Page 73 of Proof By Contradiction

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Zero mornings after. Zero dedications.

Laurence has beenhis.

This is the first time I have felt this. It’s not jealousy of Hugo, exactly. It lives a story below jealousy. The feeling you get when you realise the person in bed next to you has lived in a house you have never been inside. Rooms. Corridors. A front door that opened for someone before it opened for you.

Hugo has been in that house. The house holds the furniture Hugo chose. A hallway, Hugo walked down barefoot, on his wayto hand Laurence a book. A kitchen where Hugo made coffee without being asked. A bathroom where his aftershave sat on the shelf behind the towels for years until Laurence moved cities and put it back on a shelf behind new towels because throwing it out would mean deciding about Hugo.

I have never had a house. I have had doorways.

My empty shelf, his full one.

And the thing I can’t say out loud, the one sitting lowest in my chest, making it hard to sleep with his arm across my waist and his breath in my hair: I don’t even know what word I would put on him.

Your Ewan scrawled into the flyleaf of a book I do not own, about a country I have never visited? My mouth won’t hold it. I don’t have the weight for that word yet. I am eighteen, and my whole romantic vocabulary isyeah,nice one,cheers,see you around, and none of those are a pen pressed into the front page of a book with the wordyourat the bottom.

He has been somebody’s. I have been nobody’s.

The asymmetry is not fixable by being smarter, funnier, or better in bed. It is a thing that happened to him and did not happen to me.

I press my face into the pillow. The pillow smells of him. Specifically, of the shampoo he uses, which I could now identify blindfolded at forty paces.

The ceiling has had enough of me.

A week passes. Six days, Thursday to the following Wednesday, during whichyou could do extraordinary things with that mindthat lives in my skull like a second heartbeat: his words, my pulse.

I try to ignore it.

Can’t.

It arrives unbidden on the tram, and in the library and during a lecture on aggregate demand that I fail to take a single note in, and on Sunday night I’m sitting cross-legged on the floor of my room at Fallowfield with a notebook on my knees I haven’t opened since I left Lewisham.

The notebook is A5, hardback. Black. The edges were soft from the inside of a rucksack it lived in for three years. The cover’s got a faded ring where a can of Red Bull sat on it in Year 12 and a smaller ring where a second can sat on top of the first. I bought it with my own money at a Rymans in Catford in 2022 because the maths I was starting to do wouldn’t fit in the margins of the A-level textbook, and I needed somewhere to put it.

Inside: eighty-seven pages. My handwriting aged from fourteen to eighteen. Problems I set myself, proofs nobody read. A whole section on sequences, patterns nobody documented, because at sixteen I didn’t know you could look things up on the internet for free and check if someone else had already thought of them. I just thought of them.

The last entry is three weeks before I arrived in Manchester. I stopped opening it on results day. Couldn’t make myself write in it once I had an offer from a place I wasn’t excited about.

I open it now, turn the pages. Read myself at fifteen, at sixteen, at seventeen. The confidence of a kid who had no teachers watching and no examiners judging, and was doing this for nothing except the pleasure of following a line to somewhere true.

Page forty-three. A problem I set myself in the summer, I turned seventeen. A question about convergence, not the ones the textbook gave me, a harder one, a version with a twist in the hypothesis I couldn’t resolve and kept coming back to. Threeattempts on three pages. All wrong, I remember the frustration. I remember closing the notebook and not opening it again for weeks.

I remember the particular flavour of that frustration, too. It had a conclusion attached:maybe I’m just thick.The version of the thought that sits in a seventeen-year-old’s skull pretending to be modesty. The version that whispers that the gap between what you can do and what the actual ones can do is the gap between a boy in Lewisham and a boy who went to the right school, so stop wanking on about it. I closed the notebook and agreed with the frustration, and I haven’t opened it in twelve months because opening it would have meant risking that the frustration had a second conclusion I hadn’t reached.

I read the three attempts now, at eighteen, in a narrow bed in Fallowfield, with weeks of Laurence’s lectures on Mathematical Methods in my head and the problem sets I’ve been handing in humming in my fingers, and I see it.

I see it immediately. The mistake I was making at seventeen. The thing I was building towards. The half-step I was missing that would have taken me all the way.

I pick up a pen. My old handwriting underneath. My new handwriting on top, in a different colour, picking up where seventeen-year-old me gave up—three lines, a substitution, a reframing. Then the collapse, six steps condensed into two, and the conclusion settles.

The proof works. I write QED as I used to, with a little square at the end. I’ve always liked drawing the little square.

I look at the page. At eighteen-year-old me, finishing what seventeen-year-old me started.

A knot under my sternum. A thing I haven’t named since a sofa in Chorlton last Thursday.

I close the notebook. Put it in my bag.

Tuesday. Eight PM, his kitchen.