Ok, so this issue is about a 5-year relationship. So please try to keep that in mind with replies. I’ve invested a lot into this relationship and so has she. This relationship has actually been quite a pleasant one, for the most part.
Full disclosure: We share passwords on all our social media and email accounts. This was by our own free will, as we both figured we don’t have anything to hide.
Historically, in the first year of our relationship (way back in 2010-2011), my girlfriend had been somewhat of a compulsive liar. No physical cheating ever took place, but there were plenty of online flirting sessions and photo exchanges. This continued right up until I slept with another chick. That’s when I gave my girlfriend the ultimatum: she was either going to start being honest with me, or I’d leave her. So she chose to be honest. BOOM! Her lying problem stopped immediately and hadn’t been an issue since 2011.
Fast forward to August 2015: She made a Snapchat account (the first time). I didn’t like the idea because (1) I don’t have a Snapchat account too (I have no interest in getting one), and (2) I know the kinds of things that end up happening on Snapchat. I figured that having a Facebook, Twitter, and Instagram was enough for her, right? So I asked that she remove her Snapchat account and not to make another one. She agreed.
One month later in September, what does she do? She makes a new Snapchat account (the second time). This time, I got a bit more assertive and asked her to promise not make another one. So she promised, deleted the account, and seemed aggravated.
And NOW, just last night, I discover that she has secretly been using yet another new Snapchat account (the third one). And she has been using it since September. Only this time, the account was made with a new secret email address that she never told me about (I had no access to this email). And the password for this Snapchat account was nothing like her normal passwords. It seems that she wanted to keep me out for sure this time. On top of that, her profile photo was different than her usual sweet-looking profile photos: her tongue was out suggestively in this one.
So I created an account of my own and added her as a friend.
Then I confronted her about it, I first asked her if there was anything she wanted to tell me. She came right out and admitted that she had been using Snapchat behind my back (she had no choice because she already saw that I added her as a friend).
Then I asked her about the email address used to make the account. She instantly lied and said that the email address belonged to her friend. I called her out on her lie and asked her for the password immediately. She stalled for about 5 minutes before giving me the info, saying that she was trying to remember the password (I assumed she was deleting additional evidence from this email account). I found no smoking gun when I finally got logged in.
Regardless, I calmly told her that I was done, and would probably never trust her again. I told her flat-out that I was tired of being betrayed. When I hung up the phone, she was sobbing and begging me to please give her another chance. She’s been calling me all day long, but I haven’t answered the phone. She’s probably still holding on to some hope that things will work out, since I haven’t changed my Facebook “relationship status” yet. Ha.
I consider my feelings for this girl to be love, but I don’t have oneitis. I’ve been tempted to sleep with other women several times during this relationship (the opportunities were there), but decided to remain loyal because we agreed to an exclusive long-term relationship.
So before I go to far with this, I want to get some people’s insight. Is this whole issue about a secret Snapchat/email account even a big deal? Is it likely that she will ever be stop lying to me? (She has been honest with me 4 out of 5 years). Bottom line: is this fixable?
Thank you to everyone who took the time to read this, and for any help/advice you can offer!

^ you overreacted a bit, not really a reason to break up, but i get you that you felt like you are being lied to and disrespected.... Men are big on trust/respect and women big on security... I would advice to break up in a more adult loving way, instead of a hateful way... Mini break ups(soft next) work, as
she will experience pain and may be unlikely to repeat the action...