Arithmancy
Professor Vector had the specific energy of a mathematician who had found their ideal subject — someone for whom the pleasure of the work was entirely genuine and who had spent long enough in a field she loved that the enthusiasm had become structural rather than performed. Her classroom had a different atmosphere from Babbling's — cleaner lines, less accumulated material, the blackboard covered in the particular dense notation of someone who thought in symbols and had organized the room to support that thinking.
She opened the first lesson with theory, which was correct. Arithmancy without the theoretical foundation was numerology, and numerology was not what this class was. She covered the basic numerological framework, the correspondence tables, the way magical numbers operated differently from mathematical abstractions in ways that required both frameworks to be understood simultaneously.
He listened and took notes and did not let his face do anything that would indicate the gap between where he was and where the lesson was pitched.
The gap was significant.
He had come to Arithmancy from an engineering degree in a previous life, which meant he had arrived at it with four years of applied mathematics at a level that made the third-year curriculum feel, in certain specific respects, like familiar ground. The magical elements were genuinely new — the correspondence tables, the way number theory intersected with spell construction, the specific properties of the prime sequence in magical contexts — and he gave those his full and genuine attention. But the underlying mathematical structure was not new, and the speed at which he could move through the new material was correspondingly faster than it had any business being for someone who had never formally studied the subject.
He had spent weeks in July and August at both the Burrow and in Egypt working through the third-year to fifth year theory , and he had arrived at Hogwarts at a point that was somewhere in the middle of fifth-year material. He knew this. The Room's bookshelf had confirmed it in the first week — the third and fourth year texts had barely occupied him, and the fifth-year material was where the genuine engagement had started.
The first lesson's exercise was to work through a basic correspondence table for a given word and derive the numerological profile from it. He worked through it in eight minutes. Then he went further, without being asked, and tested the profile against the magical correspondence tables in the reference section of his workbook, cross-referencing against the prime sequence properties to check for consistency. The cross-reference produced an anomaly in the correspondence table — not significant, and one he suspected was a deliberate simplification in the introductory version — but present.
He made a note of it.
He was working through the implications when Vector arrived beside his desk.
She looked at his workbook with the same quality of attention Babbling had brought — the focused reading of someone who had found something they hadn't expected.
"You've finished the exercise," she said.
"Yes, Professor."
"And gone considerably beyond it."
"The correspondence table has a simplification in it," he said. "In the prime sequence section. I was checking whether it was intentional."
She looked at the note he had made. "It is intentional," she said. "The full table is introduced in the fourth year once the foundational framework is established. The introductory version simplifies the prime sequence interactions to make the basic correspondence tractable." She paused. "How did you identify the simplification?"
"The numbers don't close," he said. "If you run the cross-reference against the magical correspondence rather than the standard numerological one, the simplified sequence produces a remainder that the full system would require to be zero."
Vector looked at his workbook. Then at him. Then at his workbook again with the expression of someone who was deciding how to characterise what they were seeing.
"Have you studied Arithmancy before?" she said.
"Not formally," he said, which was entirely true. "I worked through the third and fourth year theory over the summer. I also spent the first week back working through the fifth-year supplementary material ." He paused. "The underlying mathematical structure is something I have more background in than most."
She looked at him for a moment. The quality of it was the productive scepticism of a mathematician — she did not believe it was the complete picture, but she was interested in the picture that was there.
"Stay after class," she said.
The conversation that followed had Vector asking questions that became progressively more specific as she identified where his knowledge was and where it wasn't. The mathematical foundations were deep — she probed the edges of them several times and found them consistent in the way of something genuinely integrated rather than recently acquired. The magical elements showed the specific texture of recent thorough work: solid, clearly laid, but without the settled quality of knowledge held for years.
She accepted this with the precision of someone who had taken an accurate reading and knew that accuracy was not always completeness.
"I'll give you the fifth-year text," she said. "Work through chapters one to three. If what I'm seeing holds, we should talk about the sixth-year material."
He thanked her and took the text and did not let his face do anything in particular.
He had been working at the fifth-year level since late August. The sixth-year material was where he expected to find something genuinely new.
He thought about the year and its shape — third-year top level by mid-November, fourth-year by Easter, fifth-year standard by the end of the following summer. These were not aspirational targets. They were the plan, worked out with the same logic he brought to everything, because he was going to enter fourth year sitting alongside students two years older in terms of magical education, and the war that was coming was not going to wait for him to catch up.
He walked out of the Arithmancy classroom with the fifth-year text under his arm and the quiet, focused satisfaction of someone who had found the work was going to be genuinely demanding.
He found that prospect straightforwardly welcome.