The Clockwork Rocket

She returned to the office to consult her sheets of paper, then she sketched a new diagram on her chest.

As Yalda contemplated the picture, it struck her just how reminiscent it was of a beam of light and its accompanying wavefronts, shown at a single moment in time. The main difference was the annoying tilt between the wavefronts and the “beam”—here, a line showing the history of the pulse.

But what did that skewed angle actually signify? By changing to different units of measurement, she could stretch or squeeze the diagram as much as she liked. Nature had no idea what a pause or a flicker was; nothing real could depend on adhering to that traditional system of units. So she chose units of time that forced the pulse and the wavefronts to trace out lines at right angles to each other.

Where did that leave her? She had a right angle between some lines… and a linear relationship between two quantities squared.

She played around with the diagram for a couple of chimes, changing the units for distance as well as those for time in such a way that the separation between the wavefronts was simply declared equal to one. Well, why not? It wasn’t just time whose units were completely artificial; a scant had once been defined as the resting width of some self-important monarch’s thumb.

When she was done, a small right triangle sat inside a larger one in the same proportions. The hypotenuse of the larger triangle was a horizontal line that joined one wavefront to another, making its length simply equal to the wavelength of the light. The small triangle’s sides—corresponding to a distance the pulse traveled and a time in which it did so—had a ratio of lengths equal to the light’s velocity. The larger triangle shared that ratio, in such a manner that the length of one of its sides was the inverse velocity.

In her chosen units, then: the inverse velocity squared, plus one squared, was equal to the wavelength squared. That simple equation corresponded to the straight line that passed through the data she’d plotted. But now this relationship didn’t need to emerge from any hypothetical properties of the hypothetical medium whose vibrations manifested as light. The sum of the squares of the sides of a right triangle equaled the square of the hypotenuse. That was it: the entire wavelength-velocity relationship that she’d extracted from all those nights of painstaking observations had turned out to be nothing but a theorem from elementary geometry in disguise.

Except… that was nonsense. Geometry was concerned with figures in space, not lines that stretched across time as well. However suggestive these results were of geometry, that could only be an analogy, at best.

Albeit a mathematically perfect one. If she pretended that she really was doing geometry in a plane, she could simply rotate the whole physical structure of the red pulse—rigidly preserving the spacing of the wavefronts—and transform it into a faster, violet pulse.

The wavelength and velocity changed, of course, but those things were just measurements that depended on the way the stack of wavefronts was disposed, relative to the person doing the measuring. The two pulses, red and violet, were no more different in essence than a pulse of light traveling north and another traveling north-east.

The message from the stars was: light is light, always the same on its own terms. Qualities such as color, direction and speed were only meaningful distinctions once the light bumped into something else, against which it could be measured. In the void, it was simply light.

Yalda was feeling disoriented; she walked in a daze to the living quarters and lay down in the bed’s slippery white sand. None of her conclusions made sense; this was just heat shock talking. If she could hallucinate Thero for a whole night, she could lose her powers of reasoning for a day. She’d sleep off her sickness, and everything would be clear in the morning.

Yalda spent the next day re-checking her calculations. All the numbers she’d relied on were correct—and her geometrical constructions were so simple that a five-year-old could have confirmed that they were right.

The only thing she could still doubt was her interpretation. Her right triangle with its wavelength-long hypotenuse might actually be nothing but a useful mnemonic, an easy way of remembering the velocity-wavelength formula. Mathematics that echoed the rules of geometry could arise anywhere, with all the lines and angles that it implied really nothing more than abstractions.

So… light was a vibration in some exotic medium that just happened to possess qualities that perfectly mimicked all the would-be geometry she’d found in the equations? As well as contriving to support shear waves and pressure waves that traveled at exactly the same speed? Was there nothing this magical material couldn’t do?

The three polarizations of light traveled at the same speed, as if they were all the same kind of thing. Yalda brought one of her diagrams of pulses and wavefronts back onto her chest. The picture projected the three dimensions of space down to just one, but in reality each wavefront was a plane, and it traced out a three-dimensional set over time. Within that set there would be three independent directions that were orthogonal to the path of the light pulse through the four dimensions that included time. The three polarizations could all be transverse waves—waves that pointed sideways, in that four-dimensional sense. There’d be no need for a miraculous coincidence to make all their velocities the same.

It was almost dusk. Yalda walked out of the building and sat at the top of the access path. Either she had lost her mind, or she had stumbled upon something that needed to be pursued much further.

She tinkered with the wavefront diagram on her chest. She’d been wondering about the significance of the inner triangle, the triangle whose hypotenuse was one. The ratio of its sides was the light’s velocity, but what exactly did the individual side lengths represent?

A simple argument with proportions established their values—which yielded a new triangular relationship, more elegant and symmetrical than the first: the sum of the squares of light’s frequencies in time and in space would equal one. Well, only her special choice of units set the sum equal to one, but the fact remained that the equivalent in cycles per scant or stride or saunter would still be independent of the color of the light.

That was really no different from saying that the true distance between furrows ploughed by a given plow did not depend on whether someone happened to walk across them askew. The wavefronts of light were all furrows from the same plow; the light’s speed, color, wavelength and frequency simply measured the angle at which you crossed the furrows.

But if light was going to play by these geometric rules, then everything it touched—every system that created or absorbed light, every substance that bent, scattered or distorted it—would have to function the same way. Ultimately, to keep the world consistent, any kind of physics that took place at one angle would have to work just as well if you picked it up and rotated it in four dimensions.

To accommodate light’s simplicity, half of science would need to be rewritten.

Yalda looked up; Sitha was starting to show against the fading gray sky. The colors were still weak, but the trail’s violet tip was as prominent as a skewer-worm’s barb.

“What have you done to me?” she said.

Then she remembered that there was no air between them, and she wrote the words across her chest instead.

6

“If time is exactly the same as space,” Giorgio asked Yalda, “why is it that I can walk to the Great Bridge, but I can’t walk to tomorrow?”

Yalda was distracted by a hubbub of exuberant buzzing and chirping from the adjoining room. In her absence, Giorgio’s co had given birth, and though the children were being cared for by their grandfather during the day, Giorgio couldn’t bear to be separated from them. He’d set up a nursery in the room beside his office.

Yalda focused on his question. “You’re already traveling toward tomorrow, along the most direct route possible. The shortest distance there is a straight line, and by standing still you’re following that line; you can’t do any better than that.”

“That makes sense,” Giorgio conceded. “But if I can’t do better, why can’t I do worse? Why can’t I dawdle and delay, and reach tomorrow later than expected? I can certainly do that if I walk to the Great Bridge.”

“And you can do it on your way to tomorrow,” Yalda replied. “If you cease standing still, if you wander around Zeugma, you will add some time to your journey. But because you can’t move very quickly, you can’t really manage much of a detour. The distance to tomorrow is vastly greater than the distance across Zeugma; the proportion by which you can increase it with any plausible peregrinations is unmeasurably small.”

Giorgio was amused, and she saw him slip out of his role for a moment to marvel openly at the sheer strangeness of these notions. Yalda knew she hadn’t convinced him that her ideas were correct, but he believed nonetheless that it was worth presenting them to the whole school of natural sciences: physicists, mathematicians, chemists and biologists. Before she spoke before so many colleagues, though, Giorgio wanted to be sure that she could defend her ideas against the inevitable barrage of objections, and he was doing his best to prepare her by anticipating every possible question and complaint.

“Exactly how far away is tomorrow?” he asked.

“As far as blue light can travel in a day.”

“Blue light? What’s so special about blue?”

“Absolutely nothing,” Yalda said firmly. “Violet is faster, and I believe there are even faster hues that we can’t perceive. But just as there’s a line in space that lies halfway between right and forward—marking equal progress in those two directions—there’s a line halfway between right and into the future. We perceive the light that reaches us at such an angle to be blue, and if we follow that light for a day, its progress marks out the equivalent distance.”

“I can’t compete with blue light,” Giorgio said, “so I can’t noticeably delay tomorrow. But why can’t I walk to yesterday?”

“For much the same reason,” Yalda replied. “Bending your path around until it’s turned backward would require an immense, sustained acceleration. In principle it ought to be possible, but it’s not something you should expect to be easy. You’re heading toward the future with a lot of inertia; you can nudge your trajectory a little with muscle power or a truck’s engine—but as you said, blue light isn’t easily outpaced.”

“But even if we only imagine it,” Giorgio persisted, “traveling toward the past would be very different from traveling toward the future. Traveling toward the future, we can shatter a stone into pieces with one blow; if we were traveling toward the past, the pieces would rise up and remake the whole before our eyes. Why is that distinction so clear… when directions in space such as north and south can barely be distinguished?”

“The same reason as we always suspected,” Yalda countered. “In the distant past, our part of the cosmos had much lower entropy; whether or not there was a single, primal world, things were certainly more orderly. The direction of increasing entropy looks radically different from the direction in which entropy decreases—but that’s not a fundamental property of space or time, it’s a happenstance of history.”

Giorgio wasn’t satisfied. “Time in either direction looks utterly different from any direction in space.”

“That’s because we’re surrounded by things that are moving almost entirely along that one axis,” Yalda said. “Not because physics decrees that they must move that way, but because they share a common history that has set them on that course. All the histories of all the worlds we can see form an almost straight bundle of lines through the four dimensions. The fastest star we know of is moving at barely one part in a gross of the speed of blue light. Living in a bundle of lines that are all so close to being parallel to each other, we shouldn’t be surprised that their common direction appears special to us.”

Giorgio changed his attack. “You say physics itself doesn’t decree that our histories are almost parallel. So according to your theory, an object could have a trajectory entirely orthogonal to our own?”

“Yes.”

“It could move with an infinite velocity?”

Yalda didn’t flinch. “Yes, that’s how we’d describe it.” It could cross what she and Giorgio thought of as a region of space in no time at all. “But that’s no stranger than saying that a vertical pole has an ‘infinite slope’: unlike a mountain road, it gets where it’s going vertically without bothering to go anywhere horizontally. An object that gets where it’s going without bothering to move across what we call time isn’t doing anything pathological; in reality, there’s nothing ‘infinite’ about it.”

“What about its kinetic energy?” Giorgio demanded. “Half its mass times its velocity squared?”

“That formula’s merely an approximation,” Yalda said. “You can’t use it for anything but small velocities.”

She summoned a diagram onto her skin. “If you want to know an object’s energy and momentum, draw an arrow whose length is the object’s mass, and point it along the line of its history. If you think the object is motionless, the arrow will point straight along the time axis; if you think it’s moving, the arrow needs to be tilted accordingly.”

“The amount by which the height of the arrow is diminished—compared to the motionless version—is its kinetic energy. For small velocities that will match the old formula, but for higher velocities it will grow much more slowly. The object’s momentum is the distance across space that the arrow spans; again, that agrees with the old formula if the object is moving slowly.”

Giorgio pretended that he hadn’t seen the picture before. “What’s this ‘true energy’?”

“The natural measure of energy is the height of the arrow in the time direction,” Yalda explained. “That way, energy is related to time in the same way momentum is related to space. Kinetic energy is a derived, secondary quantity.”

“But ‘true energy’ becomes less when you tip the arrow over,” Giorgio noted. “So when something moves… you’re now declaring that its energy is decreased?”

Yalda said, “Yes. Nothing else makes sense.”

Giorgio’s eyes widened in admiration at her effrontery. “So your theory turns the last three ages’ worth of science on its head. I suppose you’re also claiming that potential energy is upside-down in the same fashion?”

“Of course! We defined it to agree with kinetic energy, so it has the same relation to true energy.” Yalda summoned a picture of two springs accompanied by appropriate mass-length arrows: their four-dimensional momenta. “When the springs are compressed and motionless, we say they have a high potential energy. Now release them, let them fly apart, and see how things add up.”

“For true energy to be conserved, the heights of the pairs of arrows have to be identical before and after the release. But the arrows after the release are tilted, because the springs are now in motion. So those later arrows need to be longer, in order to reach the same height. That means that each relaxed spring has a slightly larger mass than it had when it was compressed—and from the point of view of someone traveling alongside it, a larger true energy. Less potential energy means more true energy. Both the old energies are upside-down.”

Giorgio let a hint of pained, Ludovico-esque weariness into his voice. “If kinetic and potential energy still agree, what can it actually mean to claim that they’re ‘upside-down’? Upside-down compared to what? When do we get to see any of this so-called true energy, to compare its direction with its alleged opposites?”

“In light,” Yalda said. “We see the direction of true energy every time we create light.”

She drew a simple diagram, line by line. “The chemists,” she said, “have been having a lot of trouble with their ladder of energies. If we’re to believe their calculations, the difference in chemical energy between fuel and the gas it becomes after burning isn’t anywhere near enough to account for the thermal energy of the gas. We kept telling them that they’d made a mistake, and that they should improve the accuracy of their measurements. But they were right, and we were wrong. The fuel itself doesn’t need to provide the energy to heat the gas… because that energy comes from the creation of light.

“Light brings its own four-dimensional momentum into the equation. It’s the need to balance that that forces the gas particles to be moving so fast. We thought that when fuel was burned, the light and the heat that was created both came from the release of chemical energy—but the truth is nothing like that! Light energy and thermal energy are opposites: creating one is what gives us the other.

“And we thought that when plants made food from soil, the light was merely an unintended by-product, a measure of inefficiency. But the energy in food isn’t extracted from the soil, and the light shining from a flower’s petals is not wasted energy escaping. Light energy and the chemical energy in food are opposites, too. If plants didn’t make light, they’d have no energy source at all.”

Yalda paused to give Giorgio a chance to respond, but he remained silent. Whatever radical notions she was proposing for the foundations of physics, these claims about food and fuel were the most shocking: the least abstract, the most tangible.

“Why can’t we cool our bodies by emitting light?” Yalda continued. “That’s what I asked myself on my way up Mount Peerless. But now it’s obvious! Emitting light can only give you more thermal energy than you started with. The very act of emitting too much light can make a living body as hot as burning sunstone.” Her grandfather’s frail body had never held enough energy to flatten a forest; rather, it had lost control of its production of light.

Giorgio said, “If emitting light generates thermal energy… why can’t we cool down by absorbing light instead? Why isn’t sunlight as good as our beds for making us cooler?”

Yalda was prepared for that. “Entropy. Light carries a certain amount of entropy—so if you absorb light, your entropy must increase. But if we cool down, our entropy decreases. What I think happens when sunlight strikes our body is that we don’t absorb it, we just scatter it. That way, we can simply take a share of its kinetic energy, and be warmed by it.”

Giorgio stopped the interrogation to take stock. “Well, you’ll certainly please the chemists,” he said. “If you’re right about this, they’ll build a statue in your honor. And the biologists will be intrigued by your ideas on energetics, even if half of them think you’re insane. There’s even something to make Ludovico happy.”

Yalda doubted that, though she knew what he meant. A wave traveling through any ordinary medium marked an increase in kinetic and potential energy, not true energy. If creating light required true energy, it could not be a ripple in some pre-existing medium; it had to be a whole new substance or entity that was created afresh in every flame. But if that brought the term “luminous corpuscles” to mind, in Yalda’s scheme light still had a wavelength—so Ludovico would call this arrogance and hypocrisy, not a triumph for his beloved Meconio.

“Now a question from the mathematicians,” Giorgio said. “You’ve shown us equations for the geometry of wavefronts, but what about an equation for the wave itself—something analogous to the wave equation on a string?”

“The geometry gives us that, easily,” Yalda replied. “For a simple wave, the sum of the squares of the frequencies in all four dimensions equals a constant. But we also know that the wave’s second rate of change in each direction will be the original wave multiplied by a negative factor proportional to the frequency squared.”

She sketched some examples, showing how doubling the frequency of a wave quadrupled its second rate of change. The square of the frequency and the second rate of change were just two ways of talking about the same thing.

“So if you sum the second rates of change of the wave along each of the four dimensions, and negate that sum, then you’ve got the original wave multiplied by a constant times the sum of the squares of the frequencies—which itself is a constant. And that’s the equation for a light wave: the sum of its second rates of change, negated, must equal a constant times the original wave.”

Giorgio contemplated this for a pause or two, then responded with a sketch of his own.

“An oscillation’s second rate of change is proportional to the opposite of the original wave,” he said. “But an exponential growth curve has a second rate of change proportional to the wave itself—there’s no negation.”

“That’s true,” Yalda said. “But—”

“If you construct a wave that oscillates rapidly as you move in one direction,” Giorgio said, “what’s to stop you from choosing a frequency in that direction so large that its square, alone, is greater than the number you’re aiming for as the sum of all four?”

“But then you’ve overshot the total,” Yalda protested. “So you won’t be able to satisfy the equation.”

“No? What if one of the other terms is negative?”

“Oh.” Yalda knew where he was heading now. “If one of the oscillations has too large a frequency, you can still satisfy the equation—by replacing the oscillation in another direction with exponential growth.” The negated second rate of change, in that case, would be a negative multiple of the original wave, allowing the sum of all four terms to be brought back down to the target.

Giorgio said, “So the question is: if light obeys the equation you’ve given us, how can it possibly be stable? Why doesn’t every tiny wrinkle in the wave explode at an exponential rate?”

7

As the crowd spilled out of the Variety Hall into the starlit square, Yalda’s feeling of delight lingered. She’d found the entire magic show enchanting, but the fact that she’d quickly guessed the trick behind the astonishing finale hadn’t detracted from the pleasure of the experience at all; instead, it had intensified it.

She turned to Tullia. “That image of the hidden assistant, projected onto a curtain of smoke… if I ever get to teach the optics course, I’m going to steal that for my first demonstration!”

“That part wasn’t bad,” Tullia conceded. “The pyrotechnics before the intermission were awfully tame, but that’s the new safety regulations for you. I suppose we should give the City Council its dues: letting off rockets inside the hall was never a good idea.”

“Antonia should have come,” Yalda said, turning sideways to squeeze through a gap in the throng. “It might have cheered her up.”

“Antonia doesn’t want to be cheered up,” Tullia replied. “She’s committed to sitting at home moping until she undergoes spontaneous fission.”

“It must be hard for her to make a decision.” Yalda had found it difficult enough trying to side-step her own family’s expectations, but growing up with a co and then walking away from him would be something else entirely.

“We could get her safely to another city in a couple of days, if she agreed to it,” Tullia said irritably. “But she’s got herself caught up in negotiations with her co—some complicated business involving four or five intermediaries. She thinks she can go back to him on her own terms.”

“Maybe she can. Maybe she’s arranging that.”

“Ha.”

“Is it so bad that she wants him to raise her children?”

“In principle, not at all,” Tullia replied. “The trouble is, he’s already proved himself incapable of taking her wishes seriously. If Antonia wanted to, she could spend five or six years living the free life in Red Towers or Jade City, and then find a nice co-stead who’d be grateful for the heirs.”

Yalda said, “You make it sound so easy, it’s a wonder everyone isn’t doing it.”

A streak of brilliant violet light appeared in the sky above the eastern horizon, spreading out rapidly from a fixed central point. The center itself remained dark, but as Yalda watched, the two dazzling threads emerging from it turned blue, then green, the new colors chasing the old in both directions. It was as if someone were dragging a giant star trail out from behind the edge of a mirror, exposing ever more of it while creating a perfect duplicate that seemed to be rushing in the opposite direction.

Yalda was transfixed; Tullia was already counting out pauses as she threaded her way through the crowd, trying to sight the nearest clock tower and fix the time of the event precisely. The two of them had never actually made plans as to what they’d do if they witnessed a Hurtler, but they’d managed to get the division of labor exactly right without a single rehearsal. Motionless, Yalda could etch the position of everything she’d seen into her memory, holding on to an image of the line of light against the stars. Tullia wouldn’t have those details, but she’d soon have the crucial timing information that would render comparisons with reports from other cities twice as valuable.

The center disgorged two red tails and faded to black; the pair of mirrored spectral worms, now fully birthed and separated, disappeared into opposite corners of the dusty haze that hung over Zeugma’s towers. Yalda had only ever seen the final part of this spectacle before: all those years ago, after the harvest, when the center must have been below the horizon for her. To date, seven reports of the same phenomenon had reached the university; the one she’d witnessed as a child was the third in that list. History and legend were full of shooting stars—some of them accompanied by all manner of implausible flourishes—but neither ancient astronomers nor the authors of the sagas had ever claimed to have seen anything like the Hurtlers.

Yalda remained still, carefully gauging the angles between her memory of the Hurtler’s trajectory and the nearest bright stars. In her rear gaze she could see the young man glaring at her, shouting, but even if she’d been wandering the square aimlessly she would have done her best to pay him no attention.

“Where’s your co?” he yelled again. Yalda marveled at his sheer boorishness; the most extraordinary event ever seen in the sky—unknown to the ages, but glimpsed by the luckiest people now once or twice in a lifetime—had just unfolded before his eyes, and all he could think to do was taunt her for her size, or her lack of a partner.

The man bent down, picked up a broken piece of cobblestone, and flung it at her; it struck her on the side of the head. Yalda couldn’t stop herself; she turned toward him.

He squealed triumphantly, “I said: where’s your co?”

Yalda squatted and retrieved the stone from the ground, feeling the heft of it and the sharp edges. This made her far angrier than when it had collided with her skull, because she knew firsthand now what the thrower should have known, what should have dissuaded him. Unusually, the man was accompanied by his own co, as well as the expected group of delighted male friends.

“Where’s your mother?” Yalda called back, and pitched the broken stone at him with all her strength.

If her choice of words stunned him, it was the impact that brought him to his knees. She’d hit him squarely in the tympanum, more by luck than intent. He cried out in agony—which could only have made the pain in his organ of speech more intense—and his hum began warbling up and down, as the need to express his suffering fought with his attempts to curtail it.

Some of his friends looked shocked; others were even more amused than before at this unexpected twist in the night’s entertainment. The man’s co bore an expression of horrified disbelief, as if she’d just seen a freight train run down an infant. Yalda felt a sudden pang of fear; she’d done the greater harm, and most potential witnesses around them were still paying more attention to the sky than the ground. Whatever they’d glimpsed in their own rear gaze, they might only be aware of half the story.

Yalda hurried away from the scene of her imprudent revenge, and caught up with Tullia on the other side of the square.

“You fixed the time?” Yalda asked her.

“Yes.” Despite her presence of mind as the event had unfolded, Tullia now appeared a bit dazed. This was her first sighting, and it would have confirmed all the scarcely believable claims that until now she’d been free to doubt.

“I’ve got the bearings,” Yalda said. “We should write up the observation now, and dispatch it tomorrow.”

“Of course.” Tullia shook herself out of her stupor. “That was, what, three and a half pauses from violet to red?”

“Sounds right.”

“Which puts it far above the atmosphere, but still close; a fraction of the distance to the sun.”

“A gross and a half severances or so,” Yalda confirmed.

The people around them were still buzzing with excitement, but Yalda detected no real sense of how extraordinary the sight had been; it was as if they’d just witnessed an elegant fireworks display to cap off the magic show.

“What if it had been closer?” Tullia asked. “What if it had hit the ground?”

Yalda had never seriously considered the possibility of a collision; with barely more than half a dozen sightings in a generation, it struck her as a remote prospect. “I wouldn’t like to be standing at the point of impact,” she conceded.

Tullia said, “I wouldn’t like to be on the same planet.”

The current thinking about the Hurtlers was that something was colliding with the tenuous gas that wafted out from the sun to occupy the surrounding region. Just as an ordinary shooting star could burn up brightly in the atmosphere, even the sun’s thin exhaust might be enough to ignite a sufficiently rapid interloper.

How rapid were the Hurtlers? If an object was moving so speedily that you might as well imagine its entire trajectory erupting with light all at once, then the closest part of that long straight line would appear to a watcher first in violet, the fastest color, with the other hues following. Each color would appear to fly out along two opposing, symmetrical trails, as the light arrived from pairs of equidistant locations ever farther from the watcher. Any measurable asymmetry in the color trails would imply a lower speed for the object itself—with light from earlier parts of the trajectory gaining a head start—but as yet nobody had observed such subtle effects with enough confidence even to be sure which way the Hurtlers were traveling.

“If you can salvage my geometrical theory of time,” Yalda bargained, “the pay-off is that something so fast won’t be carrying as much kinetic energy.”

“If I salvage your theory,” Tullia retorted, “nothing will even need kinetic energy to tear itself apart. Everything in the cosmos will be itching to turn into light and hot gas.”

“Don’t blame me if there are no happy endings; I didn’t invent entropy. Darkness and cold dust… bright light and hot gas. Does it really matter which one we end up as?”

They began making their way toward Tullia’s apartment, to put their observations onto paper.

Tullia said, “You do realize that according to your theory, someone traveling along with the Hurtler would think that half the light we just saw was coming in toward them, not going out?”

Yalda made a quick sketch on her chest.

“You’re right,” she said. “That’s eerie.” The arrow of time shared by the world and the solar atmosphere was so different from the Hurtler’s arrow of time that Tullia’s hypothetical traveler would have seen a part of the light-burst converging on them—violating the law of increasing entropy as surely as if a roomful of smoke had shrunk in on itself and turned back into fuel. Obviously entropy couldn’t increase along every direction in four-space at once, but it was unsettling to have an example of the bizarre disparity play out right in front of their eyes.

Yalda brushed the complication aside; she was having enough trouble trying to make the exponential blow-up in the light equation go away. She was scheduled to deliver a summary of her theory to the school of natural sciences in less than two stints, but if she couldn’t offer Giorgio a plausible solution to the flaw he’d uncovered, he’d cancel the talk.

When they entered the apartment, Antonia was seated on the floor with dye and paper beside her. A firestone lamp was sputtering on the shelf above, casting a forlorn shadow. She’d probably been composing another letter to Antonio, but when Yalda and Tullia approached to greet her, her skin was blank. Yalda wished she could have offered her advice or comfort, but what did a solo have to say about the choices she faced?

“How was the magic show?” Antonia asked, forcedly cheerful.

“Upstaged,” Tullia replied. She described the celestial mirror trick that had followed.

“I heard some commotion from the street,” Antonia said. “I looked out the window, but it must have been over by then.”

“Do you mind if we use the dye?” Yalda asked. She wanted to have the report on the Hurtler completed as soon as possible, ready for the couriers who’d be leaving at dawn.

“Of course not.” Antonia put the lid back on the pot and slid it toward her. “I was still gathering my thoughts; it can wait until morning.”

Yalda saw the curtain part at the entrance to the apartment. As she spun around to face the intruders one of them screamed, “Lie on the floor! All of you!” By now, four men had filed into the room, and there were more behind them. They wore police belts, and they’d unsheathed their knives.

Antonia began wailing. “I’m sorry! Tullia, I’m sorry! Someone must have—”

Tullia said, “Be quiet, you don’t know—” One of the officers stepped up to her, knife outstretched.

“Lie down, or I’ll split you open!”

Tullia knelt then lay on her chest. Yalda met her rear gaze, hoping for advice, but if there was a message she couldn’t read it.

Yalda said, “Antonia, get behind me.” She moved toward the officer who’d threatened Tullia. He was tiny; if not for the knife she could have done what she liked with him. “You want to go out the window?” she taunted him. “You’ve got no business here. Go harass someone else.”

The man raised the knife confidently, no doubt accustomed to its power to induce obedience. Yalda advanced on him, undeterred. She wouldn’t even need to extrude extra limbs for the encounter; if she seized him with both hands it wouldn’t matter if she lost an arm in the process, she could still fling him down to the street with the remaining one.

“Please, Yalda, don’t!” Antonia implored her, distraught. “I’ll go back! Don’t make trouble for yourself!”

Yalda was unmoved; what gave these buffoons the right to interfere in anyone’s life? If one of them spilled his brains on the cobblestones, the others might rethink their priorities.

Tullia addressed her calmly. “Yalda, if you resist, we’ll all get a beating. If you harm even one of them, we’ll all be killed.”

Yalda stared at the man in front of her, then forced herself to look past his triumphant sneer to the long line of colleagues waiting behind him, knives at the ready. She might be able to deal with three or four of them before she was overpowered—but if Tullia was right, it would not be worth the price.

She knelt down, then lay on the floor, subduing her rage. Her physical strength meant nothing. The rightness of her cause meant nothing. The Council had given these men the authority to capture and return runaways; Antonia’s plans for her life were irrelevant.

The officer she’d confronted put a foot on her back and held her arms behind her while someone passed him a length of hardstone chain. He slipped the loop at the end of the chain around one of her arms, then took a vial from his belt and shook a few beads of bright red resin onto her palm. It stung fiercely, but Yalda forced herself not to lash out. Then he pressed both of her palms together. Skin stuck tightly to skin, in itself no great hardship, but the resin made her body act as if these ordinary surfaces comprised a kind of internal partition, a pathological mistake that needed to be broken down.

Yalda closed her eyes for a moment, fighting not to lose consciousness. She had no right to be surprised by any of this; how many prisoners had she seen shuffling through Zeugma with their arms melded? She’d looked away, like everyone else. Murderers and thieves got what they deserved.

The officer ran the tip of his knife over her skin methodically, until he found the telltale crease of a pocket.

“Do you want me to cut it open?” he asked.

Yalda opened the pocket. He reached in and took out a handful of coins and her vial of holin.

In the corner of the room, Antonia was pleading with her own captor. Her hands had been bound with rope, but she and Tullia had been spared the melding resin, no doubt as a reward for their swift compliance. Once they were down on the street, Yalda thought, Antonia could easily slip out of those unreliable bonds and make a run for it.

Yalda’s tormentor walked over to Antonia. “You’re a runaway?”

“Yes, sir.”

“And you’re willing to return to your co?”

“Yes, sir. But my friends didn’t know; I told them he was dead. I’ll go back to him willingly, but you have to release them.”

This attempt at a bargain amused the officer. “This patrol wasn’t looking for you,” he said, “but it’s kind of you to volunteer the truth. We only came here for the fat one, the solo. She assaulted a Councilor’s son.”

He walked back to Yalda and began kicking her in the tympanum.

The room fractured, the walls collapsed into rubble. Yalda writhed and screamed, buried in shards of noise and pain.

8

“When you come before the sergeant,” Tullia whispered, “don’t argue about anything. Agree to the fine, agree to the conditions, and you’ll be out of here in a few more days.”

Yalda was bound to the wall of her cell, her own flesh the last link in the chain. She’d threaded her body through the loop of her melded arms so they were in front of her now, a minor improvement. The cell was bare and windowless, equally dark by night and by day. Twice, someone had entered unseen; the first time to beat her, the second to strew rotten grain on the floor. The loudest sounds that reached her were the thwack of wood against flesh and the hums of misery from other cells.

They’d granted her two unintended mercies, though. The floor was real soil, her favorite kind of bed; the worms that might have revolted a more fastidious guest just made her feel at home. And they’d put her next to Tullia, allowing them to whisper to each other through the wall’s porous stone. Without that, she would have lost her mind.

“I’ll be charged with sheltering a runaway, and for the holin in my room if they found it,” Tullia explained. Apparently she’d been through all of this before. “They’ll fine me a few dozen pieces, and make me swear an oath not to repeat my crimes. Your fine will probably be larger, but don’t worry: they’ll give you a chance to contact people who can help you pay it. I expect I’ll be out before you, so I’ll talk to Daria and the others at the Solo. Whatever you need, we’ll raise it.”

“He threw the stone at me!” Yalda complained. “Don’t pay them anything! Let them charge that shit-head with assault as well.”

“Can you produce a dozen witnesses against him?” Tullia asked.

“Probably not.”

“Then it doesn’t matter what he did. Stop telling yourself that it matters, or you’re going to make everything harder.”

Yalda could not accept this advice. She knew she should have restrained herself: she should have resisted lobbing back the cobblestone, sharp and heavy as she’d known it to be. But she still ached to see her assailant locked up beside her, beaten beside her, fined and humiliated and forced to promise to reform his own violent ways.

She knew that her actions had cost Antonia her life. Maybe a few years of it, maybe a few stints, but Antonia’s chance to bargain with her co had been lost the instant Yalda had brought the police into Tullia’s apartment. That was the worst of what she’d done, and she’d willingly confess her recklessness to anyone accusing her on Antonia’s behalf. But her own culpability excused no one else. Let Antonio, who was merely eager for children, let the Councilor’s son, who was only teasing a solo, let the police, who were simply doing their jobs, all line up and take their punishment beside her.

Otherwise, octofurcate them all.

Tullia grew weary of the subject, and after making her advice clear she steered their conversation elsewhere.

“Come out of this stinking prison with me for a couple of bells,” she begged Yalda. “Why live the life of the mind at all, if you’re not going to live it now?”

“I’ll lie here and hallucinate Hurtlers, shall I? That will be a real comfort.”

“Last time I checked, you had a more urgent problem,” Tullia reminded her.

“You want us to solve the exponential blow-up, here?”

“How would you rather spend your time? Plotting the dismemberment of Councilors’ sons?”

In truth, Yalda longed for a distraction, and she wished she shared Tullia’s discipline and resolve. But the problem itself seemed as intractable as their incarceration. “Giorgio was right,” she said. “The equation I found has exponential solutions. And if I try to damp them down—if I try to get rid of them by adding new terms to the equation—I just lose the original solutions.”

“It’s a strange equation for a wave,” Tullia conceded. “The nice thing about the wave equation on a string is that you can set the initial conditions and just watch them unfold: you can pluck the string into any shape you like, and give it any kind of movement, and from that the equation lets you find the shape of the string at any time in the future. What’s more, if you make a small mistake measuring the initial setup, it’s not a calamity; the errors in your prediction are equally small.

“But your light equation is more like the equation for the temperature distribution in a solid. If you have, say… a thin slab of stone and you want to know its temperature at every point, to get reliable solutions you need to specify the temperature all the way around the border of the slab. If you tried to start with the temperature along a single edge and its inward gradient, any tiny error in the data there would blow up exponentially as you moved across the slab. Your equation acts the same way.”

Yalda pondered this in the darkness. “So by analogy, to calculate the behavior of light in a certain place, over a period of time… I’d really need to know what it does on the entire border of that four-dimensional region? Not only what it’s doing at the start, but everything that happens at the boundary, and how it all ends up?”

“Precisely,” Tullia said. “The equation you’ve come up with might be said to govern the behavior of light, but practically speaking it’s no good for making predictions. Everything it tells you could be tested in retrospect, but you’d always need to know how the story ends before you can start reliably ‘predicting’ the middle.”

Yalda said, “Waves on a string can only have one velocity. We know that violet light can travel much faster than red light—and it’s not implausible that there are even faster hues, beyond our ability to detect. So why should we expect that knowing the state of the light in just one place should ever be enough to say what happens next? Some other wave we haven’t accounted for—just beyond the edge of the region we know about—could always be on the verge of crashing in and spoiling our prediction.”

“Good point,” Tullia replied. “So let’s accept the possibility of light that travels as fast as you wish… but then to compensate, I let you know about every wave that presently exists as far away as you wish. That way, you get as much warning as you could hope for; you can’t complain that some fast wave came hurtling in from a place beyond the reach of your data. And if the cosmos goes on forever, you get to know about the entire, infinite present: for one moment in time, there are no secrets from you anywhere.”

“Go ahead and grant me that!” Yalda urged her. “Wouldn’t that eliminate the problem?”

“No!” Tullia sounded both exasperated and amused that Yalda had taken the bait so readily, without thinking it through. “Your equation still has solutions that can blow up exponentially out of the tiniest mismeasurement. You still wouldn’t be able to predict what happens right in front of your eyes over the next few pauses. Does that really accord with your instincts about the way the physics of light should work?”

“No,” Yalda admitted. She adjusted her position, then cursed softly and braced herself for the sound of tearing flesh. She’d been trying to keep her arms a few scants apart within their shared sleeve of skin, hoping to make her eventual liberation less traumatic. But her body thought it knew best. Every time she dozed, or her attention wandered, she had to rip apart a fresh bundle of muscle fibers that had formed between the ends of her limbs.

When she was done, she thought back over the steps in Tullia’s argument. “What if the cosmos didn’t go on forever?” she said. “In space, or in time?”

“Then you’d still need to know what happens at its boundary,” Tullia replied. “Like the slab of stone: you need to know what’s going on at all the edges.”

Yalda considered the possibilities this offered. By declaring that the boundary of the cosmos was subject to some special rule—perhaps that the wave simply had to be zero there—she could probably keep it from blowing up in the interior. But that was an ugly resolution, an arbitrary constraint that came from nowhere and offered no deeper understanding.

“What if there are no edges?” she suggested. “What if the cosmos is like the surface of the world—finite, but borderless?”

Tullia lapsed into silence for so long that Yalda grew worried. She extruded a new, free arm and thumped the wall. “Are you all right?”

“Yes! I’m thinking!” Tullia almost sounded happy, as if Yalda had finally suggested something novel enough to be truly diverting.

Eventually Tullia declared, “I’m fairly sure that that would solve the exponential blow-up. You can wrap an oscillation around a sphere so that it joins up with itself smoothly—but you can’t do that with an exponential growth curve, which never revisits its earlier values.”

Yalda chirped with delight. “So if the cosmos is a four-dimensional version of the surface of a sphere—”

“Things would still be very strange,” Tullia warned her. “The prediction problem jumps from one extreme to the other.”

“What do you mean?”

“Think about the two-dimensional version,” Tullia said. “If you draw a circle around Zeugma, the data lying on that circle—combined with your equation—tells you everything that happens in the city. Information about the border gives you information about the interior.”

“But that’s nothing new. Where’s the problem?”

“A circle around Zeugma,” Tullia replied, “is also a circle around everything else in the world. The border of the city is also the border of everything that lies beyond. So from the data on that one circle, you can find the solution to your equation on the entire sphere.”

“Oh.”

Tullia drove the point home. “In the four-dimensional version, that’s like claiming that you can measure the light in a patch a few scants across, for a couple of pauses… and learn everything there is to know about light throughout the history of the cosmos. Because the border of your tiny region is also the border of everything else.”

Yalda buzzed wryly. “I can’t say that that accords with my instincts about the physics of light.”

“Nor mine.” Tullia’s surge of enthusiasm was gone, but she was doing her best not to sound despondent.

“We tried,” Yalda said. “And it was worth trying.”

They had managed to leave the prison for a while, but nothing was easy, even in freedom.

When it was quiet in the cells, Yalda could hear the bells from one of the city’s clock towers; she missed a few from noise, or sleep, or inattentiveness, but never so many as to lose track of time. So she knew it was the middle of the morning on their third day when the guards came and took Tullia.

This had to be her hearing with the sergeant. Yalda waited, trying to be patient. Tullia had told her that there was usually a large group of prisoners to be dealt with in each session, and the whole thing could last a bell or two.

By evening, Tullia had not returned. Either they’d set her free, or they’d moved her to a different cell while she arranged the payment of her fine.

Yalda chose to believe that she’d been freed. Tullia hadn’t resisted arrest, and she knew the system well enough to say the right things at her hearing. If the fine had been small enough she might have been released on a promissory note, rather than forced to wait until actual coins had been delivered to the sergeant. Tullia would be at the Solo Club, celebrating her freedom and looking for ways to help her friend.

Yalda blocked out the sad humming of her neighbors; she felt for them, but she didn’t have the strength to involve herself in their plights. She would have her turn before the sergeant soon; she needed to decide what she would say.

When the guards came the next morning, the light of their lamp almost blinded her. She’d planned to sneak a look at the tool they used to disconnect her chain from its place on the wall, but everything was veiled in painful brightness. As they tugged on the chain to draw her out of the cell, she quickly lengthened one of her arms and shortened the other, allowing the force to be borne by solid flesh instead of the loose tube of skin between them that she’d fought to keep empty.

She stumbled up the broad staircase into a corridor filled with searing daylight, then hurried along with slitted eyes, not wishing to drag the chain and provoke her captors. In a room full of prisoners, she was secured to the wall again. Yalda raised her gaze cautiously; there were more than a dozen men and women chained up beside her, most of them with melded limbs. Everyone looked as wretched and afraid as she was.

She felt herself shivering. No friends were permitted here to offer support. Nobody could advise her now, or speak in her defense. All she had to guide her was Tullia’s counsel, which she’d been so vehement in opposing.

The sergeant entered the room—wearing a belt much like his subordinates’, but adorned with no less than four knives—and took his place behind an impressive calmstone desk. An assistant brought in a stack of paper, smelling of fresh dye; for an instant the scent was almost comforting.

As the first case was heard, Yalda tried to concentrate on the procedure and learn what she could. A young man had stolen a loaf from the markets, and then fled from the police. He did not deny the charge.

The sergeant fined him a dozen pieces. “How will you pay this?” he asked.

“My brother might help me,” the man said, his voice soft with shame.

“Give his details to the messenger; you can wait in your cell.” A guard took the man’s chain and led him away.

The next prisoner, another young man, had trespassed in a private garden; he was not accused of stealing anything, but his fine was three times as much as the thief’s.

Everything about the process was humiliating, but Yalda prepared herself to swallow her pride. Tullia had offered to help her find the money for her fine; Daria would probably be willing to loan her a few dozen pieces. She could be out of this place by nightfall if she was suitably humble and penitent. And whatever blame fell upon her for Antonia’s fate, there was nothing to be gained by making trouble for herself. No one was going to rise up against the Council to overthrow the law on runaways because one fat solo argued with the police about an unrelated assault.

When her turn came, a guard unclamped her chain from the wall and ushered her in front of the sergeant’s desk.

“Are you Yalda, daughter of Vito?”

“Yes, sir.” The sound of her father’s name stung; Yalda had no wish to imagine him witnessing any of this.

The sergeant scanned the paper in front of him. “You are charged, firstly, with possession of a substance contrary to the order of nature and the public good. Do you dispute the charge?”

“No, sir.” In the darkness of her cell she’d rehearsed speeches on the insanity of banning a drug that spared the world fatherless children, fantasizing about the power of her impeccable logic to sway even the most hostile audience.

“On that charge, I fine you a dozen pieces.”

“Thank you, sir.”

The sergeant glanced up at her, irritated, as if her anxious tic might actually be taken to imply that the fine was lenient. “Secondly, you are charged with a grievous assault against the person of Acilio, son of Acilio, four nights ago in the square outside the Variety Hall. I have statements from six witnesses to the effect that you threw a sharpened stone that struck him and caused substantial injuries. Do you dispute the charge?”

“No, sir.”

“Do you have anything to say in mitigation?”

Yalda hesitated. Surely an honest answer would not be treated as hostility or disputation? Why ask about mitigation if you had no wish to hear the truth?

“Sir, Acilio did throw the stone at me before I assaulted him with it. It only struck me lightly, but that’s how I came to have it in my hand.”

The sergeant re-examined the paper in front of him then slid it aside and gazed up at her coldly. “What witnesses do you name for this accusation?”

“None, sir,” Yalda admitted. “Most people were looking at the sky,” she explained, “and my friend was on the other side of the square.”

“Then I fine you two dozen pieces for a gratuitous and cowardly libel,” the sergeant said, “and a further dozen for wasting my time.”

Yalda’s skin quivered, as if her body believed it could rid her of this strange insect that kept taking bites out of her flesh.

“As for the assault,” the sergeant continued, “the complainant has requested a payment of a dozen gross pieces in reparation, an amount with which I concur. Additionally, on behalf of the citizens of Zeugma, I fine you a further gross. Your total fine is a dozen-and-one gross and four dozen pieces. How will you pay this?”

Yalda couldn’t speak. Even Daria, with the fees from her public dissections, wouldn’t make that much in a year; for Tullia or Lidia it would be a lifetime’s wages.

“How will you pay?” the sergeant repeated impatiently.

“I can’t,” Yalda said. “I don’t have anything like that.”

The sergeant hummed wearily. “I don’t expect you to pluck that many coins from your pocket, you simpleton. Just give the messenger the name of someone who can organize the money for you.”

“There’s no one who can do that,” Yalda insisted. A dozen gross? She couldn’t burden Tullia with that surreal demand; she couldn’t bury all her friends in impossible debts. “Can’t you… reconsider the size of the reparation?” she pleaded.

“What I’ll do,” the sergeant said, his tone now sarcastically good-natured, “is return you to your cell for a stint, so you can reconsider the resources at your disposal.” He gestured to the guard.

As she was led back down to the basement, Yalda kept tripping on the stairs. The guard waited for her to right herself; perhaps the sheer size of her fine had impressed him to the point where any further mistreatment was superfluous.

He said, “You should pick your fights more carefully.”

Yalda said, “I didn’t even know who he was.”

The guard buzzed with mirth. “You do now.”

At first, Yalda refused to believe that things were as they seemed. A dozen gross pieces? It had to be a kind of cruel joke, a punishment for her “cowardly libel”. After a day or two she’d be hauled before the sergeant again and told what the true fine would be.

But as she heard the bells marking the end of her sixth day in prison—with the spoilt grain that she’d spurned at first, then scrabbled for blindly, now entirely gone—she experienced a moment of clarity. A part of her, she realized, had been laboring under a strange assumption: that people with the power to release her would be spending their days pondering her fate, agonizing over her hardship, questioning her punishment’s severity. And since no one was entirely devoid of feeling… anything truly intolerable to her would, in the end, be intolerable to them. Any treatment so unjust that it threatened to crush her spirit would, in the end, wear down their resolve to impose it.

But it was not like that at all. The sergeant, the guards, the Council, her accuser, buttressed by each other’s mutual approval, shared the burden of her imprisonment so equitably that it became no burden at all. No one, individually, was responsible for what they’d done to her in concert. She could die in this cell, and none of them would feel the faintest twinge of discomfort.

All she could do now was wait out the stint, then send an honest message to Tullia explaining her situation. She would not let her friends go into debt, but if they told her story to everyone at the Solo Club, perhaps some of the wealthier customers would be sympathetic to her plight. Perhaps over a year or two the money could be raised.

A narrow bridge of flesh had formed again between her sleeved arms. Yalda tugged at the fibers angrily, jerking and tearing them until the last one snapped. However long she stayed here, it wouldn’t be the guards who set her free.

On the morning of her eighth day of imprisonment, Yalda woke to find something hard on the floor when she moved her feet. She picked up the grains one by one until she had a handful, then she tipped them carefully into her mouth.

Why did she need food at all? Why not just make light, and get the energy she needed for free? She wasn’t growing, like a child; she didn’t need to add new matter to her body.

But the matter she did have was growing disordered; the microscopic building blocks of her flesh were slipping into disarray. Soil for a plant, food for an animal, offered more than materials for growth and repair: it was a source of low entropy. The rock from which it came was highly ordered—and without order, energy was useless, as likely to push you one way as another. Life rode the arrow of time that came from the world’s slow decay.

But now that she had a little order in her body, what was she going to do with it? Her captors wouldn’t let her starve to death, but how would she stay sane?

“All right, Tullia,” she whispered. “I’ll show you the life of the mind.”

Tullia had claimed that if the cosmos resembled the surface of a sphere, Yalda’s equation would render everything absurdly predictable. Her argument had sounded plausible, but Yalda wanted a deeper understanding of the problem before she abandoned the whole idea.

On a sphere, she realized, the fundamental solutions of her equation would be spherical harmonics: a kind of waveform she’d encountered once before in a course on seismology. Whatever complicated solution held across the whole surface of the sphere, it could be written as a sum of these harmonics, each multiplied by a suitable factor measuring the size of its contribution.

Yalda worked through the calculations, raising equations on her skin in the dark. First, you fixed the physical parameters: the radius of the sphere and the wavefronts’ separation. Then, as the wave’s frequency with respect to longitude rose, its frequency with respect to latitude would fall. Since you always had to wrap whole numbers of waves around the equator and around any meridian, in the end there were only a finite number of possibilities—a finite number of relevant harmonics.

She sketched a few examples to make the calculations tangible. The north and south hemispheres were identical, so she only bothered with half the sphere, drawing the waves along the various circles of latitude where they were strongest.

Around any circle of latitude, though—however large or small—each particular harmonic would execute the same number of cycles, making each one as distinctive as the harmonics of a plucked string. So if you were given the values on any circle of latitude of any wave obeying the equation, you could separate out the harmonics and determine their respective strengths—which would give you the entire, global solution. What’s more, the choice of where the “pole” lay in this scenario was completely arbitrary; in principle, you could perform the same analysis anywhere at all.

In practice, though? If the waves were already packed six dozen gross to the scant as they marched around the cosmic equator, how could you ever hope to observe their proportionately finer undulations around a circle that was just a stride or two across? And to make the problem even more acute, the higher the harmonic the faster its strength dropped as you approached the poles, so the wavefronts associated with it would be preposterously weak, as well as unmeasurably close.

Tullia’s objection, then, was purely philosophical: the idea that the entire cosmic history of light was writ small in every last corner of the world was simply too shocking to countenance—however useless it would be for aspiring fortune-tellers. Yalda was prepared to set her unease aside and see where the rest of the theory took her, but other physicists might easily view this as a flaw as fatal as Giorgio’s original complaint. What was the use in being half right, if she wasn’t even half-believed? She needed other scientists to pursue these ideas; imprisoned or free, she couldn’t begin to explore all of their ramifications herself.

Yalda slumped forward and rested her head against the aching circle of her arms. She wanted to drag the weary muscles in these limbs all the way back into her chest and replace them with rested flesh, but with no instinct or experience to guide her, she couldn’t find a safe, painless sequence of movements that would fulfill this wish. For all the many postures she’d tried since childhood, she had never before suffered a change in the topology of her skin.

She butted the gap between her arms with the top of her head, allowing the limbs to relax for a while without touching. The sense of respite was glorious, but she knew it would only be a lapse or two before her arms started to slip together.

The loose skin in the gap was puckered into folds that brushed against the top of her skull. Yalda played with the wrinkles, sliding them back and forth to massage the top of her head. To her amusement, she realized that they’d naturally arranged themselves into a set of evenly spaced “waves”, a few dozen oscillations circumnavigating the sleeve of skin. She was practically a spherical harmonic come to life—except that she was no longer even roughly spherical. She was more like a torus, now.

A torus, instead of a sphere.

What would that change?

A torus would still prevent the exponential blow-up—you couldn’t wrap an exponential growth curve around a torus, any more than you could around a sphere—but its fundamental solutions would be different.

Yalda lifted her head and stared into the darkness. A torus didn’t even need to be curved; mathematically, you could slice it open and lay it flat, turning it into a rectangle or a square. You simply had to guarantee that the wave at each edge of the square agreed with its value on the opposite edge, so the whole thing could be put back together smoothly.

The fundamental solutions would be waves that executed whole numbers of cycles as you moved across the square in either direction, bringing the wave back to its original value. The sum of the squares of those two whole numbers would have to equal a constant—fixing the relationship between the size of the cosmos and the wavefronts’ separation.

She quickly sketched some examples, picking a constant small enough to be manageable, but large enough to be broken down into a sum of squares in a few different ways.

For the kind of waves she could draw on her body, completing a few dozen oscillations at most, there would only be a handful of solutions—which was tantamount to saying that light could only travel at a handful of different velocities, equal to the ratios between its frequencies in space and in time. But in the real, cosmic, four-dimensional case, the sum of squares would be so vast that it could be written in more ways than there were grains of sand in a prison cell, and the ratios would be so numerous and closely spaced that you’d never know they were not continuous.

For each choice of the number of waves spanning the square, you could also choose to have the wave in each direction either start at zero on the edge of the square, or start at a peak. With that additional flexibility, a completely general solution—whatever its complexities and quirks—could always be written as a sum of the fundamental solutions, multiplied by various factors.

What data would you need, in order to find those factors and reconstruct the whole wave—the whole history of light for a toroidal cosmos? Unlike the spherical harmonics, the imprint of these fundamental solutions didn’t get funnelled down toward any poles. To measure their contributions you’d need to know what the wave was doing along one entire edge of the square—and not just its value, but also its rate of change in the orthogonal direction, in order to learn about the waves that had a value of zero along the chosen edge.

These requirements were almost exactly the same as those for the physicists’ beloved plucked string: you set the initial shape of the string, and its initial motion, and the equation told you what followed. The only difference was that this equation allowed waves of any speed, so you needed to gather the same information from far away—potentially, across the whole width of the cosmos. This was the offer Tullia had made—“for one moment in time, there are no secrets from you anywhere”—no longer rendered useless by exponential growth.

In a toroidal cosmos, predictions became reasonable: by knowing about your immediate surroundings, you could predict what would happen with sufficiently slow waves in the immediate future. You’d be neither helpless nor absurdly omniscient. A wave faster than anything you’d prepared for could always come along and surprise you—like a Hurtler appearing out of nowhere—but if it didn’t, things proceeded as expected.

Replace the torus with its four-dimensional equivalent, and light that followed these hypothetical rules started to behave just as it did in the real world.

Yalda lowered her head and tried to rest her arms again, but her shoulders were burning with fatigue. She couldn’t replenish those muscles, either; all of the moves that would have achieved that required her to separate her arms completely.

At least she now knew how to word her message to Tullia. “If you can’t help me pay this fine,” she’d say, “I only ask you to think carefully about the shape my body is in.”

On the eleventh day, two guards with lamps entered Yalda’s cell and unclamped her chain from the wall. She didn’t question what was happening; if the sergeant had rescheduled her appearance for a few days earlier, so much the better.

Upstairs, she was almost blinded by the glare. She didn’t realize she’d been taken to a different room than before until one of the guards made her kneel and held something in front of her face. As he turned the object, her eyes were stabbed by a glimmer of reflected sunlight.

“Are you ready?” he demanded impatiently.

“For what?” she asked, alarmed and confused.

“Someone paid your fine,” he said. “We’re cutting you free.”

Yalda tightened the skin between her arms, shrinking it down to a thumb-sized stub. She’d had fantasies about making the cut herself, or even using her teeth to tear through the skin, but at least there was still no flesh to be severed.

The guard had her place her arms on a wooden bench. The process was swift, and if it wasn’t exactly painless it hurt far less than the original melding. When the guard slipped the chain off her arm, Yalda resorbed the abused limbs completely. She rose to her feet and took a step back, then rolled her shoulders and chirped with bliss as she rearranged half the flesh in her torso. Her two small wounds ended up at the sides of her back.

The guard said irritably, “Could you groom yourself outside?”

“With pleasure.” Yalda didn’t waste time asking him who’d paid her fine; Tullia would know which businesswoman from the Solo Club had taken pity on her, and would be able to advise her on the proper way to express her thanks.

Yalda moved slowly down the corridor toward the dazzling rectangle of light that marked the entrance to the barracks. She would never have lasted a year; she could admit that now. She would have been dead or insane in a dozen stints. She needed to visit the chemistry department at the first opportunity, and come back with something volatile enough to turn this whole abomination into rubble.

She stepped out beneath the sky, shivering, humming softly to herself. For a moment she was hurt that Tullia wasn’t waiting on the street to welcome her back to freedom, but that was petty; the world hadn’t come to a halt, Tullia still needed to earn a living. She extruded two fresh arms and shielded her eyes as she looked around, trying to orient herself.

“Yalda?” A male figure approached through the haze of brightness.

“Eusebio?” Yalda had lost count of all the lessons they’d missed. First her three stints on Mount Peerless, and now this unexplained absence. “I’m sorry, I couldn’t get word to you—”

He was close enough now for Yalda to read the embarrassment on his face. Of course he would have heard exactly what had happened to her.

“May I walk with you?” he said.

“Of course.” She let Eusebio lead the way; she had yet to recover her sense of direction, let alone decide where she wanted to go.

Eusebio remained silent for some time, his gaze directed at the ground. “If you choose to end our arrangement,” he said finally, “I’ll understand your decision. And I’ll pay you for the lessons to the end of the year.”

Yalda struggled to make sense of this strange invitation. Was he trying to tell her that he was so ashamed of her scandalous behavior that he no longer wished to be her student—but he expected her to act, sparing him the unpleasantness of having to dismiss her?

“Actually, I’d rather go on tutoring you,” she said coolly. If he wanted to be rid of her, he’d have to find the courage to spell it out.

Eusebio shuddered, emitting a hum that sounded more like shame than disgust. “I can’t believe you’re not angrier,” he said wonderingly. “It was my fault; I should have warned you.”

Yalda stopped walking. “What should you have warned me about?”

“Acilio, of course. All of them—but Acilio’s the worst.”

Yalda was utterly lost now. “How could you have known that Acilio would decide to throw a rock at me?” Unless the cosmos was spherical after all, and Eusebio had sat in his apartment one night reading the harmonics for the entire future.

“I couldn’t,” he replied. “And that might well have been sheer coincidence. But once he found out who you were, that you were connected to me…”

Yalda struggled to absorb this. “You mean, he asked for that huge reparation as a way of getting at you?”

Eusebio said, “Yes. Of course you humiliated him, so he didn’t care what harm he did to you, but the penalty was chosen for my edification.”

It was Eusebio who’d paid the fine and set her free. But she’d only faced the prospect of a lifetime in the cells in the first place because of some childish dispute between him and Acilio.

And she had been the last to know about any of this. When the sergeant had urged her to reconsider the resources at her disposal, he’d been hinting that he expected her to beg her wealthy employer to come to her aid.

“So what now?” she said bitterly. “You’ve bought me, you own me?”

Eusebio recoiled, wounded. “I was remiss in not warning you about my enemies, but I’ve never treated you with anything but respect.”

Yalda could not dispute that. “I’m sorry,” she said.

“Acilio is nothing to me!” Eusebio declared. “I don’t want to fight with him! But his grandfather and my grandfather are rivals. It’s all so tedious it would merely be a tired old joke if it didn’t damage other people’s lives. All I want to do is get an education and make something of myself. But I should have warned you that I have adversaries who’ll treat anyone close to me as fair game.”

“It might have been helpful,” Yalda agreed.

“I’ll give you their names, I’ll show you their portraits,” Eusebio promised. “Everyone you should avoid.”

“I probably shouldn’t injure… anyone, really,” Yalda decided.

Eusebio said, “These are people you don’t even want to bump in a queue.”

“I see.” Yalda contemplated the situation. “Is this over now? Or will Acilio have something more in mind for me?” She wasn’t keen on being shuttled in and out of prison until Eusebio was bankrupt. Couldn’t these idiots learn to ruin each other in pointless games of chance, instead?

“I don’t think he’ll repeat himself,” Eusebio said carefully. “And it’s one thing to exploit an opportunity, but using you to bludgeon me repeatedly would be seen as rather crass.”

“Well, that’s a relief. I’m so glad there are standards to be upheld.”

Eusebio met her gaze; he was still ashamed over what had happened, but he’d done all he could to make amends. “So what do you say about our lessons?”

“I want them to continue,” Yalda said. “Draw up a guide for me, on surviving the whims of Zeugma’s ruling class, and then we can get on with the things that matter.”

The prison guards hadn’t returned the coins the police had taken from her pocket, but Yalda still had some money in the bank. The clerk looked dubious when he compared the signature she made on her chest with a print of it on paper, and insisted on asking her three of her secret questions as well.

“Give the largest proper factor of the eighth power of a gross plus five gross squared plus eleven?” The clerk interrupted her before she could answer. “What kind of question is that?”

“Too easy?” Yalda wondered. “You might be right.”

She bought a loaf in the markets, then walked past the place where Antonia’s stall had been.

She couldn’t face the university yet; she sat in a quiet park until evening, then went to Tullia’s apartment.

Tullia greeted her with a look of pure astonishment. “What happened? I heard rumors of some preposterous fine, but they wouldn’t tell me anything at the barracks. I was waiting for you to send me a message!” She ushered Yalda inside; the apartment was lit only by plants once more, but prison had given Yalda astronomer’s eyes and every sheet of paper in the room stood out clearly.

She explained what Eusebio had told her. Tullia said, “Next time I complain about my own students, you now have permission to slap me in the head.”

“Any news about Antonia?”

“I met her three days ago,” Tullia replied. “In the markets, with her co. She insisted that she was with him voluntarily; he insisted that nothing was going to happen to her by force.”

“Did you believe him?”

“Does it matter what I believe? There’s nothing we can do now.”

“I was so stupid,” Yalda said angrily. “The police weren’t even looking for her—”

“And what about Eusebio!”

“What about him?” Yalda wasn’t going to blame him for her own carelessness. “Any fool could have picked a fight with me that night; even if Eusebio had warned me about Acilio, the same thing might have happened with someone else.”

Tullia went over to one of her plants and dug into the soil with narrow fingers, finally plucking out a vial.

“Did the police find your holin?” Yalda asked her.

“Not a scrag. You should have some now; you’ve missed a lot of doses.”

“I’m no older than Antonia,” Yalda said. “And spontaneous reproduction was the least of her worries.”

“Actually, Antonia took holin when she was staying here,” Tullia replied. “I insisted. If there’s one thing worse than living with an indecisive runaway, it’s coming home to find that she’s been replaced by four screaming brats.” She handed Yalda two green cubes; Yalda didn’t want to argue anymore, so she swallowed them.

She sat on the floor and put her face in her hands. “So now it’s back to ordinary life?”

“We can’t win every battle,” Tullia said firmly. “If you want some good news, though… Rufino and Zosimo made their own observations of the Hurtler. And, strange to say, there was another one three days later.”

“Another one?”

“Not visible from here, but they saw it in Red Towers.”

Yalda was perplexed. “What does that tell us?”

“That they’re random events?” Tullia suggested. “There isn’t some cosmic slingshot out there that takes years to replenish its energy and spit the next one out. If the timing’s completely random, there’s no reason why they shouldn’t come one after the other, occasionally.”

“From exactly the same direction?” Inasmuch as the Hurtlers’ trajectories had been pinned down by people’s hasty observations, they had all been more or less parallel. “Why random in time, but not in space?”

Tullia considered this. “From the Hurtlers’ point of view, they are random in space. What we see as the time between them, they see as distance.”

“Now I’m getting a headache.”

“You know, even when he heard you were in prison, Giorgio didn’t cancel your talk?” Tullia marveled. “I wish I’d had a supervisor with that much faith in me. I was going to break the news to him that we never did solve the prediction problem—” She broke off, reading Yalda’s expression. “You didn’t?”

“No exponential blow-ups,” Yalda announced proudly, “and no seeing the cosmos in every grain of sand.”

“How?” Tullia pressed her, delighted.

Yalda shuddered, overwhelmed for a moment; she knew she wouldn’t be able to recount the discovery without reliving her imprisonment and mutilation. And after eleven days abandoned in the dark, she wasn’t ready to go and sleep beneath the markets again, surrounded by strangers who didn’t care if she lived or died.

She said, “Come closer, and I’ll write the answer on your skin.”

9

The truck dropped Yalda off in the village, then she walked the rest of the way to the farm in the mid-morning heat. After three days of traveling she’d been expecting the last leg of the journey to pass quickly, but she soon realized that her memory of the walk was a heavily edited version, concentrating on a few distinguishing features—a hill, a tree, a crossroads—while excising all the monotonous stretches in between. Halfway to the farm, she began noticing shapes among the chance arrangements of pebbles by the roadside that she could have sworn had been there since she was a child.

As she walked north along the access path, a girl she’d never seen before approached her.

“Are you Yalda?” the girl asked.

“Yes. Who are you?”

“I’m Ada.”

“Pleased to meet you,” Yalda said.

They walked along the path together. Yalda had felt herself twitching at the mites ever since she’d left the village, but now that she had company she redoubled her efforts to stop random fragments of writing from surfacing on her skin each time she dislodged one of the insects.

“My father told me to see if you were coming,” Ada explained.

“Who’s your father?”

Ada was amused that anyone could need to ask this. “Aurelio!” she said.

Yalda shed the last of her lingering nostalgia. “Do you have any cousins?”

“Of course. Lorenza and Lorenzo and Ulfa and Ulfo.” After reflecting on the depth of Yalda’s ignorance for a moment, Ada added for completeness, “Their father is Claudio. And my sister’s name is Flavia.”

“And you both have cos?”

Ada buzzed with mirth. “Everyone has a co!”

“Really?”

“Yes,” Ada confirmed. “I know your co lives in a city called Zeugma, but he wasn’t born with you, that’s why he isn’t coming to visit us.”

“You know a lot about me, considering we only just met.”

“You’re my father’s cousin,” Ada said, as if that were enough to make Yalda’s life an open book to her.

Yalda said, “Tell me about my brother.”

“Lucio? He and Lucia were going to move to their own farm. Vito was going with them. But now…” Ada stopped, unsure what she should say.

“I know about Vito,” Yalda said gently. No one had bothered to tell her when Aurelia’s life had ended, or Claudia’s. Only Vito’s demise counted as a death, worthy of mourning.

When they walked into the clearing, Yalda was overcome with sadness. Even eight exuberant new children could not make up for the three missing faces.

When she’d embraced everyone, Giusto said, “You should have brought your co-stead, he would have been welcome.”

Yalda made a sound that she hoped expressed no more than gratitude at this proposition in the abstract. Although she had never tried to correct the assumption that she was in Zeugma to hunt for a co-stead as much as to pursue her education, she had never actually lied and said that she’d found one.

Giusto led her to the pit that had been dug in a corner of the clearing. Yalda looked down; the body was wrapped in petals, it could have been anyone. She sank to her knees, humming and shaking inconsolably.

When she’d recovered her composure she turned to Giusto. “He was a good man.” Her father had done his best for her, always; she owed him her life and her sanity.

“Of course.” Giusto squeezed her shoulder awkwardly.

“What happened?”

“He went quietly,” Giusto said. “Sleeping. He’d been sick for a few days.”

Mites were swarming around the grave. Yalda said, “Should I—?”

“Yes. Everyone else has been; everyone from the village.”

Yalda shaped her hands into scoops; Giusto knelt and helped her shift the soil back into the pit. She wanted to ask him about Aurelia and Claudia, too—at least to learn how old the children were—but this wasn’t the time. Childbirth was not to be lamented like death. Any hint of a comparison would be treated as a kind of derangement.

Yalda offered to help prepare the midday meal, but there were too many hands already, all accustomed to their own tasks. She watched Aurelio and Claudio affectionately guiding their boisterous children, intervening in the worst spats, making peace without taking sides or becoming angry. Who could condemn such able, loving fathers? But while she’d never know what the children’s mothers had wanted, she could be sure that no one had allowed them the kind of choices she’d had herself.

When the meal was over, Giusto took her aside.

“I want to hear about your co-stead,” he said. “What is it that he does? I should know what kind of business my great-nephews will inherit.”

“There is no business,” Yalda said. “I study at the university. I support myself with tutoring. That’s my life: work and study. There is no co-stead.”

Giusto’s face betrayed no surprise. “So you’re free? That’s good news! I’m glad there’s no one tying you down.”

“You approve?” Yalda was confused.

Giusto said, “Without a co-stead to worry about, you can take your father’s place on the new farm. Your brother can hardly work the farm alone, with young children.”

“Young children?” Yalda gestured around the clearing. “There aren’t enough children here already?”

“It’s Lucio’s time,” Giusto said. “How long should he wait? We’ve bought the farm already. Only Vito’s death has held things up.”

Yalda said, “Here’s a plan: rent out the second farm for a few years, then once your grandchildren are a little older, either Aurelio’s family or Claudio’s can take it over, along with Lucia and Lucio.”

Giusto buzzed derisively. “You want to scramble the generations? You want your brother to be so old when his children are born that his cousins’ children have to raise them for him?”

“What Lucio and Lucia do is up to them,” Yalda replied. “But I’m not going to work on that farm.”

Giusto was growing angry now. “So you’ve forgotten your own family?”

“My family doesn’t need me,” Yalda said calmly. “I’ve told you how you can make the second farm work.”

“Your duty is to take your father’s place there.”

“I doubt that would have been his opinion.”

“What is it that you think you’re doing in Zeugma?” Giusto demanded. “I’d like to know what’s so important that everything else in your life can be neglected.”

“I’m studying light,” Yalda said. “Star trails. The Hurtlers.”

“Hurtlers?”

“They’re a bit like shooting stars. We saw one, here, years ago—”

Giusto cut her off impatiently. “I taught Aurelio and Claudio to recite the sagas, and I’m willing to do the same for you. If you want a real education, start with six ages’ worth of knowledge.”

“All of it at least six ages out of date,” Yalda retorted.

Giusto stared at her as if she’d lost her mind. As far as Yalda could tell, his whole idea of knowledge was as something static, perfected in the distant past by the great poets and philosophers. The only truth to be had was passed down from them; there was nothing new to be found.

“I’m not leaving Zeugma,” she said. “No one understands light fully, yet, but people are working toward it—in Zeugma, in Red Towers, in the other cities. You can’t ask me to walk away from that! It’s the most exciting thing happening in the world right now. And I’m part of it.”

Giusto looked away, disgusted. “That was your father’s first mistake.”

“What mistake?” Yalda demanded angrily.

“Flattering you,” Giusto replied. “Letting you think that you were something special, as a compensation for having no co. That, and sending you to school.”

Yalda hadn’t expected to find it easy to sleep, but it felt perfectly normal to be lying in the clearing again, with the soil beneath her and the stars above. Ada had taken Aurelia’s spot, but she was asleep long before Yalda settled into her own old indentation. The flowers arranged around the sleepers glowed softly in every hue, but if Yalda raised her head slightly she could see the wheatlight beyond them.

She woke well before dawn, confused for a moment to have heard no bells, but sure of the time regardless. She rose and walked over to Lucia’s bed, then crouched down and touched her sister’s shoulder.

Lucia opened her eyes; Yalda gestured for silence, holding a motionless hand in front of her tympanum. Lucia climbed to her feet and followed Yalda to the edge of the clearing.

“I’m going now,” Yalda said. “The trucks leave the village early.”

“Do you have to? I’d hoped you’d stay a few more days.” Lucia sounded disappointed, but not greatly surprised.

“Why don’t you come with me?” Yalda suggested.

“To Zeugma?”

“Why not?”

Lucia buzzed softly with mirth. “What would I do there?”

“Whatever you like,” Yalda replied. “You can look around and decide what suits you. I’ll take care of you until you find a job.”

“But there’s work for me here; I don’t need a new job.”

Yalda said, “Don’t you want to see more of the world than this?”

“That might be nice,” Lucia conceded. “But I’d miss everyone.”

“You could come back and visit, any time,” Yalda promised her.

Lucia thought for a while. “Let me wake Lucio.” She took a few steps before Yalda grabbed her arm.

“No! You can’t—”

“You’re not inviting him too?”

“No.”

“Are you crazy?” Lucia was baffled. “Why would I go to Zeugma without him?”

“That’s the whole point of leaving!” Yalda said, exasperated. “If you go by yourself, you won’t have to worry.”

“Worry about what?”

“Children.”

Lucia said, “We’re not having children for at least four more years. If we went with you to Zeugma, it wouldn’t be any different.”

“Four years?”

“Yes.”

Yalda sat on the ground, shivering, not knowing whether to believe her. “Aurelia didn’t wait. Claudia didn’t wait.”

“Well, I’m not Aurelia.”

“Don’t you miss them?”

“Of course I do,” Lucia replied. “If you missed them so much,” she added pointedly, “you should have visited more often.”

Yalda was ashamed. “I didn’t think I’d lose the chance so soon.” She searched her sister’s face, determined to uncover the family’s secrets. “What happened? Did Giusto force them?”

“He nagged them,” Lucia said. “But they had their own ideas, it wasn’t all down to him.”

“And you think he’ll let you wait four years?”

“It’s not his decision, Yalda! Lucio and I have everything planned: we’re going to work on the new farm together and save as much money as we can. Then when the time comes, he’ll hire some people to help run the farm for a couple of years while he takes care of the children. If some of the young cousins want to do it, that’s fine, but we’re not going to rely on that.”

Yalda said, “And what if you change your mind? What if you want to wait longer?”

“Then we’ll wait,” Lucia said mildly. “He’s not going to force me.”

“What makes you so sure of that?”

“He’s my co! I’ve known him my whole life.”

“Men are driven to have children,” Yalda said. “It’s in their nature, they can’t help it.” How had Daria put it? “It’s what they were invented for. There are no male insects, no male lizards—because the young of those creatures can look after themselves from birth. The only reason men exist is so they can raise children.”

Lucia said, “Women are driven to have children, too. Do you think I didn’t feel the urge myself, when I saw Aurelia’s? But if I can hold off, so can Lucio. Neither of us are helpless.”

“But you’re the only one who pays with your life.”

“Yes,” Lucia agreed. “But that’s not Lucio’s fault. It’s not in his hands, or anyone’s. However much he cares about me, he can’t take my place—that’s just impossible.”

Yalda sat in silence for a while. The stars were beginning to fade; she’d need to leave soon.

“Do you want to wake Lucio, and ask him?” If she could show them both life in Zeugma—show them both some new possibilities—that would still be worthwhile.

Lucia said, “It’s not the kind of thing we should decide in a hurry. We’ll talk about it over the next few days; if we want to come, we’ll follow you.”

“All right.”

Yalda stood and embraced her. “You won’t let Giusto pressure you?” she pleaded.

“I won’t,” Lucia promised. “Do you think Vito taught his other children nothing?” A thin gray ridge appeared on her chest; at first it was barely visible, but then it strengthened and grew, looping across the skin until it spelt out a shaky sequence of symbols: Safe journey, sister.

“You could use that skill in Zeugma,” Yalda enthused.

Lucia said, “Maybe. Go and catch your truck before someone else wakes and you have to explain why you’re sneaking out on us.”

“Write and let me know when you’re coming,” Yalda said.

“Of course.”

Yalda turned and walked away. She watched Lucia in her rear gaze until they lost sight of each other behind the dying red light of the fields.

“I have a gift for you,” Cornelio announced solemnly.

“A gift?” Yalda had accepted his invitation to the chemistry department as much out of courtesy as curiosity, but she’d hoped there’d be more to the visit than the bestowal of some token of gratitude. “Your success in your work is all the thanks I need.” With her rear gaze, she eyed the glistening vials and bottles on the shelves that lined the workshop, trying to remember how long it had been since the building had last had its roof blown off.

“That’s very gracious of you,” Cornelio said. “But have you forgotten your request?”

Her host sounded more amused than offended, but Yalda searched her memory desperately. She’d spoken to Cornelio for a chime or two after her talk to the school of natural sciences, but they’d discussed so many things that it was impossible to recall the entire conversation, ten stints later.

“I asked you what the one thing was that you’d welcome the most,” Cornelio reminded her, “if we were to repay you with something practical.”

Yalda wasn’t sure how seriously she’d taken the question, but she remembered now what her answer had been. “And you’ve made good on that offer already?”

“It’s not perfect,” Cornelio admitted. “But you might find it useful nonetheless—worth having, even short of perfection.”

“Of course.” Yalda set her anxieties aside. If Cornelio really had created what she’d asked him for, it was well worth the risk of being here.

“Let me show you.” Cornelio led her to a bench at the side of the workshop. In lieu of a heliostat, he’d set up a pair of manually adjustable mirrors that brought sunlight into the room and directed it into a box three spans or so wide.

He opened the side of the box, revealing a prism mounted within that split the beam into a spectrum that fell on a white screen. “Note the locations of the various hues, if you will,” he suggested to Yalda.

“Noted.” After witnessing three Hurtlers over Zeugma, Yalda could memorize the position of a spread of colors against any backdrop in an instant.

Cornelio covered the aperture that admitted the sunlight into the box with a card pierced by a far smaller hole. The spectrum remained visible, but it was much dimmer now. Then he slid a second, entirely opaque card into another slot, parallel with the first one, blocking the light completely.

Next, he took what appeared to be a stiff sheet of paper from a cupboard below the bench, and fastened it in place over the screen where the spectrum had been seen. Then he produced a small vial that had been divided partly in two, with one half containing an orange powder, the other a green resin. He attached the vial to a loop of cord that dangled into the interior of the box through its top face.

Cornelio closed the side of the box, carefully checking that there were no gaps along the edges. “This needs to be entirely sealed against the light,” he said. “Not a crack.”

Yalda was surprised by his diligence, but it was a good sign. “I understand.”

“First, you shake the vial,” Cornelio explained, taking hold of the cord where it protruded from the top of the box and jiggling it slightly. “That lets the ingredients react, and the gas that’s produced activates the paper.”

“Activates?”

“Sensitizes it to light. But only for a few pauses, until the gas disperses, so I shouldn’t delay—”

Cornelio pulled the opaque card most of the way out of its slot, then pushed it back in immediately.

“What’s wrong?” Yalda asked.

“Nothing,” he assured her. “That was the necessary exposure to the light: about a flicker.”

The spectrum from the smaller aperture had barely been visible, yet one flicker was long enough to cause a reaction?

Cornelio said, “The gas should have dispersed of its own accord now, but I’m thinking of adding a bellows to ensure that it’s expelled completely. Maybe we should wait a couple of pauses longer to be sure though, if you don’t mind.”

“Believe me, my patience has not been tested yet.” Yalda had seen a demonstration of an earlier version of the same idea; it had required an exposure of at least three bells to capture even the brightest star trails—after which the paper had needed to be treated with a resin that, as often as not, caused it to burst into flames.

“I think…” Cornelio opened the box, fumbling with the clasps. He peered in, then stood aside and let Yalda take a look.

The paper had been darkened very visibly in three places; three narrow black strips marked the locations of—if Yalda’s memory served her—shades of red, yellow and blue. It hadn’t captured the whole spectrum, but the very fact that the reaction was not an indiscriminate, panchromatic response would make it all the more valuable. A smudge of black that covered the entire trail of a star or a Hurtler would have been useless. This system could capture the precise locations of three specific hues at one instant, finally making it possible to quantify details of the Hurtlers that were presently just the subject of fleeting impressions.

“This is wonderful!” she declared ecstatically.

“I’m glad it meets your approval,” Cornelio said modestly.

“Does the paper ever…?”

“Start burning? No. This is a completely different reaction from the old one.”

“Then it’s perfect. I don’t know what to say.”

Cornelio had already assembled a box full of the treated paper and a rack of the activating vials. “These are yours. When you need more, just let me know.”

“Thank you.”

Yalda could already picture the device she’d build to capture data on the Hurtlers, but it would be rude to snatch up this generous gift and rush away.

She said, “I don’t know if the light recorder has occupied all of your time, but I’d be interested to hear how any of your other research is progressing.”

“I’ve been doing some theoretical work as well,” Cornelio replied. “Rotational physics vindicated our earlier measurements of chemical energy differences, but the implications need to be developed much further. In fact, we’re having to reinvent most of thermodynamics.”

Yalda was surprised. “That sounds a bit extreme.”

Cornelio said, “If I told you that your theory implies that everything in this room is hotter than infinitely hot, would that justify rewriting the textbooks?”

“Infinity is my least favorite temperature,” Yalda confessed. “If you’re serious, I might have to recant.”

Cornelio buzzed softly. “Let’s call them negative temperatures, then; that’s formally correct, though the first way of speaking has its merits too.”

Yalda found the second way much more agreeable. “True energy has the opposite sense to kinetic energy, so to be consistent I suppose you could just declare all temperatures to be negative. Since a hot gas has less true energy than a cold gas, its temperature should be less… no?”

Cornelio was regarding her with an exasperated expression, but he was too polite to articulate precisely what he was feeling.

“I’m a physicist, show some mercy!” Yalda pleaded. “Thermodynamics is your domain. All I ever studied was the ideal gas law.”

“Temperature is not a synonym for energy,” Cornelio said sternly. “It’s about the proclivity of energy to pass from one system to another, not the quantity of energy that either one contains.”

“I’m willing to believe that,” Yalda said. “But how do you make such a ‘proclivity’ precise?”

“First,” Cornelio said, “think about the range of different ways in which one system can possess the same energy. Start with a single particle of gas, under the old physics.”

He summoned a diagram onto his chest. “The particle’s kinetic energy is proportional to its momentum squared. Pick a few examples of the energy the particle might have, but don’t pin it down precisely; just say that the energy lies in some small interval. From the plot on your left, you can read off a corresponding range for the momentum in each case.”

Yalda examined the diagram. “So you follow the horizontal lines for energy across until they hit the curve, then drop them down to the momentum axis?”

“That’s correct,” Cornelio said. “But then, recall that momentum is a vector. The energy has given us a range of sizes for that vector, but no information at all about its direction. The particle might be traveling north, west, up, down; we don’t know. So, take an arrow whose length you know, more or less, and swing it around freely, without any constraints. The tip of the arrow traces out a sphere—or rather, because the length isn’t fixed exactly, a spherical shell. The volume of that shell in ‘momentum space’ represents all the possibilities open to the particle, while still having an energy that lies within the given range.”

Yalda said, “So you’ve sketched parts of these shells, and plotted their volume against the kinetic energy… which turns out to be the same kind of curve as the momentum itself.”

“In this case, yes,” Cornelio said, “but that’s not true in general! So forget the resemblance, and just concentrate on the right-hand curve on its own terms. What does it tell you?”

“The volume in momentum space gets larger as you increase the kinetic energy,” Yalda said. “That makes sense. A faster particle has its momentum lying on a bigger sphere; the shells do get thinner as the momentum grows, but the larger surface area of the sphere more than compensates for that.”

“So the volume grows,” Cornelio agreed, “but when does it grow most rapidly?”

“At the start,” Yalda said. “When the energy is low, the volume shoots up; after that, it grows ever more slowly.”

“Precisely.”

“But where does that get us?”

“Particles bounce around, collide, exchange energy,” Cornelio said. “Give a particle a little more energy when its original energy is low, and the volume in momentum space accessible to it shoots up. And if it happens to get that energy by colliding with a particle that was moving faster, the volume for the faster particle goes down—but not by as much.”

“So… you need to add the two volumes?” Yalda suggested. “And see how the sum changes when energy moves from one particle to the other?”

“Not quite,” Cornelio said. “You multiply them. Each volume measures the possibilities that are available to one particle—and each possibility for one can be accompanied by any of the possibilities for the other. So it’s the product that you need.” He produced a new diagram.

“If energy moves from one system to the other, the product of their momentum space volumes grows along one edge of this rectangle, and shrinks along the other edge. So whether there’s an overall growth in the product depends on which of those changes is the larger.”

Yalda said, “You describe one system as being hotter and one colder—but where does temperature appear in all this?”

Cornelio said, “For each system, take the volume in momentum space and divide it by its rate of change with respect to energy. That encodes all the relevant information in a single number: the temperature. Then if one system’s temperature is greater than another’s—so long as they’re either both positive or both negative—that immediately tells you that if the first system gives energy to the second, it will increase the total number of possibilities. That’s why energy flows from hot to cold: the result ends up encompassing more possibilities.”

“Whew.” Yalda summoned her own version of Cornelio’s first diagram onto her chest and performed the final stage of the calculation. “So in our simplest possible example, temperature ends up being… proportional to kinetic energy! All that work, to get back to the naïve idea that they’re really the same thing.”

Cornelio resisted rebuking her further. “Of course the true definition doesn’t contradict any of the results you were taught—for an ideal gas, under the old physics. But if you’re still clinging to some notion that temperature and energy are the same, take a look at what your own work has given us.”

Yalda gazed at his finely ridged skin, feeling suitably chastened and bamboozled. Then she began following the steps he had described for the simpler case, and the whole strange construction took on an eerie inevitability.

The true energy and momentum were linked by a circle, each simply rotating into the other. As the particle’s momentum grew from zero, its true energy began to fall—and at first, everything behaved very much like the earlier calculations, merely plotted upside-down.

But as the particle moved faster, its momentum could no longer increase without bounds. With the momentum levelling out, not only did the shells in momentum space cease growing so quickly, they became much thinner. At about two-thirds of the maximum total energy, the shells reached a peak in volume and began to shrink.

At that point, the effect of a change in energy on the number of possibilities open to the particle was reversed. A slow-moving particle could gain options by moving a bit faster… but a sufficiently fast-moving particle would lose options if it sped up. The ceiling on momentum made things cramped at the top.

The same thing showed up in the temperature, which switched sign when the shells’ volume peaked. And while negative temperatures on their own might merely have been the result of an idiosyncratic choice of conventions, Cornelio’s diagram made it clear that both negative and positive were real possibilities. You could always swap the labels for them by tinkering with the definitions, but you couldn’t banish the distinction itself.

Yalda said, “If everything in this room has a negative temperature, where are the positive ones?”

“On the surface of the sun,” Cornelio replied. “In our own burning stones.”

“I see.” A burning stone heated its surroundings, adding kinetic energy to them, so true energy would have to be flowing the other way, into the flame. Did that make sense? Cornelio had warned her that energy only flowed from the higher temperature to the lower if both had the same sign.

The mixed-sign case wasn’t hard to understand, though. A system with a positive temperature would gain possibilities if it gained energy. A system with a negative temperature would gain possibilities if it lost energy. Putting the two together, there was no subtle trade-off anymore—this was a win-win situation. Both systems could gain volume in momentum space from the same transaction.

So, any system with a negative temperature would lose true energy to any system with a positive temperature. That was why Cornelio had seen merit in calling ordinary objects “hotter than infinitely hot”; however high the positive temperature of a blazing sunstone, a “hotter than infinitely hot” cool breeze could still heap true energy upon it.

Yalda said, “But how can you know for sure that something has a positive temperature—and not just a large negative one? How do you know when things aren’t just ‘hot’ in the old-fashioned way?”

“Light,” Cornelio replied. “Whenever a system freely creates light—not in the orderly way a flower does it, but in the chaos of a flame—it’s turning true energy into something that didn’t exist before, opening up new possibilities. That’s the very definition of positive temperature.”

“So once an ordinary system with a negative temperature starts creating light,” Yalda ventured, “its temperature must change sign? Crossing infinity along the way?”

“Precisely,” Cornelio said. “Once it’s creating light, it’s lost to the ordinary world.”

Yalda couldn’t help stealing another glance at the workshop’s collection of energetically precarious concoctions. Above the shelves, the ceiling still showed signs of recent repairs.

“In the end,” Cornelio declared, “everything becomes heat and light. It’s not in our power to stop that. All we can do is slow it down a little and try to enjoy the ride.”

Yalda ended up staying in the chemistry department until dusk, then she caught a lift in the department’s truck back to the city campus, along with Cornelio and five of his students. As they drove across the dusty plain, Cornelio explained how the pressure of a gas could remain positive as its temperature changed sign, and finite as its temperature crossed infinity. The old ideal gas law—pressure times volume is proportional to temperature times quantity—was receding into the distance; it wasn’t even true within the flames of an ordinary lamp.

The back of the truck was open to the sky, so Yalda saw the Hurtler’s violet tip rushing toward them from the north, but then the driver panicked and slammed on the brakes, sending the vehicle lurching and skidding. When it came to a halt she could recall nothing but a whirl of color across a spinning bowl of stars.

Everyone clambered out, checking themselves for injuries, but it was soon clear that nobody had been hurt. Yalda was still clutching her light recording supplies; she examined the contents of the box in the starlight, but Cornelio had packed everything carefully and none of the vials appeared to have been damaged. She helped some of the students push the truck back onto the road, wasting no time fretting over the lost opportunity. At this rate, there’d be another Hurtler over Zeugma in a couple of stints.

“What do you think they are?” she asked Cornelio, as the truck lurched into motion again.

“Fragments from a big explosion,” he replied. “Something so distant that even the smallest differences in the speed of the debris could spread out its arrival over many years. My hunch is that successively later fragments will prove to be moving more slowly.”

“That’s an interesting idea.” Yalda tapped the box of goodies appreciatively. “Hopefully I’ll be able to test it soon.” A crisp image of a Hurtler’s light trail, captured at a single moment, might enable her to measure its asymmetry and finally quantify the object’s speed.

It was dark when they reached the city. At the university, Yalda bid farewell to Cornelio, stashed her new supplies in the optics workshop, then braved Ludovico’s wing of the department to see if Tullia was still around. But the place was empty. Tullia might have caught some observations of the Hurtler, then, on her way to her apartment or the Solo Club.

At the Solo, Yalda found Daria and Lidia; they hadn’t seen Tullia, but they persuaded Yalda to join them in a game of six-dice. To everyone’s amazement, Yalda won, so she stayed for a second game. Lidia beat her this time, but it was close.

Yalda was tired now, but she decided it was worth stopping by at Tullia’s apartment; it would be good to share the highlights of her trip to Amputation Alley. Tullia was planning to head up to Mount Peerless in a few stints; she could probably make use of Cornelio’s invention there herself.

When she arrived at the apartment, Yalda found the entrance unbarred but the curtain closed. She called out softly a few times, but received no reply. Tullia didn’t usually sleep so early, but if she’d dozed off after a hard day it would be unfair to wake her.

Yalda turned and started toward the stairs, but then she changed her mind; it wouldn’t hurt to sneak in quietly and check that everything was all right. She walked back, parted the curtain and stepped into the apartment.

Tullia was lying on the floor near the window. Yalda called her name but there was no response. She approached and knelt to examine her; Tullia was limbless, and her skin displayed a strange sheen. For one panicked moment Yalda thought of her grandfather, but then she realized that the patches of light she was seeing were just distorted reflections of the flowers above.

Yalda took her friend by the shoulders and shook her gently; her skin felt strange—tight, almost rigid—and she did not react at all. There was a furrow down the middle of her chest: a deep, narrow fissure, the first line of a symbol that Tullia would never have written by choice.

“No,” Yalda whispered. “That’s not happening.” She scrabbled in her pocket for her vial of holin. If Tullia had taken a weak dose, it might not be too late to augment it. Yalda tipped three of the small green cubes into her palm, then reached for Tullia’s mouth.

But Tullia had no mouth. The darker pigmentation of her lips remained visible, but the skin that bore their color and shape was part of a smooth, seamless expanse. Yalda dropped the holin and moved her fingers over Tullia’s face, gently probing one of her eyes; the eyelids were still discernible, but they’d fused together. Below her mouth, her tympanum was rigid. Her body was becoming as hard and featureless as a seed case.

Yalda was shaking now; she forced herself to be still. Who would know what to do? Daria, surely. Yalda leaned out the window and spotted a boy on the street; she threw him a coin to get his attention, then begged him to run to the restaurant under the Solo and ask the chef to “fetch Daria, urgently, for Yalda”. Two more pieces and the promise of two more on his return did the trick.

As Yalda knelt down again, her shadow fell on Tullia and she saw that there was a faint glow from her body—though it was not on the surface, like her grandfather’s affliction. This light came from deep within, and it flickered and shifted constantly: a frenzy of signaling so intense that it could now be seen through all the intervening flesh.

Yalda stroked Tullia’s forehead. “We’ll fix you up,” she promised. “It will be all right.” If Daria could get hold of some melding resin, they could glue together the sides of the furrow and let Tullia’s own body attack the dividing wall. And if Daria sent for her sunstone lamp, the same kind of flash that had set the captive arborine’s muscles twitching could disrupt the signals that were organizing Tullia’s fission. There were so many things they could do. Tullia wasn’t sick, or old, or frail. She wasn’t enslaved to an impatient co. She was a free woman, in the care of her friends.

Having reached the top of her torso, the furrow now began to divide Tullia’s tympanum. Yalda took hold of the edges and squeezed them together with all her strength. “No further,” she said. “And when this hand grows tired, there are ten more waiting.” But in fact there was no great force opposing Yalda’s intervention, just the faintest springiness in the underlying flesh.

Tullia would survive; Yalda was sure of that now. Survive, and flourish. She would enlighten her students and delight her friends for a dozen more years. She would find the light of a forest on a distant world.

The furrow itself wasn’t lengthening, but Yalda could see the walls of hard skin stretching beyond the point where she was pinching them together, growing up toward the place where Tullia’s mouth had been. The slight convexity of her blind eyes had vanished now; the organs had been resorbed and their lids subsumed into the featureless skin around them.

Yalda heard footsteps, then the curtain parted. Daria hurried in, followed by Lidia.

“You promised some boy that I’d pay him four pieces?” Daria asked irritably. “This had better be—”

As Yalda moved aside to let Daria apprehend the reason she’d been summoned, she saw that a cross-furrow was forming now, threatening to divide each of Tullia’s lateral halves. “Did you bring some melding resin?” she asked Daria. “Or should we stitch the gaps together? Clamping with my hands doesn’t work.”

Lidia approached. “There’s nothing we can do,” she told Yalda gently. “Any resin, any drug, any surgery—all that would do now is kill the children.”

Yalda looked to Daria. “That can’t be true.”

Daria said, “Once division starts, it’s irreversible.”

Lidia put a hand on Yalda’s shoulder. “Let her be.”

Yalda turned to her. “What—just let her die?”

“It’s not a choice anymore,” Daria explained sadly. “You can hold her body as tightly as you like, but her brain is already in pieces.”

“Her brain’s destroyed?” Yalda stared down at Tullia’s blank face. “It’s her brain that’s making this happen, isn’t it? Sending out the signals. Don’t tell me it’s in pieces.”

Daria walked up and knelt beside her. “Yalda, she’s gone. She would have been gone long before you found her.”

“No, no, no.” No more lives cut short. Claudia and Aurelia had been far away, out of her hands, but not Tullia.

She turned back to Lidia and pleaded, “What do we do? Tell me!”

Lidia said, “All we can do now is remember her.”

Yalda ran her hand over Tullia’s wounds; the furrow now reached the top of her skull. “There must be something.”

Daria spoke firmly now. “Yalda, this is beyond changing. She was our friend, and we loved her, but her mind is gone; she’s as dead to us now as any man in his grave. Tonight we can only grieve for her.”

Yalda felt herself shivering and humming. She didn’t believe Daria’s words, but some traitorous part of her had decided to behave as if they were true.

“And in the morning,” Daria added, “we’ll need to talk about the way we’re going to raise her children.”

10

As Yalda stood in the courtyard waiting for her guest to arrive, she counted two dozen pale streaks of color drifting in mirrored pairs across the afternoon sky. From their leisurely pace she could tell that these Hurtlers were not especially close—perhaps a little farther than the sun itself—but to be visible by day at such a distance they would have to have blazed far more brightly than the specks that only showed at night.

Brighter almost certainly meant larger.

She spotted Eusebio crossing the courtyard and held up a hand in greeting. “Hello, Councilor.”

“It’s good to see you again, Yalda,” he said.

“You too.”

He glanced up at the sky. “It seems almost normal now, doesn’t it? People can get used to the strangest things.”

“Sometimes that’s a useful trait,” Yalda said.

“But not this time,” Eusebio suggested.

“Maybe not.”

They left the courtyard and strolled across the campus; Yalda had offered to organize a meeting room, but Eusebio had wanted to avoid any suggestion that either of them were acting in an official role. Two old friends were getting together to reminisce, that was all.

“I’ve only heard third-hand versions of your theory about the Hurtlers,” he said. “But they were enough to get me worried.”

“The whole idea’s extremely speculative,” Yalda said. “I wouldn’t go jumping off any bridges yet.”

Eusebio was amused. “Yet? Believe me, that’s not where I’m headed, prematurely or otherwise.”

“But you say you’re worried.”

“Of course I’m worried. Wasn’t that the point? Why else would you raise those possibilities?”

Yalda wasn’t sure how to answer that. The truth was, when she’d first begun discussing the idea with a few colleagues, she hadn’t taken it very seriously. It had been nothing more than an audacious stab in the dark—and far too esoteric, she’d thought, to pose any risk of sowing panic.

Eusebio said, “Maybe there’s something you can clear up for me, to start with. Years ago, you gave a talk where you said the cosmos was a flat, four-dimensional torus. In which case… if you followed a bundle of histories through time, wouldn’t they remain pretty much parallel? And when they met, wouldn’t they just form a loop?” He drew a two-dimensional version, first as a square and then curled up to remove the artificial edges.

Yalda said, “The flat torus is just an idealization—the simplest case where the light equation can be solved nicely. The topology of the cosmos might be more complicated, or the geometry might not be flat. Or maybe the histories of the worlds haven’t stayed together in a nice tight bundle, the way you’ve drawn them; it’s true that everything has to join up in the end, but it doesn’t have to do it tidily. If there was a primal world that fragmented in both directions—one we’d think of as the future, and one we’d think of as the past—and the fragments themselves then broke up, and so on… we could end up colliding with the past fragments at almost any angle.”

Yalda sketched a crude illustration of the idea. “I won’t try to draw this wrapped around a torus, but you can imagine the possibilities if those two sets of fragments ever came to overlap.”

“So the primal world gets to explode both backward and forward?” Eusebio gave a chirp of delight tinged with skepticism.

“I know it sounds strange,” Yalda conceded, “but if the primal world is where entropy reaches its minimum value, it’s just as reasonable for it to explode in one direction as the other. Imagine the cosmos filled with a vast tangle of unruly threads—the histories of all the particles of matter—and then demand that somewhere they’re all packed together and perfectly aligned. I’m not sure why that has to occur at all, but unless there’s some extra rule imposed on top of it, the threads are going to break free in the same way on both sides of the constriction—creating two localized arrows of time pointing in opposite directions.”

“But whatever the fine details,” Eusebio said, “so long as the cosmos is finite—and the light equation suggests that it must be—there’s a potential for the Hurtlers to be the harbinger of something worse.”

“A potential. Exactly.” Yalda didn’t want him losing sight of that. “Admitting that there could be places in the cosmos where two sets of worlds cross paths with each other doesn’t mean that has to be what’s happening here and now.”

“Except that the alternative explanations for here and now aren’t working out too well,” Eusebio replied. “If the Hurtlers are just debris from a single monstrous explosion, shouldn’t the fragments be arriving ever more slowly?”

Yalda said, “Yes—but whether we could measure the change in velocity over just a few years is another question. It’s been tough enough quantifying the velocity at all.”

Eusebio was unpersuaded. “Why should the change be so hard to spot, though? I can imagine an exploding world sending out a blast of high-speed dust, along with a slower-moving barrage of pebbles. But if that really does explain everything, shouldn’t the difference in speed be as striking as the difference in size?”

“Perhaps,” Yalda admitted.

“My third-hand source implied that you’d more or less predicted this”—Eusebio gestured at the color trails crowding the sky—“almost two years ago. A cluster of worlds and stars, much like the one we seem to lie within ourselves, would be surrounded by a halo of fine dust. Then as you penetrated deeper into its environs you could expect to encounter larger objects.”

“It’s hard to know for sure what the structure would be,” Yalda said. “We don’t understand the breakup of worlds, let alone the long-term effects of gravity and collisions between the fragments.”

“But it’s not an unreasonable position,” Eusebio persisted, “to posit rarefied dust at the edges and more substantial material closer in?”

“No.” However much she wished to downplay the conclusions, Yalda couldn’t retreat from the entire argument she’d made.

Eusebio said, “Then if our notion of time corresponded to one of this cluster’s notions of space… we should expect to find ourselves encountering successively larger objects, all moving at similar speeds. Isn’t that right?”

He offered an illustration.

“Couldn’t you at least have us glancing the edge?” Yalda pleaded. “We don’t need to be heading in as deeply as you’ve drawn it.”

Eusebio obliged. “I knew I should have been a physicist,” he said. “If there’s something you don’t like about the world, you merely adjust a free parameter and everything’s perfect.”

“What would you have me do?” she said. “Give up hope for all of our grandchildren?”

“Not at all. I want you to imagine the worst, and then tell me how we can survive it.”

Yalda emitted a bitter, truncated buzz. “The worst? The Hurtlers will keep coming, ever larger and in ever-greater numbers, until the odds that we’re struck approach a certainty. If we survive that, we’ll probably collide with an orthogonal clump of gas—turning the world itself into something like a giant Hurtler. Somewhere along the way, there will be gravitational disruption, maybe ripping us free from the sun completely—or maybe tossing us into it. And if none of these things sound sufficiently fearsome, the encounter might scramble our arrow of time completely, leaving us with no past and no future. The world will end as a lifeless mass of thermal fluctuations in a state of maximum entropy.”

Eusebio heard her out without flinching, without disputing anything. Then he said, “So how can we survive that?”

“We can’t,” Yalda said bluntly. She pointed to his chest. “If it’s more than a glancing blow—if you’re going to deny me my choice of impact parameter—then we’re all dead.”

“Are you telling me that it’s physically impossible to protect ourselves?”

“Physically impossible?” Yalda had never heard an engineer use that phrase before. “No, of course not. It’s not physically impossible that we could shield ourselves from all of these collisions, or side-step them, or simply flee from the whole encounter. It would violate no fundamental law of physics if we built some kind of magnificent engine that carried the whole world off to a safer place. But we don’t know how to do that. And we don’t have the time to learn.”

“How long would it take?” Eusebio asked calmly. “To learn what we need to know to make ourselves safe.”

Yalda had to admire his persistence. “I can’t honestly say. An era? An age? We still don’t know the simplest things about matter! What are its basic constituents? How do they rearrange themselves in chemical reactions? What holds them together and keeps them apart? How does matter create light, or absorb it? And you want us to build a shield against collisions at infinite velocity, or an engine that can move an entire world.”

Eusebio looked around at a group of students chatting happily near the food hall, as if they might have overheard this catalog of unsolved problems and decided to rise to the challenge.

“Suppose we’d need an age, then,” he said. “A dozen gross years. How long do we actually have before the danger becomes acute?”

“I can only guess.”

“Then guess,” Eusebio insisted.

“A few dozen years,” Yalda said. “The truth is, we’re blind to whatever’s coming; a whole world, a whole blazing orthogonal star might already be disposed to strike us tomorrow. But from the progression in the size of the Hurtlers that we’ve seen so far, unless we’re especially unlucky…” She trailed off. What difference did it make? Six years, a dozen, a gross? All she could do was go on living day by day, averting her gaze from the unknowable future.

Eusebio said, “We need an age, and we don’t have it.”

“Exactly.”

“That’s what I thought,” he said, “but I had to hear it from you to be sure.”

His tone was solemn but far from despairing. Yalda stopped walking and turned to face him. “I’m sorry I had no good news for you,” she said. “Perhaps I’m wrong about all of this. Perhaps our luck will be far better than—”

Eusebio raised a hand, cutting her off. “We need an age, and we don’t have it,” he repeated. “So we find the time elsewhere.”

He wiped the colliding clusters from his chest. Then he drew two lines, one straight, one meandering, and added a few simple annotations.

“We make a rocket,” he said, “powerful enough to leave the world behind. We send it into the void and accelerate it until it matches the velocity of the Hurtlers. Once it’s done that, there’ll be very little chance that anything from the orthogonal cluster will strike it—but we might need to offset its position at the start, to keep it from colliding with gas and dust in our own cluster.

“The complete journey is as I’ve drawn it. The time that passes for the world will be the time it takes to rotate the rocket’s history by a full turn: one quarter-turn to accelerate, one half-turn to reverse, one final quarter-turn to decelerate. If the rocket accelerates at one gravity—giving the passengers no more than their ordinary weight—the time back home for each quarter-turn will be about a year, making four years in all.

“The time that passes on the rocket for those stages of the journey won’t be much greater: each curved segment is only longer than its height by a factor of pi on two. But when the rocket’s history is orthogonal to the history of the world, no time at all passes back home. So for the travelers, the journey can last as long as it needs to. If they require more time to complete their task, they can prolong the flight for another era, another age; it won’t delay their return by one flicker.”

Yalda was speechless. Their roles really had been reversed: the physics Eusebio was presenting was so gloriously simple that she was ashamed she hadn’t thought of it herself—if only in the same whimsical spirit as that in which she’d first thought of the Hurtlers as past-directed fragments of the primal world.

But when it came to practicalities, where did she begin?

“What kind of rocket can you hope to build,” she said, “in which generations can survive for an age—let alone flourish to the point where they have any chance of fulfilling their purpose? The largest rocket I’ve seen with my own eyes was the size of my arm; the largest I’ve heard of was smaller than my body. If you can send my optics workshop into the void that would be the talk of Zeugma for a generation, but I don’t know where we’d put the wheat fields.”

Eusebio hesitated, considering his reply, but he did not appear the least bit discouraged by her response.

“I believe you’ve been on Mount Peerless,” he said.

“Of course. The university has an observatory there.”

“Then you’ll know that it’s far from any permanent habitation.”

“Certainly.” Yalda thought she knew where he was heading: an isolated, high-altitude site would be the perfect place to test new kinds of rockets.

Eusebio said, “The geologists tell me that the core of Mount Peerless is pure sunstone. I plan to tunnel into it and set it alight, and blast the whole mountain into the sky.”

Yalda collected the children from school and took them back to the optics workshop. Amelia and Amelio were happy playing on the floor with a box of flawed lenses, lining them up to form impromptu telescopes and buzzing hysterically at the sight of each other’s distorted images. Valeria and Valerio were going through a stage of drawing pictures of imaginary animals that they insisted had to be preserved; Yalda gave them some old student assignments that were blank on one side of the paper, and some pots of dye that Lidia had brought home from the factory.

Then she stood at her desk watching over them while she tried to decide what to make of Eusebio’s plan.

Giorgio brought a group of students into the workshop to use the heliostat for an experiment in polarization. The children ran to greet him, and he accepted their embraces without a trace of annoyance or embarrassment before gently shooing them back to their activities. Yalda produced a stack of mechanics assignments and proceeded to mark them, marveling that she could feel guilty for taking a couple of chimes away from her conventional duties to ponder the correct means of averting the planet’s annihilation. She’d shared her ideas about the Hurtlers with Giorgio—and he’d offered his usual perceptive comments and objections—but in the end he’d still treated the whole thing as if it were an exercise in metaphysics.

Yalda arrived home with the children just as Lidia returned from her shift.

“Did you bring some more dye?” Valeria nagged her. She and Valerio had used up the last of their supplies on a series of images of giant worms with six gaping mouths arrayed along the length of their bodies.

Lidia spread her arms and jokingly opened six empty pockets. “Not today. Every batch was perfect.”

Valeria went into a sulk, which meant wrapping six arms around her co and trying to pull his head off. Yalda warned her three times, increasingly sharply, then stepped in and physically disentangled them.

“You always take his side!” Valeria screamed.

Yalda struggled to hold her still. “What side? What did Valerio do?”

At a loss for an answer, Valeria changed tactics. “We were just playing, but you had to spoil it.”

“Are you going to be sensible now?”

“I’m always sensible, you fat freak.”

Lidia made a reproving hum. “Don’t talk to your Aunty Yalda that way.”

“You’re a bigger freak!” Valeria declared, turning on her. “At least it’s not Yalda’s fault she doesn’t have a co. She might have swallowed hers by accident before she was born, but everyone knows you killed your co with a rock.”

Yalda tried to draw on her reserves of patience by reminding herself how well Valeria had behaved in the workshop.

Daria had actually predicted that the third year of school would be the time when Tullia’s children started lashing out at their adoptive parents, punishing them for the derision they received from their increasingly unsympathetic classmates. So before the school year began, Yalda had encouraged the four of them to talk about the ignorance and hostility they were facing, and tried to suggest strategies for dealing with it. At the time, the recipients of her advice had promised that nothing in the wicked world could stop them feeling the same love and loyalty for their Aunties as ever.

Amelio had been looking on impassively, but now he decided that this was the right time to announce in a tone of weary resignation, “Women are for making children, not raising them. You can’t expect them to do a good job.”

Yalda thought: Bring on the Hurtlers.

Lidia said, “Why don’t we try making our own dye? We can go to the markets and look for the ingredients.”

Valerio’s face lit up with excitement. “Yes!”

“I want to come too,” Amelia pleaded.

Valeria held out for a few pauses, then decided that the whole thing had been her idea. “I know where we should go first. I already made a list in my head.”

Yalda exchanged a glance with Lidia, thanking her for halting the descent. “Why don’t we all go,” Lidia said. “Before it gets dark.”

Yalda woke on the first bell after midnight. Daria still hadn’t come home; some nights she went straight from the university to the Solo Club, then slept in her own apartment. That left Yalda or Lidia to get the children to school in the morning, depending on Lidia’s shift, but it was hard to complain that Daria wasn’t pulling her weight when she paid more than half the rent.

There was a patch of light on the floor beside the window, a hint of color in its diffuse edges. Yalda rose and walked quietly over to it. Through the window, four pairs of trails were visible, spreading slowly but still bright enough to bury most of the stars. In the early days every Hurtler reported had been fast and close, presumably because the more distant ones had been too dim to see with the naked eye. If anything in the current crop ever came that close, it would be spectacular. Maybe that would be enough to jolt her out of her stupor—assuming it didn’t achieve something far worse.

If a part of her had trouble believing that the world could end in a barrage of rocks from the distant past, the part that did believe struggled even harder to imagine that such grand cosmic mayhem could be avoided. Maybe everyone was simply born predisposed to expect life to go on as normal—and whatever the benefits of her education, hoping to overturn that innate conviction with an argument that started with the right triangles she’d found hidden in her plots up on Mount Peerless was asking too much of her animal brain.

Yalda glanced over at the sleeping children. However much she resented them at times, she certainly wasn’t indifferent to their fate—but she could summon no deep, visceral sense that their lives, and the lives of their own children, now lay in the hands of her earnest, enthusiastic, possibly deranged ex-student.

What would Tullia have done? Joked, reasoned, mocked, argued, probed all the competing theories for weaknesses, shone some light into other people’s blind spots, then followed her own imperfect instincts like everyone else. Yalda had never stopped missing her, but she was confused enough already without resorting to begging ghosts for advice.

The light on the floor brightened; Yalda turned back to the window. A fresh, dazzling streak of violet had appeared; it spread out slowly, parallel to the older trails.

She needed to make some kind of decision, if only to let herself sleep. So… she would take an interest in Eusebio’s plans and offer him whatever guidance she could, trying to help him spot the pitfalls in his strange endeavor. She could do that much without agreeing to be a passenger on his mad flying mountain—and without abandoning hope that the cosmic pyrotechnics that obsessed them both might yet turn out to be as harmless as a swarm of mites.

Eusebio led Yalda on foot for the last few saunters across the dusty brown plain. “Sorry to make you walk so far,” he said. “But it’s not a good idea to bring the trucks too close.”

Yalda took a moment to understand what he meant. The liberator for truck fuel wasn’t identical to that used with sunstone, but it could still cross-react. A pinch of gray powder spilt from a tank and carried the wrong way on the wind could heat things up very quickly.

“So how did you get the sunstone out here in the first place?” she asked.

“The way it’s always transported: well wrapped, in small pieces.”

They reached the test rocket: a cone of hardstone about Yalda’s height. Apertures near the top revealed an intricate assembly of cogs and springs secured within. “Most of this is for attitude control,” Eusebio explained. “If the sunstone burns unevenly, that puts a torque on the rocket and the whole thing will start to swerve. The mechanism needs to detect that and respond quickly, adjusting the flow of liberator between the combustion points.”

“Detect it how?”

Eusebio took a crank from a box of tools that had been left beside the device, and began winding the main spring. “Gyroscopes. There are three wheels set spinning rapidly on gimbals; if their axes shift relative to the housing, that means the rocket is veering off a straight course.”

When he’d finished winding the spring he squatted down and gestured with the crank at the bottom of the cone, which was held half a stride above the ground on six stubby legs. “There are four dozen tapered holes drilled into the sunstone, lined with calmstone most of the way. The conical plug of sunstone that was cut out to make each hole is also lined and put back in, but with grooves carved into it that leave a gap between the pieces. The liberator flows down through the gaps and ignites the unlined part at the bottom. As the sunstone burns, the lining’s corroded, progressively exposing the fuel.”

Yalda joined him and peered up at the orderly array of not-quite-plugged holes. Sunstone lamps used the tiniest sprinkling of liberator, diluted with some kind of inert grit, to keep them blazing for a night’s performance in the Variety Hall—and they still killed a few careless operators every year. Deep inside this rocket, waiting to trickle down into the fuel, was a whole tank full of the stuff in its pure form.

“I remember when you were afraid to visit the chemistry department,” Yalda said.

“I was never afraid!” Eusebio protested. “You told me not to waste my time with chemists, because they couldn’t get their energy tables straight.”

“Yes, of course, that’s exactly what I would have said.” Yalda straightened and stepped back from the rocket.

Eusebio tossed the crank into the toolbox, then looked back across the plain toward Amando, one of the three assistants who’d been working at the site when Yalda arrived. Eusebio waved broadly with two hands, and received the same gesture in reply. Then he reached into the cone and released a lever.

“What does that do?” Yalda asked.

“One chime to ignition.”

“Thanks for the warning.”

“One chime!” Eusebio scoffed, picking up the toolbox. “We’ll be behind the barrier long before then.”

Yalda was already ten strides ahead. “Did your father really let you design his trains?” she called back to him.

“Yes, but I just tweaked someone else’s plans,” Eusebio admitted. “Trains are complicated. I wouldn’t want to have to invent them from scratch.”

When Yalda reached the truck closest to the rocket she saw the other two assistants, Silvio and Frido, lying chest-down on the tray. A sturdy timber barrier some four strides high had been attached to the side of the vehicle; the two men were propped up on their elbows behind narrow slits in which theodolites had been fitted. The theory was that the rocket would ascend vertically, run out of fuel, rise a little farther from momentum alone, then plummet back to the ground to make a crater at more or less the point from which it had risen. Knowing the height it reached would allow the team to quantify the amount of energy produced that actually ended up doing something useful. It was one thing to measure the effects of a quarter of a scrag of sunstone incinerated in a workshop experiment, where the gases produced in a sealed chamber drove a piston against a load. Expecting the same yield when gas was spilling off the sides of the rocket and fragments of heat-cracked fuel were free to sprinkle down onto the desert might be a bit optimistic.

Eusebio caught up and joined Yalda and Amando behind the shield. Amando said, “One lapse to go”; he’d synchronized his own small clock with the one Eusebio had set ticking inside the rocket. Yalda stared anxiously at the clock’s face. If the sunstone burned hotter and faster than expected, it wasn’t inconceivable that the flame could rise up and eviscerate the tank full of liberator, bringing everything together in one mighty flash when the rocket was too high for the barrier to protect them.

“What am I doing here?” she muttered, trying to decide how low to squat for safety without completely obscuring her view.

“Watching history being made,” Eusebio replied, not entirely seriously. Yalda glanced at Amando, but his face gave nothing away.

“Three, two, one,” Eusebio counted.

The barrier concealed the flash of ignition, but by the time Yalda felt the ground tremble a dazzling line of white light had risen into view. A moment later a deafening hiss arrived through the air.

Yalda shielded her eyes and sought the rocket at the top of the afterimage, but something wasn’t right; the line seared on her vision had been joined by an arc, a circle, a widening helix. Then the point of radiance that had been inscribing these curves dropped below the barrier, and the ground shuddered. She stiffened her tympanum protectively, deafening herself to the sound of the impact, then tensed for a larger explosion.

Nothing followed. Either the fuel had all been consumed, or the liberator had ended up scattered.

Yalda turned to Eusebio. He looked shaken, but he recovered his composure rapidly.

“It reached a good height,” he said. “Maybe ten strolls.”

It took them more than a bell riding around the desert before they found the remains of the rocket. If it had stayed in one piece it might have made a spectacular crater, but the fragments of hardstone casing and mirrorstone cogs strewn across the ground had barely gouged the surface, and some were already half-buried in the dust. If Yalda had stumbled upon them unawares, she would have called for an archaeologist.

“Attitude control,” Eusebio said. “It just needs some refinement.”

He left the others sorting through the debris and gave Yalda a ride back to the city.

“What does your family think about all this?” she asked him. Yalda had said nothing to Lidia and Daria, or any of her colleagues; Eusebio had asked her to hold off mentioning the project to anyone until he’d tied up various “administrative loose ends”.

“I’ve managed to convince my father that it’s worth risking the money,” he said. “Even if the world’s in no danger from the Hurtlers, anything that the travelers invent on their journey could easily double our fortune.”

“What about your co?”

“She thinks I’m crazy. But I’ve told her—and my father—that I can’t countenance bringing children into the world until this rocket’s been launched to vouchsafe their future… which seems to have made both of them happy.”

“Why?”

Eusebio buzzed amusement. “She’s glad the day’s so far away. He’s glad it’s going to be so soon.”

Yalda said nothing. Eusebio turned to her and added, “Just to be clear, she can actually wait as long as she wishes.”

Yalda fought down the urge to reply sarcastically: How generous of you. If Eusebio really was supporting his co against their father’s nagging, it wasn’t worth starting a fight with him just because he sounded smug about it.

“Are you sure this isn’t going to lose you your fortune?” she asked. “What’s the going price for a mountain these days?”

Eusebio said, “I’ve already bought mining licenses from all the Councils that claim jurisdiction. It wasn’t cheap, but it didn’t ruin me.”

Mount Peerless was almost equidistant from five different cities; the only way to ensure an undisputed title would be to pay them all off. Yalda said, “Doesn’t a mining license include some deal about a share of the profits?”

“Of course. And if I do make any money from the project, I’ll give the Councils their cut.”

“But they all think you’re planning to dig out the sunstone and sell it?”

“I haven’t disabused them of that assumption.” The truck shuddered on the stony track. “But do you really want me to try lecturing my fellow Councilors on rotational physics? My father was willing to take my word for it—having paid so much for my education—but I can’t see Acilio and his cronies patiently following the trail of evidence from the velocity-wavelength formula to the passage of time for fast-moving travelers.”

“No.” The mention of Acilio reminded Yalda of someone else she’d been trying not to think about. “How do things stand with the university?”

“I’m negotiating a payment for them to relocate the observatory,” Eusebio said. “It’s not finalized, but given the amount we’ve been discussing they’ll be able to build a new telescope twice the size.”

“But not at the same altitude.”

“You can’t have everything. Don’t you think this is more important?”

“It’s not me you’ll have to convince,” Yalda warned him. “Have you ever heard of a man called Meconio?”

“Meconio? I thought he was long dead.”

“Not in spirit.” Perhaps the university would take Eusebio’s money and accept the deal before Ludovico discovered the connection between this “mining project” and the loathsome subject of the new physics.

“How much of the mountain do you think is sunstone?” she asked.

“Maybe two-thirds, by mass.”

Yalda did some quick calculations on her back. “That might be enough for one quarter-turn in four-space, but there’s no chance at all that it will cover the whole voyage.”

Eusebio glanced at her, surprised. “You expect the yield to stay the same, after half an age working on improvements?”

“Maybe not, but if there’s barely any sunstone left over from the acceleration stage… what kind of yield are you hoping for?”

“I don’t expect the travelers to burn sunstone for the later stages,” Eusebio replied.

Yalda was startled. “You want them to turn hardstone and calmstone into fuel?”

“Either that,” he said, “or move beyond the need for fuel entirely.”

Yalda waited for a sign that he was joking; none came. “So you’re counting on this rocket riding the Eternal Flame? Is that what you told your father to expect?”

Eusebio hunched his shoulders defensively. “Just because Ninth Age charlatans wrote a lot of nonsense about a similar idea doesn’t mean it’s actually impossible.”

“A flame that consumes no fuel?”

“Tell me why it can’t exist!” he demanded. “Not the version the philosophers imagined: some magic stone that would sit on your shelf, creating light and nothing else—that would violate conservation of energy. But if light and kinetic energy are created together there’s no reason they couldn’t balance each other precisely, without any change in chemical energy to plug the gap. Fuel doesn’t need to be consumed; that’s just the way it works with the kinds of fuel we have right now.”

Yalda had no argument about the energy balance, and while she couldn’t calculate the relevant entropies on the spot, creating light generally meant an increase. In conventional flames the hot gas formed by the spent fuel also contributed to the rise in entropy, but there was no reason to think it was essential. On the face of it, then, a slab of rock could create a beam of light—balancing the energy and momentum of the beam by recoiling in the opposite direction, but suffering no other change—without violating any principle she could name.

Accepting that statement of theory was one thing. Being stranded in the void with an infinite velocity, exiled from your home until you conjured the Eternal Flame into existence, was a different proposition.

“I can’t tell you it’s impossible,” Yalda conceded, “but you still need to ensure that there’s a useful amount of sunstone left after the acceleration—even if you have to throw away half the rest of the mountain to eliminate some dead weight. Give them something they can make more efficient, not a choice between bringing a Ninth Age myth to life, or never coming back!”

“Let’s see what the detailed surveys tell us,” Eusebio said, trying to sound conciliatory. “Two thirds was just a conservative guess.”

Yalda stared out across the desert. Who was going to volunteer to ride on this folly if it looked harder to survive than the Hurtlers?

She said, “Please tell me you’re not expecting the travelers to invent their own means of dealing with waste heat.”

“Of course not.”

“So…?”

“I’m planning to divert some of the exhaust gas,” Eusebio said. “Letting it expand and drive a piston while it’s thermally isolated will cool it down and supply some useful energy—then decompressing it further while it’s circulating around the habitation will draw in heat. Most of it will then be released into the void, but some will be used to maintain the pressure in the habitation, which would otherwise decline over time as the original atmosphere leaks out.”

“So you’ll be burning some sunstone for these purposes, even when the rocket isn’t in use?”

“Yes—though compared to the amount used for propulsion it won’t be much.”

Yalda couldn’t fault this scheme, or suggest any obvious refinements, but that wasn’t good enough. “Now that you’ve proved that you have no fear of explosions,” she said, “how about a detour to Amputation Alley?”

Eusebio regarded her suspiciously. “Why?”

“There’s a man there called Cornelio who knows more about heat than either of us. You should ask his advice on this.”

“Can he keep a confidence?”

“I have no idea,” Yalda replied, irritated. Cornelio had always treated her honourably, but she wasn’t going to vouch for his willingness to go along with Eusebio’s whims.

“Never mind,” Eusebio said. “I’ll hire him as a consultant, have him sign a contract.”

Yalda lost patience. “Do you honestly think you can send a whole mountain into the void in secret? Just you, and a few dozen advisers? Maybe you could get that much dead rock off the ground through sheer trial and error, but we’re talking about risking lives! You need the best people in the world to know about this, to think about it—to criticize all your ideas, all your systems, all your strategies. And I do mean the best people, not the best you can afford to put on your payroll and subjugate to a vow of silence.”

“I have enemies,” Eusebio said pointedly. “People who, if they knew of these plans, would happily spend a good part of their own fortune just to see me fail.”

“I don’t care,” Yalda replied coolly, resisting the urge to remind him that she’d suffered far more from his enemies’ pique than he had. “If the travelers are to have any hope of surviving, you’re going to need every biologist, agronomist, geologist, chemist, physicist and engineer on the planet as worried about their fate as you are.”

“And why should they fret about the lives of a few strangers?” Eusebio retorted. “You didn’t seem too eager to spread news of the catastrophe that this trip is intended to forestall.”

“I was wrong,” Yalda admitted. “First I didn’t take my own reasoning seriously, and then I was vain enough to think that if I could see no remedy myself, there was none. You’ve shown me otherwise, and I’m grateful for that. But it can’t end there.”

Eusebio said nothing, his gaze fixed ahead.

“No more silence,” Yalda declared. “I need to make the case for the problem, and you need to make the case for the solution. Let people argue, correct us, support us, tear us down. It’s the only hope we have to get this right.”

When Yalda arrived home Daria was in the apartment, helping Valeria and Valerio infuse some anatomical realism into their sketches of giant lizards laying waste to Zeugma.

“Lizards can rearrange their flesh almost any way they like,” Daria explained, “but they have five favorite postures, which are used in different places for different tasks. If they were on the ground, smashing buildings like this, you can bet they’d have a lot of flesh in their rear legs and their tails. It’s no good drawing them the way they’d look running along a slender twig.”

The children were entranced. Yalda sat and listened, not saying too much, hoping that merely sharing their interest would be read as a sign of affection. When she tried too hard Valeria reacted with scorn, but if she kept her distance she was punished later with accusations of indifference. It was exhausting having to be so calculating about it, but whatever it was that could sometimes make the relationship between a child and their protector almost effortless, it rarely seemed in evidence with Tullia’s children and the three friends who’d agreed to raise them. This was a labor of love, but that didn’t stop it being the hardest thing Yalda had done in her life.

Lidia had taken Amelia and Amelio to the doctor, but they were expected back soon. When the children were asleep tonight, Yalda decided, she would tell Lidia and Daria everything.

They had a right to know the truth—but what if they simply doubted her sanity? Watching Valeria chirping happily as she and Valerio re-drew each other’s lizards in a cascade of jokes and refinements, Yalda felt her own conviction about the peril of the Hurtlers faltering yet again. Whenever she immersed herself in domestic life, instead of apprehending the threat to the people around her more acutely she found herself growing numb and disbelieving. It wasn’t hard to imagine a time when everyone in this household would be gone; with the passage of years that was inevitable. But picturing every woman and every girl that lived having gone the way of men, leaving not a single child to survive them, only made her mind rebel and doubt the entire chain of reasoning that could lead to such an absurd conclusion.

Daria disengaged from the anatomy lesson for a moment to speak to Yalda directly. “There’s a letter for you on the sideboard.”

“Thanks.” Yalda judged the artists sufficiently engrossed not to care if she briefly left the audience.

The letter was from Lucia. Yalda had written to her several times since her last visit to the farm, but their correspondence had been intermittent.

She uncapped the wooden tube, tugged out the rolled-up sheets and smoothed them flat. Some of the symbols were a bit shaky, as if Lucia had been unable to keep the ridges still when she pressed the paper to her skin.

My dear sister Yalda

I’m sorry that it’s been so long since I last wrote to you. I’m sorry, too, that you haven’t been able to visit us yet and see the new farm, but I understand that you must be busy, taking care of your friend Tullia’s children as well as continuing your work at the university. (You won’t be surprised to hear that when I finally told Giusto about the children he denied that such a thing was possible, and said you needed to hunt down the derelict co or co-stead responsible!)

Claudio and his children joined us on the new farm a couple of stints ago. It’s lovely to have so many people around after all these years on our own. We never stopped visiting the old place, of course, but since tradition demands that the two farms are more than a separation apart, it hasn’t been too often.

The main reason I’m writing to you now is to tell you that this will be my last letter. I remember how hurt and saddened you were that you had no news or warning about Aurelia and Claudia, and I wanted it to be different for us. So: tomorrow I will become a mother.

I would be lying if I told you that I wasn’t afraid, but I’m also filled with great hope and happiness, knowing that Lucio and I have done our best to prepare for the children’s future. The farm is very well established, and we have plenty of money saved, so while the young cousins work under their father’s watchful eye, Lucio will mostly be free to take care of the children. (And please don’t be angry with him, this is my decision as much as it is his.)

Do you have the same beautiful shooting stars there in Zeugma that we have here? I know you’re studying these things, so it must be an exciting time for you. I can’t believe how glorious they look, even during the day, and it’s rather strange and wonderful to think of my children growing up to take such sights for granted. They’ll be amazed to hear from their father that there was once a time when the sky was so much emptier!

Your sister Lucia

11

The morning after her first appearance with Eusebio in the Variety Hall, Yalda woke early and went out to see what the papers had made of it.

A boy on a nearby corner was selling City Skin, so she bought a copy, but after flicking through the sheets three times it was clear that the end of the world didn’t rate a mention here. She went back to ask the boy for Talk; he’d sold out, so she waited while he dusted his chest with dye and made a fresh one for her.

“I’ll pay you the same for just the news and entertainment sections,” she offered impatiently.

“We’re not allowed to do that,” he said, summoning the memory of another sheet onto his chest.

“Why not?”

“The advertisers don’t like it.”

When he’d finished, Yalda took the whole bundle from him and walked around the corner before discarding the financial advice, restaurant reviews and railway timetables. She had to search the remainder twice before she found what she was looking for.

Last night our spies were at the Variety Hall, where the preternaturally plump Professor Yalda regaled a non-paying (!) audience with news of civilization’s impending demise. In a performance mixing the illusionist’s art with the terrors of geometry, the bountifully bulky Professor attempted to tie fleet-footed Mother Time in knots, leaving many observers wondering at her motives.

If the audience was unpersuaded by her message of doom, the attempt that followed by Councilor Eusebio to attract supporters (or even volunteers!) for his Ride Beyond the Sky was greeted by a veritable uproar of disbelief and derision. To anyone willing to back this venture: we have a design for Mechanical Wings gathering dust in our cupboard, awaiting only the life-giving touch of a gullible investor.

In the interests of sanity, Talk consulted Professor Ludovico of Zeugma University, who explained that the Hurtlers bringing such angst to last night’s double act are in fact nothing more than Spontaneous Excitations of the Solar Miasma. Despite its alarming appearance this phenomenon can do no harm to anyone, given that said Miasma is unable to interpenetrate our own plentiful atmosphere.

Ten more nights of this unprecedented madness remain. If the customers were paying then this cheerless and unscrupulous diversion would quickly close for want of funds, but Talk urges the next best outcome: an empty hall, to shame these charlatans into silence.

When Yalda reached the apartment Daria was awake, so she showed her the piece.

“I wouldn’t pay too much attention to Talk,” Daria said loftily. “Their idea of pushing the intellectual boundaries of journalism is to cover a literary salon.”

“What’s a literary salon?”

“An event where people who can’t read or reason gather to reassure each other of their own importance.”

Yalda said, “But anyone who reads this will think the whole thing’s some kind of… investment scam!”

Daria was amused. “Anyone who’d take this uncomprehending babble seriously was already a lost cause: they were never going to help you improve the Peerless, let alone volunteer for the ride.”

“Maybe not,” Yalda conceded. “But—”

“But you want everyone in Zeugma to understand what’s at stake?” Daria suggested.

“Of course. Don’t you think they’re owed that?”

Daria said, “I’ve been in the business for ten years, and I’ve had notices far worse than this. Believe me, the people who are truly curious will still come.” She rolled the sheet deftly into a cylinder and launched it across the room. “Just forget about it.”

Lidia had worked a late shift and was still asleep; Daria agreed to take the children to school. Yalda did her best to follow Daria’s advice, but when she reached the university Giorgio had more bad news for her: the senior members of the department had voted to refuse Eusebio’s offer to fund a new observatory. What’s more, they were taking their title over the land to Zeugma’s Council, seeking an order forbidding Eusebio from interfering with their present use.

Giorgio said, “If the telescope is raised entirely above the atmosphere it would actually improve the quality of observations. So if you can get the right wording on the order…”

Yalda was in no mood for jokes. “I thought you were going to persuade your friends to vote for this! Whatever they think of Eusebio’s rocket, a bigger telescope would be a worthwhile trade-off, surely.”

“I’m sorry,” he said. “Ludovico had more favors to call in than I did.”

Yalda didn’t doubt that Giorgio had kept his word. But he was still trapped in the kind of anesthetized state from which she’d taken so long to emerge herself. When he argued the physics with her, he accepted that the orthogonal cluster theory of the Hurtlers was as plausible as any alternative—but he still couldn’t bring himself to take the threat seriously, to look at his children and imagine their extinction.

Yalda had a lecture to deliver for the introductory optics course. As her students dutifully recorded her diagrams and equations for the laws of thin lenses, she felt like someone handing out useless trinkets at the edge of a raging wildfire. But she was forbidden from discussing Eusebio’s project during her lessons, or holding recruitment meetings on the campus. If any of these bright young men and women wanted to know the truth about the matter, they’d have to make the effort to turn up for the night’s unscrupulous diversion.

At lunchtime, as Yalda entered the food hall she saw Ludovico coming out of the pantry, his arms laden with loaves for a group of fellow diners. She hung back to avoid crossing his path, but he saw her and approached, calling out to her in a booming voice pitched to attract as much attention as possible.

“Professor Yalda! I’m surprised to see you here! I thought you’d left us for the entertainment business.”

Yalda humored him with a desultory buzz, but couldn’t help adding, “It seems everyone has two jobs these days; I see you’ve gone into journalism, yourself.”

“I was consulted by journalists,” Ludovico replied stiffly. “As a noted authority on the Hurtlers, not a paid employee.”

Yalda said, “Forgive me, but I must have missed all your learned publications on ‘Spontaneous Excitations in the Solar Miasma’. Perhaps you can rectify my ignorance on the topic by explaining precisely what that phrase means?” Everyone in the hall was watching them now, through rear eyes or front.

Ludovico said, “Gladly. One particle of the solar wind expels a fast luxite, which strikes another, prompting it to do the same. And so on. Other, slower light is emitted as well. That is what the Hurtlers are: long chains of activity arising within the gas itself, mediated by fast-moving particles of light.”

Yalda bowed her head in a gesture of gratitude, then feigned deep contemplation for several pauses—contemplation that failed to dispel her puzzlement. “But why are these ‘chains of activity’ parallel to each other? Why are these ‘spontaneous excitations’, these random events, all lined up in exactly the same direction?”

Ludovico replied without hesitation, “A distant source of rapid luxites—not quite at the resonant energy that would trigger an excitation itself—illuminates the solar wind and nudges the particles into alignment. The gas spontaneously emits its own light, but it is not randomly oriented when it does so.”

Yalda was speechless for a moment, marveling at the utter shamelessness of this absurd contrivance. “That’s nonsense,” she said cheerfully. “And you know it’s nonsense.”

Ludovico replied with calm hauteur, “Refute it, then. Show me your meticulous observations establishing my theory’s falsehood.” He began to walk away, but then he paused and turned back to face her. “Oh, I’m sorry, that was thoughtless of me! To make an observation, you might need an observatory… a facility that you’d prefer to see shattered into dust by your demented co-stead. Enjoy your meal, Professor Yalda.”

On stage in the Variety Hall, Yalda tried to push the day’s setbacks out of her mind and focus on the presentation. Even her bountifully bulky body was too small for people to read from the back row, so she’d worked with the Hall’s set designers to create a contraption with a sunstone lamp and lenses that projected a series of printed images onto a large white screen behind her.

As she gazed out into the darkness that concealed the audience, she honed her message, stressing its simplicity. Time was just another direction in space: nothing else could make sense of light’s behavior, or the ferocity of burning fuel. And to keep light tame, time needed to be finite—which meant that history would wrap around and meet itself, as surely as the system of roads and railway lines that wrapped itself around the planet. But while neighboring cities worked together to plan the railway lines between them, any intersections in the histories of worlds would be haphazard and ungoverned. Spectacular as they were, the Hurtlers were mere pedestrian tracks on this map; ahead, there would be busy freight lines.

Eusebio joined her, and the screen reprised the simple sketch he’d shown her: the detour, the long slow zigzag into the future that could buy them time, and with it fresh ideas and discoveries. It would be a risky journey, daunting for anyone to contemplate, but the Peerless needed whatever Zeugma’s people could bring to it. Navigating the void was just the start; to keep the community of travelers alive and thriving would take a whole city’s worth of inspiration and expertise.

Daria had advised them not to take questions from the floor; that only invited attention-seeking hecklers. Instead, they set up two desks in a corner of the foyer and invited people to come and speak with them after the performance, quietly, face-to-face.

Yalda had braced herself for a frenzy of disparagement spurred on by the negative coverage in Talk, but the audience as a whole had been no rowdier than the night before, and the individual interlocutors who approached them after the show were, if anything, more polite and encouraging. “I don’t believe a word of your scaremongering,” one young man told Yalda amiably, “but I do wish you luck.”

“Why do you think it’s scaremongering?”

“The world has survived for eons,” he replied. “History might not mention shooting stars like these, but the world is far older than we are. Geologists say the planet has been bombarded many times before; a few more stones from the sky will hardly be a calamity. But if you can send a rocket through the void and bring it back safely, that will be something to admire.”

“I can’t interest you in being a passenger yourself?” Yalda wasn’t taunting him; it might be worth having a thoughtful, good-natured skeptic among the travelers.

He said, “I think my children will stand a better chance of surviving with solid ground beneath their feet.”

Eusebio had to leave for an appointment with a legal adviser about the observatory dispute. Yalda decided to stay on a little longer; the foyer wasn’t empty yet, even if most of the dawdlers seemed to be talking among themselves rather than waiting for the right time to approach her.

When the clock struck two chimes before midnight, she started packing away her information pamphlets. She’d gathered five more names on top of the seven from the night before, and even if these volunteers were willing to do no more than plant crops in an artificial cave inside the mountain, that would be something.

As she stepped away from the desk, a young woman hurried across the foyer toward her.

“I’m sorry,” the woman said. “I wasn’t sure if I should speak to you, but…”

Yalda put down her box of pamphlets. “What did you want to say?”

“I was thinking about your rocket. One thing worries me—” She stopped and lowered her gaze, suddenly shy, as if her words might already have been too presumptuous.

“Go on,” Yalda encouraged her. “If there’s only one thing that worries you, you’re a dozen times more confident than I am.”

The woman said, “When the rocket turns around and comes back toward us… from the point of view of the travelers on the first half of the journey, isn’t it now moving backward in time?”

“Yes it is,” Yalda agreed. “That’s exactly right.”

“And from the point of view of the Hurtlers, and the worlds you think might collide with us… the same thing is true? In the second half of the journey, the rocket will be traveling into their past as well?”

“Yes.” Yalda was impressed; though it was a simple enough observation, only Eusebio and Giorgio had raised it with her before.

The woman looked up, fidgeted anxiously. “Is that… safe?”

“We don’t know,” Yalda admitted. “To what degree the rocket will carry its own arrow of time, embodied in its passengers and cargo, and to what degree its surroundings will influence the arrow… we don’t know.”

“So you’re hoping the travelers will learn enough on the outward voyage to protect themselves on the inward leg?” the woman suggested.

“I suppose it does come down to that.” Yalda had berated Eusebio for relying on uninvented methods of propulsion, but the truth was they had no hope of preparing the travelers in advance for every hazard the journey would entail.

Gaining courage, the woman said, “I’d be satisfied if you could at least be sure that the rocket was heading into the Hurtlers’ future at the start of the trip. If it takes half an age to prepare for the clash, so be it—but having to face that problem from the very beginning would be too much.”

“Satisfied enough… to approve of the venture?”

“Satisfied enough to be a passenger myself.”

Yalda said, “Can I ask your name?”

“Benedetta.”

Yalda took her over to the desk and recorded her details, trying not to let slip that no one else had come close to making such a commitment—not even the recruiter herself.

“Have you studied somewhere?” Yalda asked her. The first passenger of the Peerless had described her profession as “factory worker”.

“In Jade City,” Benedetta admitted reluctantly, as if this were somehow shameful. “I studied engineering, but only for a year.”

“It’s not important, I was just curious.” Yalda heard the forced joviality in her own voice, and struggled to bring the tone back to normal. Asking Benedetta if she were a runaway might frighten her off completely; it was an issue they’d need to deal with at some point—in order to protect both her and the project—but for now all that mattered was that she was keen, and a quick enough thinker to have spotted a genuine problem.

They were alone in the foyer now. Yalda said, “I’m supposed to be out of here before the cleaners come in at midnight, but we can talk for a bit outside if you’re not busy.”

“There’s nowhere I need to be,” Benedetta replied.

Outside, the city was quiet. A dozen slow Hurtlers were spreading their colors across the sky. As the two of them walked away from the hall across the cobblestones, Benedetta said, “Do you really believe that time loops around on itself?”

“I can’t be certain,” Yalda replied. “But I think the evidence is strong.”

“So the future is no different from the past?”

“The real difference is a matter of what we know,” Yalda said. “What information is easily accessible to us. We can know much more about the past than the future, at least if we don’t try to look back too far. But that’s a product of the vagaries of history; there’s no absolute distinction.”

“But then… everything that’s yet to happen is fixed, just as much as everything in the past?”

“Yes.”

“So why are you striving so hard to change the future?”

Yalda buzzed with delight; she should have seen that coming. “‘Change’ isn’t quite the right word,” she suggested. “You can strive to change a bad law—because the law can be different at different times. But either we survive this encounter or we don’t. Whatever the outcome is, no one will change it.”

Benedetta accepted this, but persisted. “What word should I use then? ‘Influence’?”

“I can live with that,” Yalda said. “I’ll own up to striving to influence the future.”

“But how can you influence the future if it’s as fixed as the past? Do you try to influence what happened yesterday?”

“Not anymore,” Yalda said, “but I certainly did the day before.”

“Why, though? If you believe that what happened yesterday has always been fixed, wouldn’t it have turned out the same, regardless?” Benedetta wasn’t teasing her, or playing rhetorical games; she genuinely needed an answer.

“Ah.” Yalda hadn’t had a conversation like this since the long nights she’d spent talking with Tullia—and back then, the roles would have been reversed. “I don’t believe in the kind of predestination that says our actions are irrelevant. So I don’t accept that yesterday would have turned out the same, regardless of what I did.”

“But if your actions aren’t irrelevant, then you can’t be choosing them freely, can you?” Benedetta argued. “If the future is fixed, and your actions affect the future… then your actions themselves must be fixed, otherwise they could lead to the wrong outcome. That means you have no real choice in what you do; you’re just a puppet, steered by forces beyond your control.”

Yalda thought for a while. “Raise your right hand.”

“Why?”

“Go on, humor me.”

Benedetta complied.

“Were you free to raise it or not, as you wished?” Yalda asked her.

“I believe so.”

Yalda said, “Tell me why you should feel any differently about that, depending on whether or not time is a loop and the future is really the distant past.”

Benedetta puzzled over the question. “If it was always going to happen—if in a sense it had already happened—then when I thought I was making the decision, that was just an illusion.”

“An illusion compared to what?” Yalda pressed her. “Tell me how the world could work—how physics could function, how history could be arranged—in a way that would somehow make you ‘more free’?”

“If the future is open,” Benedetta replied. “If our actions are undetermined until we decide what to do.”

“Suppose that really is the case,” Yalda said. “Then what is it that finally determined whether you raised your arm or not?”

“I did. It was my choice.”

“But why did you make that particular choice, and not refuse me?”

Benedetta didn’t reply immediately. “The way you asked, I suppose,” she said finally.

“So I determined your action?”

“No, not completely. My mood, my state of mind played their part as well.”

Yalda said, “None of the things you’ve just referred to disappear from the world if the future is fixed rather than open. Both of us are still here. Our actions are still related in exactly the same way to our wishes, our wishes to our personal moods and histories, and so on.”

Benedetta was not convinced. “If the future is fixed, how can this conversation even mean anything? If it’s an unchangeable fact that I will say whatever I end up saying to you—as if we were just actors following a script—then how can we really be changing each other’s minds? How can we be communicating anything?”

“Do I sound as if I’m making random noises for no particular reason?” Yalda joked.

“No.”

“If there’s a script,” Yalda said, “then we’re the playwrights as well as the actors; there’s no one else who could write our lines. There’s no puppet-master rushing around coordinating everything, forcing us to act against our will—or to make choices that go against our nature—just so history will reach its pre-ordained conclusion.”

“Then how does it work?” Benedetta demanded. “How do things turn out the way they have to?”

Yalda said, “The trick is to stop thinking that it works like fate in the sagas: some tedious monarch overcomes the odds and wins a great battle, because all the bit-players are nothing but cogs whose every action is subservient to his destiny. The reality is the opposite of that: ‘the way things have to be’ is completely unspectacular, and it’s fulfilled at the lowest possible level.

“We don’t know the details for every kind of matter, but in the case of free light the basic building blocks are just cyclic waves. When you make a full circuit of the cosmos in any direction, these waves undergo a whole number of cycles, so they return smoothly to their starting values. That’s it, that’s destiny fulfilled already… because anything constructed from waves like that will automatically share the same property. However complex a pattern of light you build up, it can’t contradict itself when it comes full circle. That’s guaranteed by the lowest-level physics; it doesn’t have to be orchestrated, or scripted, or contrived.”

Benedetta considered this. “So where are we in this picture? If the matter we’re made from works the same way, where are our choices?”

“In our biology,” Yalda said. “I think there’s a degree of consistency between our desires and our actions grounded in the structure of our brains and bodies. What you want, what you do, who you are… these things might not be in perfect harmony, but we’re not prisoners trapped in our bodies while they follow some plan that has nothing to do with us.” At least not until fission took over and split you in four, but Yalda didn’t want to get into that.

Benedetta fell silent as they started across the Great Bridge. Yalda didn’t expect to change her mind on this; the important thing was for her to understand that she could raise anything with her colleagues in the project. When you planned to send a mountain flying through the void at an infinite velocity, there was no such thing as too abstruse a concern.

Finally she said, “I’ll have to think on this more deeply. I can certainly see some force in your arguments.”

Yalda could hear the reservation in her voice. “But?”

Benedetta said, “It’s one thing to argue an abstract case that the future being fixed changes nothing: that there’s really no freedom lost, because our actions are determined in the same way, regardless. The fact remains, though, that we’re accustomed to seeing the future as open. That’s how our lives appear to us, that’s how we usually feel.”

Yalda stopped walking. They were halfway across the bridge, supported by a slender arch of stonework over the blackness of the crevasse. She felt a shudder pass through the skin of her back; she’d just had an eerie sense of knowing what her shy, intense new colleague would say next.

“When our descendants turn around and travel back in time,” Benedetta wondered, “will they still have the luxury we have, of debating this in the abstract? Once past and future are no longer so clear, will they still have the choice to go on seeing things the old way?”

Eusebio counted, “Three. Two. One.”

A distant line of light split the sky, wavering in the heat haze. A pause later the bunker trembled; as the timber boards holding back the sand flexed and rattled, the air filled with fine dust. Yalda and her companions were lying flat on their backs, a stride beneath the ground, but the tilted mirror above let them watch the ascending rocket as if they were upright in the desert, while the tinted clearstone pane protected them from the glare.

Yalda was prepared for the hiss when it came, but not the crack that abruptly bisected the pane along a jagged diagonal. She reached up to support the two pieces before they could slip from the frame and decapitate someone.

Amando cursed quietly and raised his own hands to help; Eusebio did the same, exchanging a glance with Yalda expressing relief that it hadn’t been worse. They’d isolated both the mirror and the pane from ground vibrations, but the shock wave in the air had still been enough to do damage. Nereo didn’t flinch; he was still tracking the rocket with his theodolite, and probably hadn’t even noticed the crack.

Giulio, the journalist from Red Towers, turned to Eusebio chirping with excitement. “To be honest with you, I thought it would just explode on the ground. But it’s really up there!” He was too overwhelmed by the spectacle of the launch to care that he’d narrowly avoided being sliced in two by a giant stone blade.

“That’s what rockets do,” Eusebio replied modestly. “Ascend.”

“When does it fall down again?” Giulio asked.

“This one won’t,” Eusebio predicted.

Yalda wasn’t so sure. Through the protective filter the rocket had almost dimmed to invisibility; Eusebio would have gauged its progress by eye, but it would take the precise measurements of their independent observer to confirm its ultimate destination.

Giulio raised his head so he could peer over the intervening bodies and see what Nereo was up to. Nereo was propped up on a special bench that Amando had constructed; this allowed him to watch a clock with his rear gaze at the same time as he followed the rocket through the theodolite. Yalda could see a list of pairs of times and angles written along the length of Nereo’s right arm; as she watched, he added one more pair then looked away from the theodolite and began producing further columns of numbers. The rocket would still be burning fuel—so there was still a chance that it could lose stability and drive itself back toward the ground—but on the assumption that it would do no worse, now, than if it had simply cut off its engines at the last clear observation, Nereo could give a tentative verdict on its fate.

“It’s escaped the world’s gravity,” Nereo declared. “I suspect it’s heading for an eccentric orbit around the sun with a period of several dozen years.”