He saw her again on the third day.

  She was at the far end of the courtyard study tables—the ones near the eastern wall where the afternoon light came in at an angle good for reading and not much else. Three texts open. A fourth closed on top, spine-up, put aside in frustration rather than finished with.

  A notebook on her left showed lines crossed out and restarted.

  The page had the particular texture of work that kept arriving at the same wrong answer.

  She was not looking at the texts.

  She was looking at the wall.

  Kael had seen that expression before—not on her face specifically, but on the face of someone who had been running the same calculation long enough that the path had grooved into its own dead end. Not confusion. Confusion meant the problem was opaque. This meant the problem was visible and the solution kept sliding sideways at the last step. The mind was clear. The framework it was using was the problem.

  He identified the texts from ten meters away. Standard Affinity distribution series. All three: introductory, intermediate, and the applied supplement that was technically optional but that every serious student ended up needing, because the intermediate text covered theory and the applied supplement was where the theory became something you could actually use.

  The closed one on top was the applied supplement.

  He had read section four two days ago.

  He recognized the look on her face.

  He kept walking. Got four steps past her table.

  Stopped.

  The applied supplement's worked example in section four had a flaw in the derivation that produced the correct answer in standard conditions and broke down in non-standard ones. He had flagged it when he read it. He had assumed everyone who worked through the section would notice.

  She clearly hadn't noticed—or she had noticed and couldn't locate the error, which amounted to the same thing from her perspective.

  The three-hour investment she'd made in the problem was visible in the crossed-out lines on the notebook page.

  He turned and walked back.

  ?

  He had seen her twice before this, not counting the testing line.

  Once in the dining hall on the first evening—sitting alone in the manner of someone who had chosen to, not someone who'd had no other option. There was a difference in the posture: chosen solitude had a settled quality, a deliberateness. Once on the western path, the evening he'd mapped the campus, disappearing between two buildings toward the academic wing.

  He had not made an effort to place her beyond what the public records showed.

  He had filed what he observed, the way he filed everything—in the part of his awareness that tracked variables in the environment without deciding yet what weight to assign them.

  The expression she had now was the variable that resolved the weight question.

  She had Affinity 71, Capacity 68, Stability 62. She was in Block Two, which meant she'd tested at the upper tier of the cohort and been placed accordingly. She moved through the Academy with ease—not the performed ease of someone who needed the room to see them moving easily, but the genuine ease of someone for whom the room presented no particular obstacles.

  She had chosen the far end of the study tables, away from the main traffic, which meant she hadn't been there to be seen working. She'd been there to actually work.

  Three hours on the same problem.

  Not giving up.

  He turned and walked back toward the study tables.

  ?

  She noticed him when he was still six or seven steps away.

  Not startled. The straightening posture of someone returning from private thought to the social surface—the particular movement of a person who has been alone with a problem for a long time and registers another presence before consciously processing it.

  She had good peripheral awareness.

  He filed that.

  She didn't say anything. She waited.

  He liked that about her already.

  People who waited instead of filling the space with noise were people who understood that information arrived on its own timeline and that rushing it produced worse information.

  "Affinity distribution across a non-uniform field," Kael said. He glanced at the open texts to confirm what he'd identified from a distance. "The applied supplement has a worked example in section four that treats it as a boundary problem. The boundary conditions assume a uniform source."

  She looked at him. Then at the texts. "That's what I've been trying—"

  "The worked example is wrong." The same way he said everything—not as a performance, just because it was accurate. "It gets the right answer for the standard case by accident. The method breaks down as soon as the source field has any internal variation." He paused. He looked at the text.

  The ink in the supplement was clean and even—the ink of a book that had been read by many people but referenced rarely, used as a source of problems rather than as a source of understanding. The pages in section four showed wear at the edges: people had opened to this section frequently. They had worked the problems. They had gotten the right answers.

  None of them had checked whether the method that produced the right answers was actually correct.

  Or if they had checked, they hadn't had the infrastructure maintenance logs to tell them why it mattered.

  "The correct approach treats it as an integration problem, not a boundary problem," he said. "Same result, but the path is stable."

  Unauthorized reproduction: this story has been taken without approval. Report sightings.

  She looked at him for a moment.

  "Can you show me?"

  He pulled out the chair across from her and sat down.

  He took her notebook. Wrote three lines: the variable setup, the integration expression, the substitution that simplified the final step. Pushed the notebook back.

  She read the three lines.

  Then she read them again.

  Then she looked up at him.

  "That works," she said. Not with relief—with the particular flatness of someone who had just confirmed a thing they'd suspected. Which is different from discovering it. The discovery had happened three hours ago, when she'd first felt the calculation sliding sideways. She just hadn't been able to locate it.

  Now she had.

  "Why isn't this in the supplement?"

  "The supplement was written twelve years ago. The application it was designed to address used uniform field sources. The method is correct for that case." He looked at the text. "The Academy's infrastructure has been upgraded twice since then. The newer field configurations have internal variation. The supplement hasn't been updated."

  She looked at the text. Then back at him.

  "You read the infrastructure maintenance logs," she said. Not accusing—just reading the inference back to confirm it.

  "I read whatever I can access."

  She set the pen down. "Sirin Vael."

  "Kael Dorne."

  She already knew that. He could tell from the absence of any adjustment in her expression—no reorientation, no small recalibration of what she knew about him to include his name. She'd placed him before she'd spoken. Probably during the testing line, since she'd been in the first forty of the sequence and had had more time to observe the room before the day's performance anxiety crowded everything else out.

  She'd had time to learn the names.

  He filed that.

  "Block Four," she said. Not as an assessment—just confirming a fact she already had, the way you stated a coordinate to verify you were looking at the same map.

  "Block Two?" he guessed.

  "Block Two." She didn't ask how he'd known. She pulled the notebook back and began rewriting the problem from the start, using his three lines as the foundation. Her handwriting was fast and precise—the kind that came from someone who copied information as a way of thinking rather than as record-keeping. The act of writing it herself was part of how she processed it.

  Kael watched her work for a moment.

  Then he pulled the intermediate text from the stack and opened it to the section he'd already read.

  They sat in silence for twenty minutes.

  Not the silence of people who had nothing to say. Not the silence of two people who were managing the awkwardness of a new encounter.

  The silence of people who were both actually working.

  The afternoon light moved across the table as the sun shifted. A group of students passed behind them, loud enough to be noticeable and not loud enough to matter.

  Neither of them looked up.

  ?

  She stopped writing. Looked at the page she'd produced.

  Checked it against the three lines. Checked it again.

  "There are three other problems in this section," she said. "I've been doing all of them the wrong way."

  "The same error propagates. Once you have the correct substitution the others resolve quickly."

  She turned to the next problem. Worked it with the integration method. Arrived at the answer in a fraction of the time. Compared it to the supplement's answer key.

  "It matches," she said.

  "It will match. The supplement's answers are correct even when the method is wrong—in the standard case, the two approaches converge. The difference shows up when you apply them to a non-standard case."

  She looked at him. There was a quality in the look now that hadn't been there at the beginning of the conversation—not warmer exactly, but more present. The look of someone who had stopped performing evaluation and started simply being in the exchange.

  "Where do non-standard cases appear?"

  "In actual field environments. Not in examinations." He paused. "Not in any examination format that's been used here in the past eight years, based on the past exams available in the library."

  "You read the past exams."

  "I read what I can access."

  She was quiet. Not processing—she'd already processed it. The quiet had a different quality: the quality of someone who had arrived at an implication and was deciding whether to state it out loud.

  The implication was this: the Academy's examination system tested a version of the material that would never fail. The method in the supplement was wrong, and it produced the right answers on every exam ever administered here, because every exam ever administered here used the specific case for which the wrong method happened to work.

  That meant the Academy's highest-scoring students had been trained in a method that was correct for the narrow context they would be tested in, and that would break down the first time they used it in a real field environment with internal variation.

  Which meant the ranking system was selecting for performance on a test.

  Not for understanding of the material the test was supposed to measure.

  She looked at him.

  "You've been in the library," she said.

  "Every morning before class."

  "How long have you been at this?" Not what she'd asked, exactly. The real question underneath was: how long have you been thinking about what I'm now thinking about, and why haven't I seen you before?

  "Since before I arrived here," he said.

  She held that for a moment.

  "The formula for Field Response under variable load," she said. "In the testing curriculum. There's an assumption baked into the standard derivation."

  She looked at him again. A different look than the first—not evaluating a stranger, but revising a number. The revision was visible as a small shift: the expression you made when a prior estimate had been too conservative.

  "The formula for Field Response under variable load," she said. "In the testing curriculum. There's an assumption baked into the standard derivation."

  "Two," Kael said. "The second one is harder to see."

  She blinked once.

  "The second one doesn't show up in the introductory texts. It's in the calibration documentation. Restricted section."

  "You don't have access to the restricted section."

  "Not yet."

  She held his gaze for two seconds.

  He held hers.

  There was a question running in that exchange that neither of them was stating directly. He had helped her solve a problem. She was now deciding whether what she'd seen—the quality of it, the speed, the fact that he'd known where the error was before she'd told him anything about it—was worth reciprocating.

  He let her decide.

  Then she looked back at her notebook.

  "The library has a faculty liaison who processes access requests. Students with documented research applications can apply for a provisional review."

  "I know."

  "The process takes twelve days."

  "I know that too."

  She wrote a line in her notebook. Not about the problem she was working—a name and a room number. She tore the page at the seam and slid it across the table without looking up from the next problem she was starting.

  Kael looked at the paper.

  The name was an administrative contact two levels above the standard library process. Someone who handled edge cases—the kind of access requests that didn't fit the standard forms because the standard forms hadn't been designed with the requester's situation in mind.

  Someone, in other words, who could move faster than twelve days if the documentation was complete and the research application was solid.

  He picked it up and put it in his notebook.

  "The third problem in the section," Sirin said, pen already moving, "uses the same substitution?"

  "Modified for the boundary case. The variable range shifts." He turned to the problem and pointed at the line. "Here."

  She nodded and kept writing.

  Outside, the afternoon light held steady for another hour before the cloud cover moved in.

  Neither of them noticed.

  ?

  He had known, before he turned and walked back to her table, that the exchange would cost him something.

  Visibility was a cost. Sirin Vael was Block Two, Affinity 71, Capacity 68—someone who moved through the Academy with access and ease and notice. Any association with her would be noticed, would be interpreted, would feed into the social map the cohort was building in real time about who was who and what the relationships were.

  He had made the decision anyway.

  Because she had been running the same wrong calculation for three hours because the framework she'd been given was incomplete. And because the expression on her face, when she looked at the wall instead of the texts, was not the expression of someone who had given up.

  It was the expression of someone who refused to give up even when they couldn't locate the problem.

  That was a way of operating he recognized.

  And because the contact she had written on the torn page was two levels above the standard process.

  That was useful.

  He put the notebook away and returned to the integration problem.

  There were three more in the section.

  He had enough time before dinner to work through two of them.

  He intended to do exactly that.

  Whatever else the afternoon had produced—the contact on the torn page, the updated model of Sirin Vael as a variable with significant potential weight—the work in front of him was still the work. The contact would be useful when he had the documentation to accompany it. The updated model would become relevant when he had more data.

  Right now he had three problems in a section, a corrected substitution, and sufficient afternoon light.

  He turned to the next page and began.