by SL Huang
The roller coaster of emotion I’d been on the past few days rippled through me like an echo, then drained away, leaving me empty.
If she’d never had the proof, there was a god somewhere up there laughing at us.
Chapter 37
Rita Martinez died thirteen days later, without ever waking up. I went to the funeral because I had a feeling Arthur expected me to. I filed it in the category of trying to be a better person.
Professor Halliday spoke. I didn’t pay attention to what she said; I let the words sail over me and echo against the walls of the church.
I recognized a few Feds at the service, and numbly wondered if they were there to make sure no one gave anything away. The ever-helpful Agent Jones approached me afterward, out on the lawn under a copse of decorative trees.
“Ms. Russell,” she said, nodding at me. She had a paper cup of punch in her hand. There were refreshments inside, which I’d passed on in favor of the flask in my pocket and the cocktail of narcotics I was still on.
“Agent Jones,” I answered. I didn’t know or care what she wanted. I felt detached from everything. Numb.
“I just want you to know, um…” She glanced behind her and straightened her jacket. It was very odd behavior. I watched blandly. “I want you to know, I worked with one of you before, and you don’t have to worry. Whatever holes there have been in you and your colleagues’ stories, I took care of it on our end.”
I opened my mouth, but she held up a hand to forestall me.
“I know you can’t confirm it, and that’s okay. I just didn’t want you to worry. I started taking care of everything as soon as I realized. I understand how hard it can be when you’re working alone with no resources.” She straightened and gave me a sharp nod, almost as if she wanted to salute. I felt faintly ridiculous. “If you need anything else, just let me know.”
“Sure,” I said. “Thanks.”
She paused as if she were about to say something else, then nodded again and turned on her heel to stride off.
Well, that had been weird.
Professor Halliday replaced her a few minutes later, wandering over as if she hadn’t chosen to seek me out. She reached over and ran a hand along the bark of one of the decorative trees, as if she had come over expressly to do that rather than to speak to me.
A normal person probably would have made small talk about the service, or her eulogy. It felt like too much effort to me.
“Arthur told me everything,” Halliday said after a minute.
“Oh.” I didn’t know what she expected me to say to that. If she was angry with me, there wasn’t much I could do to dissuade her. I wasn’t even sure it would be right to dissuade her.
Her eyes gleamed with unshed tears. “It wasn’t ideal, but—at least she died here, among friends, instead of alone and on the run. I think I was wrong, to say we shouldn’t find her.”
What a stunningly illogical sentiment. “Professor, you do realize I was responsible for getting your friend drugged and kidnapped and threatened with torture, right?”
Her expression twitched. “You didn’t intend for any of that to happen.”
I couldn’t believe it. Her attitude was akin to telling a drunk driver she had no responsibility for plowing into a bus load of schoolchildren. Certain things were foreseeable consequences, even if you didn’t intend them.
But I was too tired to argue with her over why she should hate me.
“There’s something…” She stared intently at the tree. “I’d like to go over something with you, if you don’t mind. Can we meet tomorrow?”
I had no good reason to refuse her, and saying yes seemed like the kind of thing a better person would do, so I did.
We met in a coffee shop near the university. She’d already written out her proof again for the NSA, and with the Lancer in custody, the DHS had dropped her protective detail. It was just the two of us.
Halliday ordered a tea first and spent a long time sipping it. “Xiaohu pled guilty to espionage. Two years suspended sentence. He went home to his wife and children.”
“Oh,” I said. “Good. Your doing?”
She made a noncommittal motion. I recalled Agent Jones’s words at the funeral, her assurances that she was “taking care of things” on my behalf. It made me uncomfortable, just how much I didn’t know about what was going on.
“Do you believe…” Halliday put down her cup, straightened it, and folded her hands on the table. “Do you believe she had the proof?”
“Who knows?” For God’s sake, this better not be what she had invited me here to talk about. “Do you think so?”
“I don’t know. My intuition had always been that they are unequal, or perhaps that the question is formally independent. We in the field tend to predict—well, I had always considered inequality a near-certainty, though I would have entertained the idea of unprovability. Considering the possibility of equality is…” She raised a hand and then let it fall to her lap. “It boggles the mind.”
“She either had the proof or not,” I said. “It doesn’t matter how you feel about it.”
“I know.” She paused. “The funny part about you saying such a thing is—for all her talk of rationality, Rita didn’t want the truth in this case. She wanted a particular outcome more badly than any other. She was one of the only people I knew who felt equality was likely, let alone wanted it to be true with such fervency—she would daydream about it, and dream about being the one to find the answer herself. Discovering they were unequal or independent would change very little; discovering they were equal would shatter mathematics, and I think part of her wanted that. The sensation. To be a Gödel or Zermelo.”
“Because she wouldn’t get a sensation from any proof regarding P versus NP?” I said sarcastically.
Halliday chuckled. “True. She never could work in moderation. A person of extremes, was Rita. Either she felt the reality was quite different, or…”
Something inside me folded in on itself, twisting and tight. “You don’t think she ever solved it, not for real. You think it was her tumor talking.”
Halliday hesitated. Then, instead of answering, she reached into her bag, pulled out a stack of papers, and held it out to me.
“What’s this?”
“PDE proofs. Several.”
I took the sheaf of pages and started skimming the first one. Partial differential equations, as Halliday had said. “What does this have to do with anything?”
“They were in a safe deposit box in Rita’s name. She left most of her things to me. I’ve been going through it all, but I only found these yesterday.”
“They’re stuff she hadn’t published yet?” I didn’t understand why Halliday would feel such a need to show them to me.
“PDEs weren’t her field. Nor mine. Are they correct?”
I turned the page, kept skimming. “Yeah. So far.”
She nodded. “I thought so. But they aren’t her style, are they? They…meander. Rita’s work was always tight. Dense.”
I looked up. “Professor, stop dancing around. What are you trying to say?”
Her hands were tight against the edge of the table, her forehead knitted. “If she truly found a constructive proof showing P equals NP…it’s what she feared for mathematics. That one of the consequences would be the ability to quickly prove anything one could quickly verify, and thus large swathes of mathematics as a creative field would go obsolete.”
My heart started beating faster. “You’re suggesting she wrote an automated theorem prover.” Which would only have been possible if her proof had been correct.
Halliday’s mouth twitched upward in the slightest of smiles. “She never did like differential equations. I think it would have tickled her, to steal some of their thunder.”
Dr. Martinez had struggled with whether to share her new knowledge with the world. It made sense that she’d at least used it to spread other knowledge, even if she’d destroyed the programs she’d used to do it.
Destroyed. And Martinez herself dead. If she had ever had the proof, it was gone with her, a state functionally equivalent to one in which she’d never discovered it at all.
Even if her result was true, I would never be able to replicate it myself, and without it, I was stuck. Forever. Locked in a place bare of mathematical intuition. An idiot savant missing the one skill that counted.
Halliday stood. “I’m going to send the PDE proofs around. Maybe she stole them, too, and someone will take credit. Or maybe she just wanted to make a point, to prove something, by learning the field herself.”
Prove something. It sounded so simple, for something so entirely out of reach.
♦ ♦ ♦
I went to see Checker a few days after Martinez’s funeral. I’d spent most of the last couple weeks asleep on the good pills, until the black market prescription drugs Dr. Washington had given me had run out. My hands still got twitchy every so often, but less every day.
I hadn’t had a chance to talk to Checker yet. Or maybe I’d been avoiding it. I’d seen him at the funeral; he’d come even though he’d never met Rita Martinez—he and Pilar had been there with Arthur, dressed in black and looking suitably somber. I’d made a crack about not knowing he owned a suit, and he’d made one back about not knowing I owned soap. Touché.
I found him in the Hole, as usual, and shut the door to lean against it, my hands shoved in my jacket pockets. I’d apologized to Arthur, but Checker…he’d been mad at me, too, but for entirely different reasons.
Reasons that were a lot more complicated. Reasons that wouldn’t go away with an apology, even if Checker shared Arthur’s bizarre and stunning depth of forgiveness.
Reasons I didn’t, when it came down to it, think I should apologize for at all.
I didn’t know how to start. “You came for me,” I said finally.
Checker smiled slightly. “Of course we did.”
“Thank you.”
He pushed back from his desktop to face me, as if I were acting so strange that if he didn’t handle me carefully I might explode. “We always will, you know.”
I couldn’t remember having friends before Arthur and Checker. I wondered if I had. I had no precedent to guide me, no confidence in my ability to navigate a relationship that involved caring about someone else. “You’re right,” I said baldly. “About my memory.”
“I know.” The smile had disappeared, and his voice had gone cautious, neutral.
“I’m acknowledging it, just this once. Because you should know I made a decision.” I spoke very evenly. “I’m not going to look into it. And I’m asking you not to, either.”
“Why?” he asked after a beat.
“I don’t need a reason,” I said. “It’s my memory. My life. I’m—I’m asking you.” I wet my lips. “Leave it alone. Please.”
I couldn’t read Checker’s expression. He took his time in answering, and when he did his words were quiet and slow, as if he were placing them carefully one after the other. “I hear what you’re saying. I do. But I—I can’t.” His voice cracked. “Because—whatever happened to you might be influencing you to say that. We’ve seen it before, and even the possibility—we need to find out why this happened to you. You could be in danger. You could have other enemies out there. I’m not going to sit by and let you ignore this, even if you—even if you ask me to. I can’t.” He held my eyes, pleading, almost anguished.
I stayed leaning against the door for a long moment.
I didn’t know what to do.
“You’re going to do this even without my permission, then?” I said.
“Cas, don’t make me—”
“I’m not making you,” I said quietly.
“Can we talk about this? Please?”
“No.” I was sure. I knew what I wanted.
What I needed. What my broken, already-damaged brain needed.
To keep it all locked away.
I said to Checker, “Tell me, right now, that you’re going to let this go.”
He squeezed his eyes shut for a long moment. “I can’t.”
I felt numb.
I turned and put a hand on the doorknob. “If you’re in trouble, I’ll come,” I said, not looking back at him. “Otherwise, call me when you change your mind.”
I pushed open the door and headed out of the Hole without looking back.
Checker called after me, imploring, frantic, but I didn’t acknowledge him. His pleas echoed in my head until I fell asleep that night and dreamt of half-real monsters who smothered me in false memory and distorted realities.
When I woke only a few hours later, I stumbled for the darkened streets, seeking the strongest chemical remedies money and back alleyways could offer.
THE END
Cas Russell’s adventures will continue in
PLASTIC SMILE
coming 2016
Thank You For Reading
If you’re interested in some exciting notes on the math used for Cas’s latest adventure, turn the page! Otherwise:
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Above all, thank you so much for joining me for Root of Unity. Now turn the page for those exciting math notes, a list of my other fiction, and a whole lot more thanks that need saying…
Afterward: A Note on the Math in This Book
If P=NP, then the world would be a profoundly different place than we usually assume it to be. There would be no special value in “creative leaps,” no fundamental gap between solving a problem and recognizing the solution once it’s found. Everyone who could appreciate a symphony would be Mozart; everyone who could follow a step-by-step argument would be Gauss; everyone who could recognize a good investment strategy would be Warren Buffett. (Scott Aaronson, “Reasons to Believe”)
Even though I knew this book would focus on cryptography and complexity theory, I wasn’t sure I was going to use P vs. NP for it until I read a paper by Professor Scott Aaronson. After all, P vs. NP has been done in fictional media enough times for it to start feeling cliché, and even though the problem fascinates me—as it does many—I thought I might want to choose something a bit less overdone.
But as Professor Aaronson points out in the paper “NP-complete Problems and Physical Reality,” most people who talk about the idea of P equaling NP focus only on the most minor results of it. And though in fiction it’s much more likely for P to equal NP than the opposite—after all, as Halliday says near the end of this book, inequality would change very little, so it is somewhat less interesting for fiction—the problem has rarely been imagined in a way that explores all the possible consequences of equality:
Even many computer scientists do not seem to appreciate how different the world would be if we could solve NP-complete problems efficiently. I have heard it said, with a straight face, that a proof of P = NP would be important because it would let airlines schedule their flights better, or shipping companies pack more boxes in their trucks! One person who did understand was Gödel. In his celebrated 1956 letter to von Neumann (see [69]), in which he first raised the P versus NP question, Gödel says that a linear or quadratic-time procedure for what we now call NP-complete problems would have “consequences of the greatest magnitude.” For such an procedure “would clearly indicate that, despite the unsolvability of the Entscheidungsproblem, the mental effort of the mathematician in the case of yes-or-no questions could be completely replaced by mac
hines.”
But it would indicate even more. If such a procedure existed, then we could quickly find the smallest Boolean circuits that output (say) a table of historical stock market data, or the human genome, or the complete works of Shakespeare. It seems entirely conceivable that, by analyzing these circuits, we could make an easy fortune on Wall Street, or retrace evolution, or even generate Shakespeare’s 38th play. For broadly speaking, that which we can compress we can understand, and that which we can understand we can predict. Indeed, in a recent book [12], Eric Baum argues that much of what we call ‘insight’ or ‘intelligence’ simply means finding succinct representations for our sense data. On his view, the human mind is largely a bundle of hacks and heuristics for this succinct-representation problem, cobbled together over a billion years of evolution. So if we could solve the general case—if knowing something was tantamount to knowing the shortest efficient description of it—then we would be almost like gods.
I read this and then immediately emailed one of my critique partners. “‘Gods,’ Elaine!” I shouted through email. “GODS!”
I’m not sure I did the problem justice myself, but I certainly enjoyed writing about it, so I have no regrets.
I should point out that the reference to Dr. Martinez mathematically composing a Mozart is in direct homage to how inspired I was by Aaronson (I read all of his writing on P vs. NP after finding that paper, including the post containing the quote at the beginning of this afterward). I could have chosen any artistic field for Martinez to claim access to, but Professor Aaronson’s Mozart comparison was one of the most thrilling metaphors I’ve ever come across when it comes to the P vs. NP problem. Thrilling and terrifying!
♦ ♦ ♦
I also must give tremendous thanks to Aaron Koch, Nidhal Bouaynaya, Roman Shterenberg, and Radu F. Babiceanu for writing a paper called, “An Encryption Algorithm Based on the Prime Roots of Unity” (IPCSIT vol. 31, 2012), in which they propose an alternate form of encryption to RSA that uses prime roots of unity. In other words, a method very like the theory attributed to Sonya Halliday in this book.
