Chapter 13: The Proof
It started on Sunday night. After the pub, after the blank document, after the two hours of notes that were mostly rubbish. I woke up on Monday morning and instead of getting ready for uni I sat down at my desk and opened the document again. Read through what I'd written. Most of it was garbage. Dead ends and half-formed ideas that collapsed the moment I pushed them. But there was one thing in there that kept pulling me back.
The adjacency matrix of the hypercube. Standard stuff. Everyone who'd worked on the conjecture had looked at it. The eigenvalues were well known. They didn't give you what you needed. The bound you got from them was logarithmic and you needed polynomial. That was where every other attempt had stopped.
But what if you didn't use the standard matrix?
I'd written it down at two in the morning and barely remembered doing it. A line near the bottom of the page: What if you flip the signs on some of the edges? What does the spectrum look like then?
I started working through it. Took the adjacency matrix for a small hypercube, three dimensions, eight vertices. Wrote out all the entries. Then I started changing signs. Not randomly. Strategically. Flipping specific edges to see what happened to the eigenvalues.
The first few attempts did nothing useful. The eigenvalues moved around but not in the right direction. I tried different patterns of sign changes. Some made things worse. Some made things slightly better but not enough. I filled three pages of notebook paper with matrices and calculations and crossed most of them out.
I skipped my morning lectures. Didn't even think about it. Just sat there at my desk with paper everywhere and my pen moving and my brain running at full speed. Total Cognitive Enhancement was doing what it always did, keeping every calculation I'd done locked in, every dead end remembered so I didn't repeat it, every partial result sitting there ready to connect to the next one.
By noon I'd found something. Not the answer. But a direction.
When I flipped the signs on the edges in a specific pattern, one that followed the structure of the hypercube itself, the eigenvalues of the modified matrix were different from the original. Very different. The largest eigenvalue of the new matrix was the square root of n. Exactly what I needed.
I stopped writing. Pen frozen over the page. "Did I just...? No. Don't get excited. This might not work."
Because having the right eigenvalue wasn't enough on its own. I needed to connect it to the degree of an induced subgraph. I needed the Cauchy interlace theorem to show that a subgraph with more than half the vertices had to inherit that large eigenvalue. And then I needed to prove that the large eigenvalue forced a high-degree vertex. Three steps. Any one of them could fall apart.
I went downstairs and made a sandwich because I hadn't eaten anything all day and my hands were shaking. Mum was at work. The house was empty. I ate standing up in the kitchen, staring at the fridge without seeing it, running the argument through in my head.
Back upstairs. More paper. I wrote out the modified matrix for four dimensions. Sixteen vertices. Calculated the eigenvalues. Applied Cauchy interlacing to a subgraph with nine vertices. Checked the bound on the maximum degree.
It worked.
I did it again for five dimensions. Thirty-two vertices. Subgraph with seventeen. Same result.
My heart was going too fast. The kind where you can feel it in your ears. I put my pen down and just breathed for a minute.
"This is going to fall apart," I said. "There's going to be a step that doesn't generalise. There always is. That's why nobody's done this in twenty-four years."
I picked up the pen and started writing the general case. N dimensions. Two to the n vertices. A subgraph with more than two to the n minus one vertices. The modified adjacency matrix with the flipped signs. The eigenvalues. The interlacing argument.
It took me three hours. Writing, checking, rewriting, finding a gap and filling it only to find another one behind it. The kind of work where you think you're done and then you're not, over and over, until you stop trusting the feeling of being done at all.
At some point Mum came home. I heard the front door open and close, heard her moving around downstairs, heard Dad's voice too. He'd been away all week on a job, some rewiring contract up in Birmingham, and he must have got back today. I heard him drop his bag in the hallway and say something to Mum about the motorway being a nightmare. Then Mum called up "Liam, are you home?" and I shouted back "Yeah, just studying" without looking up from the paper.
She came upstairs about an hour later with a cup of tea. Put it on my desk next to a pile of paper covered in matrices. She picked up one of the pages and looked at it like it was written in another language. Which, to her, it basically was.
"Liam, what is all this?"
"Maths, Mum. Just maths."
"Are you sure? Because this doesn't look like anything you're supposed to be working on." She held the page up and turned it sideways like that might help. "None of this is from your course, is it?"
"No, it's something I found interesting. Something I wanted to work on."
She put the page down and looked at me properly. "You're telling me you're finding maths interesting now. You. Liam Reed. The boy who was failing his maths module three months ago. That Liam."
"I wasn't failing, I got a thirty-two."
"Out of a hundred, love."
I rubbed the back of my neck. "Yeah. Well. Things change."
"Clearly." She folded her arms and I could see her trying to figure out if I was taking the piss. "So you skipped your actual lectures today to sit in your room doing maths that isn't even for your course."
"Yes, Mum, I'll work on my coursework too," I said, and it came out more sarcastic than I meant it to. She caught it immediately. Mums always catch it.
"Don't get smart with me, Liam. I'm glad you're interested in something but you can't just ignore everything else because you've found a new hobby. You've got a degree to finish."
"I know. I know. I will."
She stood there for another few seconds. "Don't forget to eat," she said, and went back downstairs.
By eight in the evening I was on the last step. My hand was cramping from writing and my back hurt from sitting in the same position for ten hours and I didn't care about any of it because I could see the end. One more connection.
I wrote it down. Read it back. Checked the logic. Checked it again.
It held.
The proof was complete. It was sitting right there on the page and it was right and my pen was still in my hand and the house was quiet and I could hear Mum watching TV downstairs and a car going past outside and everything was completely normal except it wasn't, nothing was normal, because I'd just done it.
"YES." I stood up so fast my chair rolled back and hit the bed. "YES. YES YES YES." I was gripping the paper with both hands, reading it again, checking every line, and every line was right. Every single line. "I'VE FUCKING DONE IT. OH MY GOD. I'VE ACTUALLY DONE IT."
I was pacing around my room with the paper in my hand, reading it and laughing. Not a normal laugh. The kind that comes out when your body doesn't know what else to do. My eyes were burning and I couldn't stop moving. I sat on the bed. Stood up again. Walked to the window. Walked back.
"Twenty-four years," I said, and my voice cracked. "Twenty-four fucking years and I just did it in my bedroom. In my mum's house. What the fuck."
Mum called up from downstairs. "Liam? What on earth is going on up there? You're going to wake the whole street."
"Nothing, Mum! Sorry! I just got really excited about something, it's fine, everything's fine!"
"Excited about what? What are you even doing up there?"
"Maths, Mum! I got excited about maths!" I was standing in the middle of my room grinning like an idiot and I could hear how mental that sounded even as I said it.
There was a long pause. I could practically hear her blinking. "You got excited... about maths."
"Yeah. I know how that sounds."
"It sounds like I should be ringing that doctor after all." Another pause. "...Alright then. Keep it down though, your dad's only just got home and he's knackered."
I could hear Dad from the living room. "What's he shouting about?"
"Maths, apparently," Mum said.
"Maths?" Dad sounded like she'd told him I was excited about doing the hoovering. "Right. Okay."
I sat back down at my desk. My face hurt from smiling. I couldn't remember the last time I'd smiled this hard. I got a fresh piece of paper and wrote the proof out clean. Just the argument. No working, no dead ends. It took up less than two pages. Two pages. The whole thing. Twenty-four years of the best mathematicians in the world and the answer fit on two pages.
I read it three times. Every time I expected to find a mistake. Every time it held. The modified matrix. The eigenvalue bound. The interlacing theorem. The pigeonhole principle. Every step followed from the last one like dominos.
A nineteen-year-old kid at a shit university in a shit town who nobody had ever heard of. And I'd just solved one of the most famous open problems in combinatorics.
I picked up my phone. Almost texted Jake. Put it down again. What would I even say? "Hey mate, I just proved a twenty-four year old mathematical conjecture, fancy a pint?" He'd think I was taking the piss.
I went downstairs. Mum was on the sofa watching a cooking show. Dad was in his armchair with his feet up, half asleep already. He'd changed out of his work clothes but he still looked knackered from the drive. Biscuit was next to Mum with his head on her lap.
"You've been up there all day. What was all that noise about?" Mum said.
"Good day, Mum." I sat down on the other end of the sofa. Biscuit walked over to me, turned around three times, and lay down on my feet. "Really good day."
Dad opened one eye. "Must be a hell of a sum if you're screaming about it."
"Something like that, Dad."
He nodded, closed his eye again. Classic Dad.
She looked at me. I was still smiling. I couldn't stop.
"Have you eaten anything today? And don't lie to me because I can tell when you're lying."
"I had a sandwich earlier. A big one though. Proper sandwich. Lots of filling."
"That was this morning, Liam. It's nine o'clock at night. You've been sitting in your room for twelve hours and you've had one sandwich." She was giving me the look. The full look. "What do you want for dinner? There's leftover pasta in the fridge."
"Pasta's perfect. I'll heat it up now."
"Yes you will."
I heated it up and ate it on the sofa watching someone on TV try to make a soufflé. Normal evening. Completely normal. Except that upstairs on my desk there were two pages that were about to make Liam Reed a name every mathematician in the world would know. And I was sitting here eating pasta and I couldn't stop grinning and Mum kept giving me looks like I'd lost it.
Maybe I had. A little bit.
I went to bed at eleven. Biscuit claimed the pillow before I could get to it. I didn't argue.
I lay there in the dark. Not thinking about eigenvalues for the first time in a week. Just lying there. Still smiling.
Tomorrow I'd type it up and figure out how to submit it.