Conjecture

I

At 11:47 on a Thursday night in November, Vasudev Rao gave four years of his life to a machine, out of irritation, expecting nothing.

He had been at the board since half past seven. He had not eaten since noon. The generator had come on twice during the evening and each time the fan had stopped and the room had gone thick and close before the lights caught again, and there was a mosquito he had been failing to kill since Monday.

He typed in the definitions. He typed in the two lemmas that had survived. He started on a paragraph about the obstruction, and found, three sentences in, that he was describing his own failure more clearly than he had ever described it to a living person — because the machine did not know him, had no view about him, would not mention it to anyone at a conference in Kozhikode, and there was nothing whatsoever to protect.

He wrote for eleven minutes. When he stopped he read it back and thought, with something close to vertigo: that is the first honest account of the last four years that anybody has been given.

Then he refilled the tumbler from the flask and stood at the window while the rain came down through the light of the corridor lamp.

What came back was not a proof. It was a sketch, six lines long, and the fourth line was wrong in a way he could see from the doorway.

He was pleased about this. He was more pleased than a man of fifty-one should be about anything at midnight. He typed: line 4 fails, the sum is not absolutely convergent.

The sketch changed. The bad branch was gone, without apology and without embarrassment, and in its place there was a question.

II

There is a stage of being stuck that nobody describes properly to research students. It is not the dramatic stage, where you are still angry and still at the board at midnight and still believe the problem is withholding something from you personally. That passes. What replaces it is domestic. You come in, you do the estimate, it fails in the same place, you walk down for filter coffee in a steel tumbler and drink it standing up because sitting down would make it a break, you answer three mails about the examination schedule, you go home. Four years is long enough for that to become the shape of your days. It is long enough to write six perfectly good papers on other things and still feel, standing in a room in November with the fan going, that you have not worked on anything at all.

Arjun Sarkar had leaned in at half six that evening on his way out, helmet in hand.

“Still up there?”

“Still up there.”

“You know there are machines for this now.” He was checking his phone for the rain. “My nephew told me. We are finished, all of us. I have decided to be relieved.”

“You do PDEs, Arjun. Nobody was ever coming to save you.”

“That is the correct attitude.” He put the helmet on and did not fasten it. “Shantha says you should eat.”

A minute later Vasudev heard the cycle bell twice at the turning, which was Arjun’s idea of a farewell.

The notebooks were on the shelf in order and he could date them by season without opening them. The green one is the summer he was certain induction would work. He woke at five most mornings that summer because the problem had already started without him, and he would stand at the window of the flat in the faculty quarters with the light coming up and the first crows going over, feeling something he could not have explained to anyone at breakfast, which is that the argument wanted to close. Every mathematician knows that sensation and not one of them can defend it. You feel the two halves reaching for each other. It is not evidence, it has been wrong before, it has been wrong for other people in ways that cost them ten years, and it is the reason any of them keep going.

It was wrong that time. The blue notebook is the winter he proved it could not work, and there is a page near the back of it where he wrote one line and then nothing else for nine days. He came in on every one of those nine days. He taught, he answered his post, he wrote a referee report of eleven pages on somebody else’s paper that was the best mathematics he did that month. Then he walked home across the campus in the dark, past the tower and the statue of the old man with his hands behind his back, and did not open the notebook. He has never thrown it away.

III

K. S. Ramachandra Murthy gave him the problem in 1998.

Nobody in the department had called him anything but KSR for forty years. He was at the blackboard on the ground floor on an afternoon in May with the windows open and the fans going and the noise of the road coming in, and he had chalk on his sleeve, as he always did, and dust in the hair on the backs of his hands. He drew Halvorsen’s estimate from memory, got a constant wrong, and when Vasudev pointed it out he laughed and left it up.

“Somebody should look at Halvorsen,” he said. “It is not hard. It is only deep.”

Vasudev was twenty-three and heard a compliment. It took him ten years to understand that he had been warned, and by then the warning had been accepted and the years had been spent.

KSR never worked on Halvorsen himself. He had a problem of his own that he carried from 1971 until he retired and did not solve, and he never once spoke about it in public, and there is a folder in the departmental store room with his initials on the spine that nobody has opened since 2006.

In September, Vasudev went to see him in Malleswaram. It is a ten-minute auto from the gate, less if the driver takes the lane behind the market, and this is a fact Vasudev found himself thinking about a great deal afterwards: that the man was ten minutes away, and had been ten minutes away for years, and the distance had never been the problem.

The house is the son’s, on the second floor, with the television on in the next room and a grandson doing chemistry at the dining table and the smell of something being fried at four in the afternoon. KSR was eighty-eight and courteous and entirely himself for the first hour. They talked about a paper of his from 1984. He remembered the referee’s objection, and the referee’s name, and what he had written in reply, and he was still pleased with what he had written in reply.

Then he asked what Vasudev was working on.

Vasudev told him properly. He put down the coffee and stood up and used his hands, the way you do at a blackboard when there is no blackboard, and KSR listened with his head tipped to one side and said hm twice, both times in the right place.

Eight minutes later he asked again what Vasudev was working on.

The daughter-in-law was in the doorway with a tray. The grandson had stopped writing. It was very clear that they had all been here before and were waiting, kindly, for the guest to find his way out of it.

“The same as before, sir,” Vasudev said.

He walked back to the campus instead of taking an auto, in the heat, and stopped at the gate and stood under the trees for a while. It was not the forgetting. He had prepared himself for the forgetting on the way there. It was that in the second asking he had not been able to make himself say it again. Eight minutes, and he had not been able to give the old man eight more.

IV

Halvorsen’s conjecture is not a hard thing to say out loud.

You take a whole number, you do something arithmetic to it, and you land somewhere on a circle. Do that for every number up to n and you have n points scattered around the circle, and to the eye, and to every test anyone has thought to apply, they look thrown by hand. Halvorsen said in 1961 that they are random in one exact sense. Take any arc you like, count the points inside it, and compare that with the number you would expect if they really had been thrown. The gap never grows larger than the square root of n, times the logarithm of n.

Not twice that. That.

Vasudev could prove twice that. He had been able to prove twice that since the summer of 2021. Four years on a factor of two.

The reason is that the points can bunch at any scale — in a wide arc or a narrow one — and the bunching at one scale can quietly borrow from the bunching at another. Ruling out the conspiracy at a single fixed scale takes an afternoon. Vasudev had done it in his second month and had been briefly, stupidly happy. Ruling it out at every scale at once, with one constant that does not degrade as the scales get finer, is the whole of the problem. It is where every approach anybody has tried goes to die.

The question on the screen was this.

Can the fourth-moment bound be made uniform in the parameter?

Vasudev put the tumbler down.

It is a good question. It is the question. It is, in fact, the question he should have asked in 2023 and did not ask, because in 2023 he was certain the parameter was a nuisance rather than a door, and he was certain of it in the specific way that stops a man from checking.

I do not think it can, he types. That is the obstruction.

What goes wrong at the small scales?

He explains what goes wrong at the small scales. It takes twenty minutes and two false starts. Halfway through he stops typing, goes to the board, works out something he has not needed to work out in four years, comes back and corrects his own second sentence.

They go four rounds and it takes two hours and it is, he will admit later and only to Meghana, one of the better mathematical conversations of his life. He kills two branches. It kills one of his — the estimate he has carried since the green summer — and it is right to kill it, and he sits looking at the screen for some minutes before he types yes, agreed, which is a thing he has not typed to another person in a long time.

Then, in the third round, it reaches back into a paper he published in 2021. An obscure one. Eleven citations, three of them his own, the kind of paper you write to get a definition out of your head so that it will stop following you around. It takes the object he defined there and sets it against a theorem from a corner of harmonic analysis he has never worked in and could not name one specialist in.

He reads that line four times.

That is the move. Everything after it is bookkeeping, and the bookkeeping is eleven pages, and by half past one it is done.

V

Then he did the thing he would find hardest to explain to anyone who had not lived inside the old arrangement.

He read the eleven pages the way he had read every proof for thirty years. It is a physical activity. You go slowly. You put your thumb on the line above so that your eye cannot run ahead of you. And at every step there is a small internal event, almost muscular, when the step closes — a click, a settling, the sensation of weight coming down onto something that will take it. That is the entire pleasure of the profession. It is what Vasudev felt at twenty-two in a seminar room on the ground floor of this building, understanding somebody else’s argument for the first time and going out afterwards into the four o’clock light unable to sit down. He has felt it perhaps four hundred times since. If he were pressed, and he would never be pressed, he would tell you it is the closest thing he has to a religious life.

It closed. Every step of it closed.

Then he went home, because there was nothing left for him to do.

The proof agent worked until four. It found two gaps in the elaboration, in places he had read straight past without a flicker, and repaired both while he slept badly with the window open and the rain starting again. At seven there was a green tick and a file of three thousand nine hundred lines that nobody would read from beginning to end, including him, and he sat on the edge of the bed with the phone in his hand looking at the tick, which is the same small green tick that appears when a parcel has been delivered, and he thought, with the flat clarity that arrives at that hour before anything has been decided about the day, that for thirty years the sentence I can tell whether this is true had been the load-bearing part of what he was, it was the reason for the committees and the referee reports and the recommendation letters he wrote for other people’s careers at midnight, it was the reason a journal in Copenhagen sent him things and waited eight weeks and never complained, it was what he had instead of the other things people have, and it had gone out overnight like a bulb, and there was a green tick where it used to be, and he was still sitting there when the water heater clicked on.

VI

He was back in the building by half past seven. The thing he wanted to do, and the wanting arrived well before the thinking, was to call KSR. He got as far as the contact, and then stood in the corridor with the phone lit in his hand and worked it out in stages, the way you work out that something is over. Ten minutes away. The son would answer. KSR would be courteous. KSR would ask what he was working on, and would mean it, and would ask again eight minutes later and would mean it then too.

He put the phone back in his pocket. What he wanted was not to report a result. He wanted to say it out loud to the one man alive who had ever handed him a constant with a mistake in it and told him the thing was only deep.

There was nobody else, and that is the shape of the life he chose at twenty-three. You take a problem that four people in the world can hold in their heads. One of them is your advisor. One died in 2019 and you found out eleven days late from a mailing list, while deleting other things. One is Jon Aasheim in Trondheim, whom you have met twice and liked both times.

At ten past eight, which was half past four in Norway, he called Aasheim. It went well. That was the trouble with it.

Aasheim was delighted — immediately, without reservation, with a shout that made Vasudev hold the phone away from his ear — and wanted the file, and had two questions about page nine that were better than any question Vasudev had asked himself all night. And then he said, cheerfully, that he had been running his own problem through one of these since March. That it was very good on the tedious half and hopeless on the interesting half. That it had found an error in his 2018 paper, which he had decided to be grateful about. That he had signed the Leiden thing in June mostly so he could stop thinking about it.

“You will stop minding,” Aasheim said. “It takes about six weeks.”

“I am not sure I want to stop minding.”

“No,” Aasheim said. “Nobody does. Send me the file.”

It took Vasudev some time afterwards to identify what he was feeling, because it was not grief and it was not envy. It was that Aasheim had made his peace nine months ago and had not thought it worth mentioning, and that the world had gone on and settled without him while he was standing at a blackboard being defeated by a factor of two.

And then, in the twenty minutes before he opened the file again, he did a piece of arithmetic he would not tell anybody about for a long time.

He worked out how much of the eleven pages was defensibly his. The object was his, from 2021. The formulation was his. Four of the definitions were his and one of them had taken him a year. The machine had supplied a connection and some bookkeeping. If a paper appeared with his name on it and an acknowledgement of computational assistance, in the way people acknowledged a cluster in 2015, nobody in the world could say the sentence was false.

He got quite far into this. Far enough to think about which journal.

What stopped him was not conscience. It was that he had already sent the file to Aasheim.

He sat down at that point and was ashamed in a way he had not been for years — not the clean shame of a mistake in public, but the private kind, where you learn something about the size of yourself and there is nobody to tell and no way to give it back.

VII

He had seen the Wiles documentary at twenty-one, on a VHS somebody’s brother had carried back from England, in the seminar room downstairs with eleven other people and a monsoon going on outside. The tape was a copy of a copy. He remembered that the man had looked away from the camera because he could not go on, and that nobody had said anything for a while afterwards. What he had wanted, at twenty-one, was a proof you could not separate from a person.

Would he have found it?

He turned it over on the stairs, and in the office, and again with his hand flat on the chalk tray. The connection ran through his own paper. The analysis side was two seminars away from things he already knew. In 2019 he had sat in a talk at TIFR with the relevant theorem on the third slide and he had been thinking about lunch, and he could remember what he had for lunch.

He thought he would have found it. He thought so. He was not sure. Perhaps in 2031, in some clumsy form, and perhaps only ever the clumsy form, and then he would have died without knowing that the clean one existed.

That is the part that would not settle, and it was not that the machine had been faster. It was that he could no longer establish what he had been capable of. A man is entitled to know the size of his own ceiling. His had become a thing that could not be tested, and would stay untested now for the rest of his life.

He waited for the grief. What arrived instead, plainly and without dignity, was relief.

VIII

And then, at twenty past eight, page seven.

There is a lemma in the third line of page seven that the system had not flagged, because it had used the thing the way you use a shelf. It says that the conspiracy at one scale and the conspiracy at another are not independent — that they are two faces of a single object. Since 1994 the field has assumed the opposite. It has assumed it so completely that the assumption does not appear anywhere in the papers. It appears in the way the papers are organised.

Vasudev read it. He read it again. He stood up fast enough to knock the chair over behind him and did not pick it up.

If it is right, the obstruction was never an obstruction.

He sat with that for a long time, and it was not the good feeling he would have expected. There is a way of losing four years that is worse than failing at something hard, and it is finding out that the hard thing was not there. Every morning of those four years, the estimate at twice what he needed. Every one of them spent walking around a wall that was a shadow, thrown by a structure nobody has troubled to name.

There are at least six other problems standing in that shadow. Two of them are famous. One of them is in a folder in the store room with KSR’s initials on the spine.

The machine had found the door. It had no idea it had walked through a wall.

And here was the part he could not get past, sitting there with the chair still on its side. He had always known he was not the cleverest person in his cohort, and had never minded, because he believed the other thing about himself, the thing you cannot put in a CV: that he chose well. That out of everything he could have spent his life on, he had picked a problem that was worth it and had held on when other people let go.

Page seven says he picked a shadow. The taste was wrong too. The taste was the last part he owned.

He knew the comfortable version of all this. Proofs go cheap, understanding stays expensive, understanding is the human part, everybody keeps their office. He had sat through three of those talks in the past year and had noticed, without saying so to anyone, that they were all given by men his own age. Nobody had yet shown him why why does this belong here should be permanently harder than is this true. What he had was a wager: that noticing which consequence ought to stop your breath requires caring what happens next, and the thing on his desk did not care what happened next. The wager might hold twenty years. It might hold until March. He preferred to write it down as a wager rather than dress it up as a principle, because in 1994 the field wrote down a principle and this morning it had turned out to be a shadow.

IX

Meghana Hegde arrived at nine, shaking out an umbrella that had failed.

She took in the two crowded boards, the chair on its side, yesterday’s shirt on him, and asked whether he had done it.

“There is a proof,” Vasudev said. “It is correct. I have read it twice and I cannot tell you why it works.”

She waited, the way she had learned to.

“Page seven,” he said. “Start at page seven.”

He gave her the chalk and his hand was not steady, and she did not remark on it. The first stroke broke the chalk, the way it always did, and she took a fresh piece from the tray and stepped up to the clean half of the board.

They worked until eleven. What they had by then was wrong, and wrong in a direction so interesting that neither of them noticed the rain had stopped, and at one point Meghana said no, wait, wait and put her hand up flat to stop him talking, which is the best thing that can happen in a room. Shantha came in at ten with two coffees and went out again without saying anything, which from Shantha is a form of blessing.

At eleven Meghana put the chalk down. “Sir, can I say something.”

“Say it.”

“You have been like a funeral since I came in. And I understand, I am not saying I do not understand. But I have been standing here two hours and what I keep thinking is that this thing sat in front of you for four years and told you nothing, and last night it took two hours, and you are grieving the four years.” She stopped. Then, because she was twenty-six, she went on. “The four years are the thing that was wrong, sir. Not last night.”

He said something then that he would think about for a long time afterwards, which was: “You have never had four years.”

It landed exactly as badly as it deserved to. She went slightly still, and picked up her bag, and said she would come back at two, and he let her go without saying the other sentence, the one he had ready, because he was fifty-one and tired and had run out of whatever it is that makes a man apologise on the same morning.

She came back at two, because she was better than he was.

And at half past three, with the boards half rubbed out and the light going flat and yellow across the floor, she asked him for a problem. She had six months of coursework left. She wanted to know what she would be spending her twenties on.

Vasudev did not answer for a while. He knew four problems he could give her that afternoon, and two of them were from the list of six standing in the shadow, and any of them might be gone by the time she finished, taken in an afternoon by something that would not know it had taken anything. He knew, also, exactly what it costs to be handed a problem in this building by a man who is being generous and does not entirely understand what he is doing.

“I am afraid to give you one,” he said finally. “That is the honest answer. If I give you the good one, it may not be there in three years. If I give you a safe one you will finish it and you will never once feel what I felt at five in the morning in that summer, and you will not know what you missed, and I will know.”

“Sir,” Meghana said, “I am going to spend my twenties on something.”

So he went down to the store room and found the folder with KSR’s initials on the spine, which nobody had opened since 2006, and brought it up and put it on the desk in front of her, and told her whose it was and how long the old man had carried it and that it was standing in the shadow and might now be reachable, and that he did not know whether he was giving her a gift or a bill.

She opened it. The first page was in a hand he had spent his whole adult life learning to read.

“Then I want this one,” she said.

X

At five he took an auto to Malleswaram.

The television was on in the next room and the grandson was doing chemistry at the dining table, and the daughter-in-law brought coffee and did not stay. KSR was in the chair by the window. He was courteous. He was entirely himself.

They talked about nothing in particular for a while, and then he asked what Vasudev was working on.

Vasudev told him. He put down the coffee and stood up and used his hands. He did not mention the machine, because it was not the interesting part and there was not time for everything. He described page seven. He described the shadow, and the problems standing in it, and he said that one of them was the thing in the folder, and that a girl of twenty-six with a very good head had taken it that afternoon and would probably be difficult about it in the way the good ones are.

KSR’s face did something then that Vasudev would think about for a long time afterwards and would never be certain he had read correctly.

“Good,” he said. “Good.”

They sat. Outside, the light went off the terrace opposite and somebody started a scooter in the lane and the crows came over the way they do at that hour, all at once, and then it was quiet again.

Eight minutes later the old man turned to him and asked what he was working on.

And Vasudev Rao, who was fifty-one, and who had been given a problem in 1998 by a man with chalk on his sleeve, pulled his chair a little closer, and told him the whole thing again, from the beginning, slowly, leaving nothing out.