"But the Möbius function can only take the values zero, one, or minus one over a distributive lattice. It's only nonzero if it's evaluated on an interval isomorphic to a Boolean poset. If an interval is isomorphic to B sub n, the value of μ is minus one to the n. Otherwise, it's zero. The proof of the first of those two claims is straightforward: the endpoints of the argument interval map onto the minimum and maximum elements of B sub n, so the Möbius function is the familiar minus one to the difference in sizes of its arguments. Since it's here being evaluated from the empty set to the total set, the difference is just the size of the total set, or n. The proof in the opposite direction is more complicated, but a motivating example can be found in the poset induced by the 'divides' relation on the set of positive integers. First, note that this poset, D-sub-infinity, is the direct product of countably infinitely many chains, one corresponding to each prime number. The value of the Möbius function on this poset can be shown to depend only on the quotient of its arguments, the latter over the former, and in modding out that degree of freedom, the classical number-theoretic Möbius function is recovered. That the classical Möbius function is zero at any number divisible by the square of a prime is an extension of the fact that the value of the poset-theoretic μ on a chain is zero if the difference between the rank of its arguments is greater than one, viz. the number in D-sub-infinity is more than one prime-multiplication away from the multiplicative identity. If this is unclear, imagine actually writing down an element of D sub infinity in direct product form by enumerating its prime exponents qua component chain elements, recalling that the Möbius function distributes multiplicatively over poset direct product. And, of course, when multiplying by singular powers of new distinct primes, the sign alternates, because μ of n, n+1 is minus one in a chain. By a similar argument, we wish to show that any finite distributive lattice is vulnerable to a similar decomposition. In other words, we seek to—"

In an instant, the way forward has escaped her. She doesn't remember how to proceed. The proof is almost done. She doesn't know where to go. She's almost done, but she doesn't remember what comes next. Even though it's almost complete—

"She stopped!" she hears a voice say, like a postcard from a distant land.

What's the next step? Is it M sub three or N sub five that's the atomic nondistributive poset? Both? Is the fundamental theorem of finite distributive lattice relevant? But the claim is for an arbitrary lattice, not necessarily a finite one. She supposes it would only have to be locally finite to apply the FTFDL; possibly, she's forgetting a condition of the result. Or maybe, a simple inductive argument is strong enou—

"Can you hear us now, sugarcube?" says another.

"Yes." The numbers and symbols start fading. She feels their words crawling into her ears, squirming, pushing her thoughts down into her spine—

"Oh my goodness, I'm so glad!" A new pinprick poking through the void.

Another: "Do you wanna take the blanket off your face?" She counts no fewer than five voices swimming in the darkened waters; she does not want to remove the blanket. She tries to refocus on the one piecing together the distributive lattice theorem, but she can't remember what direction it was coming from.

"Twilight, can you still hear us?" says the pinprick.

Twilight? The asymmetric recognition is sending freezing pulses down and around her cortices. "...How do you know my name?"

"...We met today, Twilight. You used wind magic to draw back my birds."

She says nothing. Feels nothing. The connection isn't there.

"Girls, maybe all of us is a bit much for her right now." Six. "Why don't you leave her with me for a while?" says a purple dragon. How do I know he's a dragon? Or purple? After a pause: "It's okay, I know what to do." Another pause, then a murmur: "Don't take it personally, Fluttershy."

She hears the shuffling of dozens of hooves. Hundreds, maybe thousands, each of them in sync with no other. The droning clop is smothering, holding the blanket down over her mouth and nose.

She feels the dragon move fractionally closer to her, scaling his volume to match. "Hey, Twi. It's just us, now. Can you take the blanket off your face so I can see you?"

There's a new, swarming intimacy, seeming to blossom from nothing. Purple? She can, in fact, take the blanket off her face. She does. She sees him. It's Spike. It was always Spike.

"Spike? Where... What happened?" The last thing thing she can remember is trying to capture wind inside an emerald—no, that's not true, there was a white unicorn laying hoof on her spellstone—no, she received the most beautiful, ornate candle, but from where?—no, someone screamed, but who?—no, the last thing she remembers is posets, posets, posets, posets, posets, posets—

He's gazing into her eyes like he's never seen anything else. "You remember that pink pony who gasped at you in town?" No. "Her name is Pinkie Pie, and she gasped because she had never seen you before and she felt like she had to throw you a surprise party. That... didn't work out. You ran upstairs and got into bed, and by the time we caught up, you were rocking and mumbling about math."

"But you're okay now," he adds, while she thinks.

That's right, she met a pink earth pony today. In fact, she met a lot of ponies... why? She was checking their progress on... various things. Weather. Food. Decorations. Decorations... for? Twilight flails out of the bed. "What time is it?!"

"It's only been about thirty minutes, Twi." Twi? ...Oh. "Don't worry. The sun hasn't even set yet." He pushes a stool over to the window beside her bed and draws open the curtain.

Twilight breathes in the setting sunlight, and the day starts returning, piece by piece. It's time to try salvaging what's left to pick up. "Spike... who saw?"

"...Er. A couple ponies," he fidgets.

"Spike. Don't lie to me."

"Uh... everyone we met today except Tinder, and about six ponies I didn't know. But Twilight, nopony's judging you, you just got startled, it wasn't that bad..."

Twilight falls back onto the bed, hooves splayed. Her reputation in Ponyville is bucked, in one day flat. A new record. Something ripples inside, and she sighs. But at least Tinder didn't see, so maybe she can still claim to have succeeded in making a friend without making herself as much of a liar as a failure. There's nothing left to do here except wait for the Celebration, which she reminds herself she must still attend. Even the most hated pony in Ponyville has a job to do. She feels her error skitter over her side, across her flank, down into her hooves; heavy is the head of the pony who kneels beneath the crown.

But for the moment, there's nothing left to do, except read. "Who's still here?"

"The ponies you hadn't met all left, but the rest stayed with you to make sure you were okay, including Tinder, who got here a couple of minutes after we did. I dunno who's still downstairs."

Twilight never wants to find out who's still downstairs. "...Can you get me a book?"

Spike comes back up the stairs, alone excepting Twilight's personal copy of A Survey of Magical Fields and Attendant Mathematical Theory. It's her favorite book, which, quite frankly, says a lot. For many of her bedridden hours it's been an aegis between her and a pony, a thought, a world, whatever might ail her that day. Moreso than her bed, her blankets, her drawn curtains, even her closed door. To her, it not only separates, but distinguishes. It's a notoriously dense text, expounding elementary facts and overviews of countless mathematical fields and magical methods. It is (she says) the cornerstone of any serious magician's education. But she knows that every time she picks it up, she's probably the only pony in Canterlot plucking its abundant fruit; in modern estimation it is something that everypony wants to have read, but nopony actually reads. Being in Ponyville, that's all the more likely, and so nestling into it will be all the cozier and lofty. But sometimes, she doesn't even read it, just inters it in the bed with her, absorbing the sheer magnitude of magical elegance into her gentle hooves. Sometimes, even Twilight Sparkle needs company.

For the first few years of his life, Spike was not able to carry the volume for its enormity. But as in every other way, Spike has grown under Twilight's care, though perhaps not as much as the reverse. He has to trudge it up the stairs one at a time, but doesn't complain a word as she awkwardly watches. She feels such gratitude that she's begging herself to help, but the stairs are just out of range of her magic. She must get up to help by hoof, but her bed is so massive, the stairs so far away. And then he's at the top anyway. She plucks it out of his tiny, tired hands as soon as it's within her field, and pulls it the rest of the way.

"Thank you, Spike. ...Thank you."

"Sure, Twi. You know I'd do anything to help." He walks back over to the bed, looking up at the open curtains. "Let me know if there's anything else I can do, okay?"

"...Okay," she swallows. "Is there anyone left downstairs?"

"Nah, not anymore. I told them that you were okay and just wanted some alone time before the Celebration. Which, by the way..."

"...I still need to go to. I know." A shimmering, xanthous flag atop a desolate copse, billowing 3-D clouds threatening their frigid downpour... "I will." She leans from side to side, wrapping the blanket underneath her in the process.

"Okay, Twi. Make sure you get some sleep beforehoof. It's been a long day, and it'll be a long night, too."

"I'll try, in a couple hours." She's not sure whether that's true, whether it will be feasible to extinguish her mind long enough, or whether it would even be advisable to sink back into her void, but she will try to try.

"Okay. I'm gonna go get my bed and bring it up here," he says, and turns back towards the stairs.

"Spike?" Twilight sets back down the book onto the comforter.

"Yeah?"

"There is one more thing. Could you... could you bring the candle Tinder gave me up here?" Please. A single symbol of success.

Spike doesn't turn back around for a moment. "Uh... you sorta... lost concentration and dropped it, when you saw everypony."

For a second, the sun blinks out. "...Oh."

"It's okay, though. Tinder brought by another one, once he got here and heard what happened."

Twilight says nothing, but pulls the blanket back over her face, for once in full cognizance of her emotions. She starts to cry.

Spike walks to the stairs, gaze down. "It's okay, Twi. I promise, he wasn't mad. I think the second one is even prettier, anyway."

No reply.

"I'll bring it up."

As he leaves, something inside Twilight clicks into or out of place, and her mind flips over without her say. Some of her mind turns off, some of it on, and she pulls down the blanket and opens the book to the table of contents. She runs a hoof down the page, until her eyes catch the entry for infusion. She closes her eyes again. She tells herself, twice: I won't touch that section. I'll find something else to read. I'm not going to dwell on yet another failure I've made today. Then she does.