Kem had computed the trajectories for both light and flares, and sketched out the safest passage past the Wanderer's orbit. The twisted curvature of the geometry had a pernicious effect, focusing the danger into places where a simpler analysis might have anticipated safety; the Hub
did
act as a kind of shelter, but the points to which it offered the greatest protection did not lie directly opposite the Wanderer.
Kem's laborious calculations had identified the path of least danger; the only problem now was to follow it, without a glimpse of the void to confirm their position. Roi had set up systematic weight measurements, and cycling stones and Rotators in a new Null Chamber; all of this helped quantify their distance from the Hub, but the most crucial information, their angle from the Wanderer, could not be measured this way.
Sen and her team were monitoring the strength of the wind, and had done their best to calibrate a model linking the characteristics of the flow through the tunnels with the last of the solid data that the void-watchers had been able to gather on the Splinter's changing orbit. The varying density of the Incandescence could not be anticipated, but it could be measured in the wind, moment by moment, and fed into the templates to derive a range of estimates as to the effect the tunnels were having on the Splinter's position and speed.
These efforts gave them a far better chance than they would have had by merely trusting in luck, but two things remained to elevate the uncertainty. One was the Wanderer's erratic orbital shifts, which they had never been able to understand and no longer had any hope of observing. Roi had accepted that there was nothing to be done about that. The other complication was the influence of the Wanderer's curvature, and that was a flaw she could not accept without a fight.
They could never hope to track the Wanderer's proximity with a change in the weights; the differences it made from one end of the Splinter to the other were too tiny. But that lack of detectable influence did
not
mean that as the two of them swept around the Hub, the Wanderer was incapable of advancing the Splinter in its orbit, slowly dragging them out of the safe zone that Kem had delineated.
With Haf and Pel as their checkers, Roi, Kem and Nis stood with their template-frames, shift after shift, trying to merge the geometries in a way that satisfied Zak's principle. Having failed to make progress before, Roi had them try the simplest approach: imagining that both the Hub and the Wanderer possessed the twistless curvature of their first, incorrect guess for the Hub. Such an answer would not have been the true geometry, of course, but it might have opened up a crack leading in the right direction.
Roi's vision as she fell into sleep at the end of each shift was filled with images of smooth, bright surfaces colliding, grating against each other, refusing to meld. Each time she woke, the problem filled her thoughts again immediately.
How could they pass the Wanderer, blinded by the Incandescence, without slipping unknowingly into danger?
The geometry of space-time was the only guide left to them, but while their knowledge of its shape remained imperfect, that guide was uncertain, and perhaps even treacherous.
Kem put down her frame. "I can't think any more. I'm going to get some food."
"Haf can bring you food," Roi suggested.
"Pel can do it!" said Haf. "She's not doing anything!" Pel was lying on her back, gazing at the ceiling. No one had passed her a frame to check since the start of the shift.
"I can get it myself," Kem replied. She left the chamber.
Nis had paused while they were speaking; now he looked to Roi, almost accusingly.
"We're all tired," he said. "I think we're going nowhere."
"You're giving up on this?" Roi felt a stab of bitterness. "To do what?" If there'd been some other urgent task she would have sent him to it with her blessing, but Sen's team had their calculations under control; they didn't need a new recruit to train in their methods.
"I'm not giving up," Nis replied. "I'm losing my mind. These calculations aren't leading anywhere. Nothing simplifies; they just grow more tangled with every step. Someone smart enough could probably
prove
that we're never going to find this geometry."
Roi thought of Tan, who was sick now, as Zak had been. If he'd been healthy, maybe he could have done just that: proved that she was wasting her time.
"The geometry exists!" she rasped. "It's there around us. It's what we're moving through, even as we speak."
"I didn't claim it doesn't exist," Nis said wearily. "But not everything in the world fits a template. Can you write a template for the shape of the Splinter? For the shape of your own carapace?"
Roi fell silent. Nis's analogy had to become right at some point, but she had hoped that they could reach this small step further with the mathematics. Two "Hubs", two centers to the curvature; it didn't seem like such a complicated thing to capture with a template.
Nis said, "Space-time does what it does, following Zak's principle over and over, fitting together perfectly, everywhere, all the time. Without sliding a single stone along a wire. Without knowing the first thing about templates. That's how it defeats us at this game. It doesn't need to capture the details of everything it does across all of space and all of time in a few elegant symbols. It just does what it does."
He put down his frame.
Roi pushed herself against the rock, stretching her aching joints, struggling to clear her mind. There was something wise in Nis's words, but it wasn't the message of pessimism.
She said, "You're right, it's not trying to do template mathematics. It doesn't need to. But if it doesn't, then why do we?"
Nis answered dutifully, as if he were her student. "Because we need a template to distil everything that happens into a simple, compact form. How else could we calculate anything?"
"How does
space-time
calculate anything?" she replied.
"I don't understand," Nis said.
"I understand," said Haf. "We should do as Tio said."
"What?" Roi was confused now. "What did Tio say?" Tio was a friend of Haf's who had wandered from teacher to teacher among the theorists, learning a great deal but then arguing with everyone, refusing to perform the calculations they expected of him.
"I told you thirty-six shifts ago," Haf said reprovingly. "He treats space-time as lots of small, flat pieces. When you make them small, the templates describe what happens at the corners, how you join the pieces together. But the templates are easy, not like these weeds." He shook the frame of Roi's that he'd been checking.
"You just need a lot of them," Roi said. For a few heartbeats she was simply dazed, unsure if this was some false promise that her weariness had caused her to misjudge. But Haf's words made perfect sense; Tio's idea was the only way forward.
She asked Haf, "Can you find him? Can you bring him here?"
"Sure." Haf prodded Pel and they left the chamber together.
Nis said, "I still don't see it."
"Wait for Tio," Roi suggested. "If I try to explain it, the way I'm feeling right now I'll probably just end up confusing us both."
"But who does he work with? What's he been doing?"
"He's been working by himself," she said.
"A team of one?" Nis scoffed.
"Zak was a team of one," Roi said. "A long time ago."
Nis was unimpressed by the comparison. "Not everyone who thinks they're like Zak is actually correct in that perception."
"That's true," Roi conceded. "So let's just judge his ideas on their merits."
Haf and Pel returned with Tio. For a moment he seemed nervous and resentful, but when Roi addressed him respectfully and said that she needed his help, his posture softened and the words spilled out of him.
He had reformulated Zak's principle, he explained, in a way that suited a picture of space-time built up from many small, flat pieces. The result was not perfect, like Tan's geometry, which could be trusted down to the finest detail. But the calculations, though laborious and repetitive, were extremely simple. You couldn't fail to find an answer.
Roi asked, "How many divisions would we need to make, how many pieces of space-time to cover everything from our last known orbit to the Wanderer's, and a short way beyond?"
Tio fell silent, calculating something, or guessing. "Perhaps six to the eighth pieces. Six to the ninth for more accuracy."
"And what kind of team would we need? To calculate all the geometry before we reach the Wanderer's orbit?"
Tio said, "Six to the fourth, if they're good calculators. Maybe double that if everyone needs checkers."
Six to the fourth. Ruz's void-watchers, and all the theorists combined, would come to less than a quarter of that. Sen's team could not leave their work, but perhaps some of Jos's light-messengers could be diverted for a while; with no observations to bring in fresh information, there was little news to be relayed.
"Can you set up this problem in a simple way, that everyone will be able to understand?"
"Yes," Tio replied confidently.
"So that anyone who can manage arithmetic can learn what to do in half a shift?" Roi pressed him.
Tio said, "I'm sure of it."
Roi looked around the small chamber, trying to picture a much larger one nearby where they could put everyone. She noticed, distractedly, that Pel was now carrying four seed packages inside her, and that Haf's body was empty. He'd shown no signs of pain; Pel had taken the packages pre-emptively, without even needing to witness her friend's suffering.
Everything in the world was strange, but Roi didn't care. The last thing that was in their power to change had finally shown signs of yielding. She believed they would survive now.
Tan was staying in a small chamber, not far from Tio's geometry-calculators. Before starting her shift Roi brought him food, and they spoke for a while.
"Where are we now?" he asked.
"All the measurements put our orbit close to size twenty."
"Twenty!" Tan marveled. "No wonder I feel like I'm back in the Null Chamber."
"The weights are nothing," Roi agreed. "Even at the garm edge now, people can cling to the ceiling for a whole shift if they want to."
"I don't want to cling to the ceiling," Tan said. "I just want to live a few more shifts. I want to see this through."
"I understand." Roi had never asked him about his children, whether he caught himself hunting for them in the crowds. Everyone's gaze had shifted, though, whether it was focused on their own hatchlings or just the generation as a whole.
"Either that, or we'll all go out like Zak did," Tan chirped, making a joke of it.
"We've worked hard," Roi said. "There'll be ease and safety for the next generation."
"Let them have safety," Tan said, "but not ease."
"Why?"
"Do you want them to go back to the old ways?" Tan looked at her searchingly. "You do know that's what will happen, don't you? If they have nothing to push against, nothing to understand, nothing to explore."
Roi didn't know how to answer him. She knew, now, that she could never go back herself, but she wouldn't have to face that; she would not live a great deal longer than Tan. Did it matter, though, if Haf and Pel and Tio slipped gradually from their adventurous youth into a world where the generation that followed would once more live for nothing but the buzz of cooperation, whatever the team, whatever the task?
"Do I have any say in it?" she replied. "Is there anything I can do that will determine how much they'll struggle?"
Tan said, "Nothing at all. But you can still hope for the right thing."
Roi left him to start her shift, entering quietly and taking her place in the chamber from Leh; the two of them shared a position, working alternate shifts. Tio had arranged the geometry-calculators in a carefully designed pattern, so that each one exchanged information with just five of their neighbors. There was no need for messengers to weave back and forth between them; the results each person needed in order to continue always came from someone beside them.
Numbers washed back and forth across the chamber, but Roi could stay focused on her own simple tasks, ignoring the wider picture. Compared to battling the templates for a space-time connection, it was almost as mindless as weeding a crop. She let herself fall into a happy daze, thinking of nothing but the details of each calculation.
Halfway through the shift, she emerged from her trance; Tio had called a halt. A dozen people walked across the chamber, moving from calculator to calculator, asking them about the numbers associated with the piece of space-time they had just analyzed. The answer they were given decided which of the calculator's neighbors to move on to. In effect, Tio's path-walkers were letting an imaginary object fall in a straight line across each small region of space-time being modeled, and seeing where it emerged. By keeping track of a few simple details, they could follow it into the future — across a region that was effectively curved by the way the many flat pieces were joined — building up a fair approximation to a natural path.
The orbit of the Splinter. The orbit of the Wanderer. The light and flares that might pass between them.
Kem and Nis studied the results, and shaped them into instructions for Sen. The changes made in the tunnels, ever finer now, steered the Splinter closer to their best guess for the safest possible place to be.
When her shift was over, Roi went to see Kem and Nis.
"Where are we?" Roi asked. It was a question that no one could stop asking.
"Twenty and three-quarters," Kem replied. "That's the latest from the Null Chamber."
Roi echoed the number. "And now I have to sleep?" At its last sighting, the Wanderer had been in an orbit at a little more than twenty-two, but everything in its history made them believe that it would be down to less than twenty-one by now. The path they were following tried to steer down the middle of the uncertainty, putting some bounds on the danger without them being sure exactly where their nemesis lay.