“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 two 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.”