Okay so this is all very confusing when it comes to Eugeo possibly being alive in UR so I'll try to see the points for both possibilities.

A thread :
DEAD :

We'll start by trying to find all the arguments that goes on the direction of him being dead :

First, when Eu/geo died against Quinella, it was clearly mentionned in the LN that the Lightcube his fluctlight was in got freed, which means he was erased from the UW
HOWEVER his memory still lives in the Blue Rose sword. This is a case extremely similar to Yuna in Ordinal Scale as her hologram (the personality part) was made of her memories and the whole purpose of Dr Shigemura was to collect as many memories of her in order to recreate her
in a form that would obviously be a fluctlight since he was very close to Kayaba. Plus I don't see any menas of keeping her being without it being a fluctlight. Yui can't be a counterargument since she is a Top Down AI.

Another argument is that this entire part takes place
YEARS after the UW war, and unless Eugeo's life froze, he'd be dead. The weak point of this take is that he got transformed into an Integrity Knight, who have that frozen life. Yet his life reached zero and he died. Eldrie too was an IK and he died when his life points reached 0
A possibility (very unlikely but eh, let's be wild) is that Eugeo, somehow had a child. The only thing I remember being against the Taboo Index was kissing in the lips while being unmarried so people could have chilren. Maybe he had a child with Tiese but it seems too unlikely.
Back to more plausible things. We shouldn't forget Bercouli had a son and that might play a lot. Since both Bercouli and Fanatio were strong, it's possible to imagine their child being as strong as them & creating another kid even stronger. Then again, I'm not sure about this one
ALIVE :

Now comes the points that would suggest he is alive. Since his memory was still in his sword and there is a possibility t recreate someone with memories of them like it was shown with Yuna, then his memory is the most likely source that could explain his reincarnation.
I believe some sacred arts, which are just like shell commands in the end, could trigger such a "memory resurrection" process. That could also be a way to make OS canon if the process used was the one made for Yuna (although OS has some compatibility issues).
Also, (thanks @eeugeo for mentioning it) but Kirito holds the Blue Rose Sword, which is Eugeo's memory vessel. I believe he'd have heard him at least once since he is in the UW. Hence his memory might not be in the sword since he got confused when he saw that Eugeo-looking person
and started questionning himself.

Plus today is Eugeo (and the Rulid Trio to be more specific)'s birthday, hence releasing that in this specific day is not coincidential and surely highlights the fact that this is about Eugeo.
On a more technic note, SAO was meant to end with Eugeo dead, and Kawahara himself explained in the vol 14 postnote that he couldn't rewrite this scene with Eugeo surviving - as if he HAD to leave. That could be a good argument for the DEAD part but since Eugeo became popular,
it is possible that Kawahara wants to bring him back to life since he isn't human ad it is possible to think of a possible way of him surviving !
This is all I could think of right now. It's 5AM and I'm very excited !

Feel free to leave your thoughts, correct me or even dissagree with me !

Stay cool !
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