“What’s that?”
“Do you remember once, you wrote about an Oracle? A machine that could solve the halting problem?”
“Of course.” Given a device that could tell you in advance whether a given computer program would halt, or go on running forever, you’d be able to prove or disprove any theorem whatsoever about the integers: the Goldbach conjecture, Fermat’s Last Theorem, anything. You’d simply show this “Oracle” a program that would loop through all the integers, testing every possible set of values and only halting if it came to a set that violated the conjecture. You’d never need to run the program itself; the Oracle’s verdict on whether or not it halted would be enough.
Such a device might or might not be possible, but Robert had proved more than twenty years before that no ordinary computer, however ingeniously programmed, would suffice. If program H could always tell you in a finite time whether or not program X would halt, you could tack on a small addition to H to create program Z, which perversely and deliberately went into an infinite loop whenever it examined a program that halted. If Z examined itself, it would either halt eventually, or run forever. But either possibility contradicted the alleged powers of program H: if Z actually ran forever, it would be because H had claimed that it wouldn’t, and
vice versa
. Program H could not exist.
“Time travel,” Helen said, “gives me a chance to become an Oracle. There’s a way to exploit the inability to change your own past, a way to squeeze an infinite number of timelike paths – none of them closed, but some of them arbitrarily near to it – into a finite physical system. Once you do that, you can solve the halting problem.”
“How?” Robert’s mind was racing. “And once you’ve done that … what about higher cardinalities? An Oracle for Oracles, able to test conjectures about the real numbers?”
Helen smiled enigmatically. “The first problem should only take you forty or fifty years to solve. As for the rest,” she pulled away from him, moving into the darkness of the hallway, “what makes you think I know the answer myself?” She blew him a kiss, then vanished from sight.
Robert took a step toward her, but the hallway was empty.
He walked back to the car, sad and exalted, his heart pounding.
The driver asked wearily, “Where to now, sir?”
Robert said, “Further up, and further in.”
4
The night after the funeral, Jack paced the house until three a.m. When would it be bearable?
When?
She’d shown more strength and courage, dying, than he felt within himself right now. But she’d share it with him, in the weeks to come. She’d share it with them all.
In bed, in the darkness, he tried to sense her presence around him. But it was forced, it was premature. It was one thing to have faith that she was watching over him, but quite another to expect to be spared every trace of grief, every trace of pain.
He waited for sleep. He needed to get some rest before dawn, or how would he face her children in the morning?
Gradually, he became aware of someone standing in the darkness at the foot of the bed. As he examined and reexamined the shadows, he formed a clear image of the apparition’s face.
It was his own. Younger, happier, surer of himself.
Jack sat up. “What do you want?”
“I want you to come with me.” The figure approached; Jack recoiled, and it halted.
“Come with you, where?” Jack demanded.
“To a place where she’s waiting.”
Jack shook his head. “No. I don’t believe you. She said she’d come for me herself, when it was time. She said she’d guide me.”
“She didn’t understand, then,” the apparition insisted gently. “She didn’t know I could fetch you myself. Do you think I’d send her in my place? Do you think I’d shirk the task?”
Jack searched the smiling, supplicatory face. “Who are you?”
His own soul, in Heaven, remade?
Was this a gift God offered everyone? To meet, before death, the very thing you would become – if you so chose? So that even this would be an act of free will?
The apparition said, “Stoney persuaded me to let his friend treat Joyce. We lived on, together. More than a century has passed. And now we want you to join us.”
Jack choked with horror. “No! This is a trick!
You’re the Devil!
”
The thing replied mildly, “There is no Devil. And no God, either. Just people. But I promise you: people with the powers of gods are kinder than any god we ever imagined.”
Jack covered his face. “Leave me be.” He whispered fervent prayers, and waited. It was a test, a moment of vulnerability, but God wouldn’t leave him naked like this, face-to-face with the Enemy, for longer than he could endure.
He uncovered his face. The thing was still with him.
It said, “Do you remember, when your faith came to you? The sense of a shield around you melting away, like armor you’d worn to keep God at bay?”
“Yes.” Jack acknowledged the truth defiantly; he wasn’t frightened that this abomination could see into his past, into his heart.
“That took strength: to admit that you needed God. But it takes the same kind of strength, again, to understand that
some needs can never be met
. I can’t promise you Heaven. We have no disease, we have no war, we have no poverty. But we have to find our own love, our own goodness. There is no final word of comfort. We only have each other.”
Jack didn’t reply; this blasphemous fantasy wasn’t even worth challenging. He said, “I know you’re lying. Do you really imagine that I’d leave the boys alone here?”
“They’d go back to America, back to their father. How many years do you think you’d have with them, if you stay? They’ve already lost their mother. It would be easier for them now, a single clean break.”
Jack shouted angrily, “Get out of my house!”
The thing came closer, and sat on the bed. It put a hand on his shoulder. Jack sobbed, “Help me!” But he didn’t know whose aid he was invoking any more.
“Do you remember the scene in
The Seat of Oak
? When the Harpy traps everyone in her cave underground, and tries to convince them that there is no Nescia? Only this drab underworld is real, she tells them. Everything else they think they’ve seen was just make-believe.” Jack’s own young face smiled nostalgically. “And we had dear old Shrugweight reply: he didn’t think much of this so-called ‘real world’ of hers. And even if she was right, since four little children could make up a better world, he’d rather go on pretending that their imaginary one was real.
“But we had it all upside down! The real world is richer, and stranger, and more beautiful than anything ever imagined. Milton, Dante, John the Divine are the ones who trapped you in a drab, gray underworld. That’s where you are now. But if you give me your hand, I can pull you out.”
Jack’s chest was bursting.
He couldn’t lose his faith. He’d kept it through worse than this. He’d kept it through every torture and indignity God had inflicted on his wife’s frail body. No one could take it from him now.
He crooned to himself, “In my time of trouble, He will find me.”
The cool hand tightened its grip on his shoulder. “You can be with her, now. Just say the word, and you will become a part of me. I will take you inside me, and you will see through my eyes, and we will travel back to the world where she still lives.”
Jack wept openly. “Leave me in peace! Just leave me to mourn her!”
The thing nodded sadly. “If that’s what you want.”
“I do!
Go!
”
“When I’m sure.”
Suddenly, Jack thought back to the long rant Stoney had delivered in the studio. Every choice went every way, Stoney had claimed. No decision could ever be final.
“Now I know you’re lying!” he shouted triumphantly. “If you believed everything Stoney told you, how could my choice ever mean a thing? I would always say yes to you, and I would always say no! It would all be the same!”
The apparition replied solemnly, “While I’m here with you, touching you,
you can’t be divided
. Your choice will count.”
Jack wiped his eyes, and gazed into its face. It seemed to believe every word it was speaking. What if this truly was his metaphysical twin, speaking as honestly as he could, and not merely the Devil in a mask? Perhaps there was a grain of truth in Stoney’s awful vision; perhaps this was another version of himself, a living person who honestly believed that the two of them shared a history.
Then it was a visitor sent by God, to humble him. To teach him compassion toward Stoney. To show Jack that he too, with a little less faith, and a little more pride, might have been damned forever.
Jack stretched out a hand and touched the face of this poor lost soul.
There, but for the grace of God, go I.
He said, “I’ve made my choice. Now leave me.”
#
Author’s note:
Where the lives of the fictional characters of this story parallel those of real historical figures, I’ve drawn on biographies by Andrew Hodges and A.N. Wilson. The self-dual formulation of general relativity was discovered by Abhay Ashtekar in 1986, and has since led to ground-breaking developments in quantum gravity, but the implications drawn from it here are fanciful.
In the early afternoon of his fourth day out of sadness, Jamil was wandering home from the gardens at the center of Noether when he heard shouts from the playing field behind the library. On the spur of the moment, without even asking the city what game was in progress, he decided to join in.
As he rounded the corner and the field came into view, it was clear from the movements of the players that they were in the middle of a quantum soccer match. At Jamil’s request, the city painted the wave function of the hypothetical ball across his vision, and tweaked him to recognize the players as the members of two teams without changing their appearance at all. Maria had once told him that she always chose a literal perception of color-coded clothing instead; she had no desire to use pathways that had evolved for the sake of sorting people into those you defended and those you slaughtered. But almost everything that had been bequeathed to them was stained with blood, and to Jamil it seemed a far sweeter victory to adapt the worst relics to his own ends than to discard them as irretrievably tainted.
The wave function appeared as a vivid auroral light, a quicksilver plasma bright enough to be distinct in the afternoon sunlight, yet unable to dazzle the eye or conceal the players running through it. Bands of color representing the complex phase of the wave swept across the field, parting to wash over separate rising lobes of probability before hitting the boundary and bouncing back again, inverted. The match was being played by the oldest, simplest rules: semi-classical, non-relativistic. The ball was confined to the field by an infinitely high barrier, so there was no question of it tunneling out, leaking away as the match progressed. The players were treated classically: their movements pumped energy into the wave, enabling transitions from the game’s opening state – with the ball spread thinly across the entire field – into the range of higher-energy modes needed to localize it. But localization was fleeting; there was no point forming a nice sharp wave packet in the middle of the field in the hope of kicking it around like a classical object. You had to shape the wave in such a way that all of its modes – cycling at different frequencies, traveling with different velocities – would come into phase with each other, for a fraction of a second, within the goal itself. Achieving that was a matter of energy levels, and timing.
Jamil had noticed that one team was under-strength. The umpire would be skewing the field’s potential to keep the match fair, but a new participant would be especially welcome for the sake of restoring symmetry. He watched the faces of the players, most of them old friends. They were frowning with concentration, but breaking now and then into smiles of delight at their small successes, or their opponents’ ingenuity.
He was badly out of practice, but if he turned out to be dead weight he could always withdraw. And if he misjudged his skills, and lost the match with his incompetence? No one would care. The score was nil all; he could wait for a goal, but that might be an hour or more in coming. Jamil communed with the umpire, and discovered that the players had decided in advance to allow new entries at any time.
Before he could change his mind, he announced himself. The wave froze, and he ran on to the field. People nodded greetings, mostly making no fuss, though Ezequiel shouted, “Welcome back!” Jamil suddenly felt fragile again; though he’d ended his long seclusion four days before, it was well within his power, still, to be dismayed by everything the game would involve. His recovery felt like a finely balanced optical illusion, a figure and ground that could change roles in an instant, a solid cube that could evert into a hollow.
The umpire guided him to his allotted starting position, opposite a woman he hadn’t seen before. He offered her a formal bow, and she returned the gesture. This was no time for introductions, but he asked the city if she’d published a name. She had: Margit.
The umpire counted down in their heads. Jamil tensed, regretting his impulsiveness. For seven years he’d been dead to the world. After four days back, what was he good for? His muscles were incapable of atrophy, his reflexes could never be dulled, but he’d chosen to live with an unconstrained will, and at any moment his wavering resolve could desert him.
The umpire said, “Play.” The frozen light around Jamil came to life, and he sprang into motion.
Each player was responsible for a set of modes, particular harmonics of the wave that were theirs to fill, guard, or deplete as necessary. Jamil’s twelve modes cycled at between 1,000 and 1,250 milliHertz. The rules of the game endowed his body with a small, fixed potential energy, which repelled the ball slightly and allowed different modes to push and pull on each other through him, but if he stayed in one spot as the modes cycled, every influence he exerted would eventually be replaced by its opposite, and the effect would simply cancel itself out.
To drive the wave from one mode to another, you needed to move, and to drive it efficiently you needed to exploit the way the modes fell in and out of phase with each other: to take from a 1,000 milliHertz mode and give to a 1,250, you had to act in synch with the quarter-Hertz beat between them. It was like pushing a child’s swing at its natural frequency, but rather than setting a single child in motion, you were standing between two swings and acting more as an intermediary: trying to time your interventions in such a way as to speed up one child at the other’s expense. The way you pushed on the wave at a given time and place was out of your hands completely, but by changing location in just the right way you could gain control over the interaction. Every pair of modes had a spatial beat between them – like the moiré pattern formed by two sheets of woven fabric held up to the light together, shifting from transparent to opaque as the gaps between the threads fell in and out of alignment. Slicing through this cyclic landscape offered the perfect means to match the accompanying chronological beat.
Jamil sprinted across the field at a speed and angle calculated to drive two favorable transitions at once. He’d gauged the current spectrum of the wave instinctively, watching from the sidelines, and he knew which of the modes in his charge would contribute to a goal and which would detract from the probability. As he cut through the shimmering bands of color, the umpire gave him tactile feedback to supplement his visual estimates and calculations, allowing him to sense the difference between a cyclic tug, a to and fro that came to nothing, and the gentle but persistent force that meant he was successfully riding the beat.
Chusok called out to him urgently, “Take, take! Two-ten!” Everyone’s spectral territory overlapped with someone else’s, and you needed to pass amplitude from player to player as well as trying to manage it within your own range.
Two-ten
– a harmonic with two peaks across the width of the field and ten along its length, cycling at 1,160 milliHertz – was filling up as Chusok drove unwanted amplitude from various lower-energy modes into it. It was Jamil’s role to empty it, putting the amplitude somewhere useful. Any mode with an even number of peaks across the field was unfavorable for scoring, because it had a node – a zero point between the peaks – smack in the middle of both goals.
Jamil acknowledged the request with a hand signal and shifted his trajectory. It was almost a decade since he’d last played the game, but he still knew the intricate web of possibilities by heart: he could drain the two-ten harmonic into the three-ten, five-two and five-six modes – all with “good parity”, peaks along the center-line – in a single action.
As he pounded across the grass, carefully judging the correct angle by sight, increasing his speed until he felt the destructive beats give way to a steady force like a constant breeze, he suddenly recalled a time – centuries before, in another city – when he’d played with one team, week after week, for forty years. Faces and voices swam in his head. Hashim, Jamil’s ninety-eighth child, and Hashim’s granddaughter Laila had played beside him. But he’d burned his house and moved on, and when that era touched him at all now it was like an unexpected gift. The scent of the grass, the shouts of the players, the soles of his feet striking the ground, resonated with every other moment he’d spent the same way, bridging the centuries, binding his life together. He never truly felt the scale of it when he sought it out deliberately; it was always small things, tightly focused moments like this, that burst the horizon of his everyday concerns and confronted him with the astonishing vista.
The two-ten mode was draining faster than he’d expected; the see-sawing center-line dip in the wave was vanishing before his eyes. He looked around, and saw Margit performing an elaborate Lissajous maneuver, smoothly orchestrating a dozen transitions at once. Jamil froze and watched her, admiring her virtuosity while he tried to decide what to do next; there was no point competing with her when she was doing such a good job of completing the task Chusok had set him.
Margit was his opponent, but they were both aiming for exactly the same kind of spectrum. The symmetry of the field meant that any scoring wave would work equally well for either side – but only one team could be the first to reap the benefit, the first to have more than half the wave’s probability packed into their goal. So the two teams were obliged to cooperate at first, and it was only as the wave took shape from their combined efforts that it gradually became apparent which side would gain by sculpting it to perfection as rapidly as possible, and which would gain by spoiling it for the first chance, then honing it for the rebound.
Penina chided him over her shoulder as she jogged past, “You want to leave her to clean up four-six, as well?” She was smiling, but Jamil was stung; he’d been motionless for ten or fifteen seconds. It was not forbidden to drag your feet and rely on your opponents to do all the work, but it was regarded as a shamefully impoverished strategy. It was also very risky, handing them the opportunity to set up a wave that was almost impossible to exploit yourself.
He reassessed the spectrum, and quickly sorted through the alternatives. Whatever he did would have unwanted side effects; there was no magic way to avoid influencing modes in other players’ territory, and any action that would drive the transitions he needed would also trigger a multitude of others, up and down the spectrum. Finally, he made a choice that would weaken the offending mode while causing as little disruption as possible.
Jamil immersed himself in the game, planning each transition two steps in advance, switching strategy half-way through a run if he had to, but staying in motion until the sweat dripped from his body, until his calves burned, until his blood sang. He wasn’t blinded to the raw pleasures of the moment, or to memories of games past, but he let them wash over him, like the breeze that rose up and cooled his skin with no need for acknowledgment. Familiar voices shouted terse commands at him; as the wave came closer to a scoring spectrum every trace of superfluous conversation vanished, every idle glance gave way to frantic, purposeful gestures. To a bystander, this might have seemed like the height of dehumanization: twenty-two people reduced to grunting cogs in a pointless machine. Jamil smiled at the thought but refused to be distracted into a complicated imaginary rebuttal. Every step he took was the answer to that, every hoarse plea to Yann or Joracy, Chusok or Maria, Eudore or Halide. These were his friends, and he was back among them. Back in the world.
The first chance of a goal was thirty seconds away, and the opportunity would fall to Jamil’s team; a few tiny shifts in amplitude would clinch it. Margit kept her distance, but Jamil could sense her eyes on him constantly – and literally feel her at work through his skin as she slackened his contact with the wave. In theory, by mirroring your opponent’s movements at the correct position on the field you could undermine everything they did, though in practice not even the most skillful team could keep the spectrum completely frozen. Going further and spoiling was a tug of war you didn’t want to win too well: if you degraded the wave too much, your opponent’s task – spoiling your own subsequent chance at a goal – became far easier.
Jamil still had two bad-parity modes that he was hoping to weaken, but every time he changed velocity to try a new transition, Margit responded in an instant, blocking him. He gestured to Chusok for help; Chusok had his own problems with Ezequiel, but he could still make trouble for Margit by choosing where he placed unwanted amplitude. Jamil shook sweat out of his eyes; he could see the characteristic “stepping stone” pattern of lobes forming, a sign that the wave would soon converge on the goal, but from the middle of the field it was impossible to judge their shape accurately enough to know what, if anything, remained to be done.
Suddenly, Jamil felt the wave push against him. He didn’t waste time looking around for Margit; Chusok must have succeeded in distracting her. He was almost at the boundary line, but he managed to reverse smoothly, continuing to drive both the transitions he’d been aiming for.
Two long lobes of probability, each modulated by a series of oscillating mounds, raced along the sides of the field. A third, shorter lobe running along the center-line melted away, reappeared, then merged with the others as they touched the end of the field, forming an almost rectangular plateau encompassing the goal.
The plateau became a pillar of light, growing narrower and higher as dozens of modes, all finally in phase, crashed together against the impenetrable barrier of the field’s boundary. A shallow residue was still spread across the entire field, and a diminishing sequence of elliptical lobes trailed away from the goal like a staircase, but most of the wave that had started out lapping around their waists was now concentrated in a single peak that towered above their heads, nine or ten meters tall.
For an instant, it was motionless.
Then it began to fall.
The umpire said, “Forty-nine point eight.”
The wave packet had not been tight enough.