Proof by Induction

Paulie rushes out the elevator doors the moment they part, only to skid to a halt at the sight of his father’s wife. She shakes her head, but he doesn’t need the confirmation. If Tricia is out here and not in the hospital room with his father, it can only mean he has passed. He numbly accepts a hug from her.

When she releases him, a woman in a tweed jacket clears her throat. “Mr. Gifford, we are all very sorry for your loss.”

“Thank you,” he replies automatically, focusing on her crucifix. He swallows. “This is probably a dumb question, but what happens now?”

The chaplain draws herself up. “Now we all go back to the room where your father passed, unless of course you prefer not to.” She begins walking as she talks. “You can enter into his Coda and say any goodbyes you’d like to say, or ask him any questions you have about his end.”

Paulie follows her, wondering dimly if there will be fallout from the meeting he had to cancel with Professor Tappert. Paulie’s father was a professor emeritus at his same university, so certainly they should be sympathetic. He doesn’t kid himself about how this meeting was going to go, however. Tappert is on his P&T committee, and with his scant publication record and mediocre yearly reviews, his tenure prospects were already dim. They’re even dimmer now.

Inside the hospital room, Paulie stares. He isn’t sure what he expected, but he almost believes his father could open his eyes at any moment—except for the endotracheal tube stuck in his mouth. He’s never been this close to a dead body before. Is he supposed to touch it or not? Paulie puts a hand on his shoulder; it feels like his father.

He grips the bed rail.

The chaplain gestures toward Tricia. “Mrs. Gifford elected not to enter his Coda. If you would like to, you can see him there.”

Paulie eyes the console and cables behind the bed. “Is it really him?”

“Yes and no. The human mind remains aware of stimulus for up to five minutes after what we consider to be the moment of death. The Coda does for his consciousness what the rest of his telemetry does for his vital signs—takes a snapshot that we can look at later. The Coda allows you to interact with a simulacrum of your father, with his memories and personality at the end of his life.” She gesticulates awkwardly, as though the topic is distasteful. “He can tell you if he had a life insurance policy, where the will is, things like that. The Coda cannot change in the way that a person can, however; it cannot learn or grow.” Her eyes meet Paulie’s. “Your father’s soul is not in there. Your father has moved on.”

It was early morning when Paulie put the headset on, but predawn when he blinked into the virtual environment. He had only left the hospital to go home and get some sleep about five hours before the end. Now he could almost believe he had turned back around and found his father waiting here, as though the 5 AM phone call from Tricia were just a dream.

Gone was the endotracheal tube. The room was eerily silent, with none of the sounds he’d associated with the hospital from his visits over the past week.

He met his father’s eyes. “Hey.”

His father smiled ruefully. “Hey.”

“Are you—”

“Dead?” His father gestured toward the inactive monitors. “Apparently so.”

“Does it hurt?” Are you afraid, he wanted to ask, but he knew better than to talk to his father about emotions.

“Nothing hurts,” he said, picking at a scab on his leg. “I guess they have a way of turning that off.”

“Did the doctors mess up? Should I ask for an autopsy?”

His father shook his head. “Nah. I’m seventy-one, diabetic, and with a bad heart. You’re not going to win any lawsuits here.”

It occurred to Paulie that Codas could be programmed to give whatever answer benefitted the hospital.

Paulie stared out the window, over the parking lot, to the eerily empty expressway. “I really believed we were close on that Perelman proof.”

“Maybe nobody’s meant to find it.”

Easy for him to say. He’d already been beyond questions of tenure and publication; now all of that was even more meaningless for him. For Paulie, though, Perelman would have been the home run his tenure dossier needed.

He turned back toward the bed. “Okay. Well.” He put a hand on the chair he’d sat in last night while his father complained about his breathing. He should say something. Something like I love you¸ he supposed. But his father had never gone in for the mushy stuff in life, so why start now?

“Goodbye, then,” he finished instead.

“Bye, Paulie,” said his father. “Thank you for visiting.”

Thank you for visiting. The same as he’d taken to saying every time Paulie came to him since his health began to decline last year. Paulie waited, hoping this time his father would say something more, until the moment dragged on awkwardly, and then he pulled the interface off his head.

“What happens to his Coda when we leave?” he asks, leaning against a counter.

The chaplain sighs. “The equipment will be cleaned and reused, except for the actual leads that connected to his scalp, which are disposed of.”

“I don’t mean the equipment.”

“No,” she agrees. After a moment she continues. “The simulacrum itself will be digitally compressed and sent to a data storage facility.”

“Will he be… awake?”

“He’s not actually conscious now, so no, he will not be conscious in storage.”

“Okay, well I suppose that’s…”

“Mr. Gifford?”

Paulie lets his vision rest on the blinds, absent-mindedly counting. Three straight blinds. Two twisted. Five straight. The rest in a mass, discrete, but not countable from here. Three two and five. Prime numbers. Two that add to the third.

“Can he think creatively? In the, uh, simulation, I mean. Can he do math? Can he have insights?”

“Again, that’s not your father in there. That’s a slice—”

“Yes, I know, a snapshot of who he was in his last moments. Last night when I was here he was arguing with the nurse about whether or not he should have to wear that oxygen mask. He was capable of thinking critically right up until the end.”

The chaplain winces. “I hate to remind you, but he was mistaken.”

Paulie nods. “He was no doctor, but he was a mathematician. Can his Coda still think mathematically?”

“I suppose, Mr. Gifford. I’m no scientist.”

Paulie pushes off from the counter. “I’d like to take him with me. That should be possible, right?”

She bites her lip. “This hospital is affiliated with the Presbyterian Church. While we are not opposed to the Coda on a theological basis, obviously, our ethics committee has concerns when it comes to the appearance of attributing personhood to what should be a temporary means of gathering information and comfort.”

Paulie crosses his arms. “If it’s not a person, then it’s data. I’m next of kin, so the data should be my property.”

“Technically his wife is next of kin.” She holds up a hand at Paulie’s intake of breath. “It is possible to take ownership of the simulacrum, with proper paperwork, if his wife agrees. You would be billed for the computer and interface, and insurance will not cover the expense. But Mr. Gifford, I don’t recommend it. The healthiest thing you can do is move on. Let go.”

He meets her gaze. “Thanks for the advice, but my mind’s made up.”

Gina wraps him in a hug when she comes home from work. “I’m so sorry,” she murmurs. “I assume you told Maddie.”


“How did she take the news?”

He thinks back to his daughter’s return from school. How much harder she took the loss than he, even though he’s the one who lost a father. “Not well. She’s up in her room.”

Gina eyes the computer console on the coffee table. “What’s that?”

“The hospital let me take his Coda.”

“You mean—is he in there?”

“Kind of. Not really.”

She shudders. “Wow. Okay. If this helps your grieving process, then I’m all for it.”

“It’s not about grieving.”

“What, then?”

“The Perelman Hypothesis.”

She frowns. “I thought you’d given up on that when your father retired.”

“He only retired from lecturing. From office hours and meetings and committees and grantsmanship. You never retire from thinking. We were working on it together. It was going to be his last big result.”

“Paulie, people have been trying to prove that conjecture for ninety years. Whoever finally does will be some grad student in their twenties, using techniques that don’t exist yet.”

“We were close, Gina. I know it.”

She meets his eye, and holds the glance a long time before replying. “And you think you’re going to accomplish this by spending time inside a computer with your father.”

He winces at the inaccuracies, but he doesn’t correct her. “I think so,” he says instead.

“Okay, Paulie,” she says, though she shakes her head. “But do me a favor. Keep it in the den, okay? I don’t want Maddie anywhere near it. I don’t want her confused about whether Grandpa’s really gone or not. Just let her grieve.”

The hospital room was dark once again in the simulacrum.

“Hey. Thank you for visiting.”

He nodded at his father. “Do you remember me, uh, visiting you here before?”

His father seemed puzzled. “You mean last night? Yes.”

“No, I mean here in… in this thing. In your Coda.”

“The last thing I remember is not being able to breathe, and my chest hurting like a motherfucker, and then I was sitting up with all the cables and hoses off, and you walked in.”

“Do you understand that you’re dead?”

His father nodded. “Either that or I’m suddenly cured.”

“What’s the square root of i?”

Paulie’s father stared. “What?”

“The square root of i. In any form you like.”

“Paulie, why?”

“I’m trying to see if it’s really—” Paulie turned away, his fists clenched. “They say this simulacrum knows everything you knew at the last moment. This is something you could have done in your head.”

“Okay, Paulie. One over root two plus i over root two. And its negation. Or would you prefer the answer in polar form?”

Paulie breathed a sigh of relief. “Okay, so I’ve been working on Perelman. Help me find something to write with.” He started digging in drawers, but all of them were empty.

“Are you serious?”

He looked at his father. “Don’t you want this?”


“We could still have that breakthrough. One last result to rock the mathematical world. Make everybody learn your name.”

His father smiled faintly. “Your name, too.”

Paulie put a hand on the bed. “Your legacy. My career. There’s something for both of us here. Do you have anything better to do?”

“I guess I really don’t.”

He returned to searching the room, but every compartment was empty. Nothing existed in this simulation except what could be seen on the surface. Finally he hit upon the dry-erase board the shift nurses wrote their names on. He pulled a cap off a marker and tested it, half expecting it not to work as in the real world. To his relief, it left a clear line on the board.

“That’s not a lot of space,” said his father.

“No,” he agreed. “I can’t bring anything in with me or take anything out, though. Whatever we come up with has to be in small enough chunks for me to remember and replicate in the—replicate outside. So it’s just as well.”

“Okay, show me what you have.”

Paulie started filling the little board with equations. “We know how to generate particular examples—”

“Trivial solutions,” his father interrupted. “Perelman referenced a dozen himself, in his publication. We can’t enumerate an exhaustive set, though.”

Paulie nodded. “Right. Now, before you went into the hospital the first time, we had taken the approach of looking for a relationship between the cardinality of the Ricci set and the number of solutions it generates. We started by considering finite sets.”

His father rubbed his forehead. “I vaguely remember, but this was right before things went downhill.”

“That’s fine—I’ve been working on that without you, so we don’t have to repeat it, we only need to figure out the next steps. I’ve been approaching it as a series, trying to tie the value not merely to cardinality, but to its h-value. This feels right to me.”

His father perked up at that. “Not an equation,” he said. “A series.”

“Right. Call it H and see what it converges to as n approaches infinity.”

Gradually the board filled with arrows and sigmas and integrals.

“I wish we had a bigger board,” Paulie said.

“Write on the wall. What are they gonna do, yell at us?”

Paulie stared. “Goddamn that’s brilliant.”

After another hour or so they hit a dead end.

“If we had a generalized solution for hyperbolic equations,” Paulie’s father began.

“We don’t, though.”

“No, but look up Brumbaugh Manifolds. Doug Brumbaugh was working on this the last time I saw him. He may have made some progress.”

“Okay, that’s something to try. I won’t be able to hold much more in my head anyway.”

“I bet if you talk to the company that makes this, they can find a way for you to email yourself from inside or something.”

“No way,” Paulie said. “I don’t want anyone to know what we’re working on here. I don’t want someone to go find every mathematician who’s died in the last five years and hook all their Codas up in some kind of screwed up massively parallel computer and beat us to the punch.”

His father’s eyes widened. “Shit.”

“Yeah. Only a matter of time before somebody else thinks of it, though.”

“So you might as well be first?”

Paulie chewed his lip. “Do you not want to do this? Do you think this is wrong?”

He grinned ruefully. “What do I know from wrong?”

Paulie dropped into the bedside chair. “What’s it like?”


“Being dead but being conscious. Does it make you upset?”

His father shrugged. “It is what it is.”

“You had plans,” Paulie said. “You were going to remodel the house.”

“Guess now I’m not.”

Paulie gripped the bed’s footboard. “Don’t you feel anything at all?” He couldn’t remember if his father had ever had a feeling in his damned life.

“Would it change anything?”

Paulie flips through images on a tablet in the mortuary office. “Somebody told me you had an option to put a Coda interface in the niche with his ashes, but I don’t see that here.”

Next to him, Tricia winces, but she schools it quickly.

“We don’t include Coda ports in the regular lineup,” the funeral director says, “but yes, it is a choice we offer. This is not a service that has caught on yet. Many people find the idea disturbing, as though we are preventing our loved ones from moving on. Or preventing ourselves from moving on. If you elect to equip the niche with an interface, you will have to choose the special columbarium we have set aside for that. It’s, ah, not near the other niches.”

Paulie glances at Tricia, but apart from insisting on a fancier urn for her husband, she’s let him make all the decisions.

“Do it,” he says.

At the cemetery Paulie kisses the urn, and Tricia does the same. Then he watches as an employee places it into the columbarium and closes the marble cover.

A minister selected by her side of the family drones on. As far as Paulie remembers, his father wasn’t religious, but this isn’t for his benefit, after all.

On the way to the car he grabs Maddie and pulls her into a tight hug. “You know I love you, right?”

She sobs and nods against him.

“You know I’m proud of you, right?”

“Paulie,” Gina says, “you’re upsetting her.”

“I just want to make sure she knows.”


His eyes adjusted quickly to the dark. “Hey.”

His father gestured at the silent equipment by the bed. “Guess this is the end. I had an insurance policy. There isn’t much, but it should pay for a cremation. Tricia should be able to find the paperwork. You’re the beneficiary.”

“Yeah, we took care of all that.”

“Oh. How long have I been gone?”

He stepped over to the dry-erase board. “About three weeks.”

“Then… what are you still doing here?”

“We’ve been working on the Perelman Hypothesis.

“Are you serious?”

Paulie uncapped a marker. “Don’t make me go through it all again. It’s fifty degrees out, we only have so much time, and I need to walk you through what we came up with last time. Trust me, you’re on board.”

His father blinked. “Okay then. Go ahead.”

The clock on the wall ticked off seconds, while the hour and minute hand relentlessly pointed to eight minutes after five the entire time it took Paulie to run through the connection to hyperbolic equations.

“I reached out to Professor Brumbaugh like you said, but he pointed me to the Jagadish-Rajput conjecture.”

“I haven’t heard of that. Are they working on Perelman also?”

“No, they’re working on node forms, but their conjecture is that hyperbolic equations correspond to node forms. They’ve tested several hundred terms using a supercomputer and they’ve all checked out.”

His father shook his head. “How’s that help us?”

“Node forms converge. Supposing we can prove their conjecture, we can use that to prove Perelman.”

“This isn’t math. This is grasping at straws. A supercomputer says it works—so what? That’s not theory. Where’s the proof?”

Paulie capped the marker, even though he suspected it could not dry out. “Don’t you see? If the correspondence holds, then—”

“Are you trying to give me a heart attack in the afterlife? Do Jagadish and Rajput have the basis for a theorem, or just a coincidence they can’t explain? Even Euler had conjectures disproven after three hundred years!”

“Well fine then—” Paulie lowered his voice. “Fine. Help me find a counterexample, then. Or better yet, help me prove Jagadish-Rajput true, because that proof will make us both famous.”

His father crossed his arms. “Fine. This conjecture is bound to have consequences for other node forms. Maybe a proof by contradiction is our angle.”

Paulie and his father toyed with a variety of extrapolations, looking for a counterexample. At least the false starts could be erased—and Paulie wouldn’t need to remember any of them when he got out of the Coda. All he’d need to remember would be a working approach, if they found one.

“The department voted on my tenure application this week,” he said during a break. “They voted to advance it to the dean.” Paulie suspected strongly the vote was not unanimous, which boded poorly for the next level of the process, but he kept that part to himself.


Huh? That was it?

“You could congratulate me. You could wish me luck.”

“Okay. Good luck.”

“Thanks,” Paulie muttered. He added a few more lines to the board. “Maddie has a dance recital next week. She misses you a lot.”

“Wish her luck too, then.”

“It just…it reminds me of my piano recitals.”

His father leaned on his bed railing. “Is that what this is really about, Paulie? Are you here to tell me I was a shitty father? I know. I already acknowledged that, after the divorce.”

Paulie dropped into the chair by the bed. “No,” he said at last. “Sorry. I keep thinking of what other people use the Coda technology for, and I keep waiting to hear you talk about something besides math or life insurance. I keep hoping you’ll have something profound to say.”

“I’m not the mushy type.”

“You could fake it.”

“You’re the smartest person I ever met. You would see through any faking.”

Paulie blinked. A compliment.

“I wouldn’t have blamed you if you didn’t want anything to do with me,” his father went on, “after not being there for you as a kid. But then you made me a part of your life and we got along okay. You treated me like a colleague, so I tried to treat you the same. Now you’re mad at me for not acting more like a father? I didn’t think you wanted that from me.”

Paulie waited to see if he would say anything else. That was about as close to “mushy” as he’d come since the night twenty years ago when he’d apologized for abandoning him.

After a quiet eternity, he got up from the chair. “Okay, well, I think I have enough to work on for now. I’ll come back when I have some progress.”

“Bye, Paulie. Thank you for visiting.”

“Jesus, Paulie, I don’t mind driving home, but if you puke in the car, you’re cleaning it up.”

Paulie clinks his empty wine glass against Gina’s still-full one. “The free wine is the only thing that makes these parties worth attending.”

She rolls her eyes. “Our holiday party’s at the Olive Garden. You should appreciate what you’ve got.”

He smiles. “I think that’s what I just said.”

“Just pace yourself, okay?”

“It’s a deal.”

She gestures toward the food table. “I’m gonna get some crudités. You should get some food in you too.”

“I will.”

As she walks away, his phone buzzes. Paulie takes another glass of wine from a server and heads to one of the standing tables.

His pulse quickens as he reads Jagadish’s name in the Sender field. He skims the text, but the message is too long and too dense to try to absorb on a tiny screen. The sooner he can leave this stupid party and go home, the better.

“Dr. Gifford!”

He tears his eyes from the screen to meet the gaze of his colleague, Professor Hewett.

Her expression softens. “How’ve you been holding up, Paul, since, well, since your father?”

“I’m doing alright, María.”

She nods and is silent for a moment, as though considering. Finally she plunges on. “How’s your research going? Anything promising? I know a bunch of us have been hoping to see something new from you.”

“Did I hear you say Paul’s working on something new?”

Shit. Dr. Tappert. The senior professor changes course to join them as though pulled in by lasso.

Paulie chugs the rest of his wine, as much for a moment to think as for an excuse to look away from Tappert’s idiot face.

“Yeah,” he says at last. “I’m looking into Jagadish-Rajput.”

“Oh!” says Hewett. “I met a Peruvian mathematician at a conference who was working on that. His name is Segami—you should reach out to him.”

Paulie nods. “Thanks. I’ll look—”

“Wait a minute,” says Tappert. “I remember reading something about—please tell me you’re not still tilting at the Perelman Conjecture.”

Paulie’s throat tightens. “It’s a perfectly valid area of research,” he spits out. He steps away from the table and flags down a server for another glass, hoping to lose Tappert in the process.

No such luck. “Dr. Gifford,” the older professor says, resting a hand on his arm, “Perelman’s a valid area of inquiry for a young man, maybe. Or for an old man, playing at being a professor emeritus. Not for a mathematician seeking tenure.”

Hearing Tappert’s disavowal of his scholarly value is all the confirmation Paulie needs. No way had he signed off on Paulie’s tenure application.

“I disagree, Dan,” says Hewett. “I have a lot of respect for people going after tough things. After all, that’s kind of what math is about.” Turning to Paulie, she adds, “Going after Jagadish-Rajput is perfect too, because if you don’t make it all the way to Perelman, at least that’s an approach that can get you some intermediary results. You just can’t go silent for this long a time.”

Tappert shakes his head. “It’s a fool’s errand. Paul, I hated to watch your father waste his later years on this, but not nearly as much as I hate to watch you throw away your career. At least your father had tenure.”

Paulie slams his glass down on the table. “I really don’t need you to—”

A gasp goes up around him, and Hewett points at his hand. “Dr. Gifford!”

Paulie looks down to realize that he has smashed the wine glass, and lacerated his hand. The moment he sees the blood, the pain sets in.

Some police procedural natters away on the big screen in the living room, but neither of them pays much attention. Gina makes incremental progress on her cross stitch, while Paulie rubs the label off a bottle of beer and lets his mind wander.

The officers on the screen, with their private dramas and backstories, make him think of his father—alive again in the hours Paulie spends in his Coda, and nonexistent when Paulie looks away. Or maybe the experience is more like a very lucid dream. Paulie hopes not, given how many seemingly profound middle-of-the-night insights have turned out, upon waking, to be nonsense.

Then again, he’s basing all his hopes on the assumption that deduction works the same in-Coda as outside of it.

No, this is beer-fueled nonsense. The whole point of deduction is it works for any set of starting assumptions. It doesn’t matter whether space is Euclidean or not—what matters is what axioms you proceed from and whether your logic is rigorous. A theorem that’s true in the Coda is true outside of the Coda. And if it turns out this life is a simulation, as Paulie has seen posited online, Perelman is just as true in the reality outside this one. Even if it’s simulations all the way up.

Induction. Paulie is certain that if the deductive process is solid for a reality n, then it is equally true for a reality n plus one. If he can prove Perelman in-Coda, he’ll have his n equals one. He’ll have everything.

On the coffee table, his phone buzzes with an incoming notification.

“Don’t,” Gina says.

Paulie checks his screen. “It’s my work account.”

“I know. I always told you it was a mistake putting that app on your phone.”

“This’ll only take… shit.”

“What’s wrong?”

“The dean’s office updated my dossier.” He swallows. “The School of Arts and Science denied my tenure application.”

The television goes to commercials, the volume seeming to double. He can’t think.

Gina strokes his forearm. “What are you going to do?”

He sighs. “I can ask my chair to appeal, take it to the provost, but as things stand right now, I don’t see a reason why he would.”

“What then?”

“I’ve still got a year on my existing contract. After that…” He shrugs. “With my evaluations and fizzling research, I’m probably not looking at a tenure-track position. I could teach community college or high school, or somehow find a job in industry, but…hell, I wouldn’t even know where to begin. Being an academic is all I know.”

She mutes the television. “Oh God, Paulie, please don’t tell me you can’t find something around here.” Gina manages a nonprofit educational foundation. Paulie can’t even guess at what starting over would look like for her. “I want to support you, Paulie, but you have to understand that’s asking a lot.”

“We’ve still got time before we have to worry about that.” He takes a breath. “I still have one chance.”

“What do you mean?”

“If I can prove this thing. Technically I’m past the deadline to add publications to my dossier, but Perelman is such a big deal, I’m pretty sure they’d find a way to let me.”

She runs a hand through her hair. “Is this…is this about the math or is this about something else?”

“What else would it be?”

She takes a breath to answer, then stops and faces away. Paulie considers repeating his question, but then she looks back at him. “Is this about living up to your father? Or about proving yourself to him?”

He swallows. “It’s about the math, Gina. It’s always been about the math. We’re close, I know it.”

She nods slowly. “Okay. Prove your theorem then.”

He stepped into the darkened hospital room. “Hey.”


Paulie ran a hand along the back of the chair by the bed. “You got a nice, uh, write-up in the AMS Proceedings. A lot of mathematicians said some pretty amazing things about you.”

“I’m not going to see it; makes no difference to me.”

“No, I guess it wouldn’t. You never were the mushy type.”

His father chuckled. “You can say that again.”

Paulie erased the shift nurse board. “I know you don’t remember, but we’ve been trying to prove the Jagadish-Rajput conjecture.”

“The what?”

Paulie began filling the board. “I’ll catch you up on the broad strokes.”

They were approaching a point of diminishing returns. Every visit was going to have to begin with Paulie summarizing all their past conversations, as well as the work he’d done between visits. There would come a point where recap would take all the time he could reasonably spend in the Coda. Then he would really be on his own.

“We should consider a proof by contradiction, then.”

Paulie shook his head. “I tried. It hasn’t gotten me anywhere. I reached out to a mathematician named Segami who’s been working on a proof by induction, though. It’s trivial for n equals one.”

“Of course it is. Can you prove it for n equals n plus one though… Show me what you have so far.”

Paulie cleared the board again, and filled it with differential topology, Vila Groups, and half the Greek alphabet.

“What about Suárez Theory?”

“How’s that apply?”

“It’s about group automorphisms. We might be able to apply it to these Vila Groups of yours.”

“Walk me through it.”

Paulie took notes while his father dictated, stopping to ask for clarifications or to offer his own suggestions. The little board got cleared four times—each time a chance to mistranscribe something or miss an assumption. But finally Paulie capped his marker and stared at their work.

“I think—” He swallowed and tried again. “I think we just nailed down Segami.”

“Looks like.”

Paulie wandered toward the window, with its predawn view of the empty expressway. Softly, hardly daring to say it, he added, “and that gives us Jagadish-Rajput, which takes us to—” Somewhere he had raised his voice to the point where he was practically shouting. He turned back to his father. “To Perelman,” he concluded, in a more conversational tone.

“That’s good,” his father said.

Good? Holy shit, we’ve slayed the dragon, and all you can say is ‘That’s good’?”

His father shrugged. “Paulie, I’m dead. The moment you leave, I’ll forget we even had this conversation. I can’t get all emotional about this.”

Paulie sagged into the visitor chair. “What was your excuse before you died?” he muttered.


“Nothing. Fine.” Paulie met his eyes. “Anyway.”


“I was just…I mean, I should go. Try to write this up before I forget it all.”

“Makes sense.”

“Maddie misses you,” he blurted out. “And Gina. Gina sends her love.”

His father nodded.

“Maddie had her dance recital. She did great. She was graceful and confident. She didn’t get that from me. I was so proud.”

“That’s good.”

Paulie stood. “Yeah. I should go…I was wondering if there was anything you wanted to say.”

“Uh, bye, I guess? Thank you for visiting, Paulie.”

Maddie squeezes cement on a plastic wing, making the clear liquid bead up.

“Not so much!” Paulie blurts out. He reaches for a sponge. “Here, let me fix it!”

“Dad! You said you weren’t going to take over! This is my model!”

Paulie puts his hands up in surrender. “Fine, do it your way!”

Maddie frowns, chews on her lower lip, and attaches the wing.

He experiences an odd sort of reverse déjà vu, back to his first chemistry set, working through the experiments in the instruction manual—or rather, watching while his father worked through the experiments. Paulie winces and rests his hand carefully on his knee. Then he does the one thing his father never would have done. “You’re right,” he says. “I’m sorry. Keep going.”

Maddie snaps the next piece of plastic off and trims a bit of flash from it with an X-ACTO knife. “Mom showed me a vid about your, um, the math problem you solved. Are you famous now?”

He smiles. “Famous among a very small group of people.”

“That’s still something. I bet you feel super proud.”

Paulie doesn’t answer. He’s not sure what he feels. After spending decades imagining the aftermath of proving Perelman, it’s possible he burned out his ability to feel anything at all about it. The reality can’t match all he imagined.

“Maybe I could be a mathematician,” she says. “I’m good at math. Grandpa said so too.”

“You definitely are,” he says. Funny how his father could say to Maddie the things he couldn’t say to him. Maybe it was easier when it wasn’t his direct offspring he was talking to.

He squeezes her shoulder, the n plus one to his n. Just like he was the n plus one to his father’s n.

Paulie frowns. What conjecture would he be hoping to prove? That mathematical talent runs in his family? That’s trivial. He thinks instead about the things he wishes he could prove. Did his father feel anything for him like what he feels for Maddie?

Deduction is useless here.

Maddie holds two pieces together and blows on them to dry the cement. “Is it true the university gave you back your tenure?” She says the word awkwardly, like she’s testing out the concept. “Does that mean you can’t be fired?”

“It’s, ah, a little more complicated than that. Close enough, though.”

She swallows. “So we don’t have to move?” She focuses on the model with faux intensity.

Paulie shakes his head. “We never decided that we were definitely moving.”

“But now we’re definitely not?”

Paulie picks up a brush and taps the back end lightly on the table. “We’re…still talking about it.” Still avoiding the subject, if he’s being honest.

Maddie nods and attaches another piece.

He accidentally fumbles the brush. “How about you? What do you want?”

“I want to stay,” she says. “All my friends are here.”

Everything’s so simple from her perspective. Paulie doesn’t know what he wants. Since his proof—since their proof—passed through peer review, the math world has been buzzing with the laying to rest of a decades-open question. He’s gotten informal offers from schools across the country, including a couple of top-twenty departments. And, sure, his own university. Does he really want to stay someplace that hadn’t wanted him?

On the other hand, Gina has her career, and Maddie has her whole life.

He squeezes her shoulder. “I’m not sure what’s gonna happen, but I’ll make you a promise. We won’t decide without talking to you, okay?”


“I love you.”

“Love you too, Dad.”

He entered the hospital room and marveled at how unchanged it still was after all these months.

“Hey,” his father said.


He shivered, the hospital’s cool seventy degrees feeling like an ice bath compared to the warm day outside.

“You’re not going to remember this, but we proved Perelman. Here in your Coda.”

His father’s eyes widened. “Really! Now that’s something!”

Paulie nodded. “Got it published. Both our names are on it. It’s all anybody can talk about—not just the proof, but, uh…”

“Proof by simulacrum? I bet that’ll shake things up.”

“So that’s two things you’ll be remembered for. I’m not actually sure which will have the bigger impact.”

“That’s something.”


The two men fell silent.

“You don’t…I mean, you can’t remember any of the things we talked about, can you?”

“I’m sorry, Paulie, the last thing I remember is not being able to breathe.”

Paulie shook his head. “No…yeah, that…that makes sense.”

“Did you find the insurance policy?”

“Yeah. It took care of everything. Thanks for having that.”


Paulie fidgeted with the rod for the blinds.

“Is there something else?” his father asked.

“No, I guess…it’s exciting, huh?”

“I suppose. I mean, I don’t get to see all that.”

“I just thought you might be…”

His father inclined his head. “Might be what?”

Paulie walked around the bed. “No matter how many times I come back in here, you’re never going to say the things I want you to say, are you?”

“What do you want me to say?”

“Never mind. Look, it’s blazing outside. I have to get back in the car, or I’m gonna get sunstroke.”

His father nodded.

“Goodbye. Dad.” The word tasted funny on his lips; he didn’t think he’d said it once since his father came back into his life two decades ago.

“Bye, Paulie. Thank you for visiting.”

Paulie runs the air conditioner in his car for several minutes, letting it cool down inside. While he waits for the temperature to get comfortable, he checks his phone. The congratulatory emails tapered off weeks ago. In their place is a grocery list from Gina, and a drawing of a horse, against a backdrop of hearts and stars, from Maddie.

Finally he puts the car in gear and rumbles off, watching the columbarium disappear in the mirror.


(Editors’ Note: José Pablo Iriarte is interviewed by Caroline M. Yoachim in this issue.)


José Pablo Iriarte

José Pablo Iriarte is a Cuban-American writer and teacher who lives in Central Florida. Their fiction can be found in magazines such as Lightspeed, Strange Horizons, Fireside Fiction, and others, and has been featured in multiple Year’s Best volumes. José’s novelette, “The Substance of My Lives, the Accidents of Our Births,” was a Nebula Award Finalist for 2018. Learn more at, or follow José on Twitter @labyrinthrat.

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