Analysing Petra's song even though it should be obvious for people.
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"Even though you always came across, stern and cold. I knew that it all started from a kind, gentle place."
For Petra to know that despite Levi's cold nature and stern attitude he is a kind and a caring person means that she knew him for a long time, observed him for a long time and definitely spend a lot of time with him. It is indisputable that they were close.
"And from behind your every line of nonchalant words, there's a ray of warm, a gentle light that shines so brightly in the darkness!"
She knows that Levi is not an emotionless asshole all the time. She thinks highly of him and holds a deep love for him if she goes as far to say that he is a ray of hope that shines brighter than any star in the darkness.
"Many of my memories, are all the painful moments that we've been though, but despite it all we've made it this far. Without a doubt of what you could do."
Once again, Petra thinks very highly of Levi. She KNOWS that Levi can get through anything because he's strong. Not because he is humanities strongest, but because he is powerful, kind. His mere presence will make you assured that everything is going to be okay and he will take down any titan. Petra believes in him, more than anyone.
"I know you can't quite predict what will happen to us all in the future, but you had my back whatever I lacked, you were always there to give me the push I need."
Petra knows that Levi can't predict what happens to all of them, that he can't always be there to save them. She accepts that, she's ready to die if it means serving him. That's how devoted to HIM she is. The second line also implies that Petra and Levi encouraged each other, helped each other and Levi gave her the push she needed to become greater and better than she is already.
"We must choose to take the path of least regret, you always said
But when the moment comes to make the choice, who’s gonna have your back in the end?"
Petra's protective nature for Levi is often portrayed by toxic antis as "obsessive" or "annoying" however, the people who ACTUALLY know her character know that she is keeping Levi safe because not only does she know that he is the hope for humanity, she loves him (Petra is canonically in love with Levi) even though he might not love her back she will always be at his side and encourage him because she is devoted to him.
"Small feathers fluttering, joining one by one
As they take the form of a white wing in the setting sun
Fragile souls vanishing, dying one by one
As they leave behind a blackened wing of a war not yet won."
The fragile souls are the Levi squad dying one by one by Annie. They leave this world behind as the war continues, not yet won.
"Though a bird is destined to fly beyond the walls that reach the sky
I don’t quite understand, I am certain they can’t with just a single wing to take them
But you take them both upon your own back, those wings, both white and tattered black
With the strength of those wings, you would do anything to lead us all beyond this forsaken place."
I love Isayama for this. Because not only did he add this in the manga we also saw it in the anime. The bird that is destined to fly is Eren. However, Eren has just one wing, While Petra trusts Eren's abilities she can't bring herself to understand that HE is the hope for humanity. "I don't quite understand, I'm certain they can't with just a single wing to take them." When she speaks about Levi however "But YOU take them both upon your own back, those wings white and tattered black." The Wings Of Freedom Cape, in season 4 Levi is the ONLY soldier who wears the wings of freedom cape. Representing the past and the soldiers that died with them, but also representing and referencing Petra's words. And it was HE that led the Survey Corps. As a true captain he led them to Marley. "With the strength of those wings, you would do anything, to lead us all beyond this forsaken place."
"So I will die, believing I have made the right decision
Oh, how I long for this to last, but it seems that my time has finished
As I cease, I have one thing I’ve always yearned to say to you
It’s just...Never mind…
It’s not important in the end."
Petra dies knowing that she did the right thing, and that right thing is obeying Levi's orders to protect Eren at all cost. But here's the thing, she didn't do this for JUST Eren and because Levi commanded her too. She did this because she wanted to Levi AND Eren to survive. Longer than anyone, by choosing to fight the female titan not only demonstrated her bravery and willpower to fight for her friends, it demonstrated the amount of love and devotion she has for Levi.
She would give her life if it meant that she had to protect him.
At the end of the lyrics there is a parallel to Ymir and Historia. In her letter Ymir says that she has no regrets, but decides to add that her only regret is not getting married to Historia. Petra's regret was not telling Levi just how much she loved him. Ymir and Petra have parallel's with each other, both of them wrote a letter to their loved one, both of them had regrets of not telling their loved one how much they loved them, and both of them died without even having a chance to see them for the last time.
"Small feathers fluttering, joining one by one
As they take the form of a white wing in the setting sun
Fragile souls vanishing, lives lost one by one
As they leave behind a blackened wing in a war not yet won."
Same thing I said before for the Levi Squad.
"In the end of my life, was I brave in your eyes?
Did you still look at me, and see a girl willing to give everything?
Please, don’t ever forget, you have upon your back
The shining light of two wings that will guide you, both pure white and blood-spattered black."
The second lyric implies that Levi looked at her once and saw someone willing to give everything. Petra doesn't want Levi to forget about her and the lives that sacrificed themselves for humanity. That's why she's pleading that he won't leave the cape behind.
"If I were to call it something, that pure light behind you
The feeling it begets,
To Petra, Levi is a pure light of hope. Hope for humanity, hope for all the soldiers, hope for everyone.