On stage.
Li Dong turned to the first page of the main content.
He picked up a laser pointer, and the red dot landed on the first formula on the projection.
"Alright, let’s begin."
His voice betrayed no trace of nervousness.
"First, I’d like to start with the most fundamental motivation."
"The essence of Montgomery’s Conjecture on Association is to describe the spacing distribution between the non-trivial zeros of the Riemann zeta function."
"But when I was writing this paper, I didn’t start directly from the distribution of zeros."
"The path I chose was to first construct a spectral operator, allowing the statistical behavior of the zeros to emerge naturally from the operator’s spectral properties."
He paused for a moment.
"The starting point for this line of thinking was a very simple observation."
The red dot moved to the next formula.
He didn’t speak quickly.
Every step of the derivation, the meaning of every symbol, the reason for introducing every technical tool—he explained it all with perfect clarity.
The intermediate steps that had been omitted from the paper were now being filled in, one by one, through his explanation.
It was like the missing rivets on a great bridge being reinstalled in their rightful places, one by one.
And only then did you realize that the reason these rivets were omitted wasn’t because they didn’t exist.
It was because, to Li Dong, they were simply too obvious.
The first point of contention was addressed in the third minute of his lecture.
It was the first question Zhou Shenzhi had raised in the comments.
In Chapter 4, Section 2 of the paper, when transitioning from |α|∈[1, 2] to |α|∈[2, 3], Li Dong had written a single line—"by a natural analogy between the normalized contribution of second-order prime powers and the first-order case"—and then jumped straight to the conclusion.
Zhou Shenzhi had called this step "not natural."
Quite a few people on arXiv had agreed with him.
But Li Dong didn’t even mention the fact that this part had been questioned.
He simply, in the natural course of his derivation, expanded upon that logical leap.
"The contribution of the second-order prime power, p2, in the explicit formula is completely different from that of the first-order prime, p."
"But the method for handling it isn’t complex."
"You just need to first perform a precise expansion of the Ramanujan sum at p2. After expanding, you’ll find it can be written as a superposition of two convolutions..."
He drew a dividing line on the blackboard.
"One is the self-convolution of p’s first-order contribution. This part is completely parallel to the handling in the |α|∈[0, 1] interval. The only difference is that the normalization weight needs a scale transformation, replacing w(u) with w(u/2)."
"The other is a correction term purely from p2. This term’s rate of decay is O(p−3/2), which, compared to the first-order’s O(p−1), is faster by a full half-order of magnitude."
"After summation, it gets automatically absorbed into the remainder term."
"It doesn’t affect the main term."
He finished writing the last line.
"See? Not hard at all."
After that explanation, most people in the audience fell silent.
Because that intermediate step was, indeed, "not hard"—like hell it was!
Just ask the professors in attendance: if Li Dong hadn’t explained it, could any of them have thought of it?
That logical leap alone could be published as a top-tier paper!
And you just casually solved it...
Zhou Shenzhi’s expression soured even more. He could understand this logical leap, but... ’Why is the result the same, but his process is different from mine? It makes me look like an idiot!’
Then came the second point of contention, the third, the fourth...
Li Dong didn’t hesitate in the slightest.
Every logical leap he expanded upon had the same effect.
It wasn’t that there was a problem with Li Dong’s proof.
It was that the questioners simply weren’t on his level.
This fact was crueler than any rebuttal.
Time passed, second by second, minute by minute.
Ten minutes.
Fifteen minutes.
Twenty minutes.
A subtle change occurred in the atmosphere of the audience.
To be honest, although everyone believed that a paper accepted by the *Mathematical Annals* would be sound, with thirteen critical comments posted on arXiv, who wouldn’t be curious to see if any of the criticisms held water?
But gradually, that sense of scrutiny vanished.
Because they discovered that what Li Dong was saying and the paper as they had understood it... seemed to be two different things.
They hadn’t truly understood Li Dong’s logic at all; they had merely followed the paper’s derivations to understand the results.
What Li Dong was explaining now was simply too beautiful.
The lines of thought, concealed by the paper’s conciseness, were now emerging in their complete form for the first time through his oral presentation.
You could see how a nineteen-year-old youth thought about mathematics.
Where his intuition came from, what kind of geometric picture lay behind his technical choices, how he borrowed tools from a seemingly unrelated field and reshaped them into the perfect form for the task.
For example, when discussing how to control the remainder term O(log−1T), he didn’t use any existing lemmas.
Instead, he constructed a set of dynamically adaptive Fourier weight functions. This set of functions was deceptively simple in form, yet their coupling relationship was such that, in the process of tightening the error bounds, they automatically canceled out the most stubborn high-order oscillatory remainder terms.
To put it bluntly, this step alone, if written up as its own paper, would be enough to win a Ramanujan Prize.
"Is that something a human could have designed?"
An Ivy League math professor’s spirit was broken.
"It’s as if it was always meant to be this way. He wasn’t the designer; he was the discoverer!"
At this point, everyone in the audience understood.
This seminar wasn’t a nineteen-year-old Huaxia youth’s self-defense; it was a masterclass on "How a Genius Does Mathematics."
Tao Zhexuan had sat up straight, nodding slightly.
It was a heartfelt acknowledgment of a mathematician on his own level.
Zhang Yitang was rapidly jotting something down...
Not formulas, but the creative motivations Li Dong spoke of—the kind that would never appear in a textbook.
The corners of Penrose’s eyes were slightly red.
Sarah was startled and asked in a low voice,
"Professor, what’s wrong?"
Penrose shook his head, his voice a little hoarse.
"It’s nothing."
"It’s just been a long time... since I’ve listened to mathematics like this."
He was already in his sixties.
From Grothendieck’s seminars on Algebraic Geometry to the afternoon Wiles announced his proof of Fermat’s Last Theorem, he had witnessed nearly every highlight in the history of modern mathematics.
But today was different.
After listening to those truly great lectures, you would feel awestruck.
But after listening to Li Dong’s, you felt... a sense of tranquility.
It was like seeing a great river—not a raging, turbulent one, but one that was broad and steady, flowing toward a distant place you could not see.
It flowed toward the future...
Clark sat diagonally behind them, his notebook open to three questions he had prepared in advance.
His pen had been still for a long time now.
He found that the questions he’d prepared hadn’t been answered.
They had simply been "passed by" during the course of Li Dong’s derivation.
Qiu Chengtong’s expression had changed from its initial calmness to one of deep concentration.
He turned his head and glanced at Tian Gang.
Tian Gang happened to turn his head at the same moment.
Their gazes met in mid-air for an instant.
Neither looked away.
Qiu Chengtong gave a slight nod.
The movement was so small that only Tian Gang could have seen it.
Tian Gang nodded back.
Then, both men turned their heads back at the same time and continued to watch the stage.
The master and apprentice hadn’t reconciled; they had simply acknowledged the same young man.
Twenty-eight minutes.
Li Dong reached the final core theorem.
It was also the most crucial conclusion of the paper.
The validity of Montgomery’s Conjecture on Association over the complete interval of |α|∈[0, 4].
This step required unifying all the preceding elements—the stratification strategy for all intervals, the layer-by-layer separation of prime powers, and the remainder term control from the Fourier optimization framework—into a single, final inequality.
Li Dong was not flustered.
He stacked the thirteen layers of techniques together as if they were building blocks.
Each layer was placed with perfect stability.
On the projection screen, the final inequality was framed in a box.
Li Dong set down the laser pointer.
He turned to face the audience.
And smiled.
"Well, that’s about it."
"Exactly half an hour."
He glanced at the watch on his wrist.
"Mm, to be precise, twenty-eight minutes and forty-seven seconds."
"Finished with more than a minute to spare."
The audience was silent for about three seconds.
Then...
Applause.
Starting from the front row, it swept backward like a wave.
Tao Zhexuan was the first to clap.
Then Zhang Yitang, Yuan Yaxiang, Zhang Wenping, Penrose, Tian Gang, Qiu Chengtong...
And finally, the entire Sunshine Hall.
Clark’s right hand froze above his notebook, but in the end, he raised it and brought it together with his left.
He had to applaud.
Because if he didn’t, the people around him would remember him.
’Hey, look at this country bumpkin who didn’t get it...’
The applause lasted for nearly a minute.
Amid the applause, Li Dong stood on stage, still wearing that same cheerful smile.
He gave a slight bow to the audience.
"Thank you, everyone."
"Next is the Q&A session."
The applause gradually died down.
The Q&A session.
This was today’s true battlefield.
And he was ready.
"So."
Li Dong placed his hands on the lectern, his gaze level as he looked out at the several hundred people in the audience.
"Who’s first?"

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