Root of Unity (28 page)

“Okay,” Arthur said. “So, P problems you can solve quick, NP problems not so much. Yeah?”

“Well, so we thought,” I said. “We’ve never—
mathematics
has never,” I corrected, too loudly, “been able to prove, one way or another, whether P equals NP, or whether they aren’t equal, or whether it’s something that’s impossible to prove at all. It’s been one of the biggest unsolved problems in mathematics. Possibly
the
biggest—the question of whether anything we can quickly verify, we can also quickly solve.”

“Okay,” Arthur said again. “So?”

“So, most people figured P didn’t equal NP. We’d never been able to find a way to solve an NP-complete problem fast. Our whole understanding of the world…” I couldn’t explain.

“My proof threatened the economy,” Halliday managed hoarsely. “This proof, Rita’s proof—it could do so much more. It would revolutionize. Logistics, protein folding—everything would suddenly become easy. And encryption—” She made a choked sound. “A lot of encryption works because once you have the code, you have access. Which means once you have the answer…you can verify it, very fast.”

“And if P equals NP,
finding
that code is as easy as having it already and checking you’re right?” said Arthur. He let out a low whistle.

“It’s possible there’s a big enough constant in her reduction to prevent that, but the proof’s clearly constructive. She seems to have found an algorithm…” Halliday trailed off.

“Professor, even you aren’t getting this.” I spread my hands. “P equaling NP, it doesn’t just mean we can visit a bunch of cities quickly or break codes. It would mean
any
problem, any one we can put into numbers, would be near-instantly solvable. By anyone. We’re talking—we’re talking an overnight ballooning of technology into science fiction; we’re talking all of society going haywire, the basic functions of how we interact dissolving—”

“Implementation would still take some innovation; it wouldn’t quite happen overnight,” Halliday interjected, her voice firming up as she focused on the theory. “Even with a constructive proof, we’d have to translate the mathematics into programming. But, um—yes. Yes, I…I think you’re right.”

“Wait, you saying she’s right about society dissolving? From one math problem?” Arthur said. “How? Ain’t matter what Martinez found, the world’s the same place, right?”

“This one math problem rewrites our understanding of literally everything,” I said. “We can’t imagine what it might do. Everybody would suddenly be able to use a cheap desktop computer to find out—to find out
anything.
Science, medicine, economics, society—all the rules would get thrown out the window overnight, and when that happened…Arthur, I’m not exaggerating. Every piece of civilization might have to be reframed. Possibly rebuilt.”

“Rita thought so, too,” Halliday said. “Her note—it makes so much sense now.”

I’d forgotten about the note. Halliday took it out of her pocket, uncreased it in the circle of the penlight. “‘The world is dust,’” she read. “‘I made mathematics dust, your mathematics, our mathematics’—I thought she was referring to destroying my notes—”

“Keep reading,” I said.

“‘I cannot break the world. I cannot let you live in the world I see. It is too barren, too empty. No place for any mathematician. Particularly not for you, Sonya.’”

The words took on new meaning. “She wasn’t talking about the economy collapsing,” I breathed. “She was talking about just the
prospect
of knowing the reduction from NP to P, because—” My breath caught. I hadn’t realized. How had I not realized?

“What?” said Arthur.

“Her proof would make mathematicians obsolete,” I said. “Theorem-solving software—right now we can’t replicate the—the creative, the analytical leaps a human mathematician makes…” I was glad Checker wasn’t here at the moment. I would be too transparent in front of him. “But what we
can
do, if we put it in a proper logical language—”

“We can verify a proof is correct,” said Halliday. “We can do that already, Arthur. And if Rita’s proof checks out, if we can verify—”

“We can solve,” I said.

If we could understand, we could create.

Those mathematical leaps of intuition would no longer be mysterious. No longer be something unquantifiable and out of my reach. Because I wouldn’t need them anymore. Martinez’s proof might break the world, but it would also let me do math again.

Holy God. I had to find Martinez. We would find her, and I would make her tell me.

“She was trying to protect me,” said Halliday, still staring fixedly at the note. “Mathematics is…it’s everything to us. If a computer can replicate what we do, if there’s nothing special about human mathematical intuition…” I couldn’t see her face in the darkness. “She must have thought something in my own proof was getting close, that it was leading toward the breakthrough for hers. I—I think she overestimated me, as I don’t see how, but…”

“So let me get this straight,” said Arthur. “She works this out, then she suddenly cottons on to what it means, so she destroys it?”

“She thought she was saving the world,” Halliday said. “Maybe she was.”

“But what’s to stop someone else from coming along and finding out the same thing?” Arthur asked. “If it’s true, someone’s gotta find it eventually—”

“You don’t understand,” I said. I’d started to feel dizzy. “People have been trying to solve this problem forever. There’s a million-dollar prize for it, and that’s not even the reason everyone’s so obsessed. But nobody’s ever gotten close, and some mathematicians even started to suspect it couldn’t be solved at all. What Martinez came up with—it might well be hundreds of years before someone else thinks of the same breakthrough, if ever. Unless there really was something in your factoring proof,” I added to Halliday. “You two did work together; maybe something you used was the jumping-off point for her. It sounds like she was afraid you’d get there the same way.”

“I don’t know what she might have been thinking of,” Halliday answered haplessly. “Rita sometimes—she thought too well of me. She was the type of person who could make me feel slow. She would always expect I would make the leap with her, and I would have to ask her to go back, to explain—” She gave a humorless laugh. “I’m one of the top handful of people in my field, and she made me feel like a child sometimes. Often. It nearly gave me a complex.”

“It’s okay,” I said. “We won’t need to jump off your proof, because we’re not going to give up until we find her.” Figuring out the context of Martinez’s note had given me an idea. A brilliant idea. If there was one thing in the world that might be as important to Martinez as her proof, it was Sonya Halliday. “We know she cares for you, Professor, in her, uh, in her own way. We can use that to lure her out. We fake some trouble for you, make it seem like you need her.” I was gaining steam. This would work; I knew it would. “Or spread the word that you’re sick, or that you died, if you think she’d be sentimental enough to come out of hiding for that.”

“May God protect her,” murmured Halliday. “What will the NSA do to her, if they find out?”

“Sounds like we’re talking major national security stuff,” said Arthur. “Ain’t know what they would think.”

“Well,
we’ll
find her first,” I said. “And we can decide whether to hand her over or just make her cough up the proof. What she wrote to you, Professor—she was trying to save
you.
You’ll be our way in.”

“No.”

I turned sharply to Halliday. “What?”

She’d clicked off the light, leaving only her silhouette visible in the darkness. “I don’t want to go after her. Let her go.”

“
What?
What the fuck are you on?” I exploded, so harshly Arthur winced. “This is P versus NP! This is it! It’s everything! It’s—”

“And Rita made her decision.” Halliday took a breath. “It was clearly a decision she did not undertake lightly.”

“How can you let her—she screwed you over! She stole all your work. At the very least, don’t you still want it back?” If Halliday wouldn’t help—

“Well, she burned it, didn’t she?” Halliday’s voice had a caustic bite to it. “So whether or not we find her, it’s already gone. I could rewrite the factoring proof again now, thanks to you. The rest, her papers—those were hers to destroy.”

“And you’re just going to accept that?” I couldn’t believe what I was hearing. This had to be impulsiveness, the madness of a moment—Halliday would come around, she would see—“Your friend Dr. Zhang was right. This is too big to be left to one person’s whims, especially when that one person is a senile old lady!”

Halliday sniffed, hard, but when she spoke her voice was steady. “Rita was as sharp as ever. And she wanted to solve the P versus NP problem more than anything in her entire life. If the knowledge terrified her this much…” The dark shapes of her hands tightened on the papers she held, crumpling them. “I have to trust her.”

“No,” I said. “You really, really don’t.”

“Then let me rephrase. I’m choosing to trust her. If she says this broke the world for her, if she says she was only trying to protect me—”

“You’re not making any sense!” I cried. “If her proof was valid, then P equaling NP is true whether or not you’ve seen the reduction. Besides, how could you not want to know? Her emotional response is immaterial—for Christ’s sake, the Pythagoras cult thought irrational numbers were demonic and refused to accept them; that doesn’t mean—” I stuttered, out of words, out of ways to explain. “This isn’t
right,
what she’s doing. The world should know. Even if all you do is turn it over to the NSA, the knowledge should be out there. You can’t just delete it from the world; it’s wrong!”

But I didn’t care about the world, if I was honest with myself.

I needed this proof.

Without it, I was nothing. With it, with the algorithm Rita Martinez claimed she had…I could discover any result, make mathematics unfurl before me, answer the most profound questions in the universe.

With it, I could do real math.

And that was all that fucking mattered.

“Arthur, talk some sense into her,” I said, desperate and no longer caring. We’d already tried everything else to find Martinez. Now we’d caught on that the best and maybe the only thing we could use to lure her out was Halliday herself, and we knew what she
had,
and Halliday was saying
no?

“Dr. Martinez stole your work,” I added to the professor. “She stole your work, and it led to you getting kidnapped, and even then she said nothing. If she’s got a polynomial-time reduction, she might have halved the search time we took tracking you down—” We might have avoided the Feds entirely; I might have avoided getting almost killed—

“She didn’t know what you can do,” Halliday pointed out. “She didn’t know how you were searching.”

“Yeah, because a Hamiltonian cycle isn’t one of the most famous NP-complete problems of all time! She left you to die! If she’d rubbed two brain cells together, she would have known we were using some type of search algorithm and that her math could have
helped—”

“She probably didn’t think about it,” Halliday said. “Rita doesn’t…sometimes she doesn’t see the things in front of her. She’s too lost in the mathematics. I can’t fault her for that.”

“Or for ruining all your research?”

She turned away from me slightly. “She was trying to do what was best for me.”

“Arthur,” I said again, “You convince her. Convince her!”

He’d shoved his hands into his pockets. “This what you want?” he said to Halliday.

She nodded.

Arthur faced me. “I ain’t going to go against Sonya’s wishes here. Way I see it, Doc Martinez ain’t giving that proof to no one, so it ain’t like the world’s in any of that sort of danger. If Sonya wants to respect her choice, I’m on board.”

I was stunned. “And what if the Lancer finds Martinez? Is he just going to accept the fact that she doesn’t want to tell anyone about it?”

“Lot of things can go wrong in this world,” said Arthur. “Ain’t mean we can’t all make our own choices. Ain’t mean Dr. Martinez can’t make this one.” He looked down at Halliday. “Martinez wronged you, but I get why you’d forgive her. I ain’t got a beef with her beyond that.”

“You’re only saying that because you don’t understand what this means!” I accused Arthur. We couldn’t give this up. We couldn’t. I tried to temper my tone and played one last hole card. “Professor, if you’re so concerned about her, we
should
go after her. If we get to her first, then we can help her escape the Feds. She’s not going to know how to stay off the grid, but that’s one thing I’m exceptionally good at—I can help her.”

Halliday hesitated. Then she said, “No. Let it go. She’s smart, and she’s clearly figured out how not to be found. If we keep digging after her…no. Just let her go.”

No.
To find myself so close to salvation, and then to have it destroyed by people who didn’t understand…

Arthur turned toward me, his face unreadable. “Job’s over, Russell. Thank you. For your help.”

I half expected him to offer me money. I think I would have punched him if he had.

I wheeled around and stormed off, back from the lake, away from the safe house and back to my car. This job was over when I said it was fucking over. If Arthur didn’t want to help me find Martinez, if Halliday didn’t want to take advantage of her connection—well, screw them. I would do it myself.

I drove in the opposite direction I wanted to, switched cars, and made sure no one was following. Not that the NSA wouldn’t be able to pick me up again if they were interested, dammit. They knew where Checker lived, as much as Arthur had tried to keep the impression that he wasn’t on this case.

Of course, when I got to the Hole, Arthur was waiting for me. Goddammit.

“Russell,” he said.

“What?” I tried to brush past him, already checking my bug detector.

“Talked to Checker already.”

I stopped.

“Don’t pursue this. Let her go.”

“No,” I said.

“It’s the decent thing to do, Russell.”

“
Decent?”
I burst out. “Decent! She has a proof that could—” I clamped down on the words.