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Chapter 290: Chapter 269: Teacher, It’s Very Physical

Of course, Zhang Yan had no idea that he’d already slapped a new label on her in the time it took her to say one sentence.

She turned and jerked her head toward the lab bench.

”Alright, stop spacing out. Come on over.”

Li Dong followed her.

Zhang Yan spoke as she tidied up.

”East God, you finished reading that file, right?”

”I have.”

”Did you read the part about the early-stage bottleneck?”

”I did.”

”Then you know our project group has limited areas we can actually work on right now. The bigger priority is to first build the framework for the ’inversion’ side.”

Li Dong nodded.

He’d figured that out last night... no, he’d completely sorted it out just this morning.

According to the Wu Kai research group’s current technical roadmap:

On the chemical synthesis side, they were already working on the single iron atom assembly. For the terbium side, Teacher Su was already setting up the synthesis line for the Tb(pcam-Br)₃ complex.

The Au(111) single-crystal substrate had been sourced from that Russian lead metallurgy supplier.

The ultra-high vacuum chamber was already running.

The lock-in amplifier and the X-ray chopper system were undergoing ”interface alignment” under Teacher Su’s team.

As for the beamline, they had to wait for Professor Lu Mingyuan to return from the Capital City.

So, everything currently stalled in the middle was all theoretical...

The inversion algorithm, the ligand field tensor, spectral fitting, and phase calibration.

If this part didn’t move forward, there was no point in making the samples in the first part.

And no matter how well Lu Mingyuan tuned the beamline later, they would only get a set of tunneling spectra that no one could understand.

”So, your job.”

”This morning, go and run the math.”

Li Dong was slightly taken aback.

”Right now?”

”Mhm.”

Zhang Yan nodded.

”You were brought in to take the lead on the math side of our group.”

”Now that work has started, the pressure on you is going to be immense.”

”Over in Hecheng, Professor Levan is currently working on this exact mathematical inversion part.”

”If we don’t get in the game now, by the time they figure it out, all we’ll be able to do is ’reference their conclusions’.”

Zhang Yan continued.

”So this morning, don’t worry about anything else.”

”First, go to that small office next door.”

”The soundproofing in that room is better; it’s quiet.”

”You should first go over those two mathematical bottlenecks by yourself again.”

”Start by pushing through with the currently accepted ’inversion’ path.”

”Don’t worry about that ’wrong direction’ feeling you have.”

Zhang Yan glanced at him, her expression serious.

”I know you had some ideas yesterday.”

”But before your idea is fully formed, our progress here can’t stop.”

”Even if your idea is correct, you first have to prove that the ’inversion’ path is truly a dead end...”

”Otherwise, you won’t even be able to convince yourself.”

”Understand?”

Li Dong looked at her for a moment.

’So it’s not just your intuition that’s sharp, your thinking is crystal clear too,’ the thought almost slipped out.

He nodded.

”Alright.”

”I’ll go over it.”

”I’ll try to get results as fast as possible and not get stuck on the first couple of steps.”

Zhang Yan nodded.

”Right. For those first few steps, just see how far you can get.”

”Right now, so many groups, both domestic and international, are all stuck there.”

”If you can move forward half a step, our group will be half a step ahead of everyone else.”

Li Dong grunted in acknowledgment and headed toward the office.

...

Once in the room, Li Dong had his own plan.

What he needed to do was more than just ”running the math.”

He also needed to verify something along the way.

The new physics approach he’d thought of this morning needed to be backed by data.

Just because he’d come up with a beautiful picture on the physics side of things didn’t mean it was necessarily correct.

He had to use how far he could get with the math as a benchmark.

What if the mathematical direction was just a minor hiccup, with smooth sailing ahead?

Then his whole night spent on the physics approach would have been a waste of effort.

He couldn’t overturn the entire group’s project schedule based on a gut feeling.

So, he first had to push the mathematical side to its absolute limit.

Li Dong placed his bag on the desk but didn’t turn on the computer.

He took out a scratchpad and pen first.

Li Dong’s background was in pure mathematics.

Strictly speaking, inverse problems weren’t his specialty.

But in the past few days, he had already chewed through more than half of the reading list Academician E Weinan had given him.

Hadamard’s well-posedness conditions, Tikhonov regularization, spectral truncation, the Backus-Gilbert method...

All these names, along with their corresponding core theorems, were solidly filed away in the [Applied Mathematics/Inversion] slot of his Memory Palace.

Plus, his speed of comprehension was already far beyond that of an ordinary person.

Therefore, he was confident he was no worse than a typical applied mathematics scholar.

He sat down and began his derivation.

Step 1: Formal definition of the measurement operator K: T→S, where T is the ligand field tensor space and S is the tunneling spectrum space.

Step 2: Kernel function expansion, decomposing K into a sequence of eigenfunctions.

Step 3: Apply Upper Gihonov regularization, adjust parameter α.

Step 4: Map the singular value spectral distribution of K to the low-rank approximation of T.

Step 5...

He worked through it, step by step.

At each step, he checked it against the set of physical parameters he had constructed last night to see if the mathematical side could be reconciled.

For the most part, the first five or six steps presented no major problems.

He was stuck on the Tikhonov part for less than ten minutes before resolving it with a weighted norm.

For the choice of basis functions, he switched to a set of Whittaker-type local bases, avoiding some of the phase ambiguity.

...

Until Step 7.

Li Dong stared at the line he had just written and frowned.

[Perform a spectral decomposition of the phase operator on (L²(R³, dμ), 〈·, ·〉_K), requiring the transition matrix between the three sets of bases to converge to a unitary transformation with no phase ambiguity...]

”Is that right?”

”No, that’s not right.”