In the end, what we turned out to be studying was neither mathematics nor Euler. Life is not a zero sum game; even darkness and light, happiness and despair will never entirely cancel each other.

It is our fate, perhaps, to live with both.

[This post covers and contains spoilers for the first three arcs of Owarimonogatari: Ougi Formula, Sodachi Riddle, and Sodachi Lost]

Monogatari has done a whole lot of stuff in its now 60+ episode runtime, but this is undeniably the darkest it has ever been. I have never put a whole lot of stock in the ideas that “the dark makes the light shine all the brighter” or that “the inevitable consequence of light is shadow”—after all, pour enough light on anything and you can erase all shadows (and vice verse, I suppose)—and anyways the duality of happiness and despair is shown to be a false construct anyways thanks to Hanekawa (she really does see everything nowadays).

“You’re not happy because you’re not trying to be happy,” may very well be the refrain most remembered from this trio of interlocking arcs, but I think what will stick with me most is how thoroughly and consistently observant, cautious, and active Hanekawa is throughout this arc. In the wake of Tsubasa Tiger, she has most certainly become more than a cat. There’s a line in the on-screen text after she grabs the teacup Oikura throws at Araragi that indicates perhaps her thoughtfulness oversteps some sort of line, but no, Hanekawa can only wait at the finish line for the people behind her to catch up. She’s doing her best to encourage, though. Leading by example.

I wonder when Araragi will be come capable of seeing this and learning from her, the same way he indicates he can learn from Oikura. Then again, Araragi is nothing if not arrogant—taking on everything Oikura despises? He’s far better suited to speaking out of his own experience of happiness; he has some credibility there. It also shows he’s learned something. That is a kind of self-awareness that I don’t think we’ve seen from him before.

After all, we was haunted by a lost snail for a long while; recall the name of the final arc here.

The shot above best demonstrates Hanekawa’s point. There is a sort of despairing grin there. “I despise myself” has become a comfort—the on-screen text calls it something like “killing words”—but at least the constant self-hatred for her can be something. Not nothing. A formula that’s not self-negating.

But Oikura moves on more quickly, and in a much healthier way, than Araragi moves on from anything. Likely, she already knew the truth. She’s well ahead of Araragi already.

In Ougi’s own words, “If the answer’s going to be zero, there’s really no need to go out of your way to make the calculation.” So the question becomes—if Araragi cannot be Oikura’s villain, if she cannot hate her parents, and her unending emotional self-abuse must all be erased, has this all become nothing? What is left for Sodachi Oikura to be?

The case, as it stands, is that zero is better than a exponential curve all the way to negative infinity. You can build from zero. Again, life is not a zero sum game.

The irony in Ougi’s observations are that she insists the calculation be made, frequently forcing Araragi through the individual steps on her own intuition. Ougi criticizes Hanekawa for planning to tell Araragi the truth about Oikura’s mother straight out, but it’s a hypocritical thing from a girl who has been veritably handing Araragi conclusions on a silver platter. But Hanekawa won’t do things Ougi’s way. If anything, Hanekawa enables Araragi to a far greater degree that Ougi ever does. Her praise of his definitive statement on happiness is critical; Araragi might actually grow given the chance to step out of himself in a non-sacrificial way.

Ougi is careless. Ougi rushes to conclusions (correct conclusions, but even so) with the confidence of always being right. If Hanekawa gives a hint, it’s with patience—and patience communicates confience. If Ougi gives hints, they’re in a rush to the end. Ougi’s always insulting Araragi, and no matter how much he may deserve it, it still indicates and underlying contempt and lack of confidence in him.

Hair symbology is back, as Hanekawa washes out the dye she wears at school to ponder Oikura’s situation. Again, Ougi is rushing. Calling Hanekawa out for being outdated, past her prime. Hanekawa’s 10-second offer is a response to Ougi’s constant pushing for speed; she meets the challenge. But Hanekawa becomes fierce—the tiger steps on the snake. She must become fully herself, accepting every part, to be able to beat the snake that tempts with the easy path. This is Ougi’s temptation, but for this bout Hanekawa roars loud enough to beat her underclassman.

Make no mistake, Ougi is a snake in the mold of the Eden tempter. Malevolent. She tempts with knowledge, teasing Araragi and stringing him along behind her with the promise of solutions and answer. Self-knowledge. Low angle shots track towards Araragi between desks like a snake in the grass; the sounds of rattles shimmer in the background. When Araragi tries to withhold information from her, we see her hands for the first time, as if a cobra has unfurled its hood and bared its fangs. She will get what she wants, and what she wants is Araragi’s stagnation. He cannot develop with Ougi around.

Hanekawa leaves at the end of Sodachi Lost for her pre-trip around the world (from which she returns in the middle of Hitagi End). This despite her warning to Araragi that she may not be able to protect him from Ougi. Perhaps she was heartened enough by his actions to trust him on his own. Perhaps she’s aware that her continued presence can only hold him back.

She won’t become another Ougi.

At the end of this it is Oikura who closes the chapter with her actions. She again reaches out to Araragi, this time in farewell. To me, this seems emblematic of Araragi throughout this arc. He’s either pushed along by Ougi or given the platform to act by Hanekawa. But there’s not really any justice he can pursue here. After all, did Araragi really do anything wrong? Was he wrong to live a happy life with his family, thus teaching Oikura the horror of her situation? Was he wrong to not understand her unspoken plea in her house? Was he wrong to not have consciously understood the truly culprit?

No, no, no, and no. But Araragi cares too much, takes too much on, arrogantly assumes too much. For him it doesn’t matter if he was really guilty or not. He feels guilty. He sees Oikura’s misery and must respond. That’s enough.

This is, perhaps, one part of his happiness equation.