8:30 AM, in a small conference room at the Yan University School of Mathematics.
Penrose arrived first.
The professor, who was nearly sixty, seemed even more energetic today than yesterday.
He held a notebook in his hand, its pages densely packed with inspirations from the previous night.
Yang Shengguo walked in right after him.
He had dark circles under his eyes, a clear sign he hadn’t slept much last night.
"Yang, you look like you didn’t sleep a wink," Penrose said with a smile.
Yang Shengguo wasn’t very familiar with Penrose, after all. The man was a top professor from Princeton, even higher in status than Jiang Yubai, so he felt the need to explain.
"Professor Penrose, yesterday on stage, when we got to the third layer of recursion, some ideas kept running through my head. When I got back, I couldn’t help but try to push the proof further..."
Penrose’s eyes lit up.
"And what did you find?"
"I got about halfway through and got stuck."
Yang Shengguo gave a wry smile.
"I still haven’t figured out the projection direction of the weight-two Hodge-Tate on the fourth filtration layer."
Just as they were talking, Li Dong arrived.
"Teacher Yang, Professor Penrose, you’re both here early."
After a quick greeting, he sat down across from Yang Shengguo and got straight to the point.
"Today, let’s talk about the future direction of the project."
Penrose immediately perked up, leaning forward in a posture of rapt attention.
Li Dong picked up a marker and wrote a line on the whiteboard.
A General Proof Path for the Local-Global Compatibility of Automorphic Representations of GL(n)
"Yesterday on stage, while working through the GL(3) case, I said that my real goal isn’t GL(3), but GL(n)."
"But the leap from GL(3) to GL(n) can’t be solved by simple induction."
He drew a timeline on the whiteboard.
"First, we need to completely formalize Teacher Yang’s layered recursive filtration embedding scheme."
"What I wrote on stage yesterday... the core idea is correct, but the details are still lacking. We need to perform a rigorous derivation and fill in those details."
Li Dong didn’t spare his former high school teacher’s feelings in the slightest, stating it directly.
Yang Shengguo nodded. He wasn’t embarrassed, though. After all, Li Dong was no longer the high school student who scored an 87 on his test.
"But that’s all secondary. The most critical step is..."
"We need to prove that as the ramification index e_v is extended from a finite case to the general n, the GUE convergence of the pair correlation function of zeros, F_π(α), still holds within the corresponding interval."
At this, Li Dong smiled. He recalled what Jiang Yubai had said yesterday about the significance of his tools.
’Significance?’
"This step will require the complete results of my work on Montgomery’s Conjecture on Association, but it’s not a direct application."
"We need to make a fundamental generalization, extending the original statistical framework for the zeros of the Riemann ζ-function for GL(1) to general automorphic L-functions for GL(n)."
Penrose’s eyes grew brighter and brighter.
"You mean... establishing a quantitative equivalence between the GUE convergence of the pair correlation function and the upper bound of the local ramification index?"
"Exactly," Li Dong said, looking at Penrose.
"That’s the core of the zero criterion."
He wrote a framework proposition on the whiteboard.
"If we can prove that for a cuspidal automorphic representation π of GL(n), the pair correlation function of the zeros of its automorphic L-function converges strictly to the GUE predicted value in the interval |α| ∈ [0, 2/n] if and only if π satisfies local-global compatibility..."
"...then all intermediate cases from GL(2) to GL(n) will be taken care of in one fell swoop."
Penrose slapped the table.
"Brilliant!"
"Dong, this means you don’t have to tackle GL(3), GL(4), GL(5) one by one... You just need to prove that this zero criterion holds for a general n, and the entire local-global compatibility problem of the Langlands Program will be solved once and for all!"
"But the difficulty is also exponential," Li Dong said with a small smile.
"Proving the sufficiency of the zero criterion will require a completely new set of spectral analysis tools. Nothing in the existing literature is ready to use off the shelf."
Far from being daunted, Penrose only grew more excited.
"This is exactly what I want to do!"
He flipped open his notebook, pointing to a page he’d filled out the previous night.
"Yesterday on stage, you mentioned that the self-adjointness of the spectral operator guarantees the positive-definiteness of the set of weight functions. I thought about that for a long time after I got back."
"If we extend your dynamic adaptive Fourier weight functions to the GL(n) case, the self-adjointness condition needs a more general framework to guarantee it..."
"I thought of a possible direction. We could use the universality theorem for high-dimensional random matrices to directly port over the scale invariance of the GUE kernel in the GL(n) case from random matrix theory."
After listening, Li Dong gave the old-timer a long, hard look.
’Penrose’s Basic Attributes must all be above 0.2. That intuition...’
"Professor Penrose, that direction is correct."
"However, the specific technical approach will need to be carefully thought out."
"The current best results for the universality theorem of high-dimensional random matrices come from the work of Terence Tao and Ruan. But their results have restrictions on the matrix distribution types, so they can’t be directly applied to the zero statistics of automorphic L-functions."
"So we need a bridge here."
Penrose nodded emphatically.
"I’ll build that bridge."
The two of them delved deeper into the discussion, and the formulas on the whiteboard grew more and more dense.
Meanwhile, Yang Shengguo, sitting to the side, hadn’t stopped writing with the pen in his hand.
To be honest, he was having some trouble keeping up.
The discussion between Li Dong and Penrose had already ascended from specific technical details to the top-level design of the entire proof’s framework.
The mathematical tools involved spanned analytic number theory, random matrix theory, p-adic Hodge theory, the spectral decomposition of automorphic forms...
Some of these were things he had never even heard of...
But he wasn’t discouraged.
He knew where his own value lay.
The layered recursive filtration embedding—that was his contribution.
When it came to the specific details of deforming p-adic integration paths, no one present understood it better than he did.
So, what he had to do now was connect the top-level framework discussed by Li Dong and Penrose with the low-level constructions he was best at.
This would take time.
It would also require him to frantically study up when he got back.
He had to return to Jiangcheng after the winter break, and there wasn’t much time.
So, not a single minute could be wasted.
The discussion lasted for nearly two hours.
By the end, whiteboards covered the floor. Li Dong circled several unresolved problems with a red marker.
"That’s about it."
He put down the marker and looked at Yang Shengguo and Penrose.
"Professor Penrose will be in charge of constructing the bridge from the random matrix theory side. Teacher Yang will be responsible for writing the rigorous proof for the layered recursive embedding. And I’ll handle the generalization of the zero criterion to the GL(n) case."
"We’ll advance on all three fronts in parallel. If any problems come up, we can discuss them online anytime."
Penrose stood up and shook Li Dong’s hand.
"Dong, I haven’t been this excited about a project in a very long time."
Li Dong glanced at Penrose. ’Your excitement is exactly what I was hoping for.’
He remembered what Liu Ruochuan had told him.
"Keep Penrose here. The university needs him..."
At that moment, Yang Shengguo also closed his notebook, which was full of notes, and let out a long sigh of relief.
"Looks like I’ll have to pull a few all-nighters when I get back."
Li Dong smiled and patted his shoulder.
"Make sure you study up when you get back. No slacking."
Yang Shengguo was speechless.
"You’re the second author on my paper now. The standards can’t be low."
Yang Shengguo couldn’t be bothered to respond and simply turned and walked away.
After Penrose and Old Yang left, the phone in Li Dong’s pocket rang.

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