“Only vaguely,” I admitted.
“He postulated fully deterministic degrees of freedom at the Planck scale, with quantum states corresponding to equivalence classes containing many different possible configurations. What’s more, all the ordinary quantum states we prepare at an atomic level would be complex superpositions of those primordial states, which allows him to get around the Bell inequalities.” I frowned slightly; I more or less got the picture, but I’d need to go away and read ‘t Hooft’s papers.
Campbell said, “In a sense, the detailed physics isn’t all that important, so long as you accept that ‘one thing’ might not ever be exactly the same as another ‘one thing,’ regardless of the kind of objects we’re talking about. Given that supposition, physical processes that seem to be rigorously equivalent to various arithmetic operations can turn out not to be as reliable as you’d think. With scone-weighing, the flaws are obvious, but I’m talking about the potentially subtler results of misunderstanding the fundamental nature of matter.”
“Hmm.” Though it was unlikely that anyone else Campbell had confided in had taken these speculations as seriously as I did, not only did I not want to seem a pushover, I honestly had no idea whether anything he was saying bore the slightest connection to reality.
I said, “It’s an interesting idea, but I still don’t see how it could speed up the hunt for inconsistencies.”
“I have a set of models,” he said, “which are constrained by the need to agree with some of ‘t Hooft’s ideas about the physics, and also by the need to make arithmetic almost consistent for a very large range of objects. From neutrinos to clusters of galaxies, basic arithmetic involving the kinds of numbers we might encounter in ordinary situations should work out in the usual way.” He laughed. “I mean, that’s the world we’re living in, right?”
Some of us. “Yeah.”
“But the interesting thing is, I can’t make the physics work at all if the arithmetic doesn’t run askew eventually—if there aren’t trans-astronomical numbers where the physical representations no longer capture the arithmetic perfectly. And each of my models lets me predict, more or less, where those effects should begin to show up. By starting with the fundamental physical laws, I can deduce a sequence of calculations with large integers that ought to reveal an inconsistency, when performed with pretty much any computer.”
“Taking you straight to the defect, with no need to search at all.” I’d let the definite article slip out, but it hardly seemed to matter anymore.
“That’s the theory.” Campbell actually blushed slightly. “Well, when you say ‘no search,’ what’s involved really is a much smaller search. There are still free parameters in my models; there are potentially billions of possibilities to test.”
I grinned broadly, wondering if my expression looked as fake as it felt. “But no luck yet?”
“No.” He was beginning to become self-conscious again, glancing around to see who might be listening.
Was he lying to me? Keeping his results secret until he could verify them a million more times, and then decide how best to explain them to incredulous colleagues and an uncomprehending world? Or had whatever he’d done that had lobbed a small grenade into Sam’s universe somehow registered in Campbell’s own computer as arithmetic as usual, betraying no evidence of the boundary he’d crossed? After all, the offending cluster of propositions had obeyed our axioms, so perhaps Campbell had managed to force them to do so without ever realizing that they hadn’t in the past. His ideas were obviously close to the mark—and I could no longer believe this was just a coincidence—but he seemed to have no room in his theory for something that I knew for a fact: arithmetic wasn’t merely inconsistent, it was dynamic. You could take its contradictions and slide them around like bumps in a carpet.
Campbell said, “Parts of the process aren’t easy to automate; there’s some manual work to be done setting up the search for each broad class of models. I’ve only been doing this in my spare time, so it could be a while before I get around to examining all the possibilities.”
“I see.” If all of his calculations so far had produced just one hit on the far side, it was conceivable that the rest would pass without incident. He would publish a negative result ruling out an obscure class of physical theories, and life would go on as normal on both sides of the inconsistency.
What kind of weapons inspector would I be, though, to put my faith in that rosy supposition?
Campbell was looking fidgety, as if his administrative obligations were beckoning. I said, “It’d be great to talk about this a bit more while we’ve got the chance. Are you busy tonight? I’m staying at a backpacker’s down in the city, but maybe you could recommend a restaurant around here somewhere?”
He looked dubious for a moment, but then an instinctive sense of hospitality seemed to overcome his reservations. He said, “Let me check with my wife. We’re not really into restaurants, but I was cooking tonight anyway, and you’d be welcome to join us.”
Campbell’s house was a fifteen minute walk from the campus; at my request, we detoured to a liquor store so I could buy a couple of bottles of wine to accompany the meal. As I entered the house, my hand lingered on the doorframe, depositing a small device that would assist me if I needed to make an uninvited entry in the future.
Campbell’s wife, Bridget, was an organic chemist, who also taught at Victoria University. The conversation over dinner was all about department heads, budgets, and grant applications, and, despite having left academia long ago, I had no trouble relating sympathetically to the couple’s gripes. My hosts ensured that my wine glass never stayed empty for long.
When we’d finished eating, Bridget excused herself to make a call to her mother, who lived in a small town on the south island. Campbell led me into his study and switched on a laptop with fading keys that must have been twenty years old. Many households had a computer like this: the machine that could no longer run the latest trendy bloatware, but which still worked perfectly with its original OS.
Campbell turned his back to me as he typed his password, and I was careful not to be seen even trying to look. Then he opened some C++ files in an editor, and scrolled over parts of his search algorithm.
I felt giddy, and it wasn’t the wine; I’d filled my stomach with an over-the-counter sobriety aid that turned ethanol into glucose and water faster than any human being could imbibe it. I fervently hoped that Industrial Algebra really had given up their pursuit; if I could get this close to Campbell’s secrets in half a day, IA could be playing the stock market with alternative arithmetic before the month was out, and peddling inconsistency weapons to the Pentagon soon after.
I did not have a photographic memory, and Campbell was just showing me fragments anyway. I didn’t think he was deliberately taunting me; he just wanted me to see that he had something concrete, that all his claims about Planck scale physics and directed search strategies had been more than hot air.
I said, “Wait! What’s that?” He stopped hitting the PAGE DOWN key, and I pointed at a list of variable declarations in the middle of the screen:
long int i1, i2, i3;
dark d1, d2, d3;
A “long int” was a long integer, a quantity represented by twice as many bits as usual. On this vintage machine, that was likely to be a total of just sixty-four bits. “What the fuck is a ‘dark’?” I demanded. It wasn’t how I’d normally speak to someone I’d only just met, but then, I wasn’t meant to be sober.
Campbell laughed. “A dark integer. It’s a type I defined. It holds four thousand and ninety-six bits.”
“But why the name?”
“Dark matter, dark energy… dark integers. They’re all around us, but we don’t usually see them, because they don’t quite play by the rules.”
Hairs rose on the back of my neck. I could not have described the infrastructure of Sam’s world more concisely myself.
Campbell shut down the laptop. I’d been looking for an opportunity to handle the machine, however briefly, without arousing his suspicion, but that clearly wasn’t going to happen, so as we walked out of the study I went for plan B.
“I’m feeling kind of…” I sat down abruptly on the floor of the hallway. After a moment, I fished my phone out of my pocket and held it up to him. “Would you mind calling me a taxi?”
“Yeah, sure.” He accepted the phone, and I cradled my head in my arms. Before he could dial the number, I started moaning softly. There was a long pause; he was probably weighing up the embarrassment factor of various alternatives.
Finally he said, “You can sleep here on the couch if you like.” I felt a genuine pang of sympathy for him; if some clown I barely knew had pulled a stunt like this on me, I would at least have made him promise to foot the cleaning bills if he threw up in the middle of the night.
In the middle of the night, I did make a trip to the bathroom, but I kept the sound effects restrained. Halfway through, I walked quietly to the study, crossed the room in the dark, and slapped a thin, transparent patch over the adhesive label that a service company had placed on the outside of the laptop years before. My addition would be invisible to the naked eye, and it would take a scalpel to prise it off. The relay that would communicate with the patch was larger, about the size of a coat button; I stuck it behind a bookshelf. Unless Campbell was planning to paint the room or put in new carpet, it would probably remain undetected for a couple of years, and I’d already prepaid a two year account with a local wireless internet provider.
I woke not long after dawn, but this un-Bacchanalian early rising was no risk to my cover; Campbell had left the curtains open so the full force of the morning sun struck me in the face, a result that was almost certainly deliberate. I tiptoed around the house for ten minutes or so, not wanting to seem too organized if anyone was listening, then left a scrawled note of thanks and apology on the coffee table by the couch, before letting myself out and heading for the cable car stop.
Down in the city, I sat in a café opposite the backpacker’s hostel and connected to the relay, which in turn had established a successful link with the polymer circuitry of the laptop patch. When noon came and went without Campbell logging on, I sent a message to Kate telling her that I was stuck in the bank for at least another day.
I passed the time browsing the news feeds and buying overpriced snacks; half of the café’s other patrons were doing the same. Finally, just after three o’clock, Campbell started up the laptop.
The patch couldn’t read his disk drive, but it could pick up currents flowing to and from the keyboard and the display, allowing it to deduce everything he typed and everything he saw. Capturing his password was easy. Better yet, once he was logged in he set about editing one of his files, extending his search program to a new class of models. As he scrolled back and forth, it wasn’t long before the patch’s screen shots encompassed the entire contents of the file he was working on.
He labored for more than two hours, debugging what he’d written, then set the program running. This creaky old twentieth century machine, which predated the whole internet-wide search for the defect, had already scored one direct hit on the far side; I just hoped this new class of models were all incompatible with the successful ones from a few days before.
Shortly afterward, the IR sensor in the patch told me that Campbell had left the room. The patch could induce currents in the keyboard connection; I could type into the machine as if I was right there. I started a new process window. The laptop wasn’t connected to the internet at all, except through my spyware, but it took me only fifteen minutes to display and record everything there was to see: a few library and header files that the main program depended on, and the data logs listing all of the searches so far. It would not have been hard to hack into the operating system and make provisions to corrupt any future searches, but I decided to wait until I had a better grasp of the whole situation. Even once I was back in Sydney, I’d be able to eavesdrop whenever the laptop was in use, and intervene whenever it was left unattended. I’d only stayed in Wellington in case there’d been a need to return to Campbell’s house in person.
When evening fell and I found myself with nothing urgent left to do, I didn’t call Kate; it seemed wiser to let her assume that I was slaving away in a windowless computer room. I left the café and lay on my bed in the hostel. The dormitory was deserted; everyone else was out on the town.
I called Alison in Zurich and brought her up to date. In the background, I could hear her husband, Philippe, trying to comfort Laura in another room, calmly talking baby-talk in French while his daughter wailed her head off.
Alison was intrigued. “Campbell’s theory can’t be perfect, but it must be close. Maybe we’ll be able to find a way to make it fit in with the dynamics we’ve seen.” In the ten years since we’d stumbled on the defect, all our work on it had remained frustratingly empirical: running calculations and observing their effects. We’d never come close to finding any deep underlying principles.
“Do you think Sam knows all this?” she asked.
“I have no idea. If he did, I doubt he’d admit it.” Though it was Sam who had given us a taste of far-side mathematics in Shanghai, that had really just been a clip over the ear to let us know that what we were trying to wipe out with Luminous was a civilization, not a wasteland. After that near-disastrous first encounter, he had worked to establish communications with us, learning our languages and happily listening to the accounts we’d volunteered of our world, but he had not been equally forthcoming in return. We knew next to nothing about far-side physics, astronomy, biology, history, or culture. That there were living beings occupying the same space as the Earth suggested that the two universes were intimately coupled somehow, in spite of their mutual invisibility. But Sam had hinted that life was much more common on his side of the border than ours; when I’d told him that we seemed to be alone, at least in the solar system, and were surrounded by light-years of sterile vacuum, he’d taken to referring to our side as “Sparseland.”
Alison said, “Either way, I think we should keep it to ourselves. The treaty says we should do everything in our power to deal with any breach of territory of which the other side informs us. We’re doing that. But we’re not obliged to disclose the details of Campbell’s activities.”