This part of the calculation was supposed to forcefully merge the expansions from different bases.
The standard approach was to use an implicit constraint to clamp the upper bound of the transition matrix.
But when Li Dong’s calculations reached this point, he just couldn’t get past it.
He tried three different regularization methods, and each one yielded worse results than the last.
He tried to force in the "implicit constraint" that Academician E Weinan had pointed out.
It provided a constraint, but not a firm one.
The edge of the third peak could be faintly distinguished from the noise, but just as it started to emerge, it would fall back in.
Li Dong stared at the line of symbols and sighed.
’...Of course.’
’A dozen-plus research groups across the country, and every single one is stuck here.’
Just as he set down his pen for the third time, a voice suddenly spoke from behind him.
"Stuck?"
Li Dong whipped his head around.
Professor Wu Kai was standing behind him, though Li Dong hadn’t noticed him arrive.
Li Dong quickly shot to his feet.
"Teacher Wu."
"You’ve arrived."
Wu Kai waved his hand, walked over, and sat in the chair opposite him.
"I just came from a meeting at the School of Chemistry."
He glanced at the scratch paper on Li Dong’s desk.
He saw the half-finished spectral decomposition of the transition matrix on the top sheet.
Wu Kai nodded.
He could understand this part.
It wasn’t because his own math was particularly good.
It was because everyone in the entire field was stuck on this exact part.
So many people had gotten stuck that he had become familiar with it, expert or not.
Wu Kai smiled.
"Li Dong, don’t get anxious."
"This problem has been a sticking point for several years now."
"Everyone, both here in China and abroad, is stuck at the very step you’re on."
"You’ve already gotten this far in a single morning. That’s faster than the vast majority of people."
"Besides, don’t put so much pressure on yourself."
He looked at Li Dong with a kind expression.
"I never expected you to come in and solve the problem right away."
"Our group’s schedule is planned around spending one or two years grinding away at this math problem."
"We have all those samples and probe tips ready. We can work on the subsequent stages of integration for those first."
"Once you have the solution from your end, we can plug it back in. That’s also a viable path."
Wu Kai was being sincere.
He was afraid Li Dong would feel disheartened by the setback.
After all, Li Dong was still young.
He had progressed so incredibly fast in the world of pure mathematics.
For prodigies like him who achieved fame at a young age, the greatest fear was smashing their head against a brick wall in a new field.
Wu Kai hadn’t come over just to check in.
He was here to give Li Dong a safety net.
Li Dong nodded.
He understood Professor Wu Kai’s good intentions.
He was genuinely grateful.
He was about to say, ’Don’t worry, Professor, I’ll keep grinding away at it’.
But before the words left his mouth, he remembered something.
When he’d gotten stuck at step seven earlier, he hadn’t kept banging his head against that particular wall...
He had used a different method to "detour" around it, borrowing a technique from pure mathematics. He set step seven aside and jumped straight to step eight.
From step eight onward, he had worked through several more pages in one sitting.
’I don’t know if this is valid, though. After all, nobody else has used a pure math approach. What if it doesn’t apply here?’
So he asked,
"Professor, for the section after that, I used my own method. Could you take a look and see if it’s valid?"
As he spoke, Li Dong bent his head and started flipping through his scratch paper to get to the last few sheets.
Hearing this, however, Wu Kai froze solid.
’What does he mean? He got past it?’
Wu Kai wasn’t an expert in mathematics.
His career was in chemical synthesis.
But on the math side, because this project had been stuck for several years, he had started "studying along" with it for the past year or two.
Within this inversion problem, he knew where every step was, why people got stuck, and the specific point of failure...
He knew it all like the back of his hand.
The page with the spectral decomposition of the transition matrix on Li Dong’s desk...
He had seen the sticking point the moment he walked in.
But the words that had just come out of Li Dong’s mouth...
’The section after that?’
’What "after" section?’
’There is no "after" this step.’
’Everyone gets completely, irrevocably stuck at this step.’
Wu Kai spoke.
"Hold on."
"Li Dong, what did you mean by that?"
Li Dong looked up.
"Huh?"
"Oh."
He gestured toward the pages in his hand that he was flipping through.
"I mean the problem you run into after you get past this step."
"I was hoping you could take a look and see if my derivation is correct."
Li Dong said it so matter-of-factly.
Wu Kai froze for a second.
"’After’ you get past this step?"
He sounded a little uncertain.
Li Dong didn’t register the nuance in his tone.
His attention was still focused on his pages of derivations.
"Mhm."
"Look at this section... Here, I used a transition functor from the local Whittaker model to first freeze the phase differences of the three bases within a fixed algebraic structure..."
When Wu Kai heard the three words "local Whittaker model," he was completely baffled.
’What is that??’
From his dealings with applied mathematics, he had heard of basis functions, operators, and inversion, but he had never once heard of a local Whittaker model.
This wasn’t the language of inversion problems.
This was...
...the language of the Langlands Program.
Wu Kai’s hand drifted unconsciously toward Li Dong’s pile of scratch paper.
"This here..."
"It’s more than just one page?"
Only as he raised his hand did he realize that beneath the top sheet on the stack of scratch paper on Li Dong’s desk was a thick pile.
Wu Kai subconsciously pulled away the top sheet—the one with the spectral decomposition of the transition matrix—revealing the second page underneath.
He looked down.
The second page was densely packed with symbols.
The top line, he recognized...
[Let the phase ambiguity between the transition matrix elements U_ij be φ∈R/2πZ. By the dipole parity selection rule, sgn(M_dipole)∈{+1, -1} is fixed.]
When Wu Kai saw the four words "dipole parity selection rule," his eyebrow twitched.

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