The arcanophysics study hall was two floors below the main library, in a room that had been a storage space before someone decided the department needed more tables. It had no windows. The lamps were always on. Leo had been coming here since his first year because it was quiet and nobody came looking for him.
He was working on something else.
That was the thing worth noting, and he noted it himself, in the detached way he noted most things about his own behavior. He had a slate full of thermal expansion tables — ordinary coursework, second-year material he was reviewing for a student he'd agreed to tutor. The work was clean. The tables were correct. He hadn't written the word Cause in three weeks.
He hadn't opened the notebook with the crossed-out lines either. It was in his room, under a stack of other notebooks, and he hadn't needed it.
This was progress, or it was avoidance, and he'd stopped trying to tell the difference.
---
The student arrived at the fourth bell.
Her name was Petra. She was second-year, arcanophysics track, and she had the specific kind of anxiety that came from being good enough to understand most things and not quite good enough to understand all of them. Leo had tutored her twice before. She was easy to work with because she asked precise questions.
She sat down across from him and set her slate on the table. "I'm stuck on the conduit problem. The one from the assessment."
"I remember it."
"I got the answer right."
"That's good."
"I don't know why."
Leo looked up.
She was frowning at her slate. "The formula works. I put in the numbers and the answer comes out. But when I try to see why it works, I get stuck."
"Stuck where?"
"The middle term. Every derivation in the handbook shows it canceling out. But it doesn't say why it cancels. It just... does."
She looked at him.
"Is it just assumed?"
Leo set his stylus down.
"The middle term."
"Yes."
He picked up her slate and turned it toward himself. He read the derivation. He'd read it a hundred times. He knew where the assumption was hiding.
"The pressure gradient is linear," he said. "That's the assumption."
"But why?"
"Because the derivation was written for a specific case. It was generalized later without anyone checking whether the assumption still held."
"So it's wrong?"
"No. It's fine for the cases it was written for. It's approximate for everything else."
Petra frowned. "Then why doesn't the handbook say that?"
"Because the handbook is a handbook. It teaches the formula, not the history."
She looked at the slate. She was quiet for a moment.
"That's what I don't like about it," she said.
"What?"
"It works. I can use it. I get the right answer." She frowned at her own words. "That's not what I mean."
Leo waited.
"The formula doesn't care why I'm using it," she said. "It just gives me a number. And the number is right. But I don't know if it's right for the reason I think it's right."
Leo looked at her.
"You're asking whether the assumption is doing the work," he said.
Petra blinked. "Yes. That."
"It is."
She sat back. "Then how do I know when to trust the formula?"
"You don't," he said. "Not the way you want. But you can learn to notice when the assumptions are doing more work than the formula itself."
"How?"
"By finding the places where the formula stops working. They tell you more than the places where it does."
---
She worked through the problem again, with his help. He walked her through the derivation. The middle term canceled. She got the right answer. She thanked him and packed up her slate.
"You're good at this," she said.
"At what?"
"Explaining the parts nobody explains."
She left.
---
Leo sat at the table for a long time.
He hadn't lied to her, exactly. He'd told her something true. The formula worked the way he'd said. The assumption was the assumption. His answer was correct.
But he'd known the shape of her real question, and he'd walked around it. He'd given her the version that was safe to give. He'd given her the right answer.
Three weeks ago, he would have told her what he actually thought. That models fail when they stop seeing their own assumptions. That the interesting thing wasn't the formula — it was the invisible line where the formula stopped being true. That the middle term didn't cancel because the math said so. It canceled because someone had decided not to look at it.
He could have told her that. She would have understood. She was the kind of student who would have asked the next question.
He hadn't.
He looked at the slate. He looked at his stylus. He looked at the empty space across the table where Petra had been sitting.
He opened his notebook — not the one in his room, the one in his bag. He flipped past the thermal tables. He flipped past the crossed-out lines.
He looked at the word Cause.
He hadn't written it. It was still crossed out.
He looked at it for a long moment.
Then he closed the notebook and went back to the tables.