Mr. Yang grunted in acknowledgment, gesturing for Li Dong to continue.
Li Dong recounted the problem he was facing.
"On an Au(111) single-crystal substrate..."
He spoke slowly, leaving a pause after each sentence for Mr. Yang to digest the information.
"What I want to do now is to physically parameterize this effective potential barrier from its source."
"Instead of using inversion."
Mr. Yang narrowed his eyes.
He had never encountered this specific problem before.
But he was familiar with every component within it.
Li Dong continued.
"I’m stuck on one point."
"The U(1) gauge invariance of the electromagnetic field."
"The discrete parity of the atomic orbitals."
"The local C_3v symmetry of the Au(111) substrate."
"These three things must be satisfied simultaneously."
"I need to write them into a single, unified gauge structure."
"I tried taking the semidirect product of U(1) and parity, but the resulting structure was very awkward."
"I also tried going down the path of a non-Abelian Berry connection, but the resulting expression wasn’t closed under the permutation of the three indices."
"I’ve tried both paths, and neither of them worked."
He finished speaking.
Mr. Yang didn’t speak right away.
He leaned back on the sofa, his eyes closed.
After ten or so seconds, he slowly opened his eyes.
He looked at Li Dong.
"Li Dong, this understanding of yours... did you come up with it yourself?"
Li Dong nodded.
Mr. Yang said with a smile.
"Very good."
"For someone in mathematics to be able to connect these three things at the atomic scale is already quite an achievement."
"However..."
Mr. Yang looked at him and said slowly.
"Your idea of trying to fit them all into the same group... that in itself is a bit greedy."
Li Dong froze for a moment.
"In the field of gauge theory, that’s never how it should be done."
Mr. Yang’s tone was even.
"U(1), let it live on the fiber."
"C_3v, let it live on the base space."
"Parity isn’t a group. It’s an automorphism within the structure group."
"The three of them are not on the same level."
"If you force them onto one level, of course the result will be awkward."
"Let each of them return to its own level, and the overall geometry will emerge."
"That was the most critical thing Old Wu and I did back in the day, when we established the correspondence between gauge fields and fiber bundles."
"Stratification."
Having said his piece, he leaned back into the sofa, picked up his teacup, and slowly took a sip.
The room fell silent once more.
Li Dong sat on the sofa.
A BUZZ went through his mind.
It all clicked.
Everything clicked into place.
For the past two days, he had been stuck, hitting his head against a wall, constantly trying to write U(1), Z_2, and C_3v into a single, unified algebraic structure.
Whether it was a semidirect product.
Or a non-Abelian Berry connection.
All the approaches he had tried were just ways of forcing everything onto the "same level."
But Mr. Yang’s words...
’They are not on the same level.’
...had instantly resolved all the awkwardness.
U(1) is written on the fiber, performing the gauge transformation of the phase.
C_3v is written on the base space, handling the discrete symmetry of the spatial points.
And Z_2 parity is an automorphism of the structure group—the very involution that flips the U(1) on the fiber.
Three levels.
Each with its own language, its own stage.
Their interactions are naturally realized through the overall geometry of the fiber bundle.
Not crammed into a single group.
But stacked in a three-level structure.
Li Dong quickly ran through all the expressions he had been stuck on.
The transition function of the dipole operator between the three sets of bases...
It was no longer an isolated, real-valued symbol.
It was the product of the C_3v-invariant part on the base space, the U(1) gauge connection part on the fiber, and the phase flip corresponding to the structure group’s automorphism...
A product of three things.
Each component had its own clean, geometric origin.
And most crucially...
The effective potential barrier from the section with the Au(111) substrate and the Ligand Field.
It wasn’t an inexpressible "black box."
It was simply the flat part of the underlying connection on this stratified fiber bundle.
Its flat shape could be read directly from the C_3v invariant tensor.
No inversion needed.
This κ term could be written out directly using the geometry of the fiber bundle.
Parameterized physically, right from the source.
The wall in Li Dong’s mind, which had seemed utterly immovable, came crashing down without any warning.
"So that’s how it is."
He murmured under his breath.
Never before had Li Dong felt it so clearly: a string in his mind had been gently plucked, and then the entire wall had come tumbling down.
He didn’t just understand it.
He saw it.
He saw the complete geometric construction of the effective potential barrier laid out in his mind, from the probe tip to the vacuum, to the substrate, to the Ligand Field, and finally to the X-ray polarization.
And the physical path was like a line of dominoes, starting from the [vacuum section] he had written down, and toppling all the way to the κ term.
As he was immersed in the thought of the dominoes that had just toppled in his mind,
he suddenly felt that his mind was a bit clearer than usual.
’My attributes have increased?’
Li Dong really wanted to open his attribute panel to check, but it wasn’t convenient to take out his phone, so he could only judge by feel.
’Hmm, almost every attribute has increased by about 0.2...’
Li Dong’s heart skipped a beat.
’Does Mr. Yang... also have the Inheritance of the Torch skill?’
He subconsciously glanced at the centenarian sitting across from him.
The old man was leaning back on the sofa, slowly sipping his tea and looking at the green bamboo outside the window.
’No.’
Li Dong quickly realized.
’This isn’t the Inheritance of the Torch.’
’That Inheritance of the Torch thing is like a red envelope from a group chat; strictly speaking, it’s an external object.’
’This is a true master’s effect.’
’This old man, just by sitting there, could untangle the knotted string in my mind.’
’He didn’t need to explain the complete solution to me at all.’
’He only had to say that one sentence: "The three of them are not on the same level."’
’...and it reordered the thoughts I already had in my own mind.’
Li Dong stood up.
He gave Mr. Yang a slight bow.
"Thank you, Mr. Yang."
Mr. Yang glanced at him and said with a smile.
"Figured it out?"
Li Dong nodded.
"Yes. It all makes sense now."
Mr. Yang hummed in response.
He didn’t say anything more about this specific problem.
He just looked at Li Dong and said.
"Ah, Li Dong."
"You have a great talent for physics."
"In the future, if you have any questions, you can just call Fanfan directly."
Dr. Weng, who was standing nearby, heard this and nodded at Li Dong.
At that moment, Dr. Weng glanced at her wristwatch.
She leaned down, close to Mr. Yang’s ear, and spoke very softly.
"Sir."
"It’s about time."
Mr. Yang nodded to show he understood.
It wasn’t that he was unwilling to chat more.
His own mental stamina just couldn’t keep up.
Gao Yuanlin and Li Dong naturally understood and immediately stood up to take their leave.
"Mr. Yang."
"Please get some rest."
Mr. Yang gave a slight nod from the sofa.
"Alright."
"Be careful on your way out."
Dr. Weng walked them to the courtyard gate of Gui Gen Residence.
When they reached the gate, she took a small card from her pocket and handed it to Li Dong.
"This has my phone number and WeChat."
"If you need anything, just send me a message."
Li Dong took the card and said.
"Thank you, Dr. Weng."
Dr. Weng smiled, said nothing more, and turned to go back into Gui Gen Residence.

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