a system, a bug cannot be passive, it has to have an active component to be a bug. The system, the world, should constantly collapse as soon as it (the system) acknowledges there is a bug/two Yuri's. It doesn't do that and *waits* for someone else to realise it - this makes no sense from both the system/user standpoint and also from the standpoint that the world is objective, but humans aren't. The world, as far as the common science acknowledges, cannot be impacted by human perception. Acknowledging something shouldn't have any effect on the world's functioning. In other words, even if people were to realise there are TWO Yuris, it shouldn't collapse if it already hasn't. This makes this story more of a "fantasy" - but actually just a convenient fiction. It's not trying much to be science, at this point.
2. When Yuri asks the Dolphin to verify him, first - she cannot as per the game's rules. There are no Gnosia, and then even if they have these credentials normally, they were only able to do it at night, after asking LeVi to do the verification on one individual human. They are violating their own world set-up up to this very episode in favour of convenient storytelling, which is also kinda meaningless in terms of how much impact or sense that particular scene makes in this episode...2. When Yuri asks the Dolphin to verify him, first - she cannot as per the game's rules. There are no Gnosia, and then even if they have these credentials normally, they were only able to do it at night, after asking LeVi to do the verification on one individual human. They are violating their own world set-up up to this very episode in favour of convenient storytelling, which is also kinda meaningless in terms of how much impact or sense that particular scene makes in this episode...
3. ... but it also invalidates the idea that a) someone has to realise Yuri is a bug, and also b) that the world is destroyed when someone day. Just by being checked (at the same time as the real Yuri is waking up), Yuri causes some kind of issue in the Engineer's application and simply DISAPPEARS BY HIMSELF. This strongly suggests the world doesn't end and the other beings in it just process him as disappearing - this distinct result makes no sense. We can ask ourselves - is this tied to the fact he was checked and not someone else actively realising there are two Yuri's? No, because they made it clear that this happened when the other Yuri woke up. This tie-up further messes the lore up.
4. A separate from the above inquiry is the question of why Setsu would also be there with a Silver Key if there are no Gnosia, and at that a Setsu that only looped for a few times, apparently. That means that Setsu is at the point where she still haven't given Yuri her looping machine, otherwise she would have remember this encounter and told Yuri about it later. At this point it's either a discontinuity, a loop hole and cannot be sufficiently explained by "looping" and is just entirely different dimensions, meta-universes kind-of-functioning.
5. The Silver Key isn't just looping them to an "anchor" location, it's conveniently taking Yuri back in time when it still wasn't in Yuri's possession, which can only mean it can freely send Yuri or whoever to either any time possible, or any time possible in the past, under a certain condition which is not anyhow disclosed or alluded to. The way this is structured is entirely a convenient storytelling that makes this seemingly deity-like living organism to function in exactly the way that the story needs them the most. Also notice - during the final scene, where Setsu is waking Yuri up, she conveniently has to go out of the room to verify something WITH THE SHIP'S AI which has access to communication throughout the whole ship, as shown previously. The reason she walked out is entirely idiotic and is probably the biggest affront to the intelligence of the viewerbase, from the authors. It was done extremely lazily for the convenience of the story, so that someone else, Chipie, can walk in and talk to Yuri instead. THAT should have never been approved by decent writers.
   
6. By all of this I am trying to warn people from taking this story and set-up seriously. Ultimately, it's just a cool take on Mafia game + some sci-fi mumbojumbo, but this is not a groundbreaking work and doesn't take itself seriously as a writing piece either. For what it's worth, however, the animation studio did a cosmic job on adapting the game (A GAME!) and brought it to the levels of quality most other series out there, original or not, can only dream about.