Chapter 14: Submission
I was sitting at my desk staring at a blank Word document. The proof was right there next to the laptop. Two pages. Handwritten. Done. All I had to do was type it up and I couldn't do it.
"Come on, Liam. Just start. First line. Just type the first line."
I typed the title. Stared at it. Deleted it. Typed it again.
But typing it up meant making it real. Right now it was just handwritten pages on my desk that nobody had seen. Private. The moment I sent it somewhere, that changed. If I was wrong, I'd be the teenager who thought he'd solved a famous problem and was too stupid to check his own work.
I closed the laptop and picked up the proof and read it again. Line by line. Every step. The whole thing held together the same way it had the last thirty times I'd checked it.
"It's right. I know it's right. So why can't I just type the bloody thing?"
This went on for two days. I'd sit down, open the document, type a few lines, read the proof again, close the document. Go to uni, sit through lectures without hearing anything, come home, open the document, close the document. I was driving myself mental.
On Wednesday evening Mum noticed. We were eating dinner, the three of us because Dad was back from Birmingham, and I was pushing my food around the plate instead of eating it.
"Liam." Mum was looking at me. "You've barely touched your dinner. And you've been in a funny mood since Monday. What's going on with you?"
I looked up. Dad had stopped talking. They were both watching me.
"I've done something," I said. "At uni. Well, not at uni. At home. I've been working on a maths problem."
"The stuff on your desk," Mum said. "The pages everywhere." She put her fork down and folded her arms the way she does when she's been sitting on something for a while. "Liam, you know I'm glad you're doing better. I am. Since the hospital you've been more... I don't know. More here. More awake. But you don't sleep properly, you're up at all hours, and now you're locked in your room doing maths that isn't even for your course." She paused. "You came back different, love. And I'm happy about it, I really am. But sometimes I look at you and I don't quite..." She stopped herself. Shook her head. "I just want to know what's going on with my son."
She said it gently but there was something underneath it. Not suspicion. More like she'd been watching all these changes stack up and she'd been quiet about it long enough.
"It's nothing bad, Mum. I promise. It's actually good. I think. It's just hard to explain."
"Then try," she said.
I put my fork down. "There's this problem in mathematics that's been unsolved for twenty-four years. Really famous in the maths world. A lot of people have tried to crack it and nobody could. And I think I've done it."
The kitchen went quiet. Dad was holding his fork halfway to his mouth. Mum had her hands flat on the table.
"You've solved a maths problem that nobody else could solve," Dad said slowly. "You. Our Liam. Who got a thirty-two on his last assignment."
"Yeah, Mum already reminded me about that, cheers Dad."
"I'm not having a go, son. I'm just trying to understand." He put his fork down. "So what does this mean? Is this a big thing?"
"If I'm right, yeah. It's a big thing. It could get me noticed by proper universities. Cambridge, Oxford, places like that."
Mum's hand went to her mouth. Dad leaned back in his chair and looked at me like he was trying to work out if I was taking the piss.
"Cambridge," Mum said.
"Maybe. If the proof is right. I still need to send it somewhere and have someone verify it. But I don't think I've made a mistake."
"Cambridge," Mum said again, quieter this time.
Dad rubbed his face with both hands. "Right. Okay. So our son, who three months ago was failing statistics, has now solved something that the best mathematicians in the world couldn't solve. And this might get him into Cambridge."
"I wasn't failing, I got a—"
"Thirty-two, yeah, out of a hundred, we know." But he was smiling. Just a bit. That small, quiet Dad smile where his eyes change but his mouth barely moves. "If you say it's right, Liam, then send the bloody thing."
I looked at Mum. Her eyes were wet.
"Mum. Don't start."
"I'm not crying, I've got something in my eye." She wiped her face with the back of her hand. "Both eyes, apparently."
Dad reached over and squeezed her arm. She laughed, the kind that's half crying, and waved him off.
"You're not wrong," she said. And the way she said it, like there was no question about it, no room for doubt. She couldn't tell an eigenvalue from an electricity bill. But she believed me. Completely. Without needing to understand.
That was the thing that finally made me do it.
Thursday morning. Cup of tea. Handwritten pages next to the laptop. But this time I didn't stop. I typed the title and kept going. The abstract, the introduction, the definitions. My fingers were moving and I wasn't letting myself stop to recheck because I'd already rechecked thirty times and if I did it again I'd never finish.
Page three was a nightmare. I got stuck for about an hour on one step because I couldn't figure out how to write it in a way that would make sense to someone reading it cold. I could see the whole thing in my head, the matrix, the flipped signs, how the eigenvalues shifted. But getting that into words that another mathematician would follow without having to guess what I was thinking was harder than actually proving the thing in the first place.
"How did anyone write these things?" Scrolling through other papers on arXiv to see how they formatted proofs, how they bridged steps that felt obvious in your head but looked like a gap on paper. "Right. Okay. Like that. Yeah, that works."
By six in the evening I had it. Seven pages. The proof itself was still less than two pages. I scrolled through the whole document one more time, reading every line, and this time I didn't find anything wrong because there was nothing wrong.
I opened ArXiv, created an account, filled in the details, uploaded the PDF. Wrote a brief description. My hand was hovering over the submit button.
"Right. Okay. Just press it, Liam. Stop being a coward. It's one button. You've pressed harder buttons than this. That doesn't even make sense. Just press it."
I pressed it.
Submission received. My paper would be posted within twenty-four hours.
"Okay." I pushed back from the desk and just sat there for a second. My leg was still bouncing. "It's done. It's actually done. Oh God."
I went downstairs. Mum and Dad were in the living room. Dad had his feet up. Mum was on the sofa with Biscuit curled up against her leg.
"I've sent it," I said from the doorway.
Mum looked up. "The maths thing?"
"Yeah. It's out there now. Can't take it back even if I wanted to."
"How long until you know?"
"A day or two before people start reading it. Then... I don't know. Could be a week. Could be longer."
Dad looked over. "So now you wait."
"Now I wait. My favourite activity."
"Worst part of any job," Dad said. "Doing the work's easy. Waiting for someone to tell you if it's good enough is what kills you."
That night I lay in bed refreshing my email every ten minutes. Nothing. The paper wasn't even posted yet and I was already checking. I'd put my phone down, close my eyes, last about ninety seconds, pick it up again.
"Stop it. It won't be live until tomorrow. You're refreshing an empty inbox. This is pathetic."
I put the phone on the other side of the room. Got back into bed. Biscuit shifted somewhere on the landing, his claws clicking on the floorboards. I lasted about two minutes before I got up, walked across the room, checked the phone, walked back, got into bed.
Nothing. Obviously nothing.
"Right. Phone stays there. I'm not getting up again."
I got up again.
At three in the morning I finally fell asleep with the phone on my chest.
Friday. The paper went live on arXiv around noon. I was sitting in a Statistics lecture when my phone buzzed. An email from arXiv confirming the posting. My paper was now publicly available.
I put my phone back and sat through the rest of the lecture without hearing a single word. My leg was bouncing so hard the girl next to me gave me a look.
The first email came at half four. I was walking home, hood up because it had started raining, and my phone buzzed and I nearly dropped it getting it out of my pocket. Wet fingers on a wet screen. From someone at MIT. A professor. The subject line was just: "Your paper on the Sensitivity Conjecture."
I stopped walking right there on the pavement. Rain on my face. People going past. I didn't care. I opened it.
Three lines. He'd read my proof. He'd checked it. And he called it extraordinary work.
I stood there in the rain like a complete idiot with my phone getting wet and my hood falling back and I read it one more time just to make sure I wasn't making up words that weren't there.
"Oh my God." I said it out loud. An old woman with a shopping bag looked at me. "Sorry. Sorry, it's just... yeah. Sorry."
By the time I got home there were two more. I was standing in the hallway dripping rain onto the carpet and reading emails. A professor in Germany who had a question about one step but called the overall argument remarkably elegant. A PhD student in California who'd clearly written his email at full speed because there were typos everywhere and he'd apologised for being unprofessional.
I sat on my bed and I didn't know what to do with myself. My hands were shaking. I wanted to call Jake but what would I even say? I wanted to run downstairs and shove the phone in Dad's face but he wouldn't understand the emails and I'd have to explain and explaining would ruin it somehow.
So I just sat there. Reading them over and over. Three emails from mathematicians in three different countries. And they all said the same thing.
The phone buzzed again. A professor in Japan. Four now. The paper had only been live for five hours.
I put the phone face-down on the bed and pressed my hands flat on my knees. I was smiling and I couldn't stop. My face actually hurt from it.
Mum knocked on my door. "Dinner's ready. You coming down?"
"Yeah." My voice came out wrong. I cleared my throat. "Yeah, I'm coming."
I went downstairs. Dad was already eating. Mum put a plate in front of me.
"Any news on your maths thing?"
"Yeah, actually." I was trying to keep my voice level but I could hear it doing something weird, going too high and then too low like I couldn't find the right register. "A few people have read it. Professors. At MIT and some other places. They think it's right."
Mum put her hand on the table. "MIT. That's in America."
"That's the one, yeah."
"Professors in America are reading something my son wrote in his bedroom."
I couldn't help it. I grinned. "Sounds mental when you say it like that."
She looked at Dad. Dad looked at me. He had that expression where you could see him trying to figure out how to react because nothing in his life had prepared him for this particular sentence.
"Well," he said. "Alright then." He picked up his fork again and went back to his pasta like it had been nothing.
After dinner I went back upstairs. Six emails now. Four countries. The German professor had gone into proper detail about why the sign-flipping approach worked, three paragraphs breaking down the eigenvalue argument. The MIT professor had forwarded it to a colleague. The PhD student had written a second email, calmer this time, saying he'd told his entire lab and two of them were already working through the proof themselves.
People were talking about my work. Right now. In offices and labs I'd never been to, people were reading what I'd written at this desk and forwarding it to each other.
I leaned back in my chair. The grin was still there.
Three days ago I was too scared to type the first line.
I picked up the handwritten proof, the original, the two pages with all my scribbles and crossings-out and the coffee ring on the corner. Two messy pages that could've been anyone's coursework. But it wasn't anyone's coursework. It was mine. And it was right.
Biscuit was asleep on the bed, snoring gently, completely unbothered.
"Okay," I said. "Now what?"