Vacuum Diagrams

by Stephen Baxter

  Kapur stared into the flowerlike heart of the Snowflake. "I should be impressed?"

  "Damn right," Mace growled, "For a start, the whole of human civilization would be characterized by only ten to power twenty bits. Even after hundreds of Assimilations. And, just in technological terms, to get within four orders of magnitude of the theoretical limit... it's almost unimaginable.

  "Now. Look." Mace, silhouetted like a cartoon grotesque, pointed at a knot of color and activity. Kapur perceived something like a sunflower, a fist of spirals and tessellations surrounded by "petals," great sheets of information which faded into the background chatter. Pellets of data streaked into and out of the core — a little like insects, Kapur thought at first; but then he saw how the pellets embedded themselves in the sunflower, endlessly enriching and renewing it.

  "What is it?"

  "It seems to be the dominant data configuration," Mace called. "The analogue of the tetrahedral motif on the physical level. It represents a theorem. See, the heart of the structure is the core statement, the petals corollaries, endlessly thrown off and lost..."

  "What theorem?"

  "Gödel's Incompleteness. We think. We're guessing, extrapolating on hints of structure we've picked up elsewhere... But it's not really a theorem, here. It's merely a statement of the result. Like an axiom; a given."

  "I don't understand."

  Mace laughed, briefly and scornfully. Wriggling before the landscape of information he pointed again. Amid a meadow of data structures, Kapur picked out another sunflower, the characteristic Gödel shape. Mace jabbed both arms against the vast data diorama, again and again. "There, and there! What do you see, Kapur?"

  Gödel, Kapur saw, repeated over and over; there was a fractal spiral of Gödel sunflowers here, embedded in this chill web of data.

  "There's more, of course," Mace said. "We've recognized a lot of physical understanding in here, particularly representations of cosmic events. See that starburst?" A firework of red and yellow, endlessly dynamic, scattered a hundredfold through the 'Flake. "That's Mach's principle: that the inertia of an object is induced by the net gravitational attraction of the rest of the Universe—"

  "Tell me about Gödel," Kapur said patiently.

  On the low-quality laser link, Mace's voice was like a buzzing insect. "Gödel was a genius. An Austrian; a Mozart of his subject. In the middle of the twentieth century he produced a theorem on undecidability.

  "Gödel studied mathematics in the abstract. Think about that, policeman: not just the mathematics you studied at high school; not the maths I studied in the Navy college — but any sort of mathematics which it is possible to construct."

  "You have my attention," Kapur said dryly. "Go on."

  "Gödel showed that within any mathematical scheme you can write down statements which it would be impossible to prove or disprove. They are undecidable, you see. And so mathematics can never be made complete. You could never deduce everything from a finite set of axioms; there would always be new statements to make... new facts to record, if you like."

  Kapur shook his head. "I cannot imagine how it is possible even to begin to frame such a theorem, let alone to prove it."

  "It isn't that difficult," Mace said lightly. "It's rather like the standard proof that the real numbers are uncountable; you make a list of all possible statements within your general mathematical scheme — and from that list generate another statement which isn't in the list — "

  "Never mind." Kapur let the terrifying implications sink in. How could there be a hole in mathematics — in the most fundamentally abstract of human inventions? He felt as if the floor had fallen away from his Universe.

  What kind of people had these Snowmen become, to hold such an awesome, nihilistic theorem at the heart of their philosophy?

  Kapur closed his Eyes again — turned them off, in fact; the orchard of data frost-flowers melted to cold, inert iron.

  Kapur and Mace made three more trips to the iron epidermis of the Snowflake. Mace pointed out more forests of rustling data, tentatively mapped by humans. There were tigers in those forests, though, Kapur came to realize; great beasts of wisdom and understanding whose nature humans could not even guess at.

  Kapur spent several of his precious hours hanging immobile, his cold-suit barely warmer than the ancient, surrounding echo of the Big Bang. He felt old, inadequate. Assimilation — bloodless Assimilation — depended on psychology, on the determination of goals. The goal of humanity was to rise up, to grow, and ultimately to confront the Xeelee. If Kapur could determine the goals of the Snowmen, then those objectives could be subvened to serve human purposes. If not, then the 'Flake, the 'Men, had no value.

  But how could Kapur, inexpert as he was, touch the dreams of the ancient individuals frozen into this data sculpture?

  He consoled himself with the thought that failure would be no disgrace, that he could return to his home, his job, without shame.

  Kapur did not discuss his feelings openly with Mace; but, as his time wore away in the musty cage of the yacht, he sensed Mace's swelling mood of triumph. The Navy man was intelligent and endlessly fascinated by his surroundings, Kapur came to see; but he clearly felt that Assimilation was a fool's errand, a sop thrown to liberal instincts before the Navy was unleashed.

  He was probably right, Kapur thought.

  It was Mace's faint gloating, as much as a sense of outrage at the damage the Spline gravity wave planet-breakers would do to the Snowflake, which determined him to keep trying to the end of his time. He could endure failure, he decided; but not failure in front of Mace.

  He had a new idea.

  "Tell me this," he said to Mace. "How much data characterizes a human being?"

  Mace opened his mouth, closed it again.

  Kapur pressed politely, "If my thoughts were somehow transcribed, day and night for my entire life — how many bits to capture that?"

  Mace smiled and closed his Eyes. "All right, policeman; let's play games. You produce, let's say, a hundred thousand discrete thoughts per day. Each concept is — what, a hundred bits?

  "We'll give you fifty years of active adulthood, between infancy and the onset of age. That gives, ah, two times ten to power eleven bits in all." Mace pursed his lips, opened his Eyes and studied Kapur briefly. "Interesting. So there's the equivalent of something like ten to power forty-nine human individuals in the 'Flake—"

  Kapur nodded. "Isolate one of them, with your sensors. Can you do that? Pick out an island of bits. I don't want to know what happens within it; arrange it so I only perceive the inputs and outputs."

  Mace rubbed his chin. "You want to talk to a Snowman?"

  "Don't mock me," Kapur said patiently.

  "What will you talk about?"

  Kapur, feeling his way, thought quickly. "Gödel's theorem."

  Mace leaned forward, ready to scorn — then hesitated. "Well, why not?" he said at last. "You could give it a human proof of the theorem. That might be kind of interesting."

  Kapur waited, but Mace's laughter did not come. "You have to help me understand you, Mace. Are you serious?"

  "Sure... I'll code up the proof in a form compatible with their storage templates; I'll dump it into your Eyes and you can download it into the sensors when we go over there again."

  "No." Kapur held up a hand. "I want you to let me go alone."

  Mace's Eyes glinted, steel globes embedded in the flesh of his lively, amused face. "Why?"

  Kapur held his gaze. "Because you're waiting for me to fail. I don't need that; I don't consider this any kind of game, or contest between us. I don't want you around me."

  Mace laughed, uncertainly. Then, as he perceived Kapur's seriousness, a look of bafflement and hurt spread across his broad face. This, Kapur realized, could be the first time any human being had rejected Mace in any way. He searched Mace's face for remorse, for shame; but he found only wounded pride.

  "Do what you like," said Mace at last. "I'll code up the proof." There were tw
o days left.

  Kapur saw the Snowman as a dully-glowing globe of purple, miles wide, embedded beneath the planar skin of the 'Flake. Mach starbursts, Gödel sunflowers and other characteristic formations littered the globe, as still as flowers under glass. 'Flake data streams chattered softly into the Snowman, and human sensor probes ringed the 'Man like patient puppies, blocks of metal silhouetted against lurid data.

  Kapur, swaddled in his cold-suit, cowered. Here, confronting the reality of the 'Flake, his isolation scheme seemed vacuous. He had no idea, of course, if the arbitrary assemblage of data before him represented an individual — or, indeed, if consciousness itself persisted at all in the 'Flake.

  He was almost certain that it did not.

  But he had to try, he reminded himself.

  Enough. He focused his gaze on the nearest of the probes; tight laser light slid from his Eyes and into the probe's cold hide.

  When the link was secure he downloaded the human proof of Gödel to the probe.

  The proof was a string of orange beads on a wire of light; the beads splashed against the target probe and rattled into the Snowman. Finally they settled into a cubical configuration: neat and precise, although dwarfed by the richness and profusion of other forms within the 'Man.

  'Flake data slugs lanced through the human proof, copying, integrating — but changing nothing.

  Kapur opened a line to Mace, in the yacht. "I don't understand," he said. "Why don't they evaluate, interpret our proof?"

  "Are you surprised? Maybe the Snowmen aren't interested in interpretation and evaluation."

  "What do you mean?"

  "Gödel's Incompleteness, remember? No matter how much you derive from a body of data, there will always be statements you could not have deduced. Always something else to store."

  "...Ah. And Gödel is at the heart of their ancient, world-weary, philosophy."

  Mace laughed briefly. "I think you're working it out, policeman. Knowing the limitations of deduction, the Snowmen decided that to record events — and only to record — was the highest calling of life. And that's all they want to do. They took apart their world, rebuilt it as a monstrous storage system... used all the material at their disposal to freeze as much data as they could. They won't do anything with our proof; for fourteen billion years they have merely watched time unravel — "

  "There's your streak of poetry again, Mace."

  "Your Assimilation must fail," Mace said bluntly.

  Kapur sighed. "Why?"

  "Think about it. The Snowmen have no motivation we can connect with. Our actions will mean nothing to them — we, almost by definition within their Gödelian philosophy, dance meaninglessly before them. Even their own destruction would be no more than an event, a final act to be stored and noted."

  "That can't be all, Mace. There must be more. Every species wants to grow, to develop." Kapur reflected. "Even if all they wanted was a greater data storage capacity—"

  "Come in, Kapur. It's over. I'll call in the Spline."

  "No." Kapur closed his eyes, tried to keep the trembling out of his voice. "I still have time."

  With slow insolence, Mace said, "It's your mission, policeman."

  Without returning to the yacht Kapur had Mace download more human datasets and propositions; and he learned quickly how to input new material — his own reflections and feelings — into his Eye stores.

  That took most of a day.

  Kapur slept briefly, nestled within the meadow scents of his cold-suit.

  When he turned to the 'Flake once more, he had six hours left.

  The Snowman had not changed. The human proof of Gödel remained lodged within its abstraction of a belly, a cold, primitive lump.

  Kapur began to download data to the probes: more and more, as rapidly as he could. Mathematics first. He found data on an ancient, failed, experiment, a life form based on the Incompleteness theorem, a bizarre disaster which had resulted in the destruction of a moon, a loss of human life...

  Then, on a whim, music — he watched as ancient compositions frosted into veils of blue ice within the 'Man.

  Human history. He told the 'Man of the Xeelee, humanity's vast, implacable foe; and of how mankind was seeking to mobilize the resources of a Galaxy in its war.

  He told the 'Man what the humans on board the Spline ship planned to do to the Snowflake.

  He told of his own fears, doubts — his awe, here before the Snowflake, with the Galaxy a cloud beneath him; of his almost superstitious response to Gödel; of his fear of failure, and his petty relationship with Mace.

  The 'Man was like a mirror, one part of his mind told him, or like a Virtual psychoanalysis program. There was no one there to respond, he knew now, but he told it all anyway.

  He told the 'Man of his own, tenuous, qualifications for this Assimilation mission. That he was a policeman; that he specialized in the resolution of the cruel, the vicious, the most bizarre crimes. His job was to work through the sites of crimes, trying to see the smashed property, the bones and scattered flesh, through the eyes of the perpetrator.

  Kapur was qualified enough to seek the motivation of the Snowmen, after twenty-five years striving to unravel the minds of aliens within his own species.

  All of this shivered into the heart of the Snowman, without comment or reaction, without praise or disgust.

  Kapur, his time spent, grew ashamed. He fell silent, arms akimbo, before the maw of the Snowflake.

  The 'Man watched steadily.

  And, at last, Kapur understood.

  Something like a ripple passed under Kapur; it was as if space were a lake on which his encased body floated, passive.

  "Kapur." Mace's voice was strained. "The Spline."

  Kapur felt enormously tired. "What about it?"

  "...It's gone."

  Time had run out. The Spline had opened its laser-cannon orifices.

  ...The ship had been torn aside, dragged from its site like an eyeball from a socket, thrown a million miles across space; it had been left spinning, bruised and torn.

  Kapur returned to the yacht.

  "Were there injuries?"

  Mace's face was wide, blank, angry. "What do you think? But the automatics are functioning; the ship's returning to pick us up. What did you do to the damn 'Flake, Kapur?"

  "It was not I who tried to open fire on it," Kapur said softly. "What happened?"

  "Gravity waves," Mace said. "Like a tractor beam." Suddenly fear broke to the surface of Mace's hard features; his Eyes seemed even more incongruous, metal islands in a sea of human emotion. He pointed through the viewport, picking out a palm-sized patch of darkness. "From the direction of the Virgo supercluster; although that's probably coincidence..."

  "I caught an echo of the beam."

  "Kapur, I think I know how they did it."

  "The Snowmen?"

  "Mach's principle. I think they can manipulate Mach's principle."

  Kapur shook his head.

  With a kind of irritated patience, Mace said, "The Spline is embedded in a Universe of matter. That matter tugs at the Spline with gravity fields — but the fields surround the ship uniformly; they are equal in all directions, isotropic and timeless."

  Kapur frowned. "And you think the Snowmen have a way of making the field — unequal?"

  Mace laughed uneasily. "I guess you learn a lot in fourteen billion years."

  Kapur turned the concept over in his mind. The Mach beam was spectacular, he decided. But the Universe was filled with spectacular weapons and technologies.

  Gödel's theorem, though. That was something else. That was truly terrifying. Mace, young, unimaginative, had responded more to the blazing of a zap gun than to the fact of a Universe without bottom or top, without meaning, unknowable. Kapur almost envied him.

  "I think I've figured it out," he said to Mace.

  "What? Their motivation?" Through his fear, Mace looked briefly interested. "Tell me, policeman. I knew there had to be something; every sentient specie
s has goals."

  "We had the pieces of the puzzle, almost from the start," Kapur said. "In their design of the 'Flake, the Snowmen had already made near-optimal use of matter, by recording information right down to the thermodynamic limit... which is set by the background temperature of the Universe. But they knew from Gödel that there will always be more events to record."

  Mace's face crumpled sourly. "Oh. Are you telling me that they are waiting for the Universe to cool down... just so they can store more data?"

  Kapur smiled. "The idea is pleasing. In the aeons since the building of the Snowflake, they've already achieved a six-fold increase in capacity! And in another forty billion years the capacity will double again...

  "Patience, Mace. That is the key."

  Mace stared into Kapur's face, the lines around his Eyes betraying hostility. "Policeman, sometimes you frighten me."

  Kapur, obscurely pleased by this reaction, did not reply. Mace said, "Do you think there'll be another attempt?"

  "To Assimilate?" Kapur shook his head. "I doubt the 'Flake would let us come so close again."

  He turned to face the emptiness of the viewport. With eyes no more than human he looked beyond the filmy sails of the laser yacht and saw the Spline coming to collect them. It moved cautiously, all weapons orifices open.

  Over centuries and a million battles, mankind moved into a position of something like dominance over its peers. And it began to confront the Xeelee, who moved through space like ships over the surface of an ocean.

  Gradually, slowly, humans probed the great projects of the Xeelee. A hundred epic quests were undertaken, a hundred names thrown up to resonate through the long afternoon of human history...

  And over a hundred devastated human worlds, Xeelee fighters folded night-dark wings.

  Vacuum Diagrams

  A.D. 21124

  PAUL OPENED HIS EYES.

  His body ached. He lay facedown on a surface that glowed with white light. Grass, or fine hair, washed over the surface.