150: Chapter 150 What does the Great Proof look like?

When Jiang Lin walked out from the side, there was a momentary silence in the lecture hall.

This was a room crowded with the smartest people in the field, and the person walking up to the whiteboard was an eighteen-year-old youth.

He nodded slightly in greeting, skipped the pleasantries, and picked up a black whiteboard marker.

"Today, I won't expand on all the technical details of v1.0, but I will walk through the main closed-loop completely."

He wrote a line of text in the upper left corner.

[ small doubling ⇒ polynomial structural compression ]

"Intuition tells us that if a set does not expand too much after adding to itself, it must be controlled by some low-dimensional structure. The crux of the issue is whether the cost of this control can be compressed to a polynomial level."

The audience was dead silent.

The pen tips of several doctoral students in the back row hovered in mid-air.

Because this plain-spoken sentence stabilized the entire venue faster than any rigorous and complicated definition could.

...

Liang Chen's fingertips hovered above his phone screen.

His research direction was separated from Additive Combinatorics by a layer, so he could only be considered a half-relevant person.

Being able to squeeze into the lecture hall today was also because his advisor temporarily sent a message in the group chat saying: 'Go if you can, you might not understand it, but you need to see what a grand proof looks like.'

Three minutes ago, driven by the thought of recording, he casually opened a livestream with the title "PFR Calibration Meeting Live, Joining the Fun".

Due to relevant pushes, it wasn't long before some interested viewers came in.

At first, the bullet comments were mostly amusing banter.

Someone asked if there were any handsome freshmen, whether this was an opening assembly, and someone posted a message forwarded from the class group chat saying there was big news today.

Until the camera swept across that title on the whiteboard, the comment section froze stubbornly for two seconds.

"Wait, polynomial structural compression? Isn't this that PFR Conjecture?"

"Did I read that right? The speaker is that eighteen-year-old freshman?"

"Really eighteen? He looks even younger than my junior fellow apprentice, can he hold down the fort?"

...

Liang Chen didn't dare to type a response, and only quietly pushed the camera forward by half an inch.

Because for the next dozen minutes, the youth did not stop for a single moment.

Black formulas spread across both sides of the whiteboard, a red pen marked every loss entry, and a blue pen outlined the witness paths.

It was as if he was pulling out a load-bearing beam buried deep within the walls, inch by inch, and placing it right before everyone's eyes.

The first whiteboard was quickly filled up.

Many people in the audience began to give up word-for-word recording because they couldn't keep up.

...

Liang Chen's pen had long since stopped on the third page of his notebook.

His livestream room quietly crawled from just dozens of online viewers when it started to seven hundred and twenty.

And the comment section had completed its transformation from reading along line-by-line to collectively lying flat.

"I started dropping offline at the third line, can anyone explain what the loss entry marked in red is?"

"Don't ask, asking means I can't even see where the loss is."

"PhD year one reporting in, I just memorized the definition of small doubling last week, and now I feel like I haven't even entered the field."

"Person at the scene, don't understand, but deeply shocked."

One bullet comment asked if the streamer could remind the speaker to slow down, and Liang Chen silently typed a line of text on the public screen: "I wouldn't dare, he's already holding back his pace."

He looked up at the whiteboard; the framework of the third-layer heavy witness had already been laid out for the most part, and Jiang Lin's pen-dropping speed was as steady as a precision instrument, even the spacing between every two lines of formulas was almost identical without a hair's breadth of difference.

...

[ Node 38 · Boundary Visibility · Third-Layer Heavy Witness ]

"The high-dimensional boundary test proposed by Professor Voss is not a counterexample." Jiang Lin dropped his red pen in the center of the whiteboard and said, "The main proof is not broken here; the problem is that the load-bearing structure is there, but the maintenance port is not big enough."

As soon as these words fell, many auditors from outside this field understood vaguely.

...

The tip of Liang Chen's recording pen paused abruptly.

Last week, he had just meticulously read Voss's boundary test paper in his group meeting, and his advisor's conclusion was that no one would be able to fully incorporate this model into the main accounts within at least half a year.

He instinctively looked at his phone; the bullet comments in the livestream room had already exploded, scrolling so fast that the words were illegible, with only a few sporadic sentences popping out: "Voss's counterexample? Is he going to dismantle it on the spot?"

"If he can round this out, I'll eat my textbook right here."

...

Jiang Lin wrote down a line and deliberately paused.

[ witness, not new compression ]

"Pay attention, this is a witness, not a new compression step. The witness is responsible for exposing the whereabouts of the residual spectrum, and compression is responsible for paying the loss. The two cannot be mixed."

Professor Han Yanshan in the front row slowly straightened his posture.

Because Jiang Lin was not simply talking about isolated nodes.

He was forcefully pulling the most perilous load-bearing wall out of the wall in the main PFR proof, exposing the steel bars inside bluntly before everyone's eyes.

...

Liang Chen's livestream room froze right here for a full three seconds.

When refreshed again, the topmost bullet comment had been liked to the very front, whose ID was a verified university number theory assistant professor: "I retract the gimmick I mentioned half an hour ago; this move saves at least three months of detours in the field."

...

On the second whiteboard, Jiang Lin wiped away the auxiliary derivations and wrote—

[ Marton Bridge : entropy payment under probability-space switch ]

"Don't look at it as a translator." Jiang Lin drew a connection between the two nested spaces, "It is a payment system. Local degrees of freedom lose their original coordinates when entering the global assembly; this cost must be paid."

Ding Jian raised his head.

In Jiang Lin's office this morning, he had heard this trial lecture, but placed within the complete link at this moment, its weight was completely different.

It was no longer an isolated probabilistic technical explanation, but the inevitable bridge leading to global closure.

At 3:22 PM, Jiang Lin walked up to the third whiteboard and wrote the final line of title.

[ Node 47 · Global Assembly ]

Jiang Lin drew a huge global accounts matrix.

Residual spectrum, witness routes, density increments, entropy control reserves...

All variables were stuffed into the same table.

"If a payment source cannot be found for any item, the proof will not close. If an item is paid for repeatedly, the proof will not close either. If local degrees of freedom lose their coordinates and are not paid for, the proof still will not close."

He wrote a word at the very bottom.

[ closed ]

No one spoke in the lecture hall.

Accompanied by the crisp sound of his pen dropping, he switched back to the red pen and issued the final conclusion beneath the matrix.

[ finite-field PFR follows from the closed loss ledger ]

(Finite-field PFR, derived from the closed loss ledger)

After the two lines of text were written and the main beams were all laid out, the entire lecture hall seemed to be held in place.

Jiang Lin turned around and faced the audience.

"The benchmark framework is as such, you may ask questions now."

As his voice fell, the lecture hall fell into silence.

This wasn't an awkward silence, nor was it because no one had questions.

On the contrary, there were too many questions.

It was just that the brains of everyone in the audience were still chasing the storm on the whiteboard.

A dozen seconds later, Voss in the remote window raised his hand.

"Node 38, you defined the third-layer witness as an auditable path rather than a new compression step, I agree with this point."

His voice carried a slight German accent, and his speaking speed was not fast.

"But under the boundary model where K equals 8, if the third-layer residual spectrum repeatedly approaches the threshold, how do you guarantee that this witness will not re-consume the entropy reserve of the second layer in the global accounts?"

Many people instinctively lowered their heads to look at the abstracts in their hands.

This was not just an inquiry, but rather pushing the main link just now directly into a high-pressure test.

If it couldn't be defended, the accounting system would collapse instantly.

The red pen in Jiang Lin's hand spun a circle around his fingertips, and without a moment's hesitation, he turned around and drew a clear vertical line in the blank space of the whiteboard, sharply cleaving the reserve layer and the witness layer apart.

[ witness ≠ payment ]

"A witness does not equal payment." His voice was without fluctuations, yet resounding, "The third-layer route only provides audit visibility and has no payment authority. If it goes to invoke the second layer, that is recursive charging. In the global matrix, its status is only checked, and absolutely cannot be spent."

It was a crude metaphor, but it neatly and cleanly blocked all possible accounting ambiguities.

Someone in the lecture hall chuckled softly.

Voss stared at the screen, tapping his fingers slowly on the desktop a few times.

A dozen seconds later, he raised his head and nodded slightly.

"Not bad, this visibility path can withstand scrutiny."

Several old professors in the front row simultaneously relaxed their shoulders.

The most perilous first checkpoint was held.

...

The pot directly exploded in Liang Chen's livestream room.

"Holy crap, Voss said not bad?"

"Is this that Voss who threw out counterexamples last year and crippled three top-tier journal papers, he just let it pass like this?"

"I thought we were going to argue for half an hour, but it was blocked in a single sentence!"

"witness ≠ payment, how come these five words feel more useful than ten pages of derivation."

...

However, the clash left no room to breathe.

Terence Tao immediately asked whether this wrapper in the formal blueprint should be hung on the main dependency chain or outside the nodes.

A young teacher from Tsinghua University asked whether, after adding the wrapper and boundary audit structures, the 47 macro-nodes in the formal blueprint still maintained their original main dependency topology.

Professor Li from Peking University stared at the accounts matrix and asked whether the frequency domain index and the combinatorial domain threshold would misalign during boundary collapse.

For the next half hour, Jiang Lin caught them one by one.

His speaking speed was not fast, yet there was almost no hesitation.

The whiteboard was wiped and written on, written on and wiped.

He kept repeating the same thing.

Spreading out the seemingly cumbersome structures item by item for the reviewers to see.

"v1.0 does not pursue prettiness." When answering Professor Li, he said, "A single index is prettier, but a single index lies."

Someone gave a low cough.

Several professors in the front row nodded at the same time.

Only to Ding Jian's question did Jiang Lin answer exceptionally slowly.

"You said that the entropy reserve is paid locally and marked as spent once it enters the global accounts." Ding Jian flipped to the sixth page of the abstract, "I want to follow up: how do you guarantee that it won't be treated as an available degree of freedom again in the next layer of compression?"

This was precisely the most dangerous part of that bridge.

Jiang Lin turned around and wrote down—one-time entropy reserve.

"It can only be used once; once payment is complete, it enters spent." He said every word distinctly, "It cannot be available, it cannot be reusable, and even less can it be ambiguous. If it is invoked again in the next layer, then this bridge is not a bridge, but a smuggling port."

A very soft gasp sounded in the lecture hall.

This formulation was sharp and accurate.

Ding Jian's eyes lit up noticeably, and he then pursued questions about degenerate cases.

Jiang Lin did not avoid it, and wrote out the minimal state-maintenance skeleton on the whiteboard, a few lines, not long, but enough for Ding Jian to see the structure clearly.

Ding Jian lowered his head, wrote a line, and raised his head.

"The word spent will do."

"It's ugly." Jiang Lin said.

"Ugly in a good way."

Relaxed laughter rang out in the lecture hall once again.

But soon, the laughter was suppressed again by that question at 4:07 PM.

Professor Han Yanshan finally picked up the microphone.

Throughout all the previous questions, he had never made a move.

Until the entire main chain had been pressed through a round.

"I will ask the last question on the combinatorial side."

The lecture hall quieted down.

"Replace K equals 8 with that series of models where K approaches the boundary." He looked at the global accounts matrix on the whiteboard and slowly said, "The third-layer witness does not recursively charge, the spent items do not revive, and the dual-index misalignment enters degenerate archiving—if these three things happen simultaneously, is node 47 still closed?"

In fact, after asking, Professor Han Yanshan himself was very clear in his heart.

If this question cannot be answered today, it's not shameful.

It stacked Voss's boundary, Ding Jian's spent, and Professor Li's double index all together, pressing them onto the exact same global matrix of Number 47.

It was too heavy.

Anyone who has done a complete, major proof knows that this kind of worst-case combination stress test is usually not something that can be answered on the spot.

Often, one has to go back and think for a week, then send a follow-up email.

But today, he had to ask.

Professor Han Yanshan was already prepared; if Jiang Lin couldn't answer today, Professor Han Yanshan was confident that over the next two weeks, he would accompany the teenager to complete this matter.

As Professor Han Yanshan's voice trailed off, the moment everyone looked at Jiang Lin, Liang Chen caught a glimpse of the online viewer count in the livestream jumping and steadily breaking ten thousand.

New IDs kept pouring into the public chat, with prefixes bearing the tags of the Department of Mathematics from various universities.

Someone posted that their advisor made them come and squat, saying that after this question is answered, the landscape of the field is going to change.

Jiang Lin understood that what Professor Han was actually asking was whether that matrix would break when these three things happened simultaneously.

Of course he knew this matrix wouldn't break.

Because as early as the ninth Wasteland time, when he himself first wrote out the draft for Node Number 47, this was the stress test he did first.

At that time, he opened all three things simultaneously to see which cell of that matrix would leak first.

Then he saw it leaked at the second row and third column—the witness column borrowed resources from the payment column because he hadn't separated their permissions at the time.

He spent a month revising it.

He split the permissions of the three columns: witness, payment, and archive. Each column could only do one thing and could not borrow from each other.

After the revision, he ran the stress test again.

It didn't leak.

He ran it a second time.

Still didn't leak.

Yet he eventually suppressed his urge and kept running it, starting from different initial conditions, different degradation paths, and different boundary models.

He couldn't remember exactly how many times he ran it, but he remembered that on the day the last run finished, it rained heavily outside the Stone House.

He sat at the polished, oil-smooth stone table until three in the morning, looking at that matrix, and confirmed one thing.

This matrix would not break, as long as the permissions of the three columns were not mixed.

He hadn't written this matter into the outline.

It wasn't that he didn't want to write it, but that it couldn't be written in.

It wasn't a new lemma, nor was it a new proposition; it was simply a structural trade-off, a judgment that he would rather make this matrix look ugly than allow them to borrow from each other.

Such a judgment could not be written in a paper, nor could it be written in an outline.

It could only be brought out directly on the whiteboard, at the very moment when asked, for people to see.

So, at this very moment, he turned around, erased the lower half of the third whiteboard, and wrote down the title.

Combinatorial Boundary Stress Test.

Underneath, three lines listed the worst-case scenario where three things happened simultaneously.

Then on the right, he drew the minimal skeleton of the accounting matrix.

Only three columns remained: the witness column, the payment column, and the archive column.

"In the worst-case scenario, all three things come together," he said. "If these three columns borrow resources from each other, the proof is not closed."

With a red pen, he drew a vertical line between each of the three columns.

"So they cannot borrow from each other; the witness column cannot pay, the payment column cannot revive, and the archive column cannot compress."

Three sentences, three lines.

The attention of the entire lecture hall was pressed onto those three vertical lines.

[witness: visible / no charge]

[spent: paid / no reuse]

[record: tracked / no compression]

The last line—no hidden losses left outside the accounts.

Then he put down his pen, turned around, and began to speak.

"Therefore, when the three things happen simultaneously, Number 47 remains closed. The premise is that the permissions of the three columns are not mixed. All those seemingly cumbersome state flags, double indices, and degradation indices in v1.0 are designed to ensure just this one thing."

The lecture hall remained quiet for a long time.

Professor Han Yanshan stared at those three vertical lines for a long while before slowly leaning back into his chair.

In Liang Chen's livestream, the bullet comments abruptly went blank for two seconds.

It was as if the sense of suffocation from the entire hall had transmitted through the camera, and then it exploded with overwhelming speed, the scrolling subtitles moving so fast that even complete sentences were hard to make out.

"Triple worst-case scenarios stacked together and caught on the spot, and it only used three-column permission partitioning... How on earth did someone think of this approach?"

"My advisor just said in last week's group meeting that this combinatorial dead end would take at least half a year to grind through, and it turns out someone welded it shut in ten seconds standing in front of the whiteboard?"

"Our advisor woke all of us up, saying this is the most crucial half-hour in Additive Combinatorics in the past thirty years."

A bullet comment certified by the University Number Theory Research Institute was instantly pushed to the very top, steadily hanging over the scrolling public chat: "These three vertical lines are enough for the entire field of Additive Combinatorics to digest for three years."

Liang Chen glanced hastily at the top bar of the livestream; in just this brief moment, the online viewer count skyrocketed from ten thousand straight to twenty-two thousand.

New IDs in the public chat were still pouring in like a tide, with prefixes extending all the way from Tsinghua University, Peking University, Fudan, SJTU, USTC, and Shandong University to Chinese departments of mathematics in North America and Europe.

His fingertips swiped across the message page, and three pinned academic chat groups were all showing red dots of 99+, with group messages refreshing at lightning speed, repeating the exact same sentence over and over.

"Stop asking, hurry into the livestream, this is a historic moment."

At four-twenty-nine, Shing-Tung Yau raised his hand, signaling to pause here.

This leading figure in the mathematical world took the microphone and glanced at the three whiteboards filled with writing.

"Today's meeting was better than I expected."

"I have seen presentations by many young people. Some are clever, some work hard, and some have great technical skills. But what is most precious today is neither cleverness nor technique, but a sense of structure."

"Jiang Lin did not hide a major proof behind pretty language, nor did he bypass the most dangerous parts. Node Number 38, Voss's boundary test, the Marton Bridge, the Number 47 global assembly—he put all these heaviest beams onto the whiteboard. Whether it can be audited, whether it can be questioned, and whether it can be further dismantled—this is what a truly great piece of work should look like."

No one spoke in the lecture hall.

Several elderly professors in the front row sat very straight, and several overseas scholars in the remote windows did not switch away their screens.

Shing-Tung Yau's gaze shifted from the whiteboard back to Jiang Lin.

"For an eighteen-year-old young man to explain a piece of work like this to such an extent is no small feat. What is even more remarkable is that he did not treat mathematics as a show of skill. He put every single loss, every payment, and every boundary degradation into auditable accounts."

He paused.

"This is the approach of a mathematician, not that of a gifted teenager."

Many young students in the audience froze for a moment.

Jiang Lin stood beside the whiteboard, still holding that red pen in his hand.

There was no excitement, no showboating, and no deliberate modesty; he just stood there quietly, like someone who had just completed an engineering acceptance check.

The bullet comments in Liang Chen's livestream slowed down, and all the wisecracking comments disappeared.

"Mr. Qiu's words carry too much weight."

"Not a gifted teenager, but a mathematician, oh my god."

"At eighteen, to receive this evaluation makes a lifetime worthwhile."

Someone posted asking why he wasn't excited at all, and someone immediately replied: "If it were you, you'd be jumping with joy, but he probably just thinks today's acceptance check passed."

Liang Chen looked at the calm-faced young man on stage, and suddenly became clearly aware of that gap.

Most people doing mathematics chase after problems, hoping to achieve some results to prove themselves.

But this person was building a house.

He only cared about whether the house was sturdy and whether it leaked; as for whether others praised him for building a beautiful house, it didn't seem to be in his consideration at all.

"Going back tonight, everyone will continue to read this v1.0, continue to ask, and continue to dismantle." Shing-Tung Yau finally looked toward the entire room, his voice sinking a bit. "This is a good thing, because starting today, the PFR / Marton matter is no longer just his personal manuscript; it has been placed before the entire field."

As his voice fell, there was a silence for one second.

Then, applause erupted.

It wasn't out of mere etiquette, nor was it the lively bustle of a press conference.

It started from the front row and rapidly spread to the back row, the aisles, and the door, with even the people in front of the outer small screens joining in the applause.

Liang Chen was also applauding.

The mobile livestream was still on, and after a moment of thought, he posted a bullet comment.

[Up to this point in the livestream, although I'm completely in the dark from start to finish, I roughly know what a major proof looks like.]

Sent.

He raised his head; the people in the lecture hall had not yet dispersed.

Some were taking photos around the whiteboard, some were bowing their heads to flip through notes, and Jiang Lin stood beside the whiteboard, still holding that red pen, responding one by one to the scholars crowding around.

Jiang Lin was about to put his notebook back into his bag when out of the corner of his eye, he suddenly saw a very quiet figure stand up from the side-rear.

That person wore a simple T-shirt and long pants, his hair somewhat messy, and held the report summary distributed by the conference organizing committee.

He had barely spoken throughout the entire session, and his arrival had not caused any stir.

But many people in the lecture hall knew who he was.

Wei Dongyi.

Throughout most of the presentation, he kept his head down, writing extremely short symbols on paper.

Only when Jiang Lin wrote those three vertical lines did he raise his head and look for a long time.

He did not step forward to exchange pleasantries, nor did he join the crowd surrounding Jiang Lin.

He only paused for a moment as he passed the whiteboard, his gaze lingering on that global accounting matrix for a few seconds.

Then, he said very softly,

"This accounting is very clean."

His voice wasn't loud, and only a few people nearby heard it.

Clean.

In an occasion like this, this was neither polite flattery nor emotional praise.

It meant only one thing.

Losses had not been hidden, payments had not been reused, and degradation had not been disguised as compression.

Jiang Lin nodded in acknowledgement.

Wei Dongyi waved his hand slightly, as if unaccustomed to such a formal response, and turned to walk toward the door.

The crowd began to flow again.

As he stepped down from the stage, Professor Han Yanshan greeted him and gave his shoulder a heavy pat.

"Well spoken."

"There are still a few places to be added into the audit index," Jiang Lin said.

"That's for later," Professor Han Yanshan said. "Today you nailed down Number 47."

Ding Jian also walked over, smiling and nodding: "That part about the Marton Bridge was much clearer than this morning; the word spent can be kept."

"It is indeed very useful," Jiang Lin said.

Scholars around successively gathered around; some shook hands, some pursued technical details, and some wanted to join a certain branch of the subsequent formal verification.

Jiang Lin responded to each of them one by one, with a modest tone and a clear train of thought, showing none of the restlessness of youthful success.

Until the time reached five o'clock, the people gradually dispersed.

"What are you thinking about?" Professor Han Yanshan walked over and handed him a bottle of water.

"Thinking about the double index degradation formulation Professor Li mentioned." Jiang Lin unscrewed it and took a sip. "It can be added into the formal index."

"You're thinking about adding indices right after finishing your talk?" Professor Han Yanshan laughed in spite of himself.

"It's an alignment meeting after all," Jiang Lin smiled. "Figuring out which places need to be clearer is the purpose."

As they were talking, Shing-Tung Yau walked over.

"Very good, Jiang Lin."

"Thank you, Mr. Qiu."

"I heard your Low Entropy Workshop has a display at tomorrow's robotics exhibition?"

"Yes."

"If there's time, I'd actually like to see what this local-to-global aperiodic structure you mentioned looks like on a machine."

Jiang Lin was slightly startled, then nodded.

"Mr. Qiu is welcome to come and guide us."

Exiting the Mathematical Sciences Center.

The evening sunlight filtered through the phoenix tree leaves, scattering fragments of gold all over the ground.

The stream of people welcoming newcomers in the distance had not yet dispersed, and young students walked smilingly along the main thoroughfare, their faces full of longing for university life.