Chapter 11: The Book
The second day was worse than the first. I spent most of Applied Maths staring out the window. This sucks. I know I need to be happy. I can have a better life if I use what I've been given right. But still. This sucks.
After lunch I had a free period. Two hours with nothing to do. I could have gone home but Mum was there and she'd ask why I wasn't in class and I didn't feel like lying to her again.
I walked around campus for a bit. Past the science block, past the student union, past the computer lab where three people were playing games and pretending to work. I ended up at the library because it was the only building I hadn't been inside yet and it was raining again and I was cold.
The library was quiet. About fifteen people spread across three floors, most of them asleep or staring at their phones. I walked past the textbooks and the journals nobody ever touched and ended up in the maths and computer science aisle at the back.
I wasn't looking for anything specific. I ran my fingers along the spines of the books and read the titles. Most of it was basic stuff. I pulled out a few, flipped through them, put them back.
Then one caught my eye. A thick volume about computational complexity. The spine was stiff and some of the pages were still stuck together at the edges. It looked like it had been sitting there, ignored, for years.
I took it to a table by the window and started reading.
The first few chapters were standard. Boolean functions, complexity measures, the basics of how you categorise and compare different properties of binary systems. I read through it page by page, letting Total Cognitive Enhancement do its thing, locking every definition and theorem into place as I went. It was more interesting than anything I'd heard in a lecture that week but it wasn't difficult. Not yet.
Then I got to chapter nine. And something changed.
The chapter was about the sensitivity of Boolean functions. Sensitivity measured how much the output of a function changed when you flipped a single input bit. It was connected to a bunch of other complexity measures like block sensitivity, degree, and certificate complexity, and mathematicians had proven that most of these measures were all polynomially related to each other. Roughly the same size, give or take a polynomial factor. Except sensitivity. Nobody could prove that sensitivity fit into the same framework as the rest.
The Sensitivity Conjecture. Posed by Nisan and Szegedy in 1992. Twenty-four years old. Still unsolved.
I read the section twice. Not because I didn't understand it the first time but because I wanted to make sure I wasn't missing something. The conjecture was simple to state. If you had a Boolean function and you knew its degree as a real polynomial, could you bound its sensitivity by a polynomial of that degree? Everyone believed the answer was yes. Nobody could prove it.
The book laid out the history. People had tried for decades. Some of the best combinatorialists and computer scientists in the world had taken a crack at it and come up short. Partial results, special cases, bounds that got close but not close enough. The problem was famous specifically because it looked like it should be easy. It connected to things that were well understood. And it just sat there, unsolved, like a door that everyone could see but nobody could open.
The thing that got me was the Boolean functions. I worked with Boolean logic every single night. Every system I looked at with System Insight was built on it. Every hack, every vulnerability, every network structure came down to binary decisions at some level. And here was a problem about the fundamental properties of those exact functions that nobody in the world had been able to solve.
There was an equivalent formulation that caught my attention. Something about hypercubes and subgraphs. If you took more than half the vertices of an n-dimensional hypercube, there had to be a vertex with high degree. That felt more concrete. More like something you could grab hold of.
I caught myself leaning forward over the book, my forehead almost touching the paper, my pen tapping against the table without me realising. My brain was doing that thing where it starts pulling threads, connecting this to that, reaching for something just out of sight.
"Stop it," I said under my breath. I sat back and put the pen down. "You're not solving a twenty-four year old conjecture in a library on a Tuesday afternoon."
I closed the book. Sat there for a second. Opened it again. Read the equivalent formulation one more time. Closed it again.
"No. Leave it."
I put the book back on the shelf. Walked away. Got about three steps and looked back at it. The spine was sticking out slightly from where I'd pulled it.
Leave it. You've got enough going on. Null, the bounties, the money, figuring out how to get out of this uni. You don't need to add unsolved maths problems to the list.
But if I did solve it... that's the kind of thing that gets you noticed.
I walked out of the library and into the rain. Pulled my hood up and started home. But the whole way back, twenty minutes of walking, I was thinking about hypercubes. About vertices and degrees and what happens when you take more than half the corners of an n-dimensional cube. The problem was sitting in my head like a stone in a shoe. Not painful. Just there. Impossible to ignore.
Mum was home when I got in. She was on the phone with Nan, which meant she'd be busy for at least forty minutes because Nan didn't do short phone calls. I went upstairs, said hello to Biscuit, sat at my desk.
I opened my laptop and went to the forum. Looked at a few bounties, checked my wallet. The 0.45 Bitcoin from last night was still sitting there. I should have been thinking about the next job. Building the reputation. Being Null.
But I wasn't thinking about that. I was thinking about the bloody book.
What if I just read a bit more about it. Just online. Not trying to solve it. Just understanding the problem better. That's not the same as working on it.
I typed "sensitivity conjecture boolean functions" into the search bar and started reading.
Two hours later Mum called me down for dinner and I realised I hadn't looked at the forum once.