Convergent Series

by Charles Sheffield

  "They didn't say nuthin'. They clammed up and told anyone who asked to go away and stop bothering them and come back in a couple of months. You don't tell people that if you want them to stop sniffing around."

  "But you found me with no trouble." Darya was feeling very relieved. He was a pest, but he did not know anything, and it was not her fault that he was there.

  "Sure did. We found you. It wasn't hard, once we got going; there's transfer information for every Bose Transition."

  "So you followed me here. Now what do you want with me?"

  "Did I say we followed you, Professor?" He turned the title into an insult. "We didn't. You see, we were already on the way. But when we found you were here, too, I knew we really had to get together. Come on, dearie."

  Louis Nenda took Darya by the arm and led her through the undergrowth. They came to a tangled ridge of vines and horizontal woody stems, bulging up to form a long and lumpy bench. At pressure from him she sank down to a sitting position. Her legs were wobbly.

  "We had to get together," he repeated. "And you know why, don't you? You pretend you don't, Darya Lang, but you sure as hell do." He sat down next to her and patted her familiarly on the knee. "Come on, it's confessional time. You and me have things to tell each other, sweetheart. Real intimate things. Want me to go first?"

  If the results are so obvious to me, why haven't others drawn the same conclusions?

  Darya remembered thinking that, long before she ever set out for Dobelle. And finally she could answer the question. Others had drawn the same conclusions. The mystery was only that someone as crude, direct, and unintellectual as Louis Nenda could have done it.

  He had not beaten about the bush.

  "Builder artifacts, all over the spiral arm. Some in your territory, back in the Alliance, some in the Cecropia Federation, some back where I live in Zardalu-land. Yeah, and one here, too, the Umbilical.

  "Your Lang Catalog lists every one of 'em. And you use a universal galactic ephemeris to show every time there's been a change in any artifact. In appearance, size, function, anything."

  "As best I could." Darya was admitting nothing that was not written in the catalog itself. "Some times weren't recorded to enough significant figures. I'm sure other events were missed entirely. And I suspect some were logged that weren't real changes."

  "But you showed an average of thirty-seven changes per artifact, over an observation span of three thousand years—nine thousand years for artifacts in the Cecropian territory, 'cause they've been watching longer than anyone else. And no correlation of the times."

  "That's right." Darya did not like his grin. She nodded and glanced away.

  Nenda squeezed her knee with powerful fingers. His hand was thick and hairy. "Getting too close to the crucial point, am I? Don't feel bad, sweetie. Hang in—we'll be there in a minute. The event times didn't correlate, did they? But in one of your papers you made a throwaway suggestion. Remember it?"

  How long should she go on stalling? Except that Legate Pereira's instructions had been quite specific. She was not to tell anyone outside the Alliance what she had found—even if they seemed to know it already.

  She pushed his hand away from her leg. "I've made a lot of throwaway suggestions in my work."

  "So I hear. And I hear you don't forget things. But I'll refresh your memory, anyway. You said that the right way to examine possible time correlation of artifact changes was not through the examination of universal galactic event times. It was to think of the effects of a change as propagating outward from their point of origin, traveling like a radio signal, at the speed of light. So ten light-years after something happened at an artifact, information about that change would be available everywhere on the surface of a sphere, ten years in radius and with center at the artifact. Remember writing that?"

  Darya shrugged.

  "And any two spheres expand until they meet," Louis Nenda went on. "First they'll touch at one point, then as they grow they'll intersect in a circle that just gets bigger and bigger and bigger. But it gets trickier with three spheres. When they grow and meet, they'll do it at just two points. Four or more spheres don't usually have any points in common. And when you get to twelve hundred and thirty-six artifacts, with an average of thirty-seven changes for each one, you have nearly fifty thousand spheres—each one spreading out at the speed of light with an artifact as the sphere center. What's the chances that twelve hundred and thirty-six of those spheres, one from each Builder artifact, will all meet at one place? It should be negligible, too small to measure. But if they did meet, against all the odds, when would that happen?

  "Sounds like an impossible question, doesn't it? But it's not hard to program and test for intersections. And do you know the answer that program gives, Professor Lang?"

  "Why should I?" It was too late, but she stalled anyway.

  "Because you're here. Damn it, let's stop pretending. Do you want me to spell it out for you?"

  His hand was on her thigh again, but it was his tone of voice that finally made her angry enough to hit back.

  "You don't need to spell anything out for me, you—you lecherous little dwarf. And you may have followed up on it, but that's all you did—follow up! It was my original idea. And get your filthy hand off my leg!"

  He was grinning in triumph. "I never said it wasn't your idea. And if you don't want to be friendly, I won't push it. The spheres all coincide, don't they—to as many significant figures as the data permit? One place, and one time, and we both know where. The surface of Quake, at Summertide. That's why you're here, and that's why I'm here, and Atvar H'sial, and everybody but your Uncle Jack."

  He stood up. "And now the local bozos say we can't go! Any of us."

  "What?!" Darya jerked to her feet.

  "You didn't hear it yet? Old stone-head Perry came and told me an hour ago. No Quake for you, no Quake for me, no Quake for the bugs. We come a thousand light-years to sit here on our asses and miss the whole show."

  He slashed the black cane from Kallik's harness at the bole of a huge bamboo. "They say, no go. I say, then screw 'em! See now why we have to do something, Darya Lang? We have to pool our knowledge—unless you want to sit here on your ass and take orders from pipsqueaks."

  Mathematics is universal. But very little else is.

  Darya reached that conclusion after another half hour's talk with Louis Nenda. He was a horrible man, someone she would go out of her way to avoid. But when they had traded statistical analyses—grudgingly, carefully, each unwilling to offer more than was received—the agreement was uncanny. It was also in a sense inevitable. Starting from the same set of events and the same set of artifact locations, there was just one point in space and time that fitted all the data. Any small differences in the computed time and place of the final result arose from alternative criteria for minimizing the residuals of the fit, or from different tolerances in convergence of the nonlinear computations.

  They had followed near-identical approaches, and used similar tolerances and convergence factors. She and Louis Nenda agreed on results to fifteen significant figures.

  Or rather, Darya concluded after another fifteen minutes, she and whoever had done the calculations for Nenda were in agreement. It could not be his own work. He had no more than a rough grasp of the procedures. He was in charge, but someone else had done the actual analysis.

  "So we agree on the time, and it's within seconds of Summertide," he said. He was scowling again. "And all we know is that it's somewhere on Quake? Why can't you pin it closer? That's what I was hoping we could do when we compared notes."

  "You want miracles? We're dealing with distances of thousands of light-years, thousands of trillions of kilometers, and time spans of thousands of years. And we have a final uncertainty of less than two hundred kilometers in location, and less than thirty seconds in time. I think that's pretty damned good. In fact, it is a miracle, right there."

  "Maybe close enough." He slapped the cane against his own leg. "And i
t's definitely on Quake, not here on Opal. I guess that answers another question I had."

  "About the Builders?"

  "Nuts to the Builders. About the bugs. Why they want to get to Quake."

  "Atvar H'sial says she wants to study the behavior of life-forms under extreme environmental stress."

  "Yeah. Environmental stress, my ass." He started to walk back toward the cluster of buildings. "Believe that, and you'll believe in the Lost Ark. She's after the same thing as we are. She's chasing the Builders. Don't forget she's a Builder specialist, too."

  Louis Nenda was coarse, barbaric, and disgusting. But once he said it, it became obvious. Atvar H'sial had come to Dobelle too well prepared with contingency plans, just as though she had known that the requests for access to Quake would all be refused.

  "What about Julius Graves? Him too?"

  But Nenda only shook his head. "Old numb-nuts? Nah. He's a mystery. I'd normally have said, sure, he's here for the same reason as we are. But he's Council, an' even if you don't believe half of what you hear about them—I don't—I've never heard of one lying. Have you?"

  "Never. And he didn't expect to go to Quake when he came, only to Opal. He thought those twins he's after would be here."

  "So maybe he's for real. Either way, we can forget him. If he wants to go to Quake, he'll do it. The bozos can't stop him." They were back at the building, and Nenda paused just outside the door. "All right, we had our little chat. Now for the best question of all. Just what is going to happen on Quake at Summertide?"

  Darya stared at him. Did he expect her to answer that? "I don't know."

  "Come on, you're stalling again. You must know—or you wouldn't have dragged all this way."

  "You have it exactly backward. If I did know what will happen, or if I even had a halfway plausible idea of it, I'd never have left Sentinel Gate. I like it there. You dragged all this way, too. What do you think will happen?"

  He was glaring at her in frustration. "Lord knows. Hey, you're the genius. If you don't know you can be damned sure I don't. You really have no idea?"

  "Not really. It will be something significant, I believe that. It will happen on Quake. And it will tell us more about the Builders. Beyond that I can't even guess."

  "Hell." He slashed at the damp ground with the cane. Darya had the feeling that if Kallik had been there, the Hymenopt would have been the recipient of that blow. "So what now, Professor?"

  Darya Lang had been worrying the same question. Nenda seemed to want to cooperate, and she had been drawn along by her thirst for any facts and theories relevant to the Builders. But he seemed to have nothing—or at least, nothing he was willing to give. And she was already talking of working with Atvar H'sial and J'merlia. She could not work with both. And even though she had agreed to nothing definite, she could not mention her other conversations to Louis Nenda.

  "Are you proposing that we cooperate? Because if you are—"

  She did not have to finish. He had thrown his head back and was hooting with laughter. "Lady, now why would I do a thing like that? When you've just told me you don't know a damned thing!"

  "Well, we have been swapping information."

  "Sure. That's what you're good at, that's what you're famous for. Information and theories. So how are you at lying and cheating? How are you at action? Not so good, I'll bet. But that's what you'll need to get yourself over to Quake. And from what I hear, Quake won't be any picnic. I'll have my work cut out there. Think I want to baby you, sweetheart, and tell you when to run and when to hide? No thanks, dear. You arrange your own parade."

  Before she could respond he strode ahead of her, into the building and through to the interior room where they had started. Kallik and J'merlia were still there, crouched low on the floor with their multiple legs spread flat and intertwined. They were exchanging ominous whistles and grunts.

  Louis Nenda grabbed the Hymenopt roughly by the halter, attached the black cane, and pulled. "Come on, you. I told you, no fighting. We've got work to do." He turned back to Darya. "Nice to meet you, Professor. See you on Quake?"

  "You will, Louis Nenda." Darya's voice was shaking with anger. "You can count on it."

  He gave a scoffing laugh. "Fine. I'll save a drink for you there. If Perry's right, we may both need one."

  He pulled hard on the cane and dragged Kallik out.

  Seething, Darya went across to where J'merlia was slowly standing up. "How is Atvar H'sial?"

  "Much better. She will be fully ready to resume work in one more Dobelle day."

  "Good. Tell her that I have made up my mind and agree to cooperate fully with her. I will do everything we discussed. I am ready to take off for Quakeside and the Umbilical as soon as she is recovered."

  "I will tell her this at once. It is good news." J'merlia moved closer, studying Darya's face. "But you have had some bad experience, Darya Lang. Did the man seek to hurt you?"

  "No. Not a physical hurt." But he hurt me anyway. "He made me angry and upset. I'm sorry, J'merlia. He wanted to talk, and so we went outside. I thought you were asleep. I didn't realize that you would be threatened by that horrible animal of his."

  J'merlia was staring at her and shaking his thin mantis-head in a gesture he had picked up from the humans. "Threatened? By that?" He pointed to the door. "By the Hymenopt?"

  "Yes."

  "I was not threatened. Kallik and I were beginning a proto-converse—a first learning of each other's language."

  "Language?" Darya thought of the whipping cane and the halter. "Are you telling me that it can talk? It's not just a simple animal?"

  "Honored Professor Lang, Kallik can certainly talk. She never had the chance to learn more than Hymenopt speech, because she met few others and her master did not care for her to know. But she is learning. We began with less than fifty words in common; now we have more than one hundred." J'merlia moved to the door, his wounded leg still trailing. "Excuse me, honored Professor. I must leave now and find Atvar H'sial. It is a pity that Kallik is leaving this place. But maybe we will have an opportunity to talk and learn again when they arrive."

  "Arrive? Where are they going?"

  "Where everyone is going, it seems." J'merlia paused on the threshold. "To Quake. Where else?"

  CHAPTER 11

  Summertide

  minus thirteen.

  Violent resistance is a problem, but nonresistance can be harder to handle.

  Hans Rebka felt like a boxer, braced for a blow that never came. At some level he was still waiting.

  "Didn't they fight it?" he asked.

  Max Perry nodded. "Sure. At least, Louis Nenda did. But then he said he'd had it with the Dobelle system, and we could take his access request and stuff it, he was getting the hell out of here as soon as he could. And he already left."

  "What about Darya Lang and Atvar H'sial?"

  "Lang didn't say a word. There's no way of knowing what Atvar H'sial thinks, but what came out of J'merlia didn't have much steam in it. They went off to sulk on another Sling. I haven't seen them for two days—haven't had time to bother with them, to be honest. Think we ought to be worried?"

  The two men were in the final moments of waiting as the capsule taking them to Quake was coupled to the Umbilical. They were carrying their luggage, one small bag for each man. Julius Graves was over by the aircar that had brought them from Starside, fussing with his two heavy cases.

  Rebka considered Perry's question carefully. His own assignment to Dobelle involved only the rehabilitation of Max Perry. In principle it had nothing to do with members of other clades, or how they were treated. But as far everyone on Opal was concerned, he was a senior official, and he had the duties that went with the position. He had received a new coded message from Circle headquarters just before they left Starside, but he had no great hopes that it would help him much, whatever it said. Advice and direction from far away were more likely to add to problems than to solve them.

  "People ought to be protesting a lot more," he
said at last. "Especially Louis Nenda. What's the chances that he might leave Opal and try for a direct landing on Quake from space? He came in his own ship."

  "There's no way we could stop him trying. But unless his ship is designed for takeoff without spaceport facilities, he'll be in trouble. He might get down on Quake, but maybe he'd never get off it."

  "How about Darya Lang and Atvar H'sial?"

  "Impossible. They don't have a ship available, and they won't be able to rent one that will fly interplanetary. We can forget about them."

  And then Perry hesitated. He was not sure of his own statement. There was that feeling in the air, a sense of final calm before a great storm. And it was not just the cloudbursts that threatened Opal within twenty-four hours.

  It was Summertide, hanging over everything. With thirteen Dobelle days to go, Mandel and Amaranth loomed larger and brighter. Average temperatures were already up five degrees, under angry clouds like molten copper. Opal's air had changed in the last twelve hours. It was charged with a metallic taste that matched the lowering sky. Airborne dust left lips dry, eyes sore and weeping, noses itching and ready to sneeze. As the massive tides brought the seabed close to the surface, undersea earthquakes and eruptions were blowing their irritant fumes and dust high into the atmosphere.

  Julius Graves had finally stowed the cases to his satisfaction in the bottom level of the Umbilical's car. He walked over to the other two men and stared up at the lambent sky.

  "Another storm coming. A good time to be leaving Opal."

  "But a worse time to be going to Quake," Perry said.

  They climbed into the car. Perry provided his personal ID and keyed in a complex command sequence.

  The three men maintained an uneasy formality as the ascent began. When Perry had quietly informed Graves that access to Quake was denied until after Summertide, Graves had just as coolly asserted the authority of the Council. He would be going to Quake anyway.

  Perry pointed out that Graves could not prevent planetary officials from accompanying him. They had a responsibility to stop him from killing himself.