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On a ruined trail near Rawlins he followed the path they’d followed that winter, blinking his eyes into the stiff wind. On the banks of a river east of Cheyenne, where hand in hand they’d sat on a boulder tossing rocks onto the snow-dusted ice, he sat tossing them into the rushing current. Each one splashed and vanished. She vanished alongside, down into the black. On his third night on the road, just past the Nebraska border, he pulled in at the same bright diner, where the same middle-aged waitress refilled his barley soup but didn’t come back with the bread.

When he finished, he found his way to the bathroom. Took a long pull from the flask. Before the same chipped mirror he examined his face. It had been hardened. It had been turned to stone.

Something had been removed, too. What remained was ambition.

He took another pull. Back in the dining room, the waitress was already wiping the table. He stood off to the side until she looked up at him and smiled. “Where’s your friend?” she said. “You leave her somewhere?”

He lifted the jacket off the chairback and worked his arms into the sleeves. “California. Things didn’t work out.”

She swept the utensils into her palm. “You’ll find another.”

“I don’t know.”

“Well, I do,” she said, leaning to push in the chair. “You’ll find another.”

HE ARRIVED IN Cheboygan just as his parents were sitting down to Sunday dinner. His mother’s hand went to her mouth. His father reached to the china cabinet for another plate.

Almost a month remained before the university returned from break. The next morning, the phone rang. It was the registrar’s office. His sections had gone untaught. The police had been sent to his address. Did he wish to withdraw? Four days later, a certified letter, asking the same question. His fellowship money was on the line. Finally, a call from the dean. His father fielded it.

He did his work out in the woods. Frozen days. Bright snow. His thirty-second year on earth — late, in fact, for a mathematician. Wool coat. Spiral notebook. Flask. He’d brought the chain home with him, and on his first day in the woods he set it back in the maple where it belonged. After that, a month spent nearly entirely outside, in one of his old shelters under the trees. He wasn’t hungry, and he barely needed sleep.

He was going to make her sorry.

“You look terrible” was the first thing Hans Borland said when he walked back into his office. One day still remained of the break. The campus was quiet, the students just beginning to filter in.

“Kamil Malosz and I have been in battle,” said Andret.

“Good, good. I can see it.” Instead of sherry, the professor pulled a bottle of whiskey from the cabinet and cleared off his desk. Two glasses, filled to the top. “And have you defeated him?”

Andret gulped the whiskey, then laid his notebook beside the glass. “Yes,” he said. “I believe I have.”

3. Contraposition

Fine Hall

PRINCETON UNIVERSITY. MILO arrived on a day of hail — billions of white spheres the size of gumballs bouncing off the lawns and streets of central New Jersey as he carried his new briefcase across the quad. A miracle. Blue sky. Hot as a clothes dryer. And suddenly hail was bouncing up from the walk in a skittering dance higher than his belt — corn in a popper. Within a few seconds the brim of his fedora had filled with it.

Then, just like that, it stopped.

He’d bought the hat — a Borsalino — along with the dark suit, for the commencement of his position. Borland’s advice: Don’t look like the rest of them. Go against the times. And this was what he was doing, striding among the long-haired students, leaning down now and then to pick up tiny melting fragments of the universe. Around him, the dowdy-looking faculty toed back kickstands and resumed their rides. The undergraduate boys emerged from under the eaves to boot a Hacky Sack again in the electric brightness of the storm’s wake.

Dr. Milo Andret, Ph.D.

He’d had time to think. The life he’d been living — the quatrant, the deadbeat parties, Cle’s obtuse, demanding visions — it was all behind him now. From this point on he would live by different rules. He’d sold the Valiant and tossed away the old clothes. Now he wore a bespoke suit and an astringent cologne. The world parting as he stepped. He wouldn’t miss a molecule of what he’d been through.

The briefcase had been a gift from Borland on the occasion of his dissertation defense, which had been a triumph. As he reached Fine Hall, he set it down, removed the hat, and poured the hail from the brim into his palm. Little messages from the stratosphere, crenellated oblongs from the heavens.

In the departmental office he found a trio of secretaries — a pair of blondes in sweaters typing at the desks up front and a darker one in back, her head down. He held out his palm.

One of the blondes said, “You brought us candy.”

“Wish I had,” he answered. “To tell the truth, it’s hail. Heavenly mothballs. You don’t get this kind of thing in Berkeley. It’s amazing, really.”

“Relatively speaking, I guess,” said the other blonde, not looking up from her typewriter. The first one laughed brassily.

“Well, it might be if you looked at it,” said the dark one in back, still not lifting her head.

“Might be,” said one of the blondes, glancing up at the wall clock. “But I was hoping it was candy.”

A hush. It was a Friday afternoon, just a few minutes before five — he glanced up at the clock himself now for the first time. It seemed that everyone else in the building had left for the day. “Well,” he said, “I was just hoping to get the key to my office. I’m Milo Andret. The new hire. I just got here.” He dropped the melting remnants into the trash can.

The two blondes went on typing. One of them looked at the clock again.

“Well,” said the dark one in back, “if nobody’s going to help Professor Andret, then I will.”

“I’m not a professor,” he said a few minutes later, after she’d ridden up with him in the elevator and unlocked the door to his office. It was a nice-sized room with two windows over a curving footpath and a view of a sports field truncated by evergreens. Princeton had recruited him heavily. Stilclass="underline" it was more than he’d expected. “I’m an assistant professor,” he said.

“Me, too,” she said, brushing the hair from her eyes. “I mean, I’m an assistant secretary.”

A FEW MONTHS before, within hours of showing the proof to Borland, rumors of the achievement had begun to spread. Soon after, the paper had been accepted by the Annals. Publication in October: an unheard-of turnaround. At thirty-two years old, he’d found a solution to one of the great problems in the history of mathematics. The article would arrive next month in libraries around the world: the Malosz conjecture, thanks to Milo Andret, had become the Malosz theorem.

He thought briefly of Kobayashi and Timofeyev.

On his first day on campus, he walked around unrecognized. The pressed suit. The fedora. He had the feeling that he was someone else, that he’d been handed a disguise. Even in the mathematics department, only the dark-haired secretary, whose name was Helena Pierce, paid him any notice.

That Monday, his first day at work, she showed him around the building. The semester didn’t start until the following week, but his mailbox was already filled with letters. “A lot of departmental duties, I guess,” he offered.

“Yes,” she said. “Or no, actually.” She blushed. “Probably not that many, at least not now. Chairman Hay tries to give the junior faculty time for their work.” She brushed the hair from her eyes again, then pointed to the slots on either side of his name. “Not that you’re junior.” The blush deepened. “In title, maybe. I read about the Malosz theory, I confess. Congratulations, Professor Andret.”