Of course, Li Dong had no idea what was happening in Zhou Qifeng’s study.
His winter break couldn’t have been simpler.
Eat, sleep, go to the seminar room.
The only thing that changed was the ever-increasing number of whiteboards in the seminar room.
Eventually, the staff just wheeled in a dozen more from other classrooms, lining them up along the walls.
The three of them, three lines of inquiry.
Penrose was responsible for expanding the universality of the zero-point pair correlation from |α|<1 to |α|∈[0, 4].
Yang Shengguo was responsible for generalizing the recursive filtering embedding from the Sunshine Hall talk from GL2 with ramification ≤2 to GL(n) with ramification ≤n.
Li Dong himself was in charge of the most crucial step: linking the statistical properties of the zero-point pair correlation to the local ramification index to create the brand-new "zero-point criterion."
The three lines of inquiry were ultimately destined to converge at a single point.
And the "narrow-band approximation + Schwarzian derivative" scheme that Penrose had hammered out a few days ago had become an accelerator for the entire project.
The universality expansion, originally projected to take the better part of a year, was compressed to less than three weeks by this method.
It was like they’d strapped a rocket to it.
One afternoon.
After writing on the whiteboard for a good while, Penrose tossed down his marker and turned around.
"Dong."
"The cross-layer coupling terms for the second-order filtering weights—I’ve handled them."
Li Dong looked up and asked.
"Did you derive it using the Schwarzian derivative method?"
"Mhm."
"Used that standard toolkit you gave me last time."
Li Dong leaned in for a look and nodded.
"Professor, it’s clean."
Penrose smiled.
That problem was something he himself had struggled with for the better part of a month.
Now, using the new method he had proposed, he had flattened it in one go.
The old man lifted his coffee cup for a sip, his mood soaring.
In front of another whiteboard.
Yang Shengguo had been staring at his whiteboard, frowning, for nearly an hour.
"Where are you stuck, Teacher Yang?" Li Dong walked over.
Yang Shengguo sighed.
"I’ve derived it up to the (n-1)th layer."
"With the old method, you have to reconstruct the compatibility conditions for the lifting operator at every single layer."
"The workload increases exponentially."
"A couple of days ago, I was already struggling by the fifth layer."
Li Dong thought for a moment, then pulled a chart from his stack of papers and tossed it to him.
"Here, use this for now."
Yang Shengguo glanced down at it.
"...What is this?"
"It’s what I got from running my 1023-order sample of zeros," Li Dong said.
"I’ve organized the correspondence between the ramification index and the pair-correlation convergence interval into a numerical table."
"Use it to reverse-calibrate your estimate for the order of the lifting operator."
"You won’t have to construct it term-by-term for each layer anymore. You can just verify it holistically."
Yang Shengguo held the chart and studied it for two minutes.
Then he looked up.
"Kid."
"Couldn’t you have given this to me earlier?"
Li Dong blinked.
"I only just finished organizing it."
"Any earlier, and the cells in the table would’ve been empty."
Yang Shengguo sighed, then couldn’t help but let out a small laugh.
The recursive chain, originally estimated to take two months, was finished in five days.
But the truly difficult step was still the one on Li Dong’s plate.
Within the zero-point criterion, the necessity part was manageable.
The hard part was sufficiency.
There was a thick wall standing between "F_π(α) converges to the GUE predicted value n|α| on the interval |α|∈[0, 2/n]" and directly proving that "π satisfies local-global compatibility."
He and Penrose had discussed this wall more than a dozen times, approaching it from all angles, but they had never managed to break through.
That was, until Yang Shengguo’s line of inquiry reached its latter stages.
He had written the compatibility difference for his recursive lifting operator between the k-th and (k+1)th layers in a particularly clean form.
Li Dong happened to see it as he passed by Yang Shengguo’s whiteboard.
Then he suddenly looked up.
"Teacher Yang."
Yang Shengguo was erasing another whiteboard. Hearing his name, he turned around.
"Hm?"
"This line of yours," Li Dong said, pointing to the whiteboard, "is the exact dual of the term I need to eliminate in my sufficiency proof."
Yang Shengguo froze for a second.
"It can’t be that much of a coincidence, can it?"
Li Dong didn’t answer.
He snatched the marker from Yang Shengguo’s hand and drew a duality diagram in a blank space on the board.
One line was the inter-layer compatibility difference of Yang Shengguo’s recursive lifting operator.
The other line was the stubborn remainder term from his own sufficiency proof that he hadn’t been able to eliminate.
He finished drawing the two lines, and they connected.
The two of them looked at each other.
Penrose stood off to the side without a word, but the corner of his mouth twitched.
This was that mysterious, almost magical, spark of scientific collaboration.
The three lines of inquiry were supposed to be parallel.
If the three researchers were truly in sync,
you didn’t have to force them together.
They would find each other on their own.
After this breakthrough, the skeleton of the project became clear.
Penrose’s line of inquiry was already in its concluding stages.
Yang Shengguo’s had been sealed up to the final layer of GL(n).
And on Li Dong’s end, the wall in his sufficiency proof had been torn down...
All three of them knew where they stood.
The project was about eighty percent complete.
The remaining twenty percent could only be done by Li Dong alone.
"Professor."
Li Dong closed his notebook and looked at Penrose.
"Your line of inquiry... I think you can pause it for now."
Penrose nodded.
"That’s what I was thinking as well."
"If I continue any further, I’ll start encroaching on your territory."
Li Dong smiled.
"Tomorrow, I’ll help Teacher Yang go over the final layer one more time."
"After that, you can leave the rest to me."
Penrose raised his coffee cup, saying nothing.
The lights in the seminar room stayed on until two in the morning.
The writing on the whiteboards was erased and rewritten, again and again.
And just like that, winter break was over.
It was time for Yang Shengguo to return to Jiangcheng.
On the day of his departure, Li Dong and Penrose saw him off to his taxi.
"Teacher Yang, I’ll handle things from here. You can go without any worries!"
Yang Shengguo: ...
Although this kid Li Dong had a way of phrasing things poorly,
he knew he couldn’t be of any more help from this point on.
The overarching framework of the project was Li Dong’s.
The most crucial tool was also Li Dong’s.
Anyone else interfering would only be more of a hindrance than a help.
Yang Shengguo tossed his suitcase into the trunk and patted Li Dong on the shoulder with a smile.
"Kid, take care of yourself."
"Don’t dive in so deep you forget to come out."
Li Dong grinned.
"Don’t worry, Teacher."
"I’m far too fond of my life for that."
Yang Shengguo got into the car and waved.
As the car drove away, Li Dong stood by the roadside, watching until it was gone.
Penrose, standing beside him, spoke slowly.
"Dong."
"Old Yang... he’s a very endearing person."
Li Dong turned his head to glance at him.
"Professor, you’re very endearing too."
The old Professor Penrose roared with laughter.
「The first day of the new semester.」
Students from all over the country trickled back onto the Yan University campus.
At other universities, on the first day of school, you could more or less see a "haven’t had enough vacation" reluctance to study on the students’ faces.
Not at Yan University.
Walking along Wusi Road, you could overhear conversations like this everywhere.
"I’m telling you, over winter break I chewed through the second volume of Terence Tao’s *Analysis*."
"That’s a little slow, man. My roommate went through all of Hartshorne’s over the break."
"That’s nothing. A guy in the class next to mine taught himself KAM theory for the three-body problem over the break and came back to pitch a numerical project directly to his advisor."
Hearing this kind of conversation at Yan University, no one would think they were just showing off.
Everyone would just think, ’Oh, that’s normal.’

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