There was a pause. "No, it's us," said Malcolm. "Let us in." He did.

And then there they all were, Malcolm and Jude and Willem, as if they had come to see him perform a show. "Willem," he said. "You're supposed to be in Cappadocia."

"I just got back yesterday."

"But you're supposed to be gone until"-he knew this-"July sixth. That's when you said you'd be back."

"It's July seventh," Willem said, quietly.

He started to cry, then, but he was dehydrated and he didn't have any tears, just the sounds. July seventh: he had lost so many days. He couldn't remember anything.

"JB," said Jude, coming close to him, "we're going to get you out of this. Come with us. We're going to get you help."

"Okay," he said, still crying. "Okay, okay." He kept his blanket wrapped around him, he was so cold, but he allowed Malcolm to lead him to the sofa, and when Willem came over with a sweater, he held his arms up obediently, the way he had when he was a child and his mother had dressed him. "Where's Jackson?" he asked Willem.

"Jackson's not going to bother you," he heard Jude say, somewhere above him. "Don't worry, JB."

"Willem," he said, "when did you stop being my friend?"

"I've never stopped being your friend, JB," Willem said, and sat down next to him. "You know I love you."

He leaned against the sofa and closed his eyes; he could hear Jude and Malcolm talking to each other, quietly, and then Malcolm walking toward the other end of the apartment, where his bedroom was, and the plank of wood being lifted and then dropped back into place, and the flush of the toilet.

"We're ready," he heard Jude say, and he stood, and Willem stood with him, and Malcolm came over to him and put his arm across his back and they shuffled as a group toward the door, where he was gripped by a terror: if he went outside, he knew he would see Jackson, appearing as suddenly as he had that day in the cafe.

"I can't go," he said, stopping. "I don't want to go, don't make me go."

"JB," Willem began, and something about Willem's voice, about his very presence, made him in that moment irrationally furious, and he shook Malcolm's arm off of him and turned to face them, energy flooding his body. "You don't get a say in what I do, Willem," he said. "You're never here and you've never supported me and you never called me, and you don't get to come in making fun of me-poor, stupid, fucked-up JB, I'm Willem the Hero, I'm coming in to save the day-just because you want to, okay? So leave me the fuck alone."

"JB, I know you're upset," Willem said, "but no one's making fun of you, least of all me," but before he'd begun speaking, JB had seen Willem look over, quickly, and, it seemed, conspiratorially, at Jude, and for some reason this had made him even more livid. What had happened to the days when they all understood one another, when he and Willem had gone out every weekend, when they had returned the next day to share the night's stories with Malcolm and Jude, Jude who never went anywhere, who never shared stories of his own? How had it happened that he was the one who was all alone? Why had they left him for Jackson to pick over and destroy? Why hadn't they fought harder for him? Why had he ruined it all for himself? Why had they let him? He wanted to devastate them; he wanted them to feel as inhuman as he did.

"And you," he said, turning to Jude. "You like knowing how fucked up I am? You like always being the person who gets to learn everyone else's secrets, without ever telling us a single fucking thing? What do you think this is, Jude? You think you get to be a part of the club and you never have to say anything, you never have to tell us anything? Well, it doesn't fucking work like that, and we're all fucking sick of you."

"That's enough, JB," Willem said sharply, grabbing his shoulder, but he was strong suddenly, and he wrenched out of Willem's grasp, his feet unexpectedly nimble, dancing toward the bookcase like a boxer. He looked at Jude, who was standing in silence, his face very still and his eyes very large, almost as if he was waiting for him to continue, waiting for JB to hurt him further. The first time he had painted Jude's eyes, he had gone to a pet store to take photographs of a rough green snake because the colors were so similar. But in that moment they were darker, almost like a grass snake's, and he wished, ridiculously, that he had his paints, because he knew that if he had them, he'd be able to get the shade exactly right without even having to try.

"It doesn't work like that," he said to Jude again. And then, before he knew it, he was doing Jackson's imitation of Jude, the hideous parody, his mouth open as Jackson had done it, making an imbecile's moan, dragging his right leg behind him as if it were made of stone. "I'm Jude," he slurred. "I'm Jude St. Francis." For a few seconds, his was the only voice in the room, his movements the only movements, and in those seconds, he wanted to stop, but he couldn't stop. And then Willem had run at him, and the last thing he had seen was Willem drawing his fist back, and the last thing he had heard was the cracking of bone.

He woke and didn't know where he was. It was difficult to breathe. Something was on his nose, he realized. But when he tried to lift his hand to feel what it was, he couldn't. And then he had looked down and seen that his wrists were in restraints, and he knew he was in the hospital. He closed his eyes and remembered: Willem had hit him. Then he remembered why, and he shut his eyes very tightly, howling but not making a noise.

The moment passed and he opened his eyes again. He turned his head to the left, where an ugly blue curtain blocked his view of the door. And then he turned his head to the right, toward the early-morning light, and saw Jude, asleep in the chair next to his bed. The chair was too small for him to sleep in, and he had folded himself into a terrible-looking position: his knees drawn up to his chest, his cheek resting atop them, his arms wrapped around his calves.

You know you shouldn't sleep like that, Jude, he told him in his head. Your back is going to hurt when you wake up. But even if he could have reached his arm over to wake him, he wouldn't have.

Oh god, he thought. Oh god. What have I done?

I'm sorry, Jude, he said in his head, and this time he was able to cry properly, the tears running into his mouth, the mucus that he was unable to clean away bubbling over as well. But he was silent; he didn't make any noise. I'm sorry, Jude, I'm so sorry, he repeated to himself, and then he whispered the words aloud, but quietly, so quietly that he could hear only his lips opening and closing, nothing more. Forgive me, Jude. Forgive me.

Forgive me.

Forgive me.

Forgive me.

[ IV ].

The Axiom of Equality.

1.

THE NIGHT BEFORE he leaves for Boston for their friend Lionel's wedding, he gets a message from Dr. Li telling him that Dr. Kashen has died. "It was a heart attack; very fast," Dr. Li writes. The funeral is Friday afternoon.

The next morning he drives directly to the cemetery, and from the cemetery to Dr. Kashen's house, a two-story wooden structure in Newton where the professor used to host a year-end dinner for all of his current graduate students. It was understood that you weren't to discuss math at these parties. "You can talk about anything else," he'd tell them. "But we're not talking about math." Only at Dr. Kashen's parties would he be the least socially inept person in the room (he was also, not coincidentally, the least brilliant), and the professor would always make him start the conversation. "So, Jude," he'd say. "What are you interested in these days?" At least two of his fellow graduate students-both of them PhD candidates-had mild forms of autism, and he could see how hard they worked at making conversation, how hard they worked at their table manners, and prior to these dinners, he did some research into what was new in the worlds of online gaming (which one of them loved) and tennis (which the other loved), so he'd be able to ask them questions they could answer. Dr. Kashen wanted his students to someday be able to find jobs, and along with teaching them math, he also thought it his responsibility to socialize them, to teach them how to behave among others.

Sometimes Dr. Kashen's son, Leo, who was five or six years older than he, would be at dinner at well. He too had autism, but unlike Donald's and Mikhail's, his was instantly noticeable, and severe enough so that although he'd completed high school, he hadn't been able to attend more than a semester of college, and had only been able to get a job as a programmer for the phone company, where he sat in a small room day after day fixing screen after screen of code. He was Dr. Kashen's only child, and he still lived at home, along with Dr. Kashen's sister, who had moved in after his wife had died, years ago.

At the house, he speaks to Leo, who seems glazed, and mumbles, looking away from him as he does, and then to Dr. Kashen's sister, who was a math professor at Northeastern.

"Jude," she says, "it's lovely to see you. Thank you for coming." She holds his hand. "My brother always talked about you, you know."

"He was a wonderful teacher," he tells her. "He gave me so much. I'm so sorry."

"Yes," she says. "It was very sudden. And poor Leo"-they look at Leo, who is gazing at nothing-"I don't know how he's going to deal with this." She kisses him goodbye. "Thank you again."

Outside, it is fiercely cold, and the windshield is sticky with ice. He drives slowly to Harold and Julia's, letting himself in and calling their names.

"And here he is!" says Harold, materializing from the kitchen, wiping his hands on a dish towel. Harold hugs him, which he had begun doing at some point, and as uncomfortable as it makes him, he thinks it'll be more uncomfortable to try to explain why he'd like Harold to stop. "I'm so sorry about Kashen, Jude. I was shocked to hear it-I ran into him on the courts about two months back and he looked like he was in great shape."

"He was," he says, unwinding his scarf, as Harold takes his coat. "And not that old, either: seventy-four."

"Jesus," says Harold, who has just turned sixty-five. "There's a cheery thought. Go put your stuff in your room and come into the kitchen. Julia's tied up in a meeting but she'll be home in an hour or so."

He drops his bag in the guest room-"Jude's room," Harold and Julia call it; "your room"-and changes out of his suit and heads toward the kitchen, where Harold is peering into a pot on the stove, as if down a well. "I'm trying to make a bolognese," he says, without turning around, "but something's happening; it keeps separating, see?"

He looks. "How much olive oil did you use?"

"A lot."

"What's a lot?"

"A lot. Too much, obviously."

He smiles. "I'll fix it."

"Thank god," says Harold, stepping away from the stove. "I was hoping you'd say that."

Over dinner, they speak of Julia's favorite researcher, who she thinks might be trying to jump to another lab, and of the latest gossip circulating through the law school, and of the anthology of essays about Brown versus Board of Education that Harold is editing, and of one of Laurence's twin daughters, who is getting married, and then Harold says, grinning, "So, Jude, the big birthday's coming up."

"Three months away!" Julia chirps, and he groans. "What are you going to do?"

"Probably nothing," he says. He hasn't planned anything, and he has forbidden Willem from planning anything, either. Two years ago, he threw Willem a big party for his fortieth at Greene Street, and although the four of them had always said they'd go somewhere for each of their fortieth birthdays, it hasn't worked out that way. Willem had been in L.A. filming on his actual birthday, but after he had finished, they'd gone to Botswana on a safari. But it had been just the two of them: Malcolm had been working on a project in Beijing, and JB-well, Willem hadn't mentioned inviting JB, and he hadn't, either.

"You have to do something," says Harold. "We could have a dinner for you here, or in the city."

He smiles but shakes his head. "Forty's forty," he says. "It's just another year." As a child, though, he never thought he'd make it to forty: in the months after the injury, he would sometimes have dreams of himself as an adult, and although the dreams were very vague-he was never quite certain where he was living or what he was doing, though in those dreams he was usually walking, sometimes running-he was always young in them; his imagination refused to let him advance into middle age.

To change the subject, he tells them about Dr. Kashen's funeral, where Dr. Li gave a eulogy. "People who don't love math always accuse mathematicians of trying to make math complicated," Dr. Li had said. "But anyone who does love math knows it's really the opposite: math rewards simplicity, and mathematicians value it above all else. So it's no surprise that Walter's favorite axiom was also the most simple in the realm of mathematics: the axiom of the empty set.

"The axiom of the empty set is the axiom of zero. It states that there must be a concept of nothingness, that there must be the concept of zero: zero value, zero items. Math assumes there's a concept of nothingness, but is it proven? No. But it must exist.

"And if we are being philosophical-which we today are-we can say that life itself is the axiom of the empty set. It begins in zero and ends in zero. We know that both states exist, but we will not be conscious of either experience: they are states that are necessary parts of life, even as they cannot be experienced as life. We assume the concept of nothingness, but we cannot prove it. But it must exist. So I prefer to think that Walter has not died but has instead proven for himself the axiom of the empty set, that he has proven the concept of zero. I know nothing else would have made him happier. An elegant mind wants elegant endings, and Walter had the most elegant mind. So I wish him goodbye; I wish him the answer to the axiom he so loved."

They are all quiet for a while, contemplating this. "Please tell me that isn't your favorite axiom," Harold says suddenly, and he laughs. "No," he says. "It's not."

He sleeps in the next day, and that night he goes to the wedding, where because both of the grooms lived in Hood, he knows almost everyone. The non-Hood guests-Lionel's colleagues from Wellesley, and Sinclair's from Harvard, where he teaches European history-stand near one another as if for protection, looking bored and bemused. The wedding is loose-limbed and slightly chaotic-Lionel starts assigning his guests tasks as soon as they arrive, which most of them neglect: he is supposed to be making sure everyone signs the guest book; Willem is supposed to be helping people find their tables-and people walk around saying how, thanks to Lionel and Sinclair, thanks to this wedding, they won't have to go to their twentieth reunion this year. They are all here: Willem and his girlfriend, Robin; Malcolm and Sophie; and JB and his new boyfriend, whom he hasn't met, and he knows, even before checking their place cards, that they will all be assigned to the same table. "Jude!" people he hasn't seen in years say to him. "How are you? Where's JB? I just spoke to Willem! I just saw Malcolm!" And then, "Are you four all still as close as you were?"

"We all still talk," he says, "and they're doing great," which is the answer he and Willem had decided they'd give. He wonders what JB is saying, whether he is skimming over the truth, as he and Willem are, or whether he is lying outright, or whether, in a fit of JBish forthrightness, he is telling the truth: "No. We hardly ever speak anymore. I only really talk to Malcolm these days."

He hasn't seen JB in months and months. He hears of him, of course: through Malcolm, through Richard, through Black Henry Young. But he doesn't see him any longer, because even nearly three years later, he is unable to forgive him. He has tried and tried. He knows how intractable, how mean, how uncharitable he is being. But he can't. When he sees JB, he sees him doing his imitation of him, sees him confirming in that moment everything he has feared and thought he looks like, everything he has feared and thought other people think about him. But he had never thought his friends saw him like that; or at least, he never thought they would tell him. The accuracy of the imitation tears at him, but the fact that it was JB doing it devastates him. Late at night, when he can't sleep, the image he sometimes sees is JB dragging himself in a half-moon, his mouth agape and drooling, his hands held before him in claws: I'm Jude. I'm Jude St. Francis.

That night, after they had taken JB to the hospital and admitted him-JB had been stuporous and dribbling when they took him in, but then had recovered and become angry, violent, screaming wordlessly at them all, thrashing against the orderlies, wresting his body out of their arms until they had sedated him and dragged him, lolling, down the hallway-Malcolm had left in one taxi and he and Willem had gone home to Perry Street in another.

He hadn't been able to look at Willem in the cab, and without anything to distract him-no forms to fill out, no doctors to talk to-he had felt himself grow cold despite the hot, muggy night, and his hands begin to shake, and Willem had reached over and taken his right hand and held it in his left for the rest of the long, silent ride downtown.

He was there for JB's recovery. He decided he'd stay until he got better; he couldn't abandon JB then, not after all their time together. The three of them took shifts, and after work he'd sit by JB's hospital bed and read. Sometimes JB was awake, but most of the time he wasn't. He was detoxing, but the doctor had also discovered that JB had a kidney infection, and so he stayed on in the hospital's main ward, liquids dripping into his arm, his face slowly losing its bloat. When he was awake, JB would beg him for forgiveness, sometimes dramatically and pleadingly, and sometimes-when he was more lucid-quietly. These were the conversations he found most difficult.

"Jude, I'm so sorry," he'd say. "I was so messed up. Please tell me you forgive me. I was so awful. I love you, you know that. I would never want to hurt you, never."

"I know you were messed up, JB," he'd say. "I know."

"Then tell me you forgive me. Please, Jude."

He'd be silent. "It's going to be okay, JB," he'd say, but he couldn't make the words-I forgive you-leave his mouth. At night, alone, he would say them again and again: I forgive you, I forgive you. It would be so simple, he'd admonish himself. It would make JB feel better. Say it, he'd command himself as JB looked at him, the whites of his eyes smeary and yellowed. Say it. But he couldn't. He knew he was making JB feel worse; he knew it and was still unable to say it. The words were stones, held just under his tongue. He couldn't release them, he just couldn't.

Later, when JB called him nightly from rehab, strident and pedantic, he'd sat silently through his monologues on what a better person he'd become, and how he had realized he had no one to depend on but himself, and how he, Jude, needed to realize that there was more in life than just work, and to live every day in the moment and learn to love himself. He listened and breathed and said nothing. And then JB had come home and had had to readjust, and none of them heard very much from him at all for a few months. He had lost the lease on his apartment, and had moved back in with his mother while he reestablished his life.

But then one day he had called. It had been early February, almost seven months exactly after they had taken him to the hospital, and JB wanted to see him and talk. He suggested JB meet him at a cafe called Clementine that was near Willem's building, and as he inched his way past the tightly spaced tables to a seat against the back wall, he realized why he had chosen this place: because it was too small, and too cramped, for JB to do his impression of him, and recognizing that, he felt foolish and cowardly.

He hadn't seen JB in a long time, and JB leaned over the table and hugged him, lightly, carefully, before sitting down.

"You look great," he said.

"Thanks," said JB. "So do you."

For twenty minutes or so, they discussed JB's life: he had joined Crystal Meth Anonymous. He was going to live with his mother for another few months or so, and then decide what to do next. He was working again, on the same series he'd been working on before he went away.

"That's great, JB," he'd said. "I'm proud of you."

And then there was a silence, and they both stared at other people. A few tables away from him was a girl wearing a long gold necklace she kept winding and unwinding around her fingers. He watched her talk to her friend, wrapping and unwrapping her necklace, until she looked up at him and he looked away.

"Jude," JB began, "I wanted to tell you-completely sober-that I'm so sorry. It was horrible. It was-" He shook his head. "It was so cruel. I can't-" He stopped again, and there was a silence. "I'm sorry," he said. "I'm sorry."

"I know you are, JB," he said, and he felt a sort of sadness he'd never felt before. Other people had been cruel to him, had made him feel awful, but they hadn't been people he loved, they hadn't been people he had always hoped saw him as someone whole and undamaged. JB had been the first.

And yet JB had also been one of the first to be his friend. When he'd had the episode in college that had made his roommates take him to the hospital where he had met Andy, it had been JB, Andy later told him, who had carried him in, and JB who had demanded that he be seen first, who had made such an upset in the ER that he had been ejected-but not before a doctor had been summoned.

He could see JB's love for him in his paintings of him. He remembered one summer in Truro, watching JB sketch, and he had known from the expression on JB's face, his little smile, and the lingering, delicate way his large forearm moved over the page, that he was drawing something he treasured, something that was dear to him. "What're you drawing?" he'd asked, and JB had turned to him, and held up the notepad, and he had seen it was a picture of him, of his face.

Oh, JB, he thought. Oh, I will miss you.

"Can you forgive me, Jude?" JB asked, and looked at him.

He didn't have words, he could only shake his head. "I can't, JB," he said, finally. "I can't. I can't look at you without seeing-" He stopped. "I can't," he repeated. "I'm sorry, JB, I'm so sorry."

"Oh," said JB, and he swallowed. They sat there for a long time, not saying anything.

"I'll always want wonderful things for you," he said to JB, who nodded, slowly, not looking at him.

"Well," JB said, finally, and stood, and he stood as well, and held his hand out to JB, who looked at it as if it were something alien, something he'd never seen before, examining it, squinting at it. And then at last he took it, but instead of shaking it, he lowered his lips to it and held them there. And then JB returned his hand to him and bumbled, nearly ran, out of the cafe, bumping against the little tables-"Sorry, sorry"-as he went.

He still sees JB now and then, mostly at parties, always in groups, and the two of them are polite and cordial with each other. They make small talk, which is the most painful thing. JB has never tried to hug or kiss him again; he comes over to him with his hand already outstretched, and he takes it, and they shake. He sent JB flowers-but with only the briefest of notes-when "Seconds, Minutes, Hours, Days" opened, and although he skipped the opening, he had gone to the gallery the following Saturday, on his way up to work, where he had spent an hour moving slowly from one painting to the next. JB had planned on including himself in this series, but in the end he hadn't: there was just him, and Malcolm, and Willem. The paintings were beautiful, and as he looked at each, he thought not so much of the lives depicted in them, as of the life who created them-so many of these paintings were done when JB was at his most miserable, his most helpless, and yet they were self-assured, and subtle, and to see them was to imagine the empathy and tenderness and grace of the person who made them.

Malcolm has remained friends with JB, although he felt the need to apologize to him for this fact. "Oh no, Malcolm," he'd said, once Malcolm had confessed, asking him for his permission. "You should absolutely still be friends with him." He doesn't want JB to be abandoned by them all; he doesn't want Malcolm to feel he has to prove his loyalty to him by disavowing JB. He wants JB to have a friend who's known him since he was eighteen, since he was the funniest, brightest person in the school, and he and everyone else knew it.

But Willem has never spoken to JB again. Once JB returned from rehab, he called JB and said that he couldn't be friends with him any longer, and that JB knew why. And that had been the end. He had been surprised by this, and saddened, because he had always loved watching JB and Willem laugh together, and spar with each other, and loved having them tell him about their lives: they were both so fearless, so bold; they were his emissaries to a less inhibited, more joyful world. They had always known how to take pleasure from everything, and he had always admired that in them, and had been grateful that they had been willing to share it with him.

"You know, Willem," he said once, "I hope the reason you're not talking to JB isn't because of what happened with me."

"Of course it's because of what happened with you," Willem had said.

"But that's not a reason," he'd said.

"Of course it is," Willem had said. "There's no better reason than that."

He had never done it before, and so he had no real understanding of how slow, and sad, and difficult it was to end a friendship. Richard knows that he and JB and Willem and JB don't talk any longer, but he doesn't know why-or at least not from him. Now, years later, he no longer even blames JB; he simply cannot forget. He finds that some small but unignorable part of him is always wondering if JB will do it again; he finds he is scared of being left alone with him.

Two years ago, the first year JB didn't come up to Truro, Harold asked him if anything was the matter. "You never talk about him anymore," he said.

"Well," he began, not knowing how to continue. "We're not really-we're not really friends any longer, Harold."