Chapter 217 - 209: I Don’t Even Want to Leave You

Chapter 217: Chapter 209: I Don’t Even Want to Leave You

A few days later, in the basement of Building 35 at Yan University’s Yuanpei College.

The basement of Yuanpei Building 35 was even livelier than the floors above ground.

Cafeterias, coffee shops, a gym—it had everything.

Usually, when students needed to hash out a project or grind out a paper, they’d find a room here, lock themselves in, and get to work.

Li Dong had arranged to meet with Old Yang and Penrose.

Before heading out, he logged into his email and sent a PDF he had written.

Recipient: Zhou Qifeng.

Subject: Problem Set on SU(N) Gauge Field Chiral Anomaly on a Four-Dimensional Compact Euclidean Manifold.

The PDF was seventeen pages long.

It covered Fujikawa’s path integral derivation of the chiral anomaly; the topological correspondence between the second Chern number and the instanton number; the triangular closed loop connecting the index theorem, chiral zero modes, and anomaly coefficients; and a one-loop exactness proof of the Adler-Bardeen theorem. He had also added a page of his own footnote-style remarks on extending this anomaly to a curved background.

He hesitated for two seconds, then deleted that page of footnotes in the end.

’Forget it. Better not reveal my talent for physics just yet...’

He clicked send.

...

When Li Dong arrived at the seminar room, Penrose was already there.

The old professor was as excitable as ever, practically buzzing as if he’d taken an expired dose of Tadalafil.

The notebook in front of him was covered in dense writing.

Old Yang had clearly gotten a bit thinner over the past few days, and his dark circles were deeper, but his eyes were sharper than ever.

The three of them sat down at the long table.

Li Dong skipped the pleasantries and got straight to the point.

"Let’s go over the progress each of us has made these past few days."

He looked at Old Yang.

"Teacher Yang, you first."

Old Yang took a few printouts from his bag.

"For the recursive filtration part, I’ve finished the rigorization for e_v = 3."

"I built it on the foundation I laid fifteen years ago."

"The integration path preemptively embeds the information from the Hodge-Tate filtration into its parameterization, causing the universal obstruction to fall onto the zero-layer of the filtration on its own."

"In the case of e_v = 3, I added another layer."

"Specifically, I made the drift of the second-order Hodge-Tate weights compensate for the first-order orthogonality."

He looked up at Li Dong.

"Strictly speaking, I made it so the projection direction of the second-order drift was pre-absorbed by the parameterization of the integration path."

Li Dong nodded slowly.

"And what about when you try to extend it to e_v ≥ 4?"

Old Yang gave a wry smile.

"I tried. The nonlinear coupling of the fourth-order Hodge-Tate weights is too messy. With this layer-by-layer compensation approach, it starts to spin out of control by the fourth layer."

"Every time you add a layer, you have to re-tune the parameterizations of the previous three, and they no longer connect properly."

"So what’s the next step?" Li Dong’s voice was even, watching Teacher Yang without offering any hints.

"I need to abstract away this method of layer-by-layer compensation and turn it into a unified filtration embedding operator."

Old Yang flipped the printout to its back, which was covered in dense scratch work.

"The construction of this operator can’t be based on trial and error. There must be an algebraic structure to support it."

Li Dong quietly studied the page of scratch work.

Old Yang’s layer of abstraction had arrived precisely at the point Li Dong needed.

He said to Old Yang,

"Alright, then continue pushing in that direction for this part."

"Once that operator is constructed, the necessity framework on my end can be connected to it."

Old Yang nodded.

Li Dong turned to Penrose.

"Professor Penrose?"

Penrose said excitedly,

"The bridge is halfway built."

"The original conditions of the Terence Tao-Ruan high-dimensional random matrix universality theorem are too restrictive. They can’t be directly applied to the zero distribution of automorphic L-functions."

"Their conditions require the matrix elements to be independent and identically distributed."

"I’ve been trying to find a way to relax that condition these past few days."

"My approach is to directly map the scale invariance of the GUE kernel function in the GL(n) case from the second-order correlation function on the random matrix side."

"I’ve given this bridge a temporary name: the π-adaptive universality condition."

"But this is as far as I can build it."

"The other half—whether this condition can actually be applied to the zero distribution of automorphic L-functions—I can’t prove on my own."

Li Dong raised his eyebrows.

He didn’t respond to Penrose immediately. Instead, he quickly ran through his own progress from the last few days in his mind.

He had already constructed the top layer of the necessity framework for the zero criterion.

In other words, if π satisfies local-global compatibility, then the pair correlation function of the zeros of its corresponding automorphic L-function must satisfy a certain convergence within a specific interval. That direction of the proof was basically established.

The reverse direction, sufficiency, he hadn’t touched at all.

And the toughest nut to crack in this entire project was precisely the sufficiency part.

’How can you deduce the local structure of an automorphic representation from the statistical properties of its zeros?’

He had originally thought this step would be a long grind.

But now...

The π-adaptive universality condition Penrose had built gave him a perfect interface.

If that condition could be further refined, the GUE convergence of the zero distribution could be translated into a kind of "scale-invariant compactness" of a spectral operator. Then, using an auxiliary lemma he had just come up with last night, he could trace this compactness back to the ramification structure of the automorphic representation.

The path was clear.

But it was still a long way off.

Li Dong began to speak slowly.

He spoke very slowly, wanting to ensure Penrose understood every single word.

"Professor Penrose, this bridge of yours... I think you should approach building it from a different perspective."

Penrose was intrigued.

"Go on."

Li Dong didn’t reach for a pen; he just sat there.

"When you were thinking about this bridge, you were probably thinking of it as a strengthened version of a universality theorem."

"You’re trying to add and subtract conditions within the framework from Terence Tao to make it fit the case of automorphic L-functions perfectly."

"But if we look at it from another angle..."

"Imagine you are Riemann."

Hearing this, Penrose’s face flushed with excitement.

"When Riemann wrote about the Zeta function, what was on his mind?"

"He wasn’t just thinking about the properties of this complex function."

"He was thinking that the distribution of prime numbers is a piece of music, and its main melody is composed of invisible notes sitting on the critical line."

"He had the music in his head first, then the function."

Penrose was silent.

Li Dong continued.

"Now, look at this bridge you’re holding."

"It’s not a tool for ’porting a theorem from random matrix theory over to automorphic L-functions.’"

"It is a piece of music that perfectly covers all GL(n) automorphic representations."

"The main melody of this music is the GUE’s second-order correlation curve."

"Every time you change n, you’re just playing a different variation of the same piece, but the main melody remains unchanged."

"An unchanging melody means this theorem doesn’t need to be supported by the independent and identically distributed condition from the random matrix side."

"Its support comes from geometry."

"From the symmetric structure of the spectral space of the automorphic representation itself."

The seminar room fell silent for a few seconds.

Penrose’s expression slowly shifted from quiet listening to one Li Dong was very familiar with...

"My God!"

Penrose exclaimed under his breath, clasping his hands to his chest.

In his eyes, every time Li Dong spoke about mathematics, it was as if some nineteenth-century ghost was speaking through the young man’s body.

And the words Li Dong had just spoken...

Riemann, main melody, geometry...

This was Riemann’s own perspective.

"Dong..." Penrose’s voice was a little hoarse.

"If this path works, what I’m building won’t just be a bridge—it will be an entire spinal column."

"Yes," Li Dong said.

"But we haven’t even finished the bridge yet."

"One step at a time."

Penrose lowered his head, quickly scribbled two lines in his notebook, then crossed one out.

Then he looked up at Li Dong.

"Dong, you’re simply miraculous! I never want to leave you!"

Li Dong was suddenly dumbfounded...

He had indeed done that deliberately to keep Penrose around, but...

’It feels like things are developing in a strange direction, doesn’t it?’

Thankfully, Penrose added,

"Following you is like following Riemann himself."

Only then did Li Dong breathe a small sigh of relief.

’See? How could I possibly attract men?’