After taking the advice, I became a top student.

On the other side, Zhao Yiming had already taken out a draft paper from his briefcase and started deriving the fourth step when Chen Mo began to deduce it.

因为ψ是实特徵，只取±1，所以ψ(2j−1){共轭}=ψ(2j−1)，又e^{πi(2j−1)/q}{共轭}=e^{−πi(2j−1)/q}，所以右边=ψ(−1)⋅ψ(2j−1)⋅e^{−πi(2j−1)/q}。

It's exactly the same as the one on the left.

correct!

Zhao Yiming breathed a sigh of relief. Mathematics hadn't betrayed him; he could still deduce it.

But a next thought immediately popped into his head, sending a chill down his spine: "Did this student write this equation on the spot, or did he prepare it beforehand?"

He looked up at Chen Mo.

The boy was standing in front of the whiteboard, his eyes sparkling, narrating a perfectly natural fact without any pause.

It wasn't memorized.

It was something I thought of on the spot!

Zhao Yiming's fingers trembled slightly.

At this moment, the boy next to the whiteboard had already completed all the steps and written down the last formula, while he had only just completed the fifth step of the derivation.

Zhao Yiming stared blankly at the whiteboard.

"Was α chosen when constructing ψ?"

He thought of Chen Mo's first step.

At that time, Chen Mo did indeed write that the characteristic ψ modulo 2q is defined such that ψ(k) = χ(k) ⋅ (k−1)α.

He glanced at it but didn't pay any attention.

He now realized the brilliance of this choice: by adjusting α, the sign of ψ(−1) could be controlled, thus allowing the odd and even parts to form a perfect conjugate pair.

This isn't a technique; it's design.

This high school student wasn't solving a problem; he was designing a proof.

Just like an architect designing a building, the position of each brick is calculated in advance, all for the sake of symmetry and harmony at the last moment.

Zhao Yiming took a deep breath, feeling a throbbing sensation in his temples.

He taught for thirty years, mentored countless graduate students, and met many gifted students, but those people, at best, could only run faster within the existing framework.

The boy in front of us is creating a framework!

Zhao Yiming slowly raised his head and looked at the boy in front of the whiteboard.

Chen Mo was answering Zheng Mingyang's question in a calm tone, not at all like someone who had just managed to break down the defenses of both professors.

Zhao Yiming opened his mouth, wanting to say something, but found his voice was a little dry.

He cleared his throat and finally managed to squeeze out a sentence, "Chen Mo, did you come up with this proof during the exam, or did you think of it later?"

Chen Mo turned around, somewhat embarrassed: "I only wrote the result in the exam, not the proof. I only deduced it on the spot when I was explaining it to Teacher Zheng just now."

This is a newly released recommendation.

Zhao Yiming fell silent.

He recalled proving a lemma in his office last week, which took him three days, only to find out it was wrong.

He recalled that during his doctoral dissertation defense twenty years ago, his advisor asked him a question that took him ten minutes to answer.

He recalled taking the college entrance examination thirty years ago and scoring 118 points in mathematics. At that time, the full score for mathematics was 120, and he thought he was a chosen one.

Now, a high school student, in just over ten minutes, derives a proof in front of him that would normally take him half an hour to verify.

It's not because he's stupid.

It was because this student's thinking was not on the track he was familiar with. He remembered that his teacher had once discussed with him that the world looks different in the eyes of geniuses than it does to them!

Zhao Yiming closed his eyes, then opened them again.

The symbols on the whiteboard are still the same; they haven't changed.

But his mindset has changed.

Jiang Fan noticed that Professor Zhao Yiming's expression was not quite right.

From initial composure, to later solemnity, and now... bewilderment?

Jiang Fan suddenly felt a little more at peace.

So even the professor couldn't understand it.

He leaned close to Zhou Zhi's ear and whispered, "Old Zhou, look at Professor Zhao's expression."

Zhou Zhi glanced at him and whispered, "What's wrong? Is he feeling unwell?"

"No," Jiang Fan said in a low voice, suppressing a laugh, "He didn't understand either."

Zhou Zhi glared at Jiang Fan, but the shock in his heart did not diminish in the slightest. Half a month ago, he thought that this provincial competition was just to broaden the horizons of this little guy. Who would have thought that he would be the one to see the world now.

The discussion between Zheng Mingyang and Chen Mo was nearing its end. After completing this proof, there were no more difficulties in Zheng Mingyang's paper.

"Chen Mo, you've helped me a lot!"

"I hope to include this proof in my paper, and of course, you will be the third author of it."

Zheng Mingyang sincerely said that Chen Mo's contribution might not be enough to qualify him as the third author, but the proof of this lemma was extremely ingenious and could add a lot to the paper, even if it was in his own work.

This proof is also very important, so it is entirely sufficient for Chen Mo to be the third author.

He never expected that what he originally just wanted to try his luck and see if he could discover a promising talent would actually complete the final piece of his thesis puzzle.

"Of course, no problem."

Chen Mo did not refuse. At this time, he had not yet realized the importance of his proof, nor did he understand the significance of being the third author of a paper published in the Duke Mathematics Journal.

"Chen Mo, I have a question."

At this moment, Zhao Yiming also walked over.

"Please speak."

"You just said that you came up with this proof on your own. So, how did you know that this path would work before you came up with it?"

How do you know that after parity-even splitting, odd-numbered terms and even-numbered terms will form conjugate pairs?

How do you know that choosing α will make ψ(−1) = 1?

"It's like... walking in the dark, but you're stepping on steps every time. How do you know where the steps are?"

Zhao Yiming looked at Chen Mo, not questioning him, but asking for an answer, as if seeking a way to see the steps in the darkness.

The office quieted down.

Everyone looked at Chen Mo.

This problem is more difficult to answer than any mathematical problem.

Because it doesn't ask how to do it, but how to come up with it.

Chen Mo remained silent for a few seconds.

Then, he said something that Zhao Yiming would never forget: "I don't know where the steps are, but I know that if the steps exist, they must be in the most symmetrical place."

"All the beautiful results in mathematics are the shadow of symmetry."

"I'm just following my shadow."

The office fell silent again.

Zhao Yiming opened his mouth, but said nothing.

He turned around, walked back to the sofa, and sat down.

He just stared at the symbols on the whiteboard, over and over again.

"Follow your shadow..."

He muttered to himself, a bitter smile appearing on his lips.

He has been doing mathematics for thirty years, always feeling his way across the river by touching the stones.

A high school freshman has already learned to chase shadows.

This is probably the difference between geniuses and ordinary people.

No, that's probably the difference between mathematical intuition and mathematical training.

Some things can't be taught or learned.

I can only be envious.