Chapter 265 - 249: I Also Thank You on Behalf of Fields

Chapter 265: Chapter 249: I Also Thank You on Behalf of Fields

With this in mind, Li Dong pulled up the PDF of the *Annals* paper he had.

GL(n) Local-Global Compatibility.

Seventy-four pages of the core proof, plus the "Li Clan Conjecture" on the final page.

Giving this to Klein would be equivalent to handing him a formal structural diagram of the Langlands Program.

But this was far from enough.

With only this paper, what Klein would see was still just a silhouette of his Erlangen Program—a number theory version of the concept of invariants.

To let Klein see the entire program clearly, he also had to package the major achievements of the Langlands Program over the past half-century.

Li Dong thought for a moment. ’I have two file transfer opportunities anyway. I’ll save one as a backup, which means I have one to spare...’

’Fuck it. I’m doing this.’

At that, Li Dong opened his laptop.

Wang Hao, on the top bunk, heard the noise and mumbled under his breath.

"Pulling another all-nighter, Brother Dong...?"

"Just go to sleep," Li Dong replied casually.

...

For the next two-plus hours, Li Dong emptied an entire rack from his Memory Palace—the one dedicated to everything related to Langlands.

1967, Langlands’s letter to Weil.

1974, Deligne’s proof of the Weil conjectures I.

1989, that hefty tome by Arthur-Kroze on base change.

2010, Wu Baozhu’s proof of the Fundamental Lemma.

...

He took these works from the shelves of his Memory Palace one by one, organized and combined them, stripped out the derivations, and preserved only the most essential core structures.

Finally, he consolidated everything into a single folder.

Li Dong opened the group chat and clicked the file transfer function.

[Select Recipient: Felix Klein]

[University Scientific Research]: Your Excellency Klein, my associates and I have compiled the material you requested last time.

He sent the message, immediately followed by the compressed file.

[Attachment: The Langlands Program - Consolidated Collection] has been sent to Felix Klein.

[Remaining File Transfers: 1]

The group chat fell silent.

But Li Dong could see that Klein had received the file.

He wasn’t in a hurry. ’This is a huge amount of material. It will take Klein quite a while to even...’

Just as he was thinking this, he saw Klein post a message in the group.

The corner of Li Dong’s mouth twitched.

’Seriously?’

’How long has it been since I sent it??’

’Your Excellency Klein, what kind of reading speed do you and Riemann have! Does time flow differently for you than it does for me?’

After all, the materials he had just compiled would form a stack taller than a man if printed out. It had taken him over half an hour just to organize them in his Memory Palace.

And now...

Klein had taken less than five minutes...

"Sigh."

Li Dong looked at the group messages with a sense of resignation.

[Felix Klein]: Your Excellency, I never imagined that I would one day see, within a private notebook, such an edifice that is already fundamentally complete.

[Klein]: You and your associates have reunified geometry, number theory, and analysis under the action of non-commutative groups. For this achievement alone, your names deserve to be remembered by all of posterity.

[Klein]: But what shook me the most was the conjecture you included at the very end.

[Klein]: It has shown me a path.

Seeing this, Li Dong suddenly recalled the last sentence of Langlands’s commentary.

"If my half-century of work on this program has a final destination, then that destination is the direction pointed to by the author on the last page of this manuscript."

’As expected.’

’These people, who have truly stood at the summit, can see the silhouette of the yet-unbuilt edifice in a single glance.’

Langlands saw the "direction of the destination."

Klein...

[Felix Klein]: Your Excellency, I believe I now see the edifice in its entirety.

Li Dong: ???

’Hold on a second.’

’What happened to just seeing the silhouette?’

’I send over one set of documents, and in less than five minutes, you’ve grasped the entire edifice?’

Li Dong felt a bit demoralized.

Just as Li Dong was feeling a pang of jealousy.

[Felix Klein]: I would like to show this to a friend of mine.

[Klein]: His perspective on mathematics differs from mine. He is more concerned with how the next generation of young people will carry on, and he has dedicated his life to this cause.

[Klein]: Back when I was writing the Erlangen Program, he would often visit my room.

[Klein]: He would sit on that small stool, keeping me company as I smoked, listening to me ramble on about transformation groups. Later, he left Göttingen and traveled to the distant shores of North America.

[Klein]: In his letters, he often mentioned his idea of establishing a medal dedicated to mathematics, for the most brilliant young minds.

Li Dong’s breath caught in his throat.

’This is... Fields.’

The realization hit him instantly.

Historically, Fields had indeed studied across Europe in his youth.

Around 1901, he had lived in Göttingen for a time and was on excellent personal terms with Klein, Hilbert, and their circle.

Later, in his study at the University of Toronto, there was rumored to be a handwritten card that read:

"Klein’s room is the temple of mathematics."

That was no mere pleasantry; it was a heartfelt conviction from the young Fields.

Later still, when Fields was devising the medal, the first European scholar he wrote to for advice was none other than Klein.

The two exchanged many letters.

Klein had offered suggestions for its charter and had also advised him against setting the age limit too strictly.

Yet, in the end, Fields still added that "under 40" restriction to the plan.

[Klein]: His health is very poor now. He may not have much time left.

[Klein]: I had thought he would never get to see this blueprint.

[Klein]: Now, before the light of his life is extinguished, I will place the capstone on this edifice.

[Klein]: Thank you, Your Excellency "University Scientific Research." And on behalf of Fields, I thank you as well.

Li Dong stared at the series of messages, unable to snap out of his daze for a long time.

He gleaned two things from it all.

First, this titan Klein was confident he could prove the "Li Clan Conjecture."

To personally place the capstone on the edifice that was the Langlands Program.

Li Dong was speechless. ’Are the titans in this group always this unreasonable?’

After all, Terence Tao, hailed today as one of the world’s most brilliant mathematicians, had written on his blog that the "Li Clan Conjecture" was a lighthouse that would require ten, twenty, or even thirty years of sailing to reach.

And yet, Klein was saying he would go and place the capstone on it right now.

"Sigh..."

Li Dong sighed and moved on to the second point: Fields.

He quickly scrolled up through the group chat’s history.

He scrolled to a group announcement from a long time ago.

[Have Fields personally witness the first Fields Medal award ceremony]

The red countdown timer was still there.

About two years and eleven months remained.

And the extraneous comma that had been in that group announcement had vanished.

The task itself had been—corrected.

Li Dong stared at the screen, a sense of lingering fear creeping into his heart.

’That initial group announcement was a fake, just as I suspected.’

’If I had blindly followed its instructions back then, not only would I have failed to receive any reward, but I most likely would have been punished.’

’After all, it would have been the equivalent of responding to "dimensional phishing."’

’But now... everything that was happening on Klein’s end had been triggered by that one file transfer.’

’That’s why the mission has only now become real!’

Just then, another group announcement appeared.

[Group Announcement: Felix Klein, Professor Emeritus of the University of Göttingen, has reopened the final Chapter of the *Erlangen Program* thanks to the interim results of the Langlands Program recently compiled and transmitted by this group’s Group Leader.]

[He has named it the "Non-Commutative Chapter."]

[This Chapter begins with the Göttingen Private Discussion Notes, proceeds through non-commutative generalizations, and culminates in the "Zero-Point Pair Correlation—Functoriality Equivalence Conjecture" proposed by the Group Leader.]

[At the end of the Chapter, Mr. Klein wrote: "The edifice is complete. May future generations stand upon this foundation and see farther still."]

[After his passing, this Chapter was entrusted to Hermann Weyl to be compiled and published.]

[Dedication: "To my old friend, John Charles Fields."]

["May you, in your final days, see the places where the young people you have so championed are destined to go."]

[In Parallel Spacetime 1912, the technology tree has diverged. In this timeline, the convergence of non-commutative geometry and the Langlands Program has occurred forty years ahead of schedule.]

[Reward: Dimensional Phishing Mission Corrected.]