In the days that followed, while Li Dong waited for Li Qin and Old Yang to arrive in the Capital City, his mind was far from idle.
A necessary equivalence exists between the statistical properties of the pair correlation of zeros and the local structure of automorphic functions.
The last time, he had @’d Riemann in the group chat, hoping to further discuss the problem of the Hodge-Tate weight coupling structure under a High Tier ramification index. In the end, Riemann had completely ignored him.
But Li Dong had always been very interested in the problem itself.
’Pair correlation of zeros... ramification index... GL(n)...’
Li Dong sat at the desk in his dorm room, muttering to himself.
His 0.4 Logic attribute allowed him to easily deduce the next step.
Moreover, he could construct several chains of reasoning in his mind at once, advancing them step by step as if playing a game of chess.
This time, however, he ran into a rather interesting little problem.
The direction of the "equivalence" Riemann had mentioned was very clear.
It was to use the convergence behavior of the pair correlation function F_π(α) of the zeros of an automorphic L-function within a certain interval to reverse-engineer the local properties of the automorphic representation.
What made this idea interesting was that it completely bypassed traditional, purely algebraic methods.
You wouldn’t need to brute-force those mind-numbing Hodge-Tate coupling structures. As long as you could verify that the statistical properties of the zeros met specific conditions, you could directly determine whether local-global compatibility held.
From algebra to analysis, from brute-force calculation to statistics—on a methodological level, this was an attack from a higher dimension.
But the problem was...
What exactly *was* this "specific condition"?
’If the ramification index e_v doesn’t exceed n, then how long should the convergence interval for F_π(α) be?’
Li Dong scribbled a formula on his scratch paper, then crossed it out.
’That’s not right... The relationship between the interval length and n shouldn’t be linear. It should be an inverse proportion...’
He wrote another line and stared at it for half a minute.
’|α|∈[0, 2/n]?’
The thought suddenly surfaced in his mind, but he didn’t yet have enough justification to confirm it.
’So close... just a little bit more...’
Li Dong felt like he had his hand on the doorknob, but he just couldn’t turn it.
He was missing an anchor point.
A special, already-proven case that could be used for comparison and verification.
If he had a known result for n=2, e_v≤2, he could use it to work backward and deduce the interval correspondence for the general case.
’n=2... e_v≤2...’
Li Dong suddenly thought of something...
Wasn’t that Jiang Yubai and Zhou Shenzhi’s paper in the *Duke Mathematical Journal*, "On the Local-Global Compatibility of GL2 Automorphic Representations for Ramification Indices Not Exceeding 2"?
No matter how shameless the credited authors were, the conclusion itself was correct.
At this thought, Li Dong immediately logged into his university’s journal database and searched for the full text of the Duke paper.
The PDF loaded quickly.
Li Dong skipped right past the author information and started reading from the second Chapter of the main text.
’This step...’
He stared at a key equation at the bottom of page 17, reading it over three times.
This equation was the lifeblood of the entire paper.
Its purpose was...
...to, in the case of e_v=2, project the originally ineliminable universal obstruction into a specific subspace via a meticulously constructed integral path. It then used the first-order structure of the Hodge-Tate decomposition to eliminate it directly.
However, the intermediate steps of this derivation were omitted in the paper.
It simply used the phrase, "as is known from standard p-adic Hodge theory," before jumping straight to the conclusion.
’As is known from standard p-adic Hodge theory?’
The corner of Li Dong’s mouth twitched.
The phrase looked increasingly familiar to him.
He often did the same thing when writing his own papers, omitting steps he considered "obvious" because he couldn’t be bothered to write them out.
But there was a difference.
Li Dong omitted steps because he genuinely believed they weren’t worth writing down.
As for this paper omitting this step...
’It’s because the person who wrote the paper didn’t fully understand it themselves, isn’t it?’
That was Li Dong’s gut feeling.
He shook his head and started working it out himself.
His full 0.4 in all attributes engaged at once.
About fifteen minutes later...
The pen in Li Dong’s hand came to a stop.
He stared at the lines of derivation on his scratch paper.
"Holy shit..."
He finally understood.
The reason the p-adic integral path deformation scheme worked when e_v=2 wasn’t due to the construction of the path itself. The key was that during the path’s deformation, the first-order component of the Hodge-Tate weight happened to be perfectly orthogonal to the projection direction of the universal obstruction.
This orthogonality was no coincidence; it had been deliberately constructed.
The construction method was to embed the filtration structure of the Hodge-Tate decomposition into the path’s parametric equations ahead of time, when selecting the initial parameters for the integral path.
This way, when the path deformed in p-adic space, the universal obstruction would be automatically pushed onto the zero stratum of the filtration, where it was then precisely eliminated by the first-order Hodge-Tate weight.
"Old Yang... he’s really something else."
Li Dong remarked in a soft voice, filled with admiration.
The brilliance of this construction didn’t lie in the complexity of its mathematical operations.
In fact, once the logic was understood, the actual calculations weren’t difficult.
Its brilliance lay in its perspective.
’The intuition for this kind of reverse-engineering... an ordinary person would never come up with it.’
Li Dong made a silent calculation in his mind.
Old Yang’s Basic Attributes—at least his Logic—had to be 0.1 or higher.
Whether it reached 0.2, he couldn’t say.
And what did a 0.1 even mean?
It meant that in a world where an ordinary person’s Concentration, Logic, and Memory were all 0, having even one of those attributes at 0.1 already made you a superman in that dimension.
More importantly, Old Yang had come up with this scheme when he was just a master’s student.
If Jiang Yubai hadn’t robbed him of over a decade of his academic career, he would at least be a full professor at Capital University by now. He might have even established a firm foothold at the forefront of the Langlands Program long ago.
’When you get to the Capital City, we’ll see this path through to the end together.’
Li Dong thought to himself.
Then he pulled his attention back to the derivation.
Now that he understood the underlying logic of the path deformation for e_v=2, he could work backward to deduce a key piece of information.
Namely, why the scheme failed when e_v≥3.
The reason was simple.
When the ramification index rose to 3, the second-order Hodge-Tate weights would introduce additional coupling terms, shattering the original, precise orthogonality.
The universal obstruction no longer stayed put on the zero stratum of the filtration. It would drift.
And this drift was something purely algebraic methods could not control.
’So...’
The corners of Li Dong’s lips slowly curled upward.
’So that’s why I need to bypass algebra and approach this from an analytical perspective.’
’If I can prove that the convergence of F_π(α) to the GUE predicted value within the interval |α|∈[0, 2/n] is equivalent to π satisfying local-global compatibility...’
’...then I won’t have to worry about any Hodge-Tate coupling structures at all. I won’t need to brute-force a solution to the drift problem.’
’The statistical properties of the zeros will give me the answer directly.’
He picked up his pen and wrote a line of text on his scratch paper:
[Theorem (Zero Criterion): π satisfies local-global compatibility ⟺ F_π(α) converges to the GUE predicted value in |α|∈[0, 2/n]]
After writing it, Li Dong stared at the line of text for a long time.
He didn’t have a complete proof yet. Even the framework for the proof was still in its nascent stages.
But his intuition told him this was the right direction.
And that intuition had come from Riemann.

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