Li Dong was called on, but he wasn’t flustered.
He slowly stood up.
"Teacher E, I’ll just share some casual thoughts. Please correct me if I’m wrong."
E Weinan nodded.
"Go ahead."
The circle of PhD students around the U-shaped table, along with the few undergraduates in the back row, all turned to look at Li Dong.
This wasn’t just any "junior brother" speaking up.
This was East God.
They had listened for most of the class and still hadn’t fully grasped the professor’s concept of a "hidden constraint."
They were eager to see how this peer, who was about their age, understood it.
"Teacher E, the diagram you drew depicts a spectral inversion."
"But I have a slightly different perspective..."
"If we only view this as an inversion problem, the hidden constraint you mentioned can’t actually pin down the third peak."
As soon as he said this, two or three PhD students at the U-shaped table frowned.
’Can’t pin it down?’
E Weinan’s expression didn’t change.
"Continue."
"In my opinion, this shouldn’t be a simple inversion problem. It should be a duality problem."
Li Dong spoke at a measured pace, as if he were just chatting casually.
"On your diagram, the horizontal axis is energy, and the vertical axis is the density of eigenvalues."
"But have you considered that what the vertical axis truly represents isn’t the density itself, but the phase of the density?"
"If you look at it from the dual side, these three peaks aren’t three separate points..."
"But rather three sample points on a hidden curve."
"Once you find the overall geometry of this curve, the origin of the third peak and the necessary phase correction are naturally constrained by it."
"The hidden constraint is actually a global invariant of this curve."
"It’s not a number; it’s a geometric object."
No one in the room made a sound. It wasn’t because it was that difficult to understand, but because no one had ever thought of it from this angle.
E Weinan stood there, replaying Li Dong’s last few sentences in his mind twice.
He had been working on ill-posed inverse problems for nearly thirty years.
Every one of his peers in the field of inverse problems, including himself, subconsciously treated the "constraint" as a scalar quantity.
It had an upper bound and a lower bound, from which you estimated a number.
And yet, Li Dong had just told him...
’It’s not a number; it’s a geometric object.’
This perspective was on a completely different level.
It was like spending half your life on one-dimensional integrals, only to have someone suddenly point out that your integral was actually a cohomology class on a high-dimensional manifold.
The circle of PhD students around the U-shaped table couldn’t process it.
Two of them even started flipping through their notebooks, trying to confirm if the characterizations the professor had drawn earlier had truly been elevated to a higher dimension by Li Dong’s two sentences.
In the back row, the sophomore from the School of Chemistry whispered to the senior student next to him.
"...Is he borrowing this from pure math?"
"I don’t know."
The senior student shook his head.
"I’ve never seen anyone explain it like this."
E Weinan, however, was nodding to himself inwardly.
’This is indeed a valid direction.’
"Then take it one step further."
"How would you define this ’hidden curve’?"
Li Dong’s response was matter-of-fact.
"I don’t know!"
E Weinan: ???
The students: ???
Li Dong was remarkably frank.
"You mentioned Tikhonov regularization, self-adjoint extensions, and the decomposition of the spectral measure into continuous and discrete parts..."
"I can roughly follow the first two, but I haven’t even looked at the third one."
"So, to explicitly parameterize this curve... I can’t do that right now."
The circle of PhD students at the U-shaped table finally let out a quiet, collective sigh of relief.
’So even East God has things he doesn’t know.’
The one at the far end of the table, Senior Sister Zhao, who had just passed her qualifying exam at over thirty, also breathed a silent sigh of relief. She was about to flip her notebook back to the proof she had written...
...when she heard E Weinan casually add a sentence.
"Then let me fill you in."
"On the topic of spectral measures, take this down..."
Without getting up, E Weinan remained seated on the chair by the lectern. As he spoke, he used a piece of chalk to sketch a diagram in the corner of the small blackboard next to him.
The Cayley transform of self-adjoint operators, the Lebesgue-Radon decomposition of spectral types.
These were topics that took up entire Chapters in a PhD student’s textbook, yet E Weinan picked out the three or four most crucial lines and delivered them with blistering speed and precision.
"That’s the gist of it. Just get a rough idea for now."
Even the PhD students felt their scalps go numb listening to it.
The few undergraduates in the back had given up completely. They just propped up their phones and focused solely on recording.
But as for Li Dong...
The moment E Weinan put down the chalk,
he picked up right where the professor left off.
"Teacher E, in that case..."
"If we isolate the continuous part of the spectral measure and compare it to the hidden curve I mentioned, we’ll find that the blurred part of the third peak falls entirely on the edge of the continuous spectrum."
"This edge is an accumulation point of the discrete spectrum."
"So, the phase difference of π you mentioned earlier doesn’t come from a poor choice of basis functions."
"It comes from the inherent singularity of the spectral measure itself near the accumulation point. It’s an intrinsic characterization."
"Since it’s intrinsic, it must be invertible back into that hidden geometry."
"In other words..."
He looked at E Weinan.
"Your hidden constraint isn’t some contrived number. It’s the local spectral curvature at this accumulation point."
The room fell silent once more.
E Weinan didn’t speak either, instead running through what Li Dong had just said in his mind again.
’There wasn’t a single flaw in his logic.’
The material on spectral measures he had just covered was something any PhD student in inverse problems would have to struggle with for weeks just to get a basic handle on.
Li Dong had just heard it for the first time.
And immediately after, he had given him a new characterization based on it.
’This person...’
E Weinan suddenly recalled the National Power deployment meeting from a few months ago.
This kid, right in front of him, Yao Qizhi, Zhao Fangming, and Gao Wen, had given the figures: "microsecond-level response, compression by four to five orders of magnitude."
At the time, he had muttered to himself...
’What a promising prodigy for applied mathematics.’
But then the kid turned around and dove headfirst into the Langlands Program, working on the local-global compatibility for GL(n).
E Weinan respected Tian Gang, and he respected Liu Ruochuan.
But he had always felt a sense of stifled frustration.
Working on pure math was fine, of course.
But on the path of pure math, a single monumental achievement could easily take eight or ten years.
The contribution to the advancement of mathematics would be enormous.
But in the eyes of an academician like E Weinan, who came from an applied background, being able to implement things practically as soon as possible—in industry, in tackling those "bottleneck" problems...
...that was the most crucial thing in this day and age.
Look at National Power, look at Hua Wei.
There were countless places waiting for a "brain" like his.
’Such a promising talent.’
’Why did he have to go into pure math?’
He sighed inwardly and raised his hand, about to signal for Li Dong to sit down so he could continue with the lecture.
Just then...
Li Dong suddenly spoke up.
"Teacher E."
"I also encountered a problem recently, and I was hoping to take this opportunity to ask for your advice."
E Weinan was taken aback. He hadn’t expected Li Dong to have a question for him.
He subconsciously glanced at the time.
There were still over twenty minutes left before class ended.
He hesitated for a moment, then waved his hand.
"Alright, go ahead and ask."
"It’s a good opportunity for everyone else to listen in as well."
"It’s easier for you all to learn from a question posed by a student who is about your age."
The circle of PhD students: ???
’Professor, are you even speaking human language?’
Since E Weinan had given him permission, Li Dong didn’t hold back.
He directly stated the question Professor Levan had asked him in Jingchun Garden the day before.
After he finished speaking,
The PhD students below: ???
’Oh no, how come this time I can’t even understand the question?’
The undergraduates in the back were even more despondent. They didn’t even bother pretending anymore and turned to look directly at E Weinan.
’Academician E, what were you just saying? Easier for me to learn??’
E Weinan completely ignored their gazes.
He was already lost in thought.
Of course, he had read Li Dong’s paper.
In fact, he had read it more than once or twice. Just a few days ago, he had even asked Tian Gang for the official proof copy. He had taken that proof and read it straight through from two in the afternoon until nine at night.
And reading it wasn’t enough.
On scratch paper, he had tried to reproduce the technical path of "using the convergence interval of the pair correlation F_π(α) of zeros to inversely constrain the upper bound of the ramification index."
Don’t be fooled by his "applied mathematics" title.
In his early years, during his undergraduate and master’s studies, he had clawed his way up through pure mathematics. His foundation in number theory and representation theory was solid.
So, when Li Dong threw out that string of questions, the first thing that came to his mind wasn’t some "ligand field tensor."
It was the estimate of the second moment of F_π(α) from the proof copy.
’That move of Li Dong’s...’
’Can it really be used to inversely constrain a "hidden constraint"?’
’If it can...’

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