He had been a rather dreamy boy in a laboring district of Helicon. The work in fields and factories was easy enough that he could think his own shifting, abstract musings while he did it. Before the Civil Service exams changed his life, he had worked out a few simple theorems in number theory and later was crushed to find that they were already known. He lay in bed at night thinking of planes and vectors and trying to envision dimensions larger than three, listening to the distant bleat of the puff-dragons who came drifting down the mountain sides in search of prey. Bioengineered for some ancient purpose, probably hunting, they were revered beasts. He had not seen one for many years...

He had a lot of Mathist Department business to tend to, but he followed his instinct and put his calculations first. Briskly he retrieved his ideas from the bedside notepad and stared at them, doodling absently in air, stirring symbols like a pot of soup, for over an hour. When he was a teenager the rigid drills of schooling had made him think that mathematics was just felicity with a particular kind of minutiae, knowing things, a sort of high-grade coin collecting. You learned relations and theorems and put them together. Only slowly did he glimpse the soaring structures above each discipline. Great spans joined the vistas of topology to the infinitesimal intricacies of differentials, or the plodding styles of number theory to the shifting sands of group analysis. Only then did he see mathematics as a landscape, a territory of the mind to rove and scout. To traverse those expanses he worked in mind time-long stretches of uninterrupted flow when he could concentrate utterly on problems, fixing them like flies in timeless amber, turning them this way and that to his inspecting light, until they yielded their secrets.

The endless competition of gentry families had been the first and easiest psychohistorological system Hari ever modeled. He had first combined basic game theory and kin selection. Then, in a moment of inspiration, he inserted them into the equations that described sand grains skidding down the slopes of a dune. That correctly described sudden transitions: social slippages.

"I have a feeling what we're missing isn't subtle." Most collapses were not from abstruse causes. In the early days of Empire consolidation, local minor sovereignties flourished, then died. There were recurrent themes in their histories. Again and again, star-spanning realms collapsed under the weight of excessive taxation. Sometimes the taxes supported mercenary armies which defended against neighbors, or which simply kept domestic order against centrifugal forces. Whatever the ostensible cause of taxes, soon enough the great cities became depopulated, as people fled the tax collectors, seeking "rural peace." But why did they do that spontaneously?

"People." Hari sat up suddenly, "That's what we're missing."

"Huh? You proved yourself - remember? the Reductionist Theorem?... that individuals don't matter."

"They don't. But people do. Our coupled equations describe them in the mass, but we don't know the critical drivers."

"That's all hidden, down in the data."

"Maybe not. What if we were big spiders, instead of primates? Would psychohistory look the same?"

Yugo frowned. "Well... if the data were the same ...

"Data on trade, wars, population statistics? It wouldn't matter whether we were counting spiders instead of people?"

The forms steadied as the data improved. Yugo let it run, time-stepping quickly so that they saw fifteen millennia in three minutes. It was startling, the pulsing solid growing myriad offshoots, structure endlessly proliferating. The madly burgeoning patterns spoke of the Empire's complexity far more than any emperor's lofty speech.

"Now here's the overlay" Yugo said, "showing how the Seldon Equations post-dict, in yellow."

"They aren't my equations," Hari said automatically. Long ago he and Yugo had seen that to predict with psychohistory first demanded that they post-dict the past, for verification. "They were-"

"Just watch." Alongside the deep blue data-figure, a yellow lump congealed. It looked to Hari like an identical twin to the original. Each went through contortions, seething with history's energy. Each ripple and snag represented many billions of human triumphs and tragedies. Every small shudder had once been a calamity.

"They're... the same," Hari whispered.

"Damn right," Yugo said. "The theory fits."

"Yup. Psychohistory works."

Hari stared at the flexing colors. "I never thought..."

"It could work so well?" Dors had walked behind his chair and now rubbed his scalp.

"We sought out telltale groups, pivots about which events sometimes hinge."

"The homosexuals, f'instance," Yugo said.

"They're about one percent of the population, a consistent minor variant in reproductive strategies," Hari said. Socially, though, they were often masters of impro-visation, fashioning style to substance, fully at home with the arbitrary. They seemed equipped with an internal compass that pointed them at every social novelty, early on, so that they exerted leverage all out of proportion to their numbers. Often they were sensitive indicators of future turns."

Yugo went on, "So we figured, could they be a crucial indicator? Turns out they are. Helps out the equations."

Dors said severely, "Why does history smooth out?”

Hari let Yugo carry the ball. "Y'see, that same sparrow effect had a positive side. Chaotic systems could be caught at just the right instant, tilted ever so slightly in a preferred way. A well-timed nudge could drive a system, yielding benefits all out of proportion to the effort expended. "