Dorm Room 404.
When Li Dong pushed the door open, Wang Hao was still curled up on the bottom bunk, gnawing away at his copy of Vakil.
"Haozi."
"Not going over problems with Gu Ming today?"
Wang Hao turned his head to glance at him.
"Gu God has been busy with your group lately. He hasn’t had any time to hash out problems with me," Li Dong said as he sat down in his chair.
"Don’t worry."
"In a bit, your big bro will explain some problems to you."
Wang Hao shot up in his bed.
"...Brother Dong."
"What are you up to now?"
Li Dong gave him a sly grin.
"Haozi, don’t you worry."
"My explanation today will be completely different."
"I attended a lecture in Sunshine Hall today, and I’ve been enlightened."
"After you hear me out today, you’ll see that my level has truly ascended."
Wang Hao looked at his "enlightened" expression.
He silently shoved the copy of Vakil further under his pillow.
He lowered his head and sighed.
He was suddenly struck by a very ominous premonition...
Tonight’s "tutoring session" might be more terrifying than all the previous ones combined.
Two hours later, Li Dong was satisfied.
「The next morning.」
Li Dong had just crawled out of bed when his phone rang.
It was Wu Kai calling. He answered it immediately.
"Teacher Wu."
"Li Dong, Old Lu came back early this morning."
"If it’s a good time for you, come over. Let’s meet up."
Li Dong agreed.
"Okay, Teacher Wu."
"I’ll be right there."
He hurriedly washed up.
As he was getting ready, he called out to Wang Hao, who was still in a daze from last night’s "tutoring session."
"Haozi."
"I’m heading to the School of Chemistry."
"I’ll give you another lesson when I get back tonight."
Wang Hao: ????
He didn’t say a word.
He just silently buried his face back into his pillow.
「Wu Kai’s Research Group Laboratory.」
When Li Dong pushed open the lab door, Wu Kai was chatting with another professor whom he had never seen before.
The professor was in his early fifties, with a slender build, graying hair, and an unassuming gaze.
He heard the door open, looked up, and glanced at Li Dong.
He didn’t smile or speak.
He just gave a very slight nod.
When Wu Kai saw Li Dong, he waved him over.
"Li Dong."
Li Dong walked over.
Wu Kai patted the other professor’s arm.
"Old Lu, this is Li Dong."
Lu Mingyuan nodded again.
This time, he finally spoke.
"Mm, I’ve heard much."
Just three words.
Li Dong was just as concise.
He nodded back at him.
"Hello, Teacher Lu."
Wu Kai glanced at the backpack on Li Dong’s back.
He smiled.
"Li Dong, why don’t you explain your line of thinking to Old Lu."
"This is his area of expertise."
The three of them went into the small office together.
Inside the office, there was a small whiteboard.
Li Dong put down his backpack, pulled out a stack of draft paper from inside, and flipped to the top sheet.
He walked straight to the whiteboard and, without any preamble, picked up a marker.
"Teachers."
"My general line of thought is this."
He drew two vertical lines in the center of the whiteboard.
Under one, he wrote [Mathematical Inversion].
Under the other, [Chemical Synthesis and Engineering].
Between the two lines, he left a blank space.
In the very center of that blank space, he slowly wrote a single word.
[Physics].
"The two walls we’re currently stuck on."
"One on the left—the basis functions are fighting each other."
"One on the right—we’ve already tried every trick we can think of on the chemistry and engineering side."
"I discovered while reviewing the literature..."
"This space between the two walls, no one has ever walked through it."
"The path I intend to take is through this very space."
Wu Kai grunted in acknowledgment.
He had already heard this part from Li Dong before.
Hearing this, he initially thought Li Dong was about to launch into the same explanation he gave the other day.
He was even prepared to help Old Lu point out the fatal flaw in the argument when Li Dong got to the part about the "vacuum segment κ."
But when Li Dong reached that part...
He didn’t follow the previous version.
He put down the marker.
He pulled a new sheet from his draft paper and stuck it on the whiteboard.
At the very top was a single line of text.
[Three Layers].
Wu Kai let out a soft "Huh?"
He leaned in a little closer.
"U(1) on the fiber."
"C_3v on the base space."
"Z_2 parity as an automorphism on the structure group."
"These three things are not on the same layer."
"The transition function of the dipole transition matrix element under the three basis sets."
"Is equal to the product of these three things."
Li Dong spoke very slowly.
He left a slight pause between each sentence.
When he said the line, "parity as an automorphism on the structure group," Lu Mingyuan’s hand, which was holding a teacup, noticeably froze for a moment.
Wu Kai was stunned, too.
’I spent a long time in my office flipping through Li Dong’s stack of draft paper by myself.’
’In that stack, I remember that under the line [φ=0], it said "local basis change functor of GL2 + local root number of the L-factor."’
’How has it suddenly become a three-layer structure of a fiber bundle today?’
To experts in this field, these were two completely different languages.
Wu Kai subconsciously glanced at Lu Mingyuan beside him.
Lu Mingyuan didn’t look back at him.
His unassuming eyes were fixed on the whiteboard the entire time.
Li Dong continued his explanation.
Starting from the [Three Layers] line, he went back and patched up every single point in the physical pathway that had previously been a dead end.
The real representation of the dipole operator went from being "forced in" yesterday to being a "natural corollary of the parity automorphism" today.
The X-ray polarization also changed from yesterday’s "source boundary condition" to today’s "input condition for the U(1) gauge on the fiber."
And the most critical piece, κ.
To the right of [Three Layers], he wrote a new equation.
"The effective potential barrier of the vacuum segment plus the Ligand Field plus the Au(111) substrate."
"It is the flat part of the connection on the base of this stratified fiber bundle."
"Its flat shape can be read directly from the invariant tensor of C_3v."
"No inversion needed."
"This κ can be written out directly using the geometry of the fiber bundle."
"Parametrized physically, right from the source."
He put the marker down.
"Teachers, that’s the general idea."
"Do you see anything that needs improvement, or do you have any questions?"
The office was silent for a moment.
Wu Kai’s mouth opened, then closed again.
’Last night, when I was looking through that stack of draft paper alone, I had already thought Li Dong’s theory was quite beautiful.’
’I originally thought that was the limit of his ability.’
’I had even prepared my own response: "The source of this κ is empty; we should hold off for a bit."’
But now that Li Dong had finished this part of his explanation...
His κ had grown from nothing into a real form that could be naturally read from the flat part of the connection on the base of the fiber bundle.
Wu Kai took several seconds to digest this.
He turned his head to glance at Lu Mingyuan.
"Old Lu."
"What do you think?"
Lu Mingyuan did not speak immediately.
He had originally been brought over by Wu Kai to "offer some pointers" to Li Dong. After all, the area of vacuum segments and effective potential barriers had been his domain for nearly twenty years.
But as this "lecture" progressed...
He himself had gone from an initial ’Mm, I’m listening,’ to ’Wait a minute, I have to write this down.’
He felt around in his pocket.
He pulled out a small notebook.
He lowered his head, not looking at anyone.
He said,
"I have two small questions."
"Don’t get flustered. Hear me out before you answer."
Li Dong nodded.
"Please, go ahead."
Lu Mingyuan’s first question targeted the section on the vacuum segment plus the Ligand Field.
"You just said that the invariant tensor of C_3v directly gives the flat shape."
"However, the local potential distribution of the top-site, bridge-site, and face-centered-site gold atoms on the Au(111) surface is not strictly C_3v."
"It only approaches C_3v under a long-range average."
"How does your ’flat shape’ handle this local deviation?"
His question was a sharp one.
It was the kind of question only someone who had actually conducted experiments in this field, who had really prepared samples on Au(111), would ask.
Even Wu Kai nodded slightly in approval upon hearing it.
Li Dong thought for a moment.
He didn’t use the marker, nor did he flip through his draft paper.
He simply began to speak.
"Teacher Lu, I’m not treating it as a strict C_3v."
"I’m treating it as ’C_3v plus a small symmetry breaking’."
"This breaking component is written as an external field on the base space within the stratified structure."
"It doesn’t enter the U(1) on the fiber, nor does it enter the Z_2 automorphism of the structure group."
"It just gives the ’average flatness’ of the base space a slight push toward the local side."
"In the expansion of κ, it only contributes a High-Tier small quantity."
Lu Mingyuan said nothing.
He lowered his head and carefully copied down this explanation.
Wu Kai was taken aback.
He glanced at Lu Mingyuan.
’Old Lu was an old-timer who had been in this field for over twenty years, and he was exceptionally rigorous when it came to academics.’
’Usually, in his own research group, if a student reported data that was off by a single decimal point, he would chew them out.’
’A man like him never asked a question expecting it to be answered so easily.’
’For every question he posed, he already had a clear answer in his own mind.’
’But now, he was actually taking notes...’
’That could only mean one thing.’
’The answer this student gave was even more beautiful than the one he himself had come up with.’

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