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#But that's up to Eolyne to decide
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I believe that Eolyne is Eugeo, but not in the way that you think he is. {{An in depth Theory}} -This is either 1000 IQ, or I’m losing my God Damn mind. You pick.
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Eolyne is a reincarnation of Eugeo, but they’re not exactly the same person. Eugeo is the core of who Eolyne is, but Eolyne is also a new fraction of Eugeo. He’s his own person. Because people change, they don’t remain the same. Eugeo might be the core of Eolyne, but Eolyne is also his own unique person and he’s lived a life all his own.
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Think of it like this, Eolyne is essentially an Integrity Knight in a way. His memories and feelings of Kirito have been locked away, but within Eolyne, Eugeo is still there - just dormant.
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Let’s talk about their fluctlight ids.
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Eugeo’s ID is NND7-6361
and
Eolyne’s ID is NCD1-13091
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Okay, so they’re very different, so like there’s no way they could be one in the same right? 
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Well interestingly enough, in the novel the Blue Rose Sword’s object id is:   DI:WSM:0131. 
_ _ {{The other id for the Blue Rose Sword we see is WLSS703, but in the light novel it’s: DI:WSM:0131. And since the novel is technically the true canon -because the anime did change things around and omit events, and such- we are going to go with the end all be all that the true ID for the Blue Rose Sword is  DI:WSM:0131.}}
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If you recall Eugeo became one with the sword so technically his fluctlight id would change, right? Meaning that his fluctlight would become one with the sword, I would think. 
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So wouldn’t that change some of Eugeo’s id to: DI:WSM:0131? -for the time he was merged with the sword, before and after his death.-
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Now if you look at Eolyne’s id you see it has the numbers: 13091
If you remove the 9 you get 1301.
If you move the numbers around from 1301 and you put 0 in the front, follow it up with 1, then 3, then 1 again you get
0131. -This matches the Blue Rose Sword’s ID. 
Now if you take the two 6′s in Eugeo’s fluctlight id and you add them together, you get 12! And if you subtract 12 by the number 3 in Eugeo’s fluctlight you get, you guessed it: 9!
And the number 9 is in Eolyne’s id.
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Now you ask: Well what about the 1 in Eugeo’s fluctlight at the end, if you subtract or add that to 9, then wouldn’t the number change to 8 or 10, instead of 9? Well there’s a simple explanation for that. 
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Eugeo and the Blue Rose Sword’s ids come together to make Eolyne’s id. 
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The system would take the two ids and merge them together, which means that some numbers could be dropped completely or added on, so the 1 in Eugeo’s fluctlight id was dropped and replaced with the 9 after adding 6 to become 12 and subtract the 3. Then the dropped 1 from Eugeo’s fluctlight id replaced the number 7 in NND7 to become NCD1.  _ _
So now you’ wonder, well what about how the middle N in NND7? How did that N change to a C to make NCD1?
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To answer your question...I actually don’t know. There’s no C in Eugeo’s fluctlight id or the object id of the Blue Rose Sword. My only guess is that the C is unique to Eolyne and Eolyne alone. It’s a unique difference, one single letter changed, to differentiate Eolyne from Eugeo.
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So... to get to the point:
Basically the system merged Eugeo and The Blue Rose Sword together as one, and that entity became Eolyne and the N to C letter change represents that even though Eugeo is the core base of Eolyne, Eolyne is his own person as well.
And I realize I might be reaching or looking too far into this, but it’s really too peculiar of a coincidence that taking Eugeo’s Fluctlight id and The Blue Rose Sword’s id and putting them together matches somewhat up with Eolyne’s fluctlight id - just that the we had to rearrange numbers, add and subtract some and swap out a letter to create Eolyne’s id.
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There’s also far too many coincidences to overlook that Eolyne and Eugeo are one in the same, but also different.
For one Kirito keeps remarking how similar Eugeo and Eolyne are. They both have blonde hair, their eyes are both green, it’s confirmed they share the same voice actor for crying out loud. And I bet once the mask comes off, we will see that Eolyne’s face is identical to Eugeo’s or nearly identical.
Now back to explaining how Eugeo and Eolyne can be one in the same, but also not the same and different. You see, Eolyne is his own person too. He’s lived his own life separate from the life he lived as Eugeo. But he will still have Eugeo’s memories and feelings from his past life.
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{{He might recover those memories or feelings sometime during The Novel and Anime, or they might remain dormant forever. But let’s just assume he’s going to regain those memories and feelings, and go from there.}}
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So
Eolyne is technically both Eugeo and not Eugeo at the same time, which is very contradictory. But if you take into the fact people change and become someone new every day, then it’s easy to overlook the contradiction. 
Eolyne is basically an anomaly in this way. The core of his fluctlight was built up from the remains of whatever was left of Eugeo and the Blue Rose Sword - but the other part of him is his very own unique part, not relating to Eugeo at all.
So I believe if/when Eolyne recalls his memories of his past life and the feelings that go along with it, he will either decide to live as he always has - as Eolyne, or Eugeo, or even both.
And that no matter who Eolyne chooses to live as, Kirito will accept and support his choice. Because Eugeo/Eolyne will always be Kirito’s best friend, no matter what life they’re in. And even if Eolyne decides to live just as Eolyne, he will still cherish and love Kirito just as Eugeo would have.
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{{Also it’s kind of ironically amazing isn’t it? That this theory here sort of parallels the theory that Kirito’s dual wielding is a metaphor for him being bi, while Eugeo becoming Eolyne can be a metaphor for a trans person transitioning. AKA Eugeo/Eolyne deciding to live as either Eugeo, Eolyne or both. Just like how a trans person can choose to live as a male, a female, both, or even niether.}}
{{I know a lot of people will argue with me on this front, but like you cannot deny that Kirito’s dual wielding can be a metaphor for bisexuality, and Eugeo’s {{possible}} reincarnation could be a metaphor for transitioning. Even if they’re not exact in terms of actual bisexuality and transitioning, the implications are definitely still there.}}
I know I’m probably reaching. I know this is probably just my desperate shipper heart of both romantic Yujikiri and platonic Yujikiri because I enjoy both - trying to confirm it to myself that Eugeo and Eolyne are the same, even though different. I am well aware of it. There’s probably several holes in my theory, and I accept that.
Just please let me have this, if nothing else. Please just let me believe that the creator Reki Kawahara knows what he’s doing and that he planned this all along, and I’ve cracked the case. _ _ To me it’s just too much of a coincidence, that you can mess around with the arrangement of the numbers, and the addition and subtraction of Eugeo’s fluctlight id and the Blue Rose Sword’s id to get Eolyne’s fluctlight id. Like that cannot just be by accident, if it is, then that's one hell of an accident. This would make Eolyne an entity that was created by fusing Eugeo and The Blue Rose Sword together as one living and breathing entity. {{But seriously, what are the chances of all this being just an accident?!}} 
In the words of Master Oogway:
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Also let’s not forget the fact that Eolyne basically said the very same thing Eugeo did when he met {{or rather was reunited with}} Kirito.
Which was: 
“Nice to meet you, Kirito-kun.”
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{{This seriously cannot just be a whole bunch of miraculous coincidences. You miss me with that!}} 
{{You gonna TELL ME? That Eolyne just happened to say the same thing Eugeo did when Kirito reunited with Eugeo? You’re gonna say that Kirito reuniting with Eolyne isn’t also him reuniting with Eugeo?! How could it be anything else?}}
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Nevertheless...
It’s just like the question asked in Avatar The Last Air Bender: Do you believe friendships can transcend lifetimes?
If you want my answer: Yes, they can. Even if the second time around -or even the 3rd or 4th, or so on-, that person is not exactly the same person in their past life or lives, the truth of the matter is that the core part of them will always be the same - no matter how different they are from one life to the next. And it’s up to them to choose who they want to live life as.
 The very idea that Kirito and Eugeo’s friendship/relationship with one another could {{and possibly has}} transcended a lifetime...?
That my friends...
Is heartbreakingly beautiful in all the right ways.
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