"At that time I was in my early thirties, and I imagined these conjectures could be partially proven within my lifetime."
"In the past half-century, Deligne, Dalinfield, Laforgue, Arthur, Kloze, Wu Baozhu, Wiles..."
"Every one of my colleagues has added a beautiful brick to this blueprint, and I thank them all."
"But I must honestly admit, I never expected to see the foundation of this blueprint truly laid in my lifetime."
"Today, I have seen it."
"The zero-pair correlation criterion constructed by the author is not a technique, nor is it a shortcut."
"It is a fundamental change in perspective on the Langlands Program."
"It re-measures a purely algebraic conjecture with an analytic ruler."
"And what is astonishing is that the new ruler and the old one are compatible."
"This compatibility itself is more precious than any specific theorem in the paper."
"Because it means that the deep structure this program involves is not a wall, but a door."
"The author has opened this door for us."
"Before this, we did not realize it was a door."
In the final paragraph of his commentary, Langlands addressed the conjecture at the end of the paper.
He did not say, "I am unable to judge," nor did he say, "This requires more discussion."
He used only three sentences.
"Regarding the conjecture appended at the end of this paper..."
"I do not intend to make any comment."
"If my life’s work on this program has a final destination, then that destination is the direction the author points to on the last page of this manuscript."
—Robert P. Langlands
After this commentary was posted.
Twitter, MathOverflow, Weibo, and Moments almost exploded at the same time.
The top-voted comment came from a young professor at MIT who specializes in representation theory, posted at two in the morning.
Robert Langlands, reviewing a paper under his own name for the first time in over twenty years.
And that paper was from Huaxia.
Shuimu University BBS.
The "Mathematics" board was in a complete meltdown.
A post was pinned to the top: "I know why Professor Wiles is coming to Yan University."
Inside was a single sentence.
"I don’t need to say it, everyone already knows. It’s because of East God."
It was followed by over three hundred replies.
The name "East God" was no longer just circulating within Yan University.
Like a storm cloud, his influence began to spread over the young mathematicians of Huaxia, and indeed, the world.
"You guys only see East God flexing."
"But have you ever thought about... what East God’s paper will change?"
A serious reply quickly appeared below.
The poster’s ID read "Shuimu Mathematics College · Pure Math PhD Track."
"It changes a lot."
"The most direct impact is that all domestic research groups working on the Langlands Program will have to completely overhaul their research plans for the next year or two."
"Those few groups that have been grinding away at GL(2) with ramification ≤ 2 for years can probably just scrap their first two Chapters and jump straight to the GL(n) Chapter to continue their work."
"Of course, that’s a bit far removed from us. Let me talk about something closer to home..."
"This semester, we just started ’Automorphic Forms and L-functions’, and we’re using Bump as the textbook."
"...I’m guessing we’ll have to switch textbooks next semester."
A string of wails followed.
"I have a small question. Will this new material be added to the graduate school entrance exam next semester?"
Someone replied.
"’Automorphic Forms and L-functions’ is already a specialized graduate course. It won’t be made a required course that quickly."
"But the section on ’Li Dong’s Zero Criterion’ will most likely be added to elective textbooks as supplementary reading."
"To be honest, I can’t even understand it now. How am I supposed to learn it then?"
"Dude above, don’t worry. You’re not alone."
At this moment, this group of bright-eyed university students finally, belatedly, realized something.
’Fuck, I’m going to fail the class.’
Yan University, Men’s Dorm 404.
Li Dong was sitting on the bottom bunk, an A4-sized blank notebook open in front of him.
It was densely filled with notes:
[List of Derivative Research Directions from Theorem 1: Local-Global Compatibility of GL(n) Automorphic Representations]
Direction 1: Generalization of the e_v criterion on general reductive algebraic groups G
Direction 2... (omitted)
There were over twenty entries in total.
Beneath each entry were his rough ideas.
Some were already fleshed out to the third step, while for others, the direction wasn’t even set in stone—just an arrow pointing forward, followed by two question marks.
He wasn’t writing a paper; he was drawing a map for his next two or three years.
The "Li Clan Conjecture" in the paper had only been proposed.
’How to prove it? Where to start? What’s the fallback if I get stuck halfway through a proof?’ Li Dong had to think through these things himself first.
Otherwise, what would he talk about when he met Professor Wiles later?
He took photos of the few pages he had organized that morning, then cross-referenced them with the contents of several related reference books he had just gone through in his "Memory Palace" the previous night.
(I’ll put the related literature and books in the author’s notes.)
After organizing everything, Li Dong glanced at the time on his phone.
Half an hour to go.
He closed his notebook, looked up, and... froze.
From the top bunks, Wang Hao, Chen Nan, and Liu Qiang were silently watching him.
Wang Hao sighed.
Chen Nan also sighed.
Liu Qiang pushed his glasses up his nose and sighed again.
Li Dong: "...Are you three having a fit?"
Wang Hao didn’t answer him, turning instead to ask Chen Nan and Liu Qiang.
"You two, tell me, how much longer do you think Brother Dong will be living here?"
Chen Nan and Liu Qiang sighed again in perfect unison.
Chen Nan spoke first. "Brother Dong, let’s be honest. How much longer do we get to be roommates?"

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