I Was the Clown All Along!

Qi Hao returned to the dorm, spread out his scratch paper, and wrote out the complete proof of the lemma following Old Wang's line of reasoning.

Sure enough, it worked.

Once he had broken through the most critical technical bottleneck in the paper, the rest of the derivation flowed as smoothly as if his Ren and Du meridians had been unblocked. He wrote another four pages in one go, stopping only when the lights went out.

His life had followed much the same routine these past few days: if he wasn't in class, he was writing his paper, and if he wasn't doing that, he was looking for Old Wang.

Today, he had hit another snag.

It concerned estimating the asymptotic convergence rate during game-tree pruning and involved the relationship between a graph's spectral radius and the width of a partially ordered set.

He thought about it for two hours and looked through several English papers he could find, but he still couldn't verify whether his idea was correct.

Forget it. I'll just ask Old Wang again.

I might as well make use of my teacher!

Qi Hao gathered up his scratch paper and found Old Wang's QQ account, which he had added at the beginning of the semester. He sent him a message.

Qi Hao: Hello, Teacher Wang. I've run into another problem. Would it be convenient to ask you about it in person?

Old Wang was evidently used to it by now and replied quickly.

Old Wang: I'll be in seminar room B503 in the Liren Building at three thirty this afternoon. Just come over.

Qi Hao: Okay, I'll be right there.

At three thirty that afternoon, Qi Hao stood outside B503.

The door was half open, and Old Wang's voice came from inside. He sounded stern.

"Did you leave your brains at home when you came to school today? Do I really have to repeat this over and over..."

Good heavens, was Old Wang really this fierce?

How had he never known?

Qi Hao hesitated over whether to knock, but remembering Old Wang's message, he reached out.

He knocked softly twice, and the voice inside stopped.

"Come in," Old Wang called from inside.

Qi Hao pushed the door open and went in.

B503 was a medium-sized seminar room with a long conference table in the middle. Seven or eight people sat around it, all seemingly graduate students.

Some had laptops open in front of them, some were writing in notebooks, and some looked utterly lost.

Old Wang stood at one end of the conference table, a whiteboard covered in formulas and diagrams beside him. When he saw Qi Hao, he nodded and said, "Come in and have a seat. Wait for me a moment."

Qi Hao found an empty chair and sat down in the corner.

Old Wang wrote a few more lines on the whiteboard, then turned around.

"All right, I'll go over the problem again."

"Let G be a directed acyclic graph derived from a finite partially ordered set, with adjacency matrix... satisfying the boundary conditions. Now we want to use the extremal properties of the Rayleigh quotient to prove the convergent mapping relationship between the graph's minimum path cover number and its eigenvalue spectral gap."

Old Wang's gaze roamed over the students, whose heads were nearly buried in the table, before settling on a young man to his left. "Liang Wenhao, stand up and tell us: in this paper, how should the asymptotic convergence rate of the spectral gap be bounded in the next step?"

Called on by name, Liang Wenhao braced himself and stood up. He stammered for quite a while. "Teacher Wang... if we apply the Cauchy inequality here to expand the norm, then... then use the second-smallest eigenvalue of the Laplacian matrix to approximate..."

"Nonsense!" Old Wang cut him off mercilessly. "Apply the Cauchy inequality? Take a close look at the partial order width term on the right. The norm expansion diverges immediately, so what the hell are you approximating? Sit down!"

Liang Wenhao sat back down as if he had been granted a pardon.

Old Wang shook his head, frustrated that his students wouldn't live up to his expectations, then pointed at another young man on his right who was trying to hide his face behind his laptop. "You, tell us!"

The student he called on didn't even bother struggling. He stood up and lowered his head, looking ashamed. "Teacher, I haven't fully grasped the derivation in this cutting-edge paper yet... I can't find the boundary conditions for the bounds."

After calling on two people in a row who knew nothing, Old Wang's face turned completely black.

He tossed the whiteboard marker in his hand onto the conference table, and it rolled a long way.

"None of you know? I've been teaching you for a year or two. Even a pig should have learned how to find a matrix's spectral radius by now!"

Old Wang gave them an earful from the front of the room. The students stared at their noses, their noses at their hearts, not daring to breathe too loudly.

In the oppressive atmosphere, Qi Hao, sitting on the empty chair in the corner, silently raised his hand.

He hadn't wanted to show off, but the asymptotic convergence rate estimate he had been thinking about followed exactly the same logic as the derivation on the whiteboard. Since Old Wang was already discussing it, he could take the opportunity to check his idea.

However, Old Wang, still in a temper, frowned slightly when he saw the hand raised in the corner.

As he saw it, although his graduate students had shaky foundations and were stupid, at least they knew how to seek truth from facts, to say they knew something when they knew it and admit it when they didn't. But what was this freshman Qi Hao doing now?

Any normal teacher was afraid of encountering an impetuous student who only half understood something but liked to force an appearance of knowing everything in front of a crowd.

Students so eager for grand achievements were often ten times more of a headache than those who were simply stupid!

Besides, Qi Hao was only a freshman who had barely scratched the surface of basic graph theory and combinatorial mathematics. What could he understand about spectral gap convergence?

This was simply ridiculous!

"Qi Hao, you're only a freshman. Do you understand the assumptions and the objective of the derivation for the problem on the whiteboard?" Old Wang held back his temper as he asked, thinking that Qi Hao had been doing fairly well lately.

"I understand, Teacher Wang." Qi Hao paid no mind to Old Wang's stern expression and stood up confidently. "Teacher Wang, the general approach of using the Rayleigh quotient to approximate the extremum here is entirely correct. But the reason it diverges when estimating the spectral gap is that the asymmetry of the weighted directed graph derived from the partially ordered set has been overlooked."

"If we abandon the usual norm expansion here and instead introduce the left eigenvector from the Perron-Frobenius theorem to reweight the width of the partially ordered set, then apply the Courant-Fischer minimax theorem, we can directly bridge the graph's spectral radius and the partial order width perfectly within a convergent similarity transformation of the matrix."

The impatience and severity on Old Wang's face gradually began to fade.

Qi Hao had actually given the correct approach to the problem in such a short time!

He realized that Qi Hao reacted quickly and had definitely put in a tremendous amount of hard work!

The reason Qi Hao reacted quickly was that, for him, these theorems were more than things he merely knew about. He had truly learned how to use them!

An undergraduate, not even finished with his first semester as a freshman, had solved the problem Old Wang posed entirely through self-study, working through textbooks and puzzling things out on his own?

This kid had hidden depths!

He had originally thought Qi Hao was only a little smarter than an ordinary student, but now it seemed he had misjudged him.

Qi Hao's potential seemed even greater than he had anticipated!

Meanwhile, Liang Wenhao and the person beside him had now recognized Qi Hao as the weirdo from the library who wanted to study Soviet mathematics!

They had assumed Qi Hao was at least a graduate student, but he turned out to be only a freshman!

When they had seen Qi Hao's derivation in the study room, they had confidently assumed he was some genius graduate student from the school. They had even put on the airs of experienced seniors, earnestly advised him to study academic circles in depth, and tried to be magnanimous by bringing him into their own low-quality publication research group.

Now, recalling the pretentious "your senior brother will show you how to soar" act he had put on, Liang Wenhao was so embarrassed his toes could have dug a three-bedroom apartment in the soles of his shoes. He wished he could buy a ticket to hang onto the outside of a vehicle and flee the Solar System overnight.

So, in the end, he was the clown all along!

P.S. Thank you all for your support, so I posted an extra three thousand characters. There will be more bonus updates from time to time. Please keep supporting the story by following along and with monthly votes and recommendation votes!