(Originally appeared in Bloody Fabulous, published by Prime Books)

Incomplete Proofs

by John Chu

Next autumn's proof of Gödel's incompleteness theorem tailored itself to Grant's body. A three section proof, the trousers grew snug around his waist and shortened to break against his feet. The shirt buttoned itself as it tightened against his chest, arms and shoulders. Jacket sleeves shrank to reveal his hands. The proof looked retro rather than timeless, not at all what Grant expected from Duncan. Grant wondered what the buyers and journal editors on the other side of the curtain would make of it. He hadn't verified a proof for an audience this important in years.

His own jacket, shirt and trousers pooled around his feet. Duncan's stylists stopped fussing with Grant's face and hair, pronouncing him fit for the runway. They scooped up his clothes, patted him on the back for luck, then left him alone to focus.

The cutting edge mathematics that held the proof together permeated Grant's brain. He felt its structure, how each lemma and proposition stitched together to support the conclusion that no axiomatic system could be both consistent and complete. Either some truths were unprovable or the system could erroneously prove falsehoods. This time, Duncan had proved the theorem through computability theory.

The audience's quiet murmur bled through the curtain. Grant took a deep breath, then cursed himself for letting himself become a cog in Duncan's machinations again. His grad students had been having the time of their lives watching mathematicians verify proof after proof. Otherwise, he'd have told Duncan's stylists to go stuff themselves when they asked him to verify the final proof of the new Duncan Banks autumn collection.

Grant exhaled. His feet tested the runway's sprung floor as he stepped through the curtain. Where other theorem houses placed safety nets for their mathematicians, a trench of spikes lay on either side between Grant and the audience. Nothing was too over the top for Duncan. Journal editors thought he was potentially the best mathematician since Gauss or Euler. People had worn their proofs, or ready-to-wear copies thereof, for over a century. Editors expected the same from Duncan.

The audience hushed except for his students: Marc and Lisa. They stood, cheering and waving their arms in the air. The silence surrounding them made their excitement sound ironic. He resisted the urge to bury his face in his hands. Instead, he launched into the first steps of the proof: a tumbling pass down the length of the runway.

The jacket, shirt and trousers exploded apart at their seams into their constituent lemmas and propositions. They swirled in wide arcs around him as he twisted and spun through the air. Pain spiked his knees and shocked through the rest of his body each time he landed. Air whooshed past him and flapping lemmas surrounded him on all sides. Each somersault, jump, and handstand evoked the logic and reasoning that stitched pieces of proof together. The body canvas and chest canvas slipped inside the jacket's shell and gave the jacket its retro shape.

The proof danced in counterpoint exactly as he expected his logic and reasoning to animate them. As he flipped through the air in a pike position, the trouser slid onto his legs. One section proved, two to go. He stepped to the far end of the runway. The shirt weaved around his arms then settled on his torso. It buttoned itself as Grant ran, building up speed for the proof's final steps. Focused on proving the theorem, he danced with the jacket.

It flew toward him from the side rather than from the back. Its sleeves reached out as if it wanted a hug. Rather than sliding onto his body, it was about to tangle him in mid-air, knocking him into the pit of spikes. If he lined himself up with the jacket, it'd slide onto his body but, in the process, he'd tumble off the side of the runway into the pit of spikes. No valid proof took a mathematician off the runway. Either he repaired Duncan's proof right now, or he'd be impaled by rows of sharp spikes.

Grant stretched his mind out to the jacket. He'd already started his front triple layout when he realized the jacket's shape was subtly off. Duncan hadn't intended the jacket to feel retro. Its chest was prone to collapse and the lapel rolled too easily. The proof's linch pins, the body canvass and chest canvass inside the jacket's shell, were fine by themselves, but they didn't hold this proof together. Grant need stronger intermediate results.

The jacket sideswiped him as he started the second revolution of his layout. If this were a valid proof, he'd be wearing the jacket now. Instead, he thought back to where the proof had gone wrong. The jacket split on its seams into pieces. It flowed around him rather than tangling him and knocking him into the pit of spikes. He landed, then tumbled an extra pass, flipping and twisting in the air. Through that reasoning, he proved stronger versions of the lemmas Duncan had used. The body and chest canvasses morphed in response from what Grant was given into their proper shapes.

The math was so complicated that the reasoning took longer than the length of the runway. As he hit the end, he front tumbled towards the curtain he'd started from. His lungs burned with each breath. His heart pound not from nerves or even fear of death but from exhaustion. His legs wanted to crumble each time they hit the runway. The jacket reformed, now swooshing towards him from the front. He dove and the jacket rushed onto his body just before he rolled to stand next to the curtain.

Grant stood, his arms stretched overhead, the jacket, shirt and trousers crisp on his body. His students stood again and hollered, their arms pumping. This time, applause did fill the space. He nodded to half the audience, pivot turned, then nodded to the other half. Disconcertingly, the applause seemed to be growing. They had to have noticed the jacket he wore now wasn't the jacket he'd started with, but that didn't stop anyone from cheering.

Duncan strode onto the runway to receive the applause as Grant left. The two passed each other. Grant glared. Letting this proof onto the runway was no accident. At the latest, the theorem house should have caught its flaws during the run through. Duncan trapped Grant in his arms. He whispered into Grant's ear.

"Hi, Tsai." Duncan always called Grant by his last name. "I know you're angry at me. Meet me in the dressing room after the show. I'll explain everything."

#

Grant hung the proof of Gödel's incompleteness theorem on a rack with the other proofs verified tonight. His own clothes lay in a heap on the dressing room table next to some proof that hadn't been put away. His wallet and cellphone fell out as he pulled out his shirt. Wisps of thread jutted where there should have been buttons. One sleeve dangled from its seam. His trousers had been rent into strips. That explained how the stylists had undressed him so quickly. They'd assumed he'd worn a proof to the show and could fix the damage. His clothes, though, were just clothes, ready-to-wear.

The proof still on the table had an apology in Duncan's handwriting pinned to it. Grant recognized the work, a proof of Fermat's last theorem Duncan had created during grad school. Its asymmetrical curved seams emphasized Grant's musculature. He didn't feel clothed as much as he felt like an anatomy chart. Spent, he slumped into the chair next to the table and waited.

Duncan strode into the dressing room wearing the proof of the first significant problem they'd solved together: ten was a solitary number. A critical triumph, the proof never sold well. Too few people had the body to pull it off. Duncan had, and damn it, he still did.

To Grant's dismay, Duncan wore the proof better now than ever. His brawn no longer fought to burst out of the proof. Rather, the proof now exposed his beautiful proportions. He was still the mutant spawn of the sun and lightning. If the sun had passed its zenith and the lightning now the lament of distant thunder, he still made any room feel too cramped to contain him.

"You couldn't have asked me for help before you put the proof on the runway?" Grant was determined to stay angry despite Duncan's smile. "You invited Marc and Lisa here, all expenses paid. It's not like you didn't know how to find me."

"You wouldn't have come, much less helped, if I hadn't brought your students here." Duncan sat on the table. "The proof of Gödel's incompleteness theorem is the signature piece of the fall collection. No one understood its flaw much less how to fix it. I brought here, the only way I could, the one person who could fix it."

Grant stood. He folded his arms across his chest. With Duncan sitting on the table, they saw each other eye to eye. Duncan's gaze burned, but Grant met it.

Duncan was right, as usual. Grant wasn't above refusing to help just to spite him.

"The one person who could fix the proof? Give me a break. And what were you going to do if I'd failed or died trying? Let the fiasco destroy your theorem house?"

Buyers and editors were a fickle lot. One tumbling pass that wasn't parallel to the sides of the runway was enough to cancel orders and deny publication.

"Like you could have failed." Duncan shook his head. "I don't risk my theorem house on just anyone." He pulled out his cellphone, tapped at it, then handed it to Grant. "I'm about to offer the man in this video a job. Tell me what you think of him."

The screen filled with a beefy guy jumping, spinning and twisting up and down a runway. He used computability theory to prove Gödel's incompleteness theorem, a corrected version of Duncan's proof. The man reasoned with a strength and incisiveness that made Grant's jaw drop.

Grant might quibble with the handful of moves that were not textbook perfect, but the result was superior to Grant's attempt. He wanted Lisa and Marc to see this version.

"You should have gotten him to verify your proof tonight. If you already had this, why did you show the flawed proof?" Grant handed back the cellphone. "Never mind. I no longer have to understand or care about your machinations."

Duncan stood. He bore down on Grant. His head and shoulders blocked the room from view. The musk and leather of his cologne enveloped Grant.

"Tsai, this is video from tonight. That's you correcting my proof." He held the phone in front of Grant's eyes. "When I say you're the one person who could fix the proof, it's not flattery. As hard as you are on everyone else's work, you're even harder on your own. God knows how you sell yourself in interviews. Are you surprised you can't find another job?"

Grant's face burned. He pushed the phone away. His gaze fell to the floor as he sat.

"What makes you think I'm looking?" Grant slid the away from Duncan.

"Your department is eliminating its graduate program and some of its non-tenured faculty. Your student evaluations are... bi-modal. A small number of students will register for anything you teach. Everyone else writes comments like 'Dr. Tsai can't teach a cow how to moo.'"

Being a force of nature had its advantages. By the time Duncan had worked over the department's administrators, they probably thought that Duncan was doing them a favor by letting them show him Grant's evaluations.

"You used to be charming, Duncan." Once, Grant would have done, hell, he had done anything Duncan wanted.

"I still am." Duncan flashed a quick smile. "But you no longer trust me when I'm charming and I need you to work with me."

"Yeah, right." Grant forced himself to match Duncan's gaze. "The first Duncan Banks collection got published in all the major journals and sold to all the major buyers. Everyone wanted to work with you. You didn't need me anymore and I might as well not have existed."

"You're never going to forgive me." Duncan seemed to deflate a little. "I'm not who I--"

"Make your damn offer."

When the semester ended in a few months, Grant would be out of a job. Besides, his legs felt like marble. Otherwise, he'd have walked out.

"I have the outline of a solution for P=NP. Flesh it out with me. Please?"

Grant sighed. Whether P=NP was one of the remaining great unsolved problems. Proving that P=NP meant biologists could quickly compute the structure of a protein rather than guessing its structure then checking for correctness. It meant computationally tractable ways to find optimal solutions to all sorts of packing and scheduling problems. No industry would be unaffected. Grant and Duncan would be heroes for the ages.

"Show me." Grant tried to sound bored.

The proof of ten as a solitary number transformed into pieces of muslin. They changed shape as they slid around Duncan's body. A lemma around Duncan's back fortified two results on his shoulders. What covered his chest seemed to stay there out of sheer faith that someday something might hold it in place. He'd built it on conjectures Grant didn't recognize. After a minute, Duncan wore something that fit roughly on him, pinned together by hope and determination more than it was stitched together by mathematical theorems and logic.

"Well?" Duncan showed his palms to Grant. Rather than casting his light on the world, Duncan looked as if he were in eclipse.

Grant let the outline inhabit his mind. Not enough hung on Duncan to prove anything. Grant wasn't even sure what it actually proved, but the bits that were actually stitched together dazzled. The intermediate results, if verifiable, would advance mathematics nearly as much as the conclusion.

Grant's hands gripped the chair. He forced himself not to engage with proof. Math hadn't excited him this much in years. But was it worth being burned by the sun and shocked by lightning again? They had never been, and could never be, just about the math.

"By the age of thirty-five, most mathematicians have already done their best work." Grant didn't see any reason to be harsh, not when Duncan had been thoroughly, if bewilderingly, non-toxic. "You'll want someone in his or her prime."

The outline transformed back into the proof of ten as a solitary number. It's austere elegance replaced the buzz of the outline in Grant's mind.

"Do you need to see your video again? This has nothing to do with age." Duncan frowned. "Tsai, I can't change how I treated you but--"

"I have grad students now." Grant's thighs burned as he stood. "I can't just abandon my kids."

Maybe that wasn't the real reason either, but like his age, it was the truth. Grant pushed himself to the door.

"Wait." Duncan's voice swung away from Grant, not towards. "The proof, it's as much yours as mine. You should write it up."

Duncan reached Grant with long strides. He offered the proof to Grant.

"You need the proof as a template for the ready-to-wear version." Grant could write it up from memory. "Send me a copy of the video instead?"

The video had its flaws, but it was a proof verified at a major show. More importantly for the job search, it was available right now.

"I uploaded it just after the show ended." Duncan patted Grant's back. "Within a week, anyone with even half an interest in math will have seen it."

"I'll send you a draft of the paper to review in a few days." Grant forced himself away from the intoxicating heat of Duncan's desire.

"When you're swamped with job offers, don't forget who asked you first, ok?"

Grant wouldn't dignify that ridiculous thought. He just grunted a hoarse laugh as he walked out the door.

#

Stacks of boxes took the place of reference books, papers and office supplies on Marc and Lisa's office shelves. The two grad students sat at their desk reading their tablets. What made Grant suspicious though was the missing junk food.

Either Marc and Lisa gotten better at hiding their stockpile of chips, cookies and whatnot, or they'd removed it. Hyperactive metabolisms be damned. They couldn't eat like crap forever if they expected to be able to verify their proofs at conferences. However, Grant had never expected them to believe him much less do anything about it.

"Not that I'm complaining, but why is the office suspiciously clean?"

"We're leaving school, Grant." Lisa could make anything sound reasonable. "Everyone's seen that video of you in Dr. Banks's show. The word is you're going back to him. Nice of you to tell us yourself that you're dumping us."

Lisa looked hurt. Grant hadn't realized that was possible.

"One, I'm not going to make any plans without letting you know first." Grant put his hands at his waist. "Two, I can't believe neither of you bothered to check with me."

"You've been sending everything to voice mail." Lisa swiped then tapped on her tablet. It chimed a few times. She thrust her sent mail folder in Grant's face. "And you haven't answered any of my emails."

Grant fished his cellphone out of his pocket. The screen stayed blank when he tried to wake it. "Oh, I forgot to turn it back on."

His phone vibrated as it booted up. His eyebrows raised at the number of voicemails waiting for him. He checked his email. His jaw dropped at the backlog. He recognized the sender addresses: every theorem house of note, editors of all the important journals, and all of the best mathematics schools in the world, except one, Duncan's. Its mathematics department would have been formidable even without him.

"I may have other options." Grant leaned against a shelf. He swiped through his email, sorting his job offers. Duncan was right again, damn him.

"Why are you so surprised?" Lisa never let him get away with anything. "You fixing the proof on the fly is all anyone has talked about for days."

"I always disconnect my router when I write up a proof." Grant's gaze met her dismay. "Otherwise, I get distracted."

"Why the hell are you teaching here in the first place, Grant?" It was Marc's turn to rip into him. "You're not just good for your age--"

"Hey, I'm not even twenty years older than you."

"You're good." Marc held out the last word. "That last pass up and down the runway--" His eyes widened and his voice broke with awe. "Why don't you have your own collections?"

"That's why you let your funding lapse." Lisa's face lit with revelation. "You'd planned to abandon us."

"No, we're losing funding because I write really crappy grant proposals."

"Look." Marc stood, opening his palms to Grant. "Go back to Dr. Banks. If he wanted me to work with him, I'd ditch you in a second. It'll be fine. I've started auditions for Project Prove It."

"No." Grant straightened up. He breathed deeply expanding his chest as he crossed his armed and subtly flared his lats. Stealing a page from the Duncan dominance display playbook was shameless, but he hadn't found another language Marc understood yet. "Auditioning to get on the reality show will eat up the rest of the semester. If you get in, all being on Project Prove It will demonstrate is that you can create trivial proofs within twenty-four hours for people with more money than aesthetic sense. You can't build a career on that."

Actually, Marc probably could. Grant just wasn't about to let him settle. Marc sat down, a bit stunned.

Lisa wasn't the least bit impressed with dominance displays. If anything, she grew louder. "I can complete all the math for my dissertation by the end of the semester even while looking for a job. After that, it's just writing. A full time job won't get in the way of that. I have a whole year after you leave when you can still sign my dissertation."

"Oh, Lisa." Grant had expected no less than her full-throttled self confidence. "You have no idea how much work you have ahead of you."

Grant's cellphone buzzed. He shut it off again.

"If you ever answered your phone or read your email, you'd probably find some way to fund us." Lisa shrugged. "All I know is if I had a research assistantship next semester, I could afford to stay in school."

And that was why Grant never ever dismissed Lisa. She had more sense than he did.

"Give me a few days. I'll think of something."

Grant locked himself in his own office. Towers of books surround him. He sped through his email and voicemail. When he'd caught up, he stretched away the back strain then hid his face in his hands. After that video, the universities clearly expected him to bring money in, not need support for his kids. Wherever he ended up, they could follow him, but they wouldn't if he had no way to support them.

People who distracted themselves from their dissertations never finished them. If he didn't secure a future for his kids, they'd secure one for themselves and their hard work would never come to fruition. Lisa's plan to finish was unworkable and Marc wouldn't even bother.

He called Duncan and left a message with his assistant. Maybe Grant was just capitulating to Duncan's master plan, but he couldn't think how else to secure funding before his kids did anything stupid. He'd get them their doctorates before Duncan could discard him again. Security, job or otherwise, could wait.

#

Grant's flip-flops squeaked against the just-mopped floor. Water that clung to him from the shower chilled off his skin. He dropped his towel in the middle of the bench then unjammed his locker door. The pounding reverberated through the locker room. He pulled his gym bag out then dropped it on the floor.

"Tsai, are you in here?" Duncan's voice echoed off the metal. "I'm sorry my assistant wouldn't put you through to me. Never happen again. I've fired him. Your kids said that you work out every night but I didn't see anyone in the gym."

"Over here." Grant took a deep breath. "Fund my kids until they complete their dissertations and I'll do anything you want."

When Grant looked up again, Duncan stood just in front of the gym bag. No footfalls, much less squeaking.

Rows of lockers dutifully closed in on Duncan. The ceiling lowered and the walls collapsed as he sucked up all the space in the room. Maxwell's equations in differential form covered the front of his T-shirt. The symbols rippled as they curved around his body. The T-shirt caressed his beautifully powerful shoulders, chest, arms and back. A wrinkled leather belt held faded jeans on his body. The slight stretch across his thighs did the same. Dust and wear had ground his boots gray. The messenger bag Grant had given him long ago hung off a shoulder. Every tear on the bag had been expertly mended.

"Hi, Tsai." Duncan's face registered Grant's gaze sweeping through him. "Something wrong?"

"No. I'd just forgotten what you looked like." That sentence had made more sense in his head. "Do we have a deal?"

"Tsai, work with me because you want to, not because you have to." He handed Grant a manila folder from his messenger bag. "The advance for the Gödel's incompleteness theorem proof will support your grad students long enough for you to line up proper funding."

Grant skimmed the contract in the folder as he dressed. Signing it wouldn't make him Duncan's slave or anything. He patted his pockets. Empty. He tucked the contract under one arm then found the pen stowed in his gym bag's outer pocket.

"Whoa." Duncan squeezed Grant's shoulder. "Have the contract looked at first. You'll find it fair, but I could be scamming you."

"You're not that kind of asshole." Grant shoved the folder into the bag.

"Thank you." Duncan sounded as if Grant had paid him a compliment. "Now we can focus on what you really want."

"No, I'm good." Grant felt bad about leave right after getting what he needed from Duncan, but not that bad. "I'll send the contract back tomorrow."

Grant slid into his coat. He slammed the locker shut, picked up his gym bag then waved goodbye.

"No one knows you better than I do. You'll settle for funding, but that's not all you want."

Grant's soles squeaked on the floor as he turned and with each step away. So much for the graceful exit, not that it mattered. Duncan froze Grant with a single word.

"Tenure."

Grant knew the scene behind him. A slight smile leavened Duncan's face. His messenger bag slumped on the floor against a rusty bench leg. Duncan straddled the bench, leaning forward. His hands gripped one end of the bench. His arms braced his torso as if he were about to lift into a handstand. He presented the illusion of being perfectly relaxed while his T-shirt exposed every muscle of his torso.

"Where?"

"With me. Same university."

Grant turned around. His shoes squeaked again. He forced himself not to wince. Duncan looked as exactly as he imagined.

"Isn't that overkill?" Grant glared down at Duncan. He had so few chances to do that. "With tenure, how will you get rid of me when you don't need me anymore?"

Duncan lifted his torso and legs parallel to the bench. His smile faded as his gaze focused onto the bench's graffiti. "You know, you get older, you realize the first guy you first bounced ideas off of has ruined you for everyone else since. So you change." His hands thumped against the bench as he walked himself away from Grant. "You become what that first guy wants."

Duncan straightened into a handstand, lowered his legs, then stood upright on the bench facing one end. He backflipped off the bench landing to one side. The control required to avoid crashing into the lockers was intimidating. His hands spread in front him as if to ask, "Well?"

The day Duncan didn't look confident was the day the world would end. However, the years had abraded the smugness from his demeanor. His blistering gaze had always been inquisitive, but it now also yearned. Grant had walked away from the smug Duncan, but this one taunted him with possibilities he'd long convinced himself didn't exist.

Grant pursed his lips. His hands gripped his gym bag. He'd ask for the ridiculous. That ought to cool Duncan's ardor.

"For a start, the university will matriculate my grad students."

Not only did he want the university to fund his students, but he wanted enough funding from Duncan's theorem house that he could buy off his teaching responsibilities. No reason why undergrads should suffer. Grant set down his gym bag and for what felt like minutes detailed his dream job.

"Done." Duncan unzipped Grant's gym bag.

"What?" Grant felt sucker punched. "My demands aren't reasonable."

"Done." Duncan teased the manila folder out of Grant's gym bag then handed it to Grant. "I told you. No one knows you better than I do."

Grant studied the pages he'd glazed past the first time. They met his conditions term for term.

Duncan had changed, but he hadn't become what Grant wanted. Life hadn't taught Duncan any humility. Rather than scaling his self-assurance down to match his achievements, he'd scaled his achievements up to match his self-assurance.

"It's up to you, Grant." Duncan walked towards the locker room exit. "Refuse my offer and I'll never bother you again. I promise."

Grant stuffed the folder back into his bag. "Tell me this, Duncan. What are you really trying to prove?"

Duncan turned around. His gaze pressed against Grant. He looked as if he were intuiting the right response from how Grant's bag pressed against his back, how Grant hadn't tucked in his T-shirt or how off-kilter Grant had knotted his boot laces.

"I should have known you'd see it right away even if I hid the conclusion from my outline." Duncan shrugged. "But I knew you'd find proving P=NP sexier."

"You've jumped ahead a few steps again. Back it up, Duncan." For years, saying that might as well have been Grant's full time job. If Duncan expected mere humans to understand him, he needed to take it step by step.

This time, Duncan had skipped past surprise and straight to wistful. As he sighed, he seemed to deflate. Grant had never seen him look this mortal before.

"P≠NP. Or maybe I'm wrong." Duncan took a deep breath. "Rehashing old results in ever more elegant ways has done so well for me, I don't have to be practical. I can do real math now. You know, throw yourself into unsolvable problems. Get lost in every twist and leap the way young mathematicians say they'll do until they realize they need to eat. It's time to tackle the impossible and... I just thought you'd want to do that too. With me." Duncan showed his palms to Grant again. "Like I said, it's up to you."

P≠NP meant that computationally intractable problems would always be intractable. The best anyone could do was recognize that then focus on heuristics and other approximate solutions. Mathematicians would care about that result, but no one else. The proof might become the most elegant anyone has ever seen but his theorem house would never sell it. No one, not even Duncan, had a body perfect enough to wear it in public without embarrassing themselves.

They held each other's gaze for what seemed like days before Duncan turned around. He started towards the door, his motion so perfectly controlled, Grant couldn't tell how Duncan felt.

"I think when we flesh out your outline, we may find that P=NP is undecidable." Grant allowed a small smile on his face. Duncan was at least capable of the truth on occasion. That was a start.

Duncan stopped, then pivoted around. Grant made a note to ask Duncan someday how he did that without squeaking.

The puzzled look on Duncan's face melted into one of realization then resolve. "Perhaps." He shrugged. "Let's discuss it over dinner. My treat."

Wind swept across the parking lot. A hoodie coalesced around Duncan. His giddy smile outshone the stars and the moon. The light poles seemed bunched together, corralled by the encroaching Jersey barriers. Not even the parking lot could contain Duncan tonight.

No wonder Duncan was so happy. Grant had done exactly what Duncan wanted. Maybe Duncan had discovered the virtue of telling Grant the truth. Maybe Duncan had maneuvered Grant here the way he'd maneuvered Grant back onto the runway. Grant only knew one way to find out what was true. Take Duncan on. They'd tumble and swirl around each other until either they covered each other or Grant fell into the trenches of spikes.

Grant suspected he could tumble on the runway forever and never really know. Not everything that's true had a proof. No consistent formal system was complete. He wouldn't be a mathematician though if he didn't want to find out.

END