by Greg Egan
Campbell said, “I want to know exactly what was so important to you that you crossed the Tasman, lied your way into my house, abused my hospitality, and stole my files. I don’t think it was simply curiosity, or jealousy. I think you found something ten years ago, and now you’re afraid my work is going to put it at risk.”
I sat down again. The rush of adrenaline I’d experienced at being cornered had dissipated. I could almost hear Alison whispering in my ear, “Either you kill him, Bruno, or you recruit him.” I had no intention of killing anyone, but I wasn’t yet certain that these were the only two choices.
I said, “And if I tell you to mind your own business?”
He shrugged. “Then I’ll work harder. I know you’ve screwed that laptop, and maybe the other computers in my house, but I’m not so broke that I can’t get a new machine.”
Which would be a hundred times faster. He’d re-run every search, probably with wider parameter ranges. The suitcase nuke from Sparseland that had started this whole mess would detonate again, and for all I knew it could be ten times, a hundred times, more powerful.
I said, “Have you ever wanted to join a secret society?”
Campbell gave an incredulous laugh. “No!”
“Neither did I. Too bad.”
I told him everything. The discovery of the defect. Industrial Algebra’s pursuit of the result. The epiphany in Shanghai. Sam establishing contact. The treaty, the ten quiet years. Then the sudden jolt of his own work, and the still-unfolding consequences.
Campbell was clearly shaken, but despite the fact that I’d confirmed his original suspicion he wasn’t ready to take my word for the whole story.
I knew better than to invite him into my office for a demonstration; faking it there would have been trivial. We walked to the local shopping center, and I handed him two hundred dollars to buy a new notebook. I told him the kind of software he’d need to download, without limiting his choice to any particular package. Then I gave him some further instructions. Within half an hour, he had seen the defect for himself, and nudged the border a short distance in each direction.
We were sitting in the food hall, surrounded by boisterous teenagers who’d just got out from school. Campbell was looking at me as if I’d seized a toy machine gun from his hands, transformed it into solid metal, then bashed him over the head with it.
I said, “Cheer up. There was no war of the worlds after Shanghai; I think we’re going to survive this, too.” After all these years, the chance to share the burden with someone new was actually making me feel much more optimistic.
“The defect is dynamic,” he muttered. “That changes everything.”
“You don’t say.”
Campbell scowled. “I don’t just mean the politics, the dangers. I’m talking about the underlying physical model.”
“Yeah?” I hadn’t come close to examining that issue seriously; it had been enough of a struggle coming to terms with his original calculations.
“All along, I’ve assumed that there were exact symmetries in the Planck scale physics that accounted for a stable boundary between macroscopic arithmetics. It was an artificial restriction, but I took it for granted, because anything else seemed ... ”
“Unbelievable?”
“Yes.” He blinked and looked away, surveying the crowd of diners as if he had no idea how he’d ended up among them. “I’m flying back in a few hours.”
“Does Bridget know why you came?”
“Not exactly.”
I said, “No one else can know what I’ve told you. Not yet. The risks are too great, everything’s too fluid.”
“Yeah.” He met my gaze. He wasn’t just humoring me; he understood what people like IA might do.
“In the long term,” I said, “we’re going to have to find a way to make this safe. To make everyone safe.” I’d never quite articulated that goal before, but I was only just beginning to absorb the ramifications of Campbell’s insights.
“How?” he wondered. “Do we want to build a wall, or do we want to tear one down?”
“I don’t know. The first thing we need is a better map, a better feel for the whole territory.”
He’d hired a car at the airport in order to drive here and confront me; it was parked in a side street close to my house. I walked him to it.
We shook hands before parting. I said, “Welcome to the reluctant cabal.”
Campbell winced. “Let’s find a way to change it from reluctant to redundant.”
#
In the weeks that followed, Campbell worked on refinements to his theory, emailing Alison and me every few days. Alison had taken my unilateral decision to recruit Campbell with much more equanimity than I’d expected. “Better to have him inside the tent,” was all she’d said.
This proved to be an understatement. While the two of us soon caught up with him on all the technicalities, it was clear that his intuition on the subject, hard-won over many years of trial-and-error, was the key to his spectacular progress now. Merely stealing his notes and his algorithms would never have brought us so far.
Gradually, the dynamic version of the theory took shape. As far as macroscopic objects were concerned – and in this context, “macroscopic” stretched all the way down to the quantum states of subatomic particles – all traces of Platonic mathematics were banished. A “proof” concerning the integers was just a class of physical processes, and the result of that proof was neither read from, nor written to, any universal book of truths. Rather, the agreement between proofs was simply a strong, but imperfect, correlation between the different processes that counted as proofs of the same thing. Those correlations arose from the way that the primordial states of Planck-scale physics were carved up – imperfectly – into subsystems that appeared to be distinct objects.
The truths of mathematics appeared to be enduring and universal because they persisted with great efficiency within the states of matter and space-time. But there was a built-in flaw in the whole idealization of distinct objects, and the point where the concept finally cracked open was the defect Alison and I had found in our volunteers’ data, which appeared to any macroscopic test as the border between contradictory mathematical systems.
We’d derived a crude empirical rule which said that the border shifted when a proposition’s neighbors outvoted it. If you managed to prove that x+1=y+1 and x–1=y–1, then x=y became a sitting duck, even if it hadn’t been true before. The consequences of Campbell’s search had shown that the reality was more complex, and in his new model, the old border rule became an approximation for a more subtle process, anchored in the dynamics of primordial states that knew nothing of the arithmetic of electrons and apples. The near-side arithmetic Campbell had blasted into the far side hadn’t got there by besieging the target with syllogisms; it had got there because he’d gone straight for a far deeper failure in the whole idea of “integers” than Alison and I had ever dreamed of.
Had Sam dreamed of it? I waited for his next contact, but as the weeks passed he remained silent, and the last thing I felt like doing was calling him myself. I had enough people to lie to without adding him to the list.
Kate asked me how work was going, and I waffled about the details of the three uninspiring contracts I’d started recently. When I stopped talking, she looked at me as if I’d just stammered my way through an unconvincing denial of some unspoken crime. I wondered how my mixture of concealed elation and fear was coming across to her. Was that how the most passionate, conflicted adulterer would appear? I didn’t actually reach the brink of confession, but I pictured myself approaching it. I had less reason now to think that the secret would bring her harm than when I’d first made my decision to keep her in the dark. But then, what if I told her everything, and the next day Campbell was kidnapped and tortured? If we were all being watched, and the people doing it were good at their jobs, we’d only know about it when it was too late.
Campbell’s emails dropped off for a while, and I assumed he’d
hit a roadblock. Sam had offered no further complaints. Perhaps, I thought, this was the new status quo, the start of another quiet decade. I could live with that.
Then Campbell flung his second grenade. He reached me by IM and said, “I’ve started making maps.”
“Of the defect?” I replied.
“Of the planets.”
I stared at his image, uncomprehending.
“The far-side planets,” he said. “The physical worlds.”
He’d bought himself some time on a geographically scattered set of processor clusters. He was no longer repeating his dangerous incursions, of course, but by playing around in the natural ebb and flow at the border, he’d made some extraordinary discoveries.
Alison and I had realized long ago that random “proofs” in the natural world would influence what happened at the border, but Campbell’s theory made that notion more precise. By looking at the exact timing of changes to propositions at the border, measured in a dozen different computers world-wide, he had set up a kind of ... radar? CT machine? Whatever you called it, it allowed him to deduce the locations where the relevant natural processes were occurring, and his model allowed him to distinguish between both near-side and far-side processes, and processes in matter and those in vacuum. He could measure the density of far-side matter, out to a distance of several light-hours, and crudely image nearby planets.
“Not just on the far side,” he said. “I validated the technique by imaging our own planets.” He sent me a data log, with comparisons to an online almanac. For Jupiter, the farthest of the planets he’d located, the positions were out by as much as a hundred thousand kilometers; not exactly GPS quality, but that was a bit like complaining that your abacus couldn’t tell north from north-west.
“Maybe that’s how Sam found us in Shanghai?” I wondered. “The same kind of thing, only more refined?”
Campbell said, “Possibly.”
“So what about the far-side planets?”
“Well, here’s the first interesting thing. None of the planets coincide with ours. Nor does their sun with our sun.” He sent me an image of the far-side system, one star and its six planets, overlaid on our own.
“But Sam’s time lags,” I protested, “when we communicate—”
“Make no sense if he’s too far away. Exactly. So he is not living on any of these planets, and he’s not even in a natural orbit around their star. He’s in powered flight, moving with the Earth. Which suggests to me that they’ve known about us for much longer than Shanghai.”
“Known about us,” I said, “but maybe they still didn’t anticipate anything like Shanghai.” When we’d set Luminous on to the task of eliminating the defect – not knowing that we were threatening anyone – it had taken several minutes before the far side had responded. Computers on board a spacecraft moving with the Earth would have detected the assault quickly, but it might have taken the recruitment of larger, planet-bound machines, minutes away at lightspeed, to repel it.
Until I’d encountered Campbell’s theories, my working assumption had been that Sam’s world was like a hidden message encoded in the Earth, with the different arithmetic giving different meanings to all the air, water and rock around us. But their matter was not bound to our matter; they didn’t need our specks of dust or molecules of air to represent the dark integers. The two worlds split apart at a much lower level; vacuum could be rock, and rock, vacuum.
I said, “So do you want the Nobel for physics, or peace?”
Campbell smiled modestly. “Can I hold out for both?”
“That’s the answer I was looking for.” I couldn’t get the stupid Cold War metaphors out of my brain: what would Sam’s hot-headed colleagues think, if they knew that we were now flying spy planes over their territory? Saying “screw them, they were doing it first!” might have been a fair response, but it was not a particularly helpful one.
I said, “We’re never going to match their Sputnik, unless you happen to know a trustworthy billionaire who wants to help us launch a space probe on a very strange trajectory. Everything we want to do has to work from Earth.”
“I’ll tear up my letter to Richard Branson then, shall I?”
I stared at the map of the far-side solar system. “There must be some relative motion between their star and ours. It can’t have been this close for all that long.”
“I don’t have enough accuracy in my measurements to make a meaningful estimate of the velocity,” Campbell said. “But I’ve done some crude estimates of the distances between their stars, and it’s much smaller than ours. So it’s not all that unlikely to find some star this close to us, even if it’s unlikely to be the same one that was close a thousand years ago. Then again, there might be a selection effect at work here: the whole reason Sam’s civilization managed to notice us at all was because we weren’t shooting past them at a substantial fraction of lightspeed.”
“OK. So maybe this is their home system, but it could just as easily be an expeditionary base for a team that’s been following our sun for thousands of years.”
“Yes.”
I said, “Where do we go with this?”
“I can’t increase the resolution much,” Campbell replied, “without buying time on a lot more clusters.” It wasn’t that he needed much processing power for the calculations, but there were minimum prices to be paid to do anything at all, and what would give us clearer pictures would be more computers, not more time on each one.
I said, “We can’t risk asking for volunteers, like the old days. We’d have to lie about what the download was for, and you can be certain that somebody would reverse-engineer it and catch us out.”
“Absolutely.”
I slept on the problem, then woke with an idea at four a.m. and went to my office, trying to flesh out the details before Campbell responded to my email. He was bleary-eyed when the messenger window opened; it was later in Wellington than in Sydney, but it looked as if he’d had as little sleep as I had.
I said, “We use the internet.”
“I thought we decided that was too risky.”
“Not screen-savers for volunteers; I’m talking about the internet itself. We work out a way to do the calculations using nothing but data packets and network routers. We bounce traffic all around the world, and we get the geographical resolution for free.”
“You’ve got to be joking, Bruno—”
“Why? Any computing circuit can be built by stringing together enough NAND gates; you think we can’t leverage packet switching into a NAND gate? But that’s just the proof that it’s possible; I expect we can actually make it a thousand times tighter.”
Campbell said, “I’m going to get some aspirin and come back.”
We roped in Alison to help, but it still took us six weeks to get a workable design, and another month to get it functioning. We ended up exploiting authentication and error-correction protocols built into the internet at several different layers; the heterogeneous approach not only helped us do all the calculations we needed, but made our gentle siphoning of computing power less likely to be detected and mistaken for anything malicious. In fact we were “stealing” far less from the routers and servers of the net than if we’d sat down for a hardcore 3D multiplayer gaming session, but security systems had their own ideas about what constituted fair use and what was suspicious. The most important thing was not the size of the burden we imposed, but the signature of our behavior.
Our new globe-spanning arithmetical telescope generated pictures far sharper than before, with kilometer-scale resolution out to a billion kilometers. This gave us crude relief-maps of the far-side planets, revealing mountains on four of them, and what might have been oceans on two of those four. If there were any artificial structures, they were either too small to see, or too subtle in their artificiality.
The relative motion of our sun and the star these planets orbited turned out to be about six kilometers per second. In the decade since Shanghai, the two solar systems had c
hanged their relative location by about two billion kilometers. Wherever the computers were now that had fought with Luminous to control the border, they certainly hadn’t been on any of these planets at the time. Perhaps there were two ships, with one following the Earth, and the other, heavier one saving fuel by merely following the sun.
Yuen had finally recovered his health, and the full cabal held an IM-conference to discuss these results.
“We should be showing these to geologists, xenobiologists ... everyone,” Yuen lamented. He wasn’t making a serious proposal, but I shared his sense of frustration.
Alison said, “What I regret most is that we can’t rub Sam’s face in these pictures, just to show him that we’re not as stupid as he thinks.”
“I imagine his own pictures are sharper,” Campbell replied.
“Which is as you’d expect,” Alison retorted, “given a head start of a few centuries. If they’re so brilliant on the far side, why do they need us to tell them what you did to jump the border?”
“They might have guessed precisely what I did,” he countered, “but they could still be seeking confirmation. Perhaps what they really want is to rule out the possibility that we’ve discovered something different, something they’ve never even thought of.”
I gazed at the false colors of one contoured sphere, imagining gray-blue oceans, snow-topped mountains with alien forests, strange cities, wondrous machines. Even if that was pure fantasy and this temporary neighbor was barren, there had to be a living home world from which the ships that pursued us had been launched.
After Shanghai, Sam and his colleagues had chosen to keep us in the dark for ten years, but it had been our own decision to cement the mistrust by holding on to the secret of our accidental weapon. If they’d already guessed its nature, then they might already have found a defense against it, in which case our silence bought us no advantage at all to compensate for the suspicion it engendered.
If that assumption was wrong, though? Then handing over the details of Campbell’s work could be just what the far-side hawks were waiting for, before raising their shields and crushing us.