Chapter 292 - 269: Professor, It’s Very Physical

Chapter 292: Chapter 269: Professor, It’s Very Physical

This approach was from the physics side of things.

But right below it was another line...

[Under the local basis transformation functor of GL₂, the phase difference of the three basis sets is reduced to an isomorphism class of a finite-dimensional algebraic representation, determined by the local root number of the L-factor.]

Below this line, Li Dong used a pair correlation function symbol he often used, F_π(α), to pin down the possible values of φ to a discrete set.

The final step...

He combined the zeros of F_π(α) with the sgn from the parity selection rule to arrive at a clean, simple conclusion:

[φ=0]

Wu Kai stared at the line [φ=0].

He stared for three seconds.

Five seconds.

...

He didn’t even realize that the hand holding his thermos had trembled, spilling water onto his cuff.

The third page.

The fourth.

...

On each subsequent page, Li Dong began to develop his work on the third coordination shell.

Strangely, as Li Dong worked on the third shell, he didn’t use any form of regularization.

He was using a different language entirely.

Wu Kai couldn’t quite understand it...

But he recognized the most crucial terms.

"WKB."

"Potential barrier decay."

"Gamov factor."

...

Wu Kai’s head snapped up, and he stared at Li Dong.

His Adam’s apple bobbed. He took a sharp breath.

"Li Dong."

"You..."

Wu Kai subconsciously wanted to ask...

’Did you also complete the inversion that comes after this?’

But he swallowed the words back, because he had a feeling this wasn’t the inversion.

So he settled on a question that was closest to his intuition.

"You... pushed past this step?"

...

Li Dong looked up.

He blinked, clearly not understanding which step Wu Kai was talking about.

"Pushed past what?"

"I haven’t."

Li Dong shook his head earnestly.

Wu Kai was speechless.

Wu Kai slapped the draft paper in his hand down on the desk, pointing to the line [φ=0] on the second page.

"Then what’s this?"

Li Dong glanced down.

"Oh, this."

"I just resolved the phase ambiguity of the basis function back in step seven."

"Then in steps eight and nine, I continued pushing forward from the part about the third coordination shell."

"After that, I couldn’t get any further."

...

Wu Kai stared at Li Dong’s calm face, stunned for a full three seconds.

It finally dawned on him.

The "couldn’t get any further" that Li Dong had just mentioned...

...wasn’t *the* wall—no, not the one that had stumped researchers both at home and abroad!

When Li Dong said he "couldn’t get any further,"...

...he meant he couldn’t get past the entire spectral inversion problem as a whole!

As for this wall that everyone had been beating their heads bloody against, he...

Wu Kai simply held up the page with the [φ=0] line and pointed directly at the derivation at the top.

"This step..."

"Look here, this exact step!"

"Did you get past it?"

Li Dong glanced down.

He looked up and said to Wu Kai,

"Yeah."

"This step wasn’t very hard."

Wu Kai: ???

Wu Kai immediately started ranting in his head.

’Not very hard?’

’If it wasn’t very hard, why have over a dozen research groups across the country been stuck on it for three whole years?’

Li Dong was completely oblivious to the storm raging in Wu Kai’s mind.

He pointed to the derivation below his [φ=0] line.

"I used the local basis transformation functor of GL₂ here."

"Basically, I first treated the three basis sets as Whittaker models of the same global automorphic representation at different local places."

"Then, I used the local root number of the L-factor to lock the phase difference of the transition matrix within a finite-dimensional algebraic representation."

"Finally, I wrapped it up using the real sign provided by the dipole parity selection rule."

"It’s not too convoluted, basically."

Li Dong said it all very naturally.

Wu Kai froze as he listened.

He could follow the first part, "GL₂."

But starting from the next phrase, "local basis transformation functor"... he was completely lost.

Pure mathematics—the Langlands Program, automorphic representations, Whittaker models...

All of that was a million miles away from his field.

Wu Kai was speechless for a long time.

Finally, he managed to squeeze out a sentence.

"Li Dong."

"Speak English."

Li Dong was speechless.

Only then did it click for Li Dong.

So *this* was the wall where over a dozen research groups across the country were stuck.

They weren’t stuck on the entirety of the "inversion" problem.

They were stuck on one specific step *within* the inversion.

The step with the basis function’s phase difference.

He had assumed everyone was stuck at the main bottleneck of the entire inversion issue.

So when Professor Wu Kai said they’d been "stuck for several years," he had subconsciously applied that "several years" to the grand problem of the entire inversion.

But that wasn’t it.

Their "several years" had been spent on the one step he had bypassed in half an hour.

...

Li Dong couldn’t exactly laugh at them, though.

He knew full well.

He had managed to bypass it in half an hour because the tools he was using weren’t borrowed from the field of inversion at all.

His tools came from the Langlands Program.

For a scholar in applied mathematics, it would never even occur to them to approach a spectral inversion problem from the direction of GL₂’s local basis transformation functor.

They were two different languages.

Two different circles.

If it weren’t for Li Dong—someone who was literally "laying the foundation" for the Langlands Program—happening to join this project team...

...these two paths would have never crossed.

Because who would expect a pure mathematics scholar to show up and do applied mathematics?

That’s why this step, which seemed like an insurmountable chasm to outsiders...

...was a simple, convenient task for him.

Li Dong figured this all out.

He looked up at Wu Kai.

He took a deep breath.

"Teacher Wu."

He said, in all seriousness,

"This step isn’t the real problem."

Wu Kai, still reeling from the shock of Li Dong’s progress, asked subconsciously,

"What did you say?"

Li Dong’s tone was calm.

"The real place where our project, and indeed all dozen or so groups around the world, are truly stuck..."