[3 Days After the Second Collapse - Answers and Questions]

When we returned to our office, we found Sanae there, sitting at Renko's desk, sipping tea and nibbling rice crackers with a bored expression on her face.

"Okay Sanae, time for you to head home," Renko said as she took off her shoes.

"Oh! Sanae, are you still here?" I said with surprise as I crowded into the entryway behind Renko.

"What? You leave me here for almost an hour then come back and tell me to get lost? Are you two going to make out now or something? Can you really not wait?"

"Nope, not one more second. You're still a minor, so I'm afraid it's time for all good little children to head home."

"Seriously? It's not even dinner time yet!"

"It is for me and I'm going to gobble Merry up, so vamoose." Renko said, turning and latching her arms around my waist as I stepped out of the entryway.

"Ah! Renko get off! You're going to knock me over." I reached down to pinch her cheek but she snuggled her face into my ribs, twisting her head to avoid my fingers as I tried to regain my balance.

"Should I make a schedule or something? Or could you at least hang a sign on the door so I know when its safe to come over?" Sanae said, rising up with a disgusted expression.

"Alright, look Sanae. I know this is a bit of a jerk move on my part and I'm sorry. I actually do have something I have to discuss with Merry right now though, and it is actually is a private matter. The sort of thing that really does fall into the category of 'classified information' for us time-travelers. I'm sorry about this, and I'll make it up to you the next time we see you, but can you please give us some time?"

Sanae looked from Renko's face to mine and back again, trying to determine how much of either story to believe. "Alright, fine. You do whatever you're going to do, but the next time you come by the shrine make an offering to Lady Kanako and Lady Suwako." She walked past me into the entryway to put on her boots then looked back at us one last time before heading out the door. "See you later, I guess."

Renko watched her fly away, then slid the door shut and propped a stick diagonally against the corner of the frame, wedging it closed. She let out a breath before turning to face me.

"I take it you've come to a conclusion?" I asked.

Renko walked past me and pulled out two cushions, then sat down cross-legged on the floor, holding her hat in her hands and running her fingers over the brim. "I think so. Normally I would run my theories past the mastermind of an Incident and ask them to disprove them, but the person I need to talk to in this case is avoiding me."

"You mean the youkai sage?"

"Exactly. I'll run my theories by you this time. If you should ever happen to see her again, maybe you can do me the favour of asking her about them."

"I'll try to remember, but your reasoning can be a little convoluted."

"Well, this will be one to remember for sure. I'm certain you'll manage, Merry. Now where to start... In order to explain this one I'm going to have to get a little bit technical. How much do you know about dimensions, Merry?"

"Dimensions? In the sense of physics? Not as much as you. I won't be able to follow along if you get into string theory and the like."

"That's okay, I'll keep it simple." She stood up for a moment and looked around the office then went over to the corner where I kept my writing desk and supplies and grabbed a few things before coming back and sitting down. "Okay, Merry. How many dimensions are there in this world we live in?" she asked.

"Three, right? Up/down, left/right and front/back."

"There's more than that, but in particular, I'd like you to add one more dimension to your mental model: time. For the purpose of our discussion let's say there's three spatial dimensions and one chronal one. You can move forward or backward through any of the three spatial ones, but only forward through time, right?"

"Excluding the fact that it's 2008 now, and we shouldn't even have been born yet, yes, I'd say that's generally right."

"I'm sure I've mentioned this before, but time and space are actually inseparable. You can't move through one without also moving through the others as well."

"Yes, you mentioned that when we were first at the Scarlet Devil Mansion, when we were talking about Sakuya's abilities. She can get anywhere she wants to in the mansion quickly, and we can too when we're with her, but if we're alone it takes forever to find anything in that place."

"Right. It takes time to move distance and because Sakuya can mess with time, she can mess with distance too, make things seem to take less time that they should, to the point where, to an outside observer, it looks like she's doing things instantly or making things appear from nowhere."

"I think the last time we had this discussion you were theorizing that Sakuya might have been responsible for making the Scarlet Devil Mansion appear to be so big. You were thinking she had expanded space by expanding time, but it turns out it really is that big inside and she had mainly been shrinking it when she was with us, right?"

"Yes, that's right, but the specifics of that don't matter now. The important thing is that by controlling time she was able to manipulate space. The same is true in reverse. If A=B then B must equal A as well right? If you can control time you can manipulate space and if you can manipulate space you can control time."

"I understand in theory, but I have a hard time imaging what that would mean in the real world."

"Well, it would probably be something like what we saw Reisen do during the Incident at Eientei."

"Ah, you mean the corridor that stretched on forever? She wasn't manipulating time though."

"In a way she was, actually. If you look at the time it requires to move down that corridor normally, and the time it would take to move down that corridor with the way Reisen had warped it, you can see that one amount of time is far greater than the other, maybe infinitely so. If she had done the opposite and made the space more compressed, then she would be achieving the same effect as Sakuya via the opposite method. By manipulating space, she would also manipulate time."

"Okay, I follow that. You're thinking that the youkai sage has a similar ability?"

"Actually, no. I think the youkai sage is far more powerful than that. What she does doesn't follow these rules. Sakuya or Reisen can both shorten the amount of time it takes to get somewhere, but they can't make it zero. Even when Sakuya seems to move instantly to us, time is still flowing normally for her. Neither of those two can ever be truly free from the yoke of time. The youkai sage, from what I've heard, anyway, doesn't seem to have any such restrictions. She can move across any distance instantaneously without affecting the flow of time outside of herself. She's not like Lady Yasaka either, where she's in two places at once or only restricted to appearing near shrines. She doesn't even have to be involved in the gaps she creates herself, as evidenced by her sending us to Hakugyokuro. It's complete nonsense from the point of view of physics."

"Alright then, so why all this talk of dimensions, if the rules don't apply to her then?"

"Ah, you're jumping ahead. Let me get there, it'll make more sense if you see the steps in order. Now back to the planned lecture. Merry, do you know why time moves in only one direction? Always flowing from the past to the future? Our own inexplicable situation excepted, of course."

"What do you mean, 'why'? That's just the way time works, right?"

"In our experience, yes. But the laws of the universe would still hold if the opposite is true. Even in the 2080s, scientists had never found an ironclad explanation for why things have to be the way they are. If time flowed backwards, then the other three dimensions would still be able to work normally, but that never happens, as far as we've observed. It seems to be an arbitrarily one-way street. So in a way you're right. That's just the way things work. Time flows in one direction, even though in any other dimension things could go one of two ways. It also never stands still. There doesn't seem to be any valid state in the time dimension other than forward movement, from the past to the future. Now, keep that idea in the corner of you mind."

Saying that, Renko produced one of the items she had grabbed off of my desk. A scrap sheet of stained paper. "Now, how many dimensions are there on this sheet of paper, Merry?"

"Three technically, right? The thickness of paper counts as a dimension."

"Quite right. Let's say this only had two dimensions though, just for argument's sake. Like a perfectly straight line in math, it's technically impossible, but useful to illustrate a point. Speaking of which, how would you calculate the length of this line?" Renko said, drawing her finger across the paper diagonally, from one corner to the other.

"I've heard you give this lecture enough times to remember. The length is the square root of the sum of the squares of length of the base and the height, right?"

"Exactly. But more importantly, that length, between those two corners is a number. It's not zero and it's not even really close. It's a finite length. As it so happens though, I exist in a three-dimensional space. That extra dimension let's me manipulate things on lower dimensional planes in some interesting ways. For example, I can cheat like this." she said, folding the paper into a triangle with the opposite corners touching. Now the two dimensional distance is just the same, but if I were to travel between a point on one corner to a point on the other corner in the third dimension, the distance would be almost zero."

There was a mystery I had read where such a trick was the key to solving the crime, but I'll not mention it's name here because I remember seeing it in Suzunaan.

"So, as you can see," Renko continued, "higher dimensional spaces can interfere with lower dimensional ones. My theory is that Reisen and Sakuya's abilities work the same way. Warping three dimensional space by manipulating something in the fourth dimension. I know Reisen says she does it by manipulating 'wavelengths', but I don't believe those waves she's seeing are any sort of matter or energy in our three dimensions."

"So since space and time are linked, you're thinking the youkai sage can manipulate time too?"

"You're jumping ahead again, but sure, let's go there. I think the youkai sage can see and manipulate things from an even higher dimension, beyond those four, and in so doing she bends the four dimensional space-time in which we exist. Someone like Reisen or Sakuya has the power to essentially fold the paper of our universe. But the Administrator..." Renko carefully rolled the sheet of paper into a tube, arranging it so the top and bottom just barely overlapped. "If I did this and then asked you what the distance from the top of the sheet to the bottom is, the answer would be infinite. If you proceed from the top toward the bottom via any path within the dimensions of the paper, you eventually find yourself at the top again. If this were a longer strip of paper, I could even join the ends with a twist to create..."

"A moebius strip."

"Exactly. A two dimensional shape in which you can transition from one side of the plane to other just by going in a straight line."

Somehow when Renko said that, it sent a chill down my spine. I felt like we were approaching the heart of the matter, but I still wasn't sure where she was leading.

"So, higher dimensions can influence lower dimensions, and, like I said earlier, time is unidirectional. It only moves forward, and it always moves forward. It never stops, and it can't go back. Everything we know about physics points to that being the fundamental nature of reality. But..." she said, putting the paper aside and fishing out a different object, a length of string this time, pulled from a fray in the lining of one of her trenchcoat pockets. "If a higher dimensional being could take that unidirectional dimension of time and do this..." she said, tying the two ends of the string together. "Then time, even moving relentlessly forward, would be a loop."