The theorem itself was expressed as a commuting hypercube, one of the Niah’s favorite forms. You could think of a square with four different sets of mathematical objects associated with each of its corners, and a way of mapping one set into another associated with each edge of the square. If the maps commuted, then going across the top of the square, then down, had exactly the same effect as going down the left edge of the square, then across: either way, you mapped each element from the top-left set into the same element of the bottom-right set. A similar kind of result might hold for sets and maps that could naturally be placed at the corners and edges of a cube, or a hypercube of any dimension. It was also possible for the square faces in these structures to stand for relationships that held between the maps between sets, and for cubes to describe relationships between those relationships, and so on.
That a theorem took this form didn’t guarantee its importance; it was easy to cook up trivial examples of sets and maps that commuted. The Niah didn’t carve trivia into their timeless ceramic, though, and this theorem was no exception. The seven-dimensional commuting hypercube established a dazzlingly elegant correspondence between seven distinct, major branches of Niah mathematics, intertwining their most important concepts into a unified whole. It was a result Joan had never seen before: no mathematician anywhere in the Amalgam, or in any ancestral culture she had studied, had reached the same insight.
She explained as much of this as she could to the three archaeologists; they couldn’t take in all the details, but their faces became orange with fascination when she sketched what she thought the result would have meant to the Niah themselves.
“This isn’t quite the Big Crunch,” she joked, “but it must have made them think they were getting closer.” The Big Crunch was her nickname for the mythical result that the Niah had aspired to reach: a unification of every field of mathematics that they considered significant. To find such a thing would not have meant the end of mathematics—it would not have subsumed every last conceivable, interesting mathematical truth—but it would certainly have marked a point of closure for the Niah’s own style of investigation.
“I’m sure they found it,” Surat insisted. “They reached the Big Crunch, then they had nothing more to live for.”
Rali was scathing. “So the whole culture committed collective suicide?”
“Not actively, no,” Surat replied. “But it was the search that had kept them going.”
“Entire cultures don’t lose the will to live,” Rali said. “They get wiped out by external forces: disease, invasion, changes in climate.”
“The Niah survived for three million years,” Surat countered. “They had the means to weather all of those forces. Unless they were wiped out by alien invaders with vastly superior technology.” She turned to Joan. “What do you think?”
“About aliens destroying the Niah?”
“I was joking about the aliens. But what about the mathematics? What if they found the Big Crunch?”
“There’s more to life than mathematics,” Joan said. “But not much more.”
Sando said, “And there’s more to this find than one tablet. If we get back to work, we might have the proof in our hands before sunset.”
5
Joan briefed Halzoun by video link while Sando prepared the evening meal. Halzoun was the mathematician Pirit had appointed to supervise her, but apparently his day job was far too important to allow him to travel. Joan was grateful; Halzoun was the most tedious Noudah she had encountered. He could understand the Niah’s work when she explained it to him, but he seemed to have no interest in it for its own sake. He spent most of their conversations trying to catch her out in some deception or contradiction, and the rest pressing her to imagine military or commercial applications of the Niah’s gloriously useless insights. Sometimes she played along with this infantile fantasy, hinting at potential superweapons based on exotic physics that might come tumbling out of the vacuum, if only one possessed the right Niah theorems to coax them into existence.
Sando was her minder too, but at least he was more subtle about it. Pirit had insisted that she stay in his shelter, rather than sharing Rali and Surat’s; Joan didn’t mind, because with Sando she didn’t have the stress of having to keep quiet about everything. Privacy and modesty were nonissues for the Noudah, and Joan had become Noudah enough not to care herself. Nor was there any danger of their proximity leading to a sexual bond; the Noudah had a complex system of biochemical cues that meant desire only arose in couples with a suitable mixture of genetic differences and similarities. She would have had to search a crowded Noudah city for a week to find someone to lust after, though at least it would have been guaranteed to be mutual.
After they’d eaten, Sando said, “You should be happy. That was our best find yet.”
“I am happy.” Joan made a conscious effort to exhibit a viridian tinge. “It was the first new result I’ve seen on this planet. It was the reason I came here, the reason I traveled so far.”
“Something’s wrong, though, I think.”
“I wish I could have shared the news with my friend,” Joan admitted. Pirit claimed to be negotiating with the Tirans to allow Anne to communicate with her, but Joan was not convinced that she was genuinely trying. She was sure that she would have relished the thought of listening in on a conversation between the two of them—while forcing them to speak Noudah, of course—in the hope that they’d slip up and reveal something useful, but at the same time she would have had to face the fact that the Tirans would be listening too. What an excruciating dilemma.
“You should have brought a communications link with you,” Sando suggested. “A homestyle one, I mean. Nothing we could eavesdrop on.”
“We couldn’t do that,” Joan said.
He pondered this. “You really are afraid of us, aren’t you? You think the smallest technological trinket will be enough to send us straight to the stars, and then you’ll have a horde of rampaging barbarians to deal with.”
“We know how to deal with barbarians,” Joan said coolly.
Sando’s face grew dark with mirth. “Now I’m afraid.”
“I just wish I knew what was happening to her,” Joan said. “What she was doing, how they were treating her.”
“Probably much the same as we’re treating you,” Sando suggested. “We’re really not that different.” He thought for a moment. “There was something I wanted to show you.” He brought over his portable console, and summoned up an article from a Tiran journal. “See what a borderless world we live in,” he joked.
The article was entitled “Seekers and Spreaders: What We Must Learn From the Niah.” Sando said, “This might give you some idea of how they’re thinking over there. Jaqad is an academic archaeologist, but she’s also very close to the people in power.”
Joan read from the console while Sando made repairs to their shelter, secreting a molasseslike substance from a gland at the tip of his tail and spreading it over the cracks in the walls.
There were two main routes a culture could take, Jaqad argued, once it satisfied its basic material needs. One was to think and study: to stand back and observe, to seek knowledge and insight from the world around it. The other was to invest its energy in entrenching its good fortune.
The Niah had learned a great deal in three million years, but in the end it had not been enough to save them. Exactly what had killed them was still a matter of speculation, but it was hard to believe that if they had colonized other worlds they would have vanished on all of them. “Had the Niah been Spreaders,” Jaqad wrote, “we might expect a visit from them, or them from us, sometime in the coming centuries.”
The Noudah, in contrast, were determined Spreaders. Once they had the means, they would plant colonies across the galaxy. They would, Jaqad was sure, create new biospheres, reengineer stars, and even alter space and time to guarantee their survival. The growth of their empire would come first; any knowledge that failed to serve that purpose would be a mere
distraction. “In any competition between Seekers and Spreaders, it is a Law of History that the Spreaders must win out in the end. Seekers, such as the Niah, might hog resources and block the way, but in the long run their own nature will be their downfall.”
Joan stopped reading. “When you look out into the galaxy with your telescopes,” she asked Sando, “how many reengineered stars do you see?”
“Would we recognize them?”
“Yes. Natural stellar processes aren’t that complicated; your scientists already know everything there is to know about the subject.”
“I’ll take your word for that. So … you’re saying Jaqad is wrong? The Niah themselves never left this world, but the galaxy already belongs to creatures more like them than like us?”
“It’s not Noudah versus Niah,” Joan said. “It’s a matter of how a culture’s perspective changes with time. Once a species conquers disease, modifies their biology, and spreads even a short distance beyond their home world, they usually start to relax a bit. The territorial imperative isn’t some timeless law of history; it belongs to a certain phase.”
“What if it persists, though? Into a later phase?”
“That can cause friction,” Joan admitted.
“Nevertheless, no Spreaders have conquered the galaxy?”
“Not yet.”
Sando went back to his repairs; Joan read the rest of the article. She’d thought she’d already grasped the lesson demanded by the subtitle, but it turned out that Jaqad had something more specific in mind.
“Having argued this way, how can I defend my own field of study from the very same charges as I have brought against the Niah? Having grasped the essential character of this doomed race, why should we waste our time and resources studying them further?
“The answer is simple. We still do not know exactly how and why the Niah died, but when we do, that could turn out to be the most important discovery in history. When we finally leave our world behind, we should not expect to find only other Spreaders to compete with us, as honorable opponents in battle. There will be Seekers as well, blocking the way: tired, old races squatting uselessly on their hoards of knowledge and wealth.
“Time will defeat them in the end, but we already waited three million years to be born; we should have no patience to wait again. If we can learn how the Niah died, that will be our key, that will be our weapon. If we know the Seekers’ weakness, we can find a way to hasten their demise.”
6
The proof of the Niah’s theorem turned out to be buried deep in the hillside, but over the following days they extracted it all.
It was as beautiful and satisfying as Joan could have wished, merging six earlier, simpler theorems while extending the techniques used in their proofs. She could even see hints at how the same methods might be stretched further to yield still stronger results. “The Big Crunch” had always been a slightly mocking, irreverent term, but now she was struck anew by how little justice it did to the real trend that had fascinated the Niah. It was not a matter of everything in mathematics collapsing in on itself, with one branch turning out to have been merely a recapitulation of another under a different guise. Rather, the principle was that every sufficiently beautiful mathematical system was rich enough to mirror in part—and sometimes in a complex and distorted fashion—every other sufficiently beautiful system. Nothing became sterile and redundant, nothing proved to have been a waste of time, but everything was shown to be magnificently intertwined.
After briefing Halzoun, Joan used the satellite dish to transmit the theorem and its proof to the decoy node. That had been the deal with Pirit: anything she learned from the Niah belonged to the whole galaxy, as long as she explained it to her hosts first.
The archaeologists moved across the hillside, hunting for more artifacts in the same layer of sediment. Joan was eager to see what else the same group of Niah might have published. One possible eight-dimensional hypercube was hovering in her mind; if she’d sat down and thought about it for a few decades she might have worked out the details herself, but the Niah did what they did so well that it would have seemed crass to try to follow clumsily in their footsteps when their own immaculately polished results might simply be lying in the ground, waiting to be uncovered.
A month after the discovery, Joan was woken by the sound of an intruder moving through the shelter. She knew it wasn’t Sando; even as she slept an ancient part of her Noudah brain was listening to his heartbeat. The stranger’s heart was too quiet to hear, which required great discipline, but the shelter’s flexible adhesive made the floor emit a characteristic squeak beneath even the gentlest footsteps. As she rose from her couch she heard Sando waking, and she turned in his direction.
Bright torchlight on his face dazzled her for a moment. The intruder held two knives to Sando’s respiration membranes; a deep enough cut there would mean choking to death, in excruciating pain. The nanomachines that had built Joan’s body had wired extensive skills in unarmed combat into her brain, and one scenario involving a feigned escape attempt followed by a sideways flick of her powerful tail was already playing out in the back of her mind, but as yet she could see no way to guarantee that Sando came through it all unharmed.
She said, “What do you want?”
The intruder remained in darkness. “Tell me about the ship that brought you to Baneth.”
“Why?”
“Because it would be a shame to shred your colleague here, just when his work was going so well.” Sando refused to show any emotion on his face, but the blank pallor itself was as stark an expression of fear as anything Joan could imagine.
She said, “There’s a coherent state that can be prepared for a quark-gluon plasma in which virtual black holes catalyze baryon decay. In effect, you can turn all of your fuel’s rest mass into photons, yielding the most efficient exhaust stream possible.” She recited a long list of technical details. The claimed baryon decay process didn’t actually exist, but the pseudophysics underpinning it was mathematically consistent, and could not be ruled out by anything the Noudah had yet observed. She and Anne had prepared an entire fictitious science and technology, and even a fictitious history of their culture, precisely for emergencies like this; they could spout red herrings for a decade if necessary, and never get caught out contradicting themselves.
“That wasn’t so hard, was it?” the intruder gloated.
“What now?”
“You’re going to take a trip with me. If you do this nicely, nobody needs to get hurt.”
Something moved in the shadows, and the intruder screamed in pain. Joan leaped forward and knocked one of the knives out of his hand with her tail; the other knife grazed Sando’s membrane, but a second tail whipped out of the darkness and intervened. As the intruder fell backward, the beam of his torch revealed Surat and Rali tensed beside him, and a pick buried deep in his side.
Joan’s rush of combat hormones suddenly faded, and she let out a long, deep wail of anguish. Sando was unscathed, but a stream of dark liquid was pumping out of the intruder’s wound.
Surat was annoyed. “Stop blubbing, and help us tie up this Tiran cousin-fucker.”
“Tie him up? You’ve killed him!”
“Don’t be stupid, that’s just sheath fluid.” Joan recalled her Noudah anatomy; sheath fluid was like oil in a hydraulic machine. You could lose it all and it would cost you most of the strength in your limbs and tail, but you wouldn’t die, and your body would make more eventually.
Rali found some cable and they trussed up the intruder. Sando was shaken, but he seemed to be recovering. He took Joan aside. “I’m going to have to call Pirit.”
“I understand. But what will she do to these two?” She wasn’t sure exactly how much Rali and Surat had heard, but it was certain to have been more than Pirit wanted them to know.
“Don’t worry about that, I can protect them.”
Just before dawn someone sent by Pirit arrived in a truck to take the intruder away. Sando declared
a rest day, and Rali and Surat went back to their shelter to sleep. Joan went for a walk along the hillside; she didn’t feel like sleeping.
Sando caught up with her. He said, “I told them you’d been working on a military research project, and you were exiled here for some political misdemeanor.”
“And they believed you?”
“All they heard was half of a conversation full of incomprehensible physics. All they know is that someone thought you were worth kidnapping.”
Joan said, “I’m sorry about what happened.”
Sando hesitated. “What did you expect?”
Joan was stung. “One of us went to Tira, one of us came here. We thought that would keep everyone happy!”
“We’re Spreaders,” said Sando. “Give us one of anything, and we want two. Especially if our enemy has the other one. Did you really think you could come here, do a bit of fossicking, and then simply fly away without changing a thing?”
“Your culture has always believed there were other civilizations in the galaxy. Our existence hardly came as a shock.”
Sando’s face became yellow, an expression of almost parental reproach. “Believing in something in the abstract is not the same as having it dangled in front of you. We were never going to have an existential crisis at finding out that we’re not unique; the Niah might be related to us, but they were still alien enough to get us used to the idea. But did you really think we were just going to relax and accept your refusal to share your technology? That one of you went to the Tirans only makes it worse for the Ghahari, and vice versa. Both governments are going absolutely crazy, each one terrified that the other has found a way to make its alien talk.”
Joan stopped walking. “The war games, the border skirmishes? You’re blaming all of that on Anne and me?”
Sando’s body sagged wearily. “To be honest, I don’t know all the details. And if it’s any consolation, I’m sure we would have found another reason if you hadn’t come along.”
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