135: Chapter 135 Something Bigger Than the Gold Medal

After the award ceremony at Zijin Villa, the crowd density outside the main venue experienced a brief dispersal and flow before regrouping in various public areas of the hotel.

In the tea break area, some people held paper cups while exchanging business cards with unfamiliar peers next to them, trying to expand their academic networking circles.

Others looked down, quickly swiping across their phone screens to discuss the dense agenda of the afternoon's various sub-venues with their advisors or lab mates, calculating whose keynote report they should attend.

But more of them gathered in groups of three or five, lowering their voices to exchange various versions of what they had just heard in the venue about that jaw-dropping gold medal.

"There was completely no news beforehand."

"I didn't expect the gold medal to be given to a high school student either."

"What high school student, that's an independent researcher."

...

Such whispering spread to every corner of the venue.

Jiang Lin did not linger in the tea break area at all.

Just as he walked to the relatively open area outside the periphery of the main venue, Tang Yue from the organizing committee spotted him with sharp eyes.

She wove through the crowd and walked quickly in front of Jiang Lin.

"Teacher Jiang, this way please. The media meet-and-greet is arranged in the small conference room on the second floor. This is a unified process arranged by the organizing committee, featuring a fifteen-minute group interview. Afterwards, if the awardee is willing, the organizing committee will coordinate subsequent individual interviews."

"Alright, then sorry to trouble you to lead the way." Jiang Lin nodded.

As he followed Tang Yue toward the side elevator, he put the medal and the red certificate into the canvas document bag issued by the organizing committee.

The space of the media meeting room was not large, and the front row was densely packed with cameras and fill lights from various media outlets.

The back row was filled with senior journalists from professional mathematics journals, university press publications of major universities, and several leading domestic technology media outlets.

Everyone's face was filled with excitement, tightly grasping voice recorders and interview outlines in their hands.

Behind the long table, the nameplates of the awardees were placed in order.

Ding Jian and Jiang Lin sat in the middle, flanked by several other equally accomplished silver award winners such as Chen Gao and Miao Shuang.

The host briefly delivered an opening remark, setting the tone for this interview.

Afterwards, to show respect for senior scholars and to slightly dampen the overly fervent atmosphere at the scene, he first gave the first questioning opportunity to Ding Jian.

The question was quite conventional, ranging from personal impressions after winning the award to the future development trends of probability theory and mathematical Physics fields domestically.

Ding Jian truly lived up to being a battle-tested top scholar; his answers were calm, poised, and graceful, possessing both a mathematician's rigor and depth, as well as maintaining demeanor when facing mass media.

But everyone knew who the true focus of today's media meet-and-greet was.

The topic quickly and inevitably fell upon Jiang Lin.

The first person to stand up and ask a question was a senior journalist from an official media outlet.

The question was very routine, and also the version the public most wanted to hear.

"Hello Teacher Jiang, first of all, congratulations on becoming the youngest gold award winner in the history of the ICCM Mathematics Awards. This is definitely a history-making moment. May I ask what your biggest inner feeling is right now standing on this podium holding the gold medal? Is there anything special you want to say to your peers?"

Accompanied by the movement of the camera lenses, the gazes of the whole audience swished and all focused on Jiang Lin.

Jiang Lin slightly raised his eyelids, looking at the camera: "Thanks to the review committee for the recognition. The award is an affirmation of the work done over the past period, but it is not the purpose of our research itself. Mathematics has no end; once the Jiangs Brick problem is resolved, there are still many more complex derivative problems to deal with in the subsequent stage, just continue to peaceably do research."

Speaking of this, he paused slightly.

Just when the questioning journalist thought he had finished answering and was preparing to take over with a smile, Jiang Lin spoke again.

"As for what I want to say to my peers, do not treat me as a reference point, nor treat this gold medal as some sort of inspirational filter. Go find a field that you are truly willing to stay up late for, willing to endure repeated failures, and not demanding short-term returns. Everyone has their own coordinate system, and there is no need to forcibly solve someone else's equation."

The questioning journalist froze for a moment, then quickly followed up based on professional instinct: "Then can you talk about how, as an independent researcher without the support of top university platforms or guidance from famous teacher teams, you managed to make such a major academic breakthrough? We are all very curious, in this process, what was the most difficult thing you encountered?"

This question was very common and fit communication patterns very well.

However, Jiang Lin only pondered slightly, then replied in a nearly plain-descriptive manner: "Difficulties have always been there, and getting stuck is the norm in theoretical research. There is nothing special about this. When encountering an impassable place, just change the train of thought and slowly push downward."

A brief silence appeared at the scene.

This answer simply could not be written into that press release titled 《The Counterattack Path of a Genius from a Humble Background》.

Someone in the back row was unwilling to give up and wanted to continue digging up some sensational material on this teenage genius persona.

A tech media journalist raised their hand and just opened their mouth to ask a sentence: "Teacher Jiang Lin, everyone is very curious, how did you balance the heavy Gaokao studies and such profound frontier mathematical research during high school? Usually you..."

Seeing that the topic was about to lean toward educational experience sharing, the experienced host smiled, took over the conversation, and turned the direction of questioning to the next silver award winner.

"Alright, thank you Teacher Jiang Lin for sharing. Next, we leave the time to Teacher Miao Shuang, regarding the field of geometric analysis..."

The fifteen-minute group interview ended quickly in such an atmosphere where the questioners were passionate and enthusiastic, while the respondent was calm and restrained.

Jiang Lin had just stood up and prepared to leave when he was blocked by two sharp-eyed professional media journalists.

"Teacher Jiang, Teacher Jiang, please stay." One of the journalists held a printed piece of literature data in hand, "We are from the 《Mathematics Frontiers》 magazine, can we take two minutes of your time to ask you to talk about the PFR Conjecture..."

Jiang Lin stopped his footsteps and looked up at these two professional journalists who had obviously done their homework.

"Sorry, you two. Regarding that question, it is currently inconvenient to reveal more details. I still have some personal matters to deal with on my end, so please excuse me first."

After speaking, he slightly nodded in greeting and walked straight out of the door of the media room.

Behind him came the low-voiced exclamations of the two journalists.

"This temperament is too indifferent, winning such a big award and acting like nothing happened."

The other slightly older journalist shook their head and put the voice recorder back into the bag: "This kind of person's mind is not on exposure. If you ask him about his life story, he will only feel that you delayed him from going back to revise proofs."

...

Coming out of the media meeting room and glancing at the phone, the time was approaching 12:30.

The morning agenda was basically over, and dense crowds of people surged out from various large and small conference halls.

The attendees followed the blue-and-white indicator signs along the way, walking in twos and threes towards the direction of Jinling Cafe and the cafeteria.

The hotel's cafeteria was located on the first floor of the main building.

Outside the huge floor-to-ceiling windows was a meticulously designed courtyard water feature.

Rockeries, flowing water, and cruising koi fish made the scenery pleasant.

To cater to scholars from all corners of the country, the food pickup area was carefully divided into two major zones: Chinese and Western.

On the Chinese food counter, it was filled with authentic Huaiyang style delicacies such as salted duck, stir-fried wild rice shoots, crab roe lion's head, and shredded dried tofu in broth, steaming hot.

The Western zone had various whole wheat breads, vegetable salads, cold cut meat platters, and two imported freshly-ground coffee machines humming away.

In the corner, a special Cantonese soup bar and seasonal fruit area were specially set up.

Jiang Lin took a white porcelain dinner plate and walked to the Chinese zone.

He was not a picky eater; with nimble actions, he picked a few dishes combining meat and vegetables, asked the chef to dish up two large bowls of tightly packed plain white rice, and finally took a bottle of room-temperature mineral water.

Holding the fully loaded dinner plate and turning around, Jiang Lin scanned the bustling restaurant for an empty seat.

"Jiang Lin, over here."

Not far away, Professor Gu Nanzhou was sitting in a four-person seat by the window, waving at him.

In the seat opposite him, Professor Lin Zhaoye was lowering his head and leisurely drinking a bowl of spare rib soup.

Jiang Lin walked over, pulled out a chair next to Professor Lin Zhaoye, and sat down.

"This appetite is not bad," Professor Gu Nanzhou looked at the mountain-like pile of carbohydrates on Jiang Lin's plate and joked with a smile, "Judging by your posture, dealing with those reporters this morning took more physical energy than deriving formulas on the blackboard?"

"Deriving formulas taxes the brain, while handling interviews costs both brainpower and physical energy." Jiang Lin picked up his chopsticks and began shoveling rice into his mouth in big bites, "The food here by the organizing committee is made quite well, just right to make up for the consumption."

Professor Lin Zhaoye put down the soup bowl, pulled a tissue to wipe his mouth, and obvious amusement filled his eyes behind the lenses: "Eating big bowls of rice and pursuing great learning, isn't this just fittingly appropriate."

"Hahaha."

...

Ten minutes to two in the afternoon.

After lunch and taking a brief twenty-minute rest in his room, Jiang Lin once again followed the blue-and-white indicator signs back to the main venue of the Zijin Conference Center.

Compared to the solemnity and dignity of the morning's opening ceremony gala, the afternoon's forum venue noticeably had a bit more relaxation of cross-disciplinary free communication.

This was the core main forum of this ICCM—[Basic Science and International Cooperation].

As a fixed and extremely important segment on the day of the opening ceremony, this forum was not limited to the pure mathematics track.

The organizing committee had an extremely broad vision, deliberately inviting heavyweight scholars and practitioners from theoretical Physics, artificial intelligence, complex systems, and the domestic top tier industrial sector.

This was an event with the broadest interdisciplinary vision and the most intense collision of the entire conference.

The forum adopted a synchronous online and offline live broadcast method, with hundreds of people still sitting in the audience. In addition to participating scholars, there were also many corporate representatives who came here by reputation.

Jiang Lin was still sitting in that forward position of the guest seats from the morning.

As soon as he sat down, a middle-aged researcher next to him wearing a badge from a certain research institute of the Chinese Academy of Sciences sharply noticed him.

The researcher turned slightly sideways, with a kind smile on his face, and took the initiative to exchange pleasantries: "Jiang Lin, are you also interested in this kind of cross-disciplinary topic?"

"Come over to listen to ideas from different directions." Jiang Lin responded politely and indifferently.

That person was also quite shrewd, nodding knowingly: "That's true, that's true. Now that science has developed to this stage, the bottlenecks of many single disciplines are at the intersection zones, and current theoretical tools are getting closer and closer to AI algorithms and statistical Physics."

At two o'clock sharp, the forum started on time.

Serving as the host was a vice president of Southeast University.

The opening remarks were concise and straight to the core.

"Basic science is the bottom-most source of all human technological breakthroughs. And international cooperation is the inevitable path of development for all frontier disciplines. The reason why this conference has for the first time established Basic Science and International Cooperation as the core forum theme on the opening day is precisely the hope of breaking down barriers, opening up boundaries between disciplines, and promoting our Chinese academic community to achieve overall advancement on a higher dimension."

The first person invited to the stage to deliver a keynote speech was Xue Qikun, President of Southern University of Science and Technology and a leading figure in the Physics world.

He wore a crisp dark suit with his hair combed meticulously, standing in front of the podium with a steady and infectious speaking pace.

Academician Xue started from the experimental breakthrough of the quantum anomalous Hall effect, and talked in a simple and easy-to-understand way about the underlying supporting role of basic mathematical tools like topology on physical models.

Subsequently, changing the topic, he candidly talked about the various experiences, stumbles, and unspeakable dilemmas experienced in multinational scientific research cooperation.

"Everyone, without solid and rigorous mathematical tools backing us up behind the scenes, many wonderful physical conjectures proposed by us physicists can only ever remain as pretty formulas on paper and can never land on the ground."

Speaking of this, Academician Xue swept his gaze across the mathematicians in the front row, half-jokingly sighing.

"To be honest, we who do experimental Physics sometimes really envy you who do mathematics. For a piece of data, we have to build platforms costing tens of millions, wait for liquid helium to cool down, and prevent shock and magnetism. What about you? A pen, a sheet of draft paper, as long as the logic is correct, you can deduce the laws of the entire universe in your mind."

An appreciative soft chuckle erupted from the audience, and the atmosphere instantly relaxed a lot.

Jiang Lin sat in the front row, listening intently.

Regarding the intersection boundary of geometric analysis, partial differential equations, and quantum Physics, he had only made scattered contact in literature previously and had not deeply studied it.

But at this moment, listening to a front-line experimental physicist standing at the international forefront tell of the pain points under real application scenarios with such straightforward words felt instead like a gust of wind blowing away certain inherent fogs in his mind, giving him a three-dimensional brand-new understanding of the universality of high-dimensional construction and the projection of mathematics in the real world.

The second person to take the stage to speak was Professor Zhou Zhihua, Academician of the Academia Europaea.

He was one of the absolute leading figures in the domestic artificial intelligence field.

Academician Zhou's topic was closer to current tech industry hotspots.

Cutting directly from the theoretical bottleneck of machine learning, it pointed straight to the most core shortcomings of current AI development.

"Right now, many deep learning models, especially large models, have shockingly good results in engineering applications. However, everyone, we must soberly recognize that the mathematical interpretability of these models is extremely poor."

"We are just like a group of blindfolded children building extremely complex building blocks, stacking them very high and magnificent, but we have no idea what their underlying mechanical principles are. Once a block is pulled out from the bottom, the entire building could collapse at any time."

He waved his hand slightly and deepened his tone: "Therefore, if future artificial intelligence wants to achieve truly revolutionary breakthroughs, it cannot rely solely on stacking computing power and data. Ultimately, it must still return to basic mathematics, to the deepest roots of statistical learning theory and optimization theory to find answers."

Hearing this, Jiang Lin's gaze shifted slightly.

The set of proof frameworks regarding the PFR Conjecture that had completed closure surfaced in his mind.

What is the essence of deep learning large models?

In a sense, it is performing feature extraction and dimensionality reduction compression on massive, seemingly disordered data in high-dimensional space.

Mathematically, it can be roughly abstracted as equivalent to the approximation and mapping of a high-dimensional discrete structure.

One of the reasons why AI algorithms are opaque black boxes is how the internal information entropy of these data evolves after layer-by-layer network compression, and why the core feature support is not lost during the dimensionality reduction process.

And the core weapon Jiang Lin had just constructed in the PFR proof, the third-layer loss recovery mechanism and the entropy control function on the Finite Field Model, solved precisely the local error amortization problem of high-dimensional structures during the compression process.

This was originally just to solve pure Additive Combinatorics difficult problems.

But this mathematical tool used to constrain residual spectra and control the morphological collapse of finite field entropy, if it could undergo a proper layer of discretization and modeling, might be able to migrate an analytical framework into the high-dimensional tensor networks of machine learning, providing theoretical guidance for that massive black box.

...

This high-quality forum lasted for a full two and a half hours.

It was only around four-thirty in the afternoon that the curtain fell amidst thunderous applause.

As the venue dispersed, the flow of people inside began to slowly churn.

Many unwilling participants, young scholars, and corporate representatives left their seats one after another, surrounding the front of the podium like a tide, hoping to seize the last opportunity to exchange WeChat with the guests who had just spoken or to ask a couple more questions.

Jiang Lin feared this kind of meaningless social engagement the most, so taking advantage of the crowd not yet fully congested, he walked quickly outward along the side path at the edge of the venue.

He was about to walk out of the thick, heavy doors of the main venue.

"Jiang Lin, wait a moment."

A deep voice came from behind.

Jiang Lin stopped in his tracks and turned around.

Professor Zhou Zhihua, who had given the keynote speech on stage just now, was parting the crowd surrounding him and walking quickly toward this side.

His steps were very large, and his face wore the unabashed appreciative smile of an elder seeing an outstanding younger generation.

"Professor Zhou." Jiang Lin greeted politely.

"I've heard your name long ago, and today I finally get to see a live person." Professor Zhou Zhihua walked up close, looked Jiang Lin up and down, and went straight to the point, "The set of mathematical methods you used in the Jiangs Brick construction was carefully studied by several of our AI theory PhD students at the group meeting yesterday. Your train of thought is simply too brilliant."

Saying this, he smoothly extended an olive branch.

"Jiang Lin, are you interested in spending some time trying to work on some problems of the underlying mathematical foundations of machine learning? If you are willing to try migrating the train of thought of finite-state compression into learning theory, there might be some very interesting intersections."

"Thank you for your recognition and invitation, Professor Zhou." Jiang Lin's reply remained airtight and courteous, "If there are suitable opportunities later, or if I encounter problems in cross-fields, I will definitely consult and communicate with you."

"Alright, it's a deal." Professor Zhou Zhihua knew that geniuses of this level all had their own ideas, so he didn't force it.

...

"Jiang Lin."

Just as Professor Zhou Zhihua turned and left, another voice came from behind the pillar in the rest area on the right side of the venue.

Jiang Lin turned his head.

Professor Han Yanshan was standing between two rows of empty seats.

He was wearing an old plaid shirt that had been washed somewhat white, with his sleeves casually rolled up above his elbows.

Only a short two months had passed, and the white hair on Professor Han Yanshan's head seemed to be a bit more than when he was at the Jiangda University lecture hall.

But his eyes at this moment were brighter than ever.

Clearly, he had been waiting here for a very long time.

He waited through the full two-and-a-half-hour forum, waited for the surging crowd to recede at the end of the session, and waited until all the congratulations, group photos, and business card exchanges rushing toward Jiang Lin were completely over.

"Professor Han." Jiang Lin looked at him.

Professor Han Yanshan didn't come up to congratulate him on winning the gold award first like others did, nor did he engage in any small talk.

He reached his hand into his somewhat faded trouser pocket and pulled out an A4 sheet of paper that had been folded multiple times and whose edges were already somewhat frayed.

He handed the paper in front of Jiang Lin.

Jiang Lin lowered his gaze and glanced at it.

That was a printed screenshot of the last ten minutes of the ICM report video.

On the blank space beside the picture, it was densely filled with notes taken by Professor Han Yanshan.

Heavy red circles were drawn next to some formulas, huge question marks were placed in some places, and beside the words finite field bridging, Professor Han Yanshan used a red marker to heavily mark three words: "Not a metaphor."

Professor Han Yanshan took a deep breath, gazed steadily at the eighteen-year-old youth in front of him, and asked the question that had lingered in his mind for fifty days.

"If it's convenient, regarding the PFR Conjecture under the Finite Field Model, can you talk about where exactly it has reached now?"

Jiang Lin could understand Professor Han Yanshan's mood at this moment.

He had stubbornly grinded away in this direction for a full twelve years.

He had done high-dimensional generalizations of the Freiman-type structure theorem, and had long struggled in the quagmire of approximate groups and small sumsets.

Without any exaggeration, looking across the entire domestic mathematics community, Professor Han Yanshan was one of the very few people who could deduce the extremely steep technical route of PFR / Marton from beginning to end without any stumbles.

Telling someone like him that the main proof had closed was like telling a mountaineer who had endured ten years at the Mount Everest base camp: "Hey, someone has already reached the summit of that hardest north face route."

Upon hearing this news, the mountaineer would definitely not open champagne to celebrate first.

He would immediately rush up, with red eyes demanding of you:

"What is your roadmap?"

"At which altitude did you build the assault camp?"

"What equipment did you use?"

"What date was the weather window you waited for?"

"How was your oxygen reserve planned?"

Jiang Lin pondered slightly and asked: "Professor Han, after you returned from Jiangda University in June, did you continue to deduce that analogy?"

Professor Han Yanshan was clearly stunned for a moment, not expecting Jiang Lin to ask a counter-question.

But he still answered directly: "I deduced it, I deduced it for a full fifty days."

His tone inevitably carried a bit of self-deprecation and defeat.

"Deducing to the end, I found that the four words you said at the time, 'indescribable', every single word was correct. From local structural compulsion to crossing over to global group approximation, that theoretical chasm in the middle simply cannot be jumped across by intuition."

"I later deduced past it."

Those seven words, without any embellishment or fluctuation in tone, falling into Professor Han Yanshan's ears, were no different from detonating a nuclear bomb in his brain.

Professor Han Yanshan froze entirely, and even his breathing seemed to pause.

His Adam's apple rolled up and down with difficulty, and he stared closely at Jiang Lin, as if trying his best to confirm whether the words he had just heard truly meant what they literally said.

"The key lemma has been proved out?" Professor Han Yanshan's voice was extremely hoarse.

"Connected." Jiang Lin replied.

"What is the bound of the covering number?"

"Polynomial level, didn't drop back to the despairing exponential level."

Jiang Lin methodically threw out the core technical details.

"The core lies in the third-layer loss recovery mechanism. I designed a local alternative construction to contain the truncation terms that would otherwise expand exponentially during the derivation process. This alternative hierarchy shares the same mathematical structure at the bottom level as the boundary compression framework in Jiangs Brick. Only here, what's being played is no longer geometric tiling, but the Fourier support contraction on the Finite Field Model."

Professor Han Yanshan looked Jiang Lin over for two seconds, and suddenly asked: "What you contained is the major spectrum, or the residual spectrum remaining after the energy increment?"

This sentence was simply piercingly insightful.

If Jiang Lin had just talked in generalities, or merely borrowed a few nice-sounding words, he would immediately expose himself in front of this question.

Because under the Finite Field Model, the real difficulty is not the spectrum itself, but those residual spectra that continue to leak outward along boundary conditions after each round of local structure stripping.

Every time they leaked once, the covering number expanded by one layer, ultimately dragging all polynomial hopes back into exponential hell.

Jiang Lin answered without hesitation: "Strip the residual spectrum first, then recover the energy loss. What is truly contained is not the spectrum, but the expansion of the covering number caused by spectrum leakage."

Professor Han Yanshan's pupils slightly constricted.

Jiang Lin continued to talk with ease: "The role of the third-layer alternative construction is not to prove that the residual spectrum disappears, but to cast it back into a controllable low-dimensional model after each local truncation. The residual spectrum is still there, but it no longer diffuses freely. As long as it doesn't diffuse freely, the covering number will not explode exponentially."

Professor Han Yanshan stared motionlessly at Jiang Lin, as if trying to see from his face whether these few sentences were some sort of carefully packaged illusion.

A moment later, he suddenly took out pen and paper from the briefcase he carried with him.

"Wait a moment, don't go on yet."

His voice was so low, carrying an almost brute urgency.

Jiang Lin nodded.

Professor Han Yanshan didn't start deducing from definitions.

People who had truly worked in this direction didn't need to start writing from what weak doubling was, what the major spectrum was, or what the Finite Field Model was.

Those things had long been repeatedly disassembled, crushed, and reassembled in his failures over the past twelve years, turning into a kind of near-instinctive muscle memory.

He wrote directly to the most dangerous step.

Let $A \subset \mathbb F_2^n$, satisfying the weak doubling condition.

Normalized indicator, local model, Fourier major spectrum.

Spectrum stripping, energy increment, covering number recurrence.

Line after line of symbols quickly unfolded.

At first, he wrote very fast.

Because the preceding path had no suspense.

After pressing $A$ into the Finite Field Model, first take the normalized indicator function, and then find the first-layer major spectrum according to the energy distribution.

Through the low-codimension subspace spanned by the major spectrum, make the first structured slice of $A$.

If the density increment on the slice is sufficient, enter the next round.

If insufficient, switch to the covering estimation of the approximate subspace model.

These were all standard tools.

The real problem was never in the first layer.

The loss of the first layer could still be tolerated.

The second layer could also barely be maintained within an acceptable range.

But when it reached the third layer, all beautiful structures began to deform.

When Professor Han Yanshan wrote to the third spectrum stripping, the tip of his pen noticeably slowed down.

He drew an extremely narrow parenthesis on the paper surface and circled a group of residual frequencies.

They were in the most troublesome position, neither the main spectrum nor noise that could be directly discarded.

Each residual term looked very small individually, small enough not to support a new structure theorem.

But they weren't small enough to be completely swallowed by the error term.

Once the recurrence continued, these residual spectra would be forced to enter the next round of local models and re-participate in covering estimation.

And as long as they re-participated in covering estimation, they must pay for a spectrum dimension loss once again.

Professor Han Yanshan stared at that recurrence formula, his expression sinking bit by bit.

"These fifty days of mine have always been stuck on this term."

He heavily clicked the tip of his pen on $d_j$.

"After the third layer, as long as the residual spectrum re-enters the next round of local model, it must pay for a spectrum dimension loss once again. This is neither a constant error nor a tail term that can be stuffed into $O(1)$."

His voice lowered a bit.

"It will directly rewrite the growth order of the covering number."

Jiang Lin glanced at that piece of paper, stretched out his finger, and tapped lightly beside the recurrence formula written by Professor Han Yanshan.

"You multiplied an extra time here."

Professor Han Yanshan's brows frowned fiercely.

Jiang Lin said calmly: "The residual spectrum should not re-enter the recurrence as a free spectrum for the next round."

Professor Han Yanshan did not speak.

Jiang Lin continued: "You regarded the residual frequencies after the third layer as a new spectral object, so every round you had to repay a Chang-type dimension loss. My treatment is not to continue stripping spectra, but to retain indices for the residual spectrum clusters left behind after the second layer stripping, putting them into a fixed low-dimensional alternative model."

Professor Han Yanshan's pupils slightly contracted.

The moment these words fell, he had already heard the key to the problem.

Not continuing to strip, but fixing indices.

Not allowing the residual spectra to regain degrees of freedom in the next round, but locking them into the spectral cluster structure for which the price had already been paid in the previous layer.

Jiang Lin's finger moved back half an inch along that recurrence formula.

"The role of the third-layer alternative construction is not to prove that the residual spectrum disappears. It only does one thing: preventing the residual spectrum from regenerating new covering directions."

"Can you talk about it in detail?" Professor Han Yanshan hurriedly pursued with a question.

Jiang Lin said: "After every local truncation, the residual terms do not enter the new covering recursion, but are cast back into the low-dimensional model corresponding to the spectral cluster of the previous layer. The error caused by it is paid for by the unfilled gap of the energy increment from the previous layer."

The voices at the edge of the venue were still mixed and noisy.

Some people dragged chairs, some packed up materials, and others walked out of the main venue in twos and threes.

Yet at that instant, those sounds seemed to be suddenly isolated to a very distant place.

Professor Han Yanshan lowered his head and stared at that recurrence formula.

Paid by the energy gap of the previous layer, the residual spectral clusters have fixed indices.

The covering directions are not regenerated.

The next moment, he put pen to paper again, not continuing to write down along the original recurrence, but starting a new line beside it to tear apart that multiplicative loss from just now.

The term that originally entered ( M _ { j+1 } ) was forcibly stripped out of the covering recursion by him.

Then he wrote down a new local error term beside it.

\ mathcal E _ j ^ { bad }

Then he wrote below it: \ mathcal E _ j ^ { bad } \ leq \ Delta E _ j

( \ Delta E _ j ).

The remaining gap of the energy increment.

If the loss caused by the residual spectrum no longer enters the next round of recursion as a new covering direction, but is absorbed by the energy gap not completely consumed by the previous layer, then the originally most fatal layer-by-layer multiplicative loss will be cut off.

The covering number no longer multiplies by a term anew at every layer.

It has turned into local amortization.

Not: M _ { j+1 } \ leq M _ j \ cdot K ^ { C d _ j }

Instead: M _ { j+1 } \ leq M _ j \ cdot K ^ { C d_0 } + O ( \ mathcal E _ j ^ { bad } )

More precisely, what truly grows is not the spectral dimension regenerated in each round, but the complexity of the spectral clusters already fixed in the first and second layers.

The bad terms are still there, the residual spectrum is still there, and the errors are still there.

Yet they have all been amortized into the structural layers that have already paid the price, no longer possessing the qualification to independently manufacture covering directions.

Professor Han Yanshan began to reorganize that failed path from just now.

First layer large spectrum, paying a dimension loss once.

Second layer localization, fixing the spectral cluster indices.

Third layer no longer performs free spectrum stripping, but performs surrogate model projection.

The residual spectral terms enter the energy gap.

The covering recursion closes.

He wrote faster and faster.

Beside the densely packed failed derivations from before, a new loss flow chart gradually appeared.

Of course, there were no boundary conditions, no threshold selections, no extreme degeneration cases, no error control after each localization, and no fine estimation of whether all constants could eventually be squeezed into polynomial bounds.

It was not yet a complete proof.

Far from it.

Yet it was no longer a vague idea.

Because that most fatal recurrence had changed its shape.

Professor Han Yanshan stared at the paper surface, finally understanding what Jiang Lin's so-called third-layer loss recycling meant.

It was not wiping out the third-layer loss out of thin air.

Nor was it forcibly holding down all residual terms with a stronger structural assumption.

Instead, it re-regulated the ledger attribution of the residual spectral terms.

In all past failed routes, the residual spectrum would be treated as newly generated free spectrum for the next round, so every time the covering number advanced a layer, it had to repeatedly pay the spectral dimension loss once.

Jiang Lin did not do this.

He fixed the residual spectral clusters in the previous layer's structure, paid for them with the remaining gap of the energy increment, and rewrote the originally uncontrollable multiplicative loss into amortizable local errors.

This was by no means a technical tweak; it entirely belonged to a rewriting of the recurrence framework.

Professor Han Yanshan slowly took off the glasses on the bridge of his nose.

The pair of eyes behind the lenses were severely bloodshot.

"The key is not in the spectrum set itself; the residual spectral terms can no longer be sent into the covering recursion as a new round of free variables."

As he muttered to himself, he continued looking at the newly written loss flow, as if confirming over and over again that he was not hallucinating due to fatigue.

"Instead, the residual spectrum is plucked out of the covering recursion and changed to energy gap amortization."

Having said this, Professor Han Yanshan abruptly raised his head.

He revealed an expression bordering on loss of control, the kind that would only appear when someone who had walked down the same failed path for twelve years suddenly discovered that a recurrence he had always thought unclosable could actually be re-closed by changing the loss attribution method.

"Jiang Lin, if this step can be made rigorous, the covering number will not fall back to the exponential level."

Professor Han Yanshan's voice was terribly hoarse.

Jiang Lin nodded.

Professor Han Yanshan finally completely lost his composure.

"Damn it."

The voice was very light, without any rude intention.

There was only a tremendous sense of absurdity after being repeatedly crushed by twelve years of failure, suddenly seeing the recurrence formula close again.

"This is not something that a local lemma can explain; if the ledger attribution of the residual spectrum can really be changed like this, then the subsequent covering estimates, entropy control functions, and even the entry conditions on Marton's side will all have to be rewritten."

He raised his head and looked at Jiang Lin.

"So, what you filled in is not a lemma, but a rewriting of the entire loss recurrence."

"Just rewriting the recurrence is not enough."

Jiang Lin glanced at the symbols on that sheet of paper, which were so crowded they were nearly chaotic, and shook his head.

"The real trouble is that Marton's side cannot be pushed hard along the common Finite Field Model translation path. To ensure that the entropy structure does not collapse, that path usually has to additionally stuff in very strong non-degeneration conditions, so strong that they can hardly accommodate weak-doubling objects."

Professor Han Yanshan abruptly raised his head.

"So you bypassed the non-degeneration assumption?"

"Not entirely bypassed," Jiang Lin said. "More accurately, it is not taking it as an entry condition."

Professor Han Yanshan's breathing noticeably missed a beat.

Jiang Lin continued: "I start directly from the weak-doubling structure and construct an entropy control function. Instead of first assuming the object is sufficiently non-degenerate before doing the covering, I first use the third-layer recycling mechanism to suppress the covering recurrence, and then let the non-degeneration emerge as a result within the Finite Field Model."

"If this holds true, Marton's bridging part will no longer be an extra piece of proof hanging behind the PFR Conjecture, but will grow together with the weak-doubling structure itself."

This time, Professor Han Yanshan did not immediately follow up with questions.

Because that rewritten recurrence formula was no longer just dealing with the covering number problem of the PFR Conjecture, but was simultaneously changing the entry conditions of Marton's bridging part.

Once that recurrence from just now closed, what changed was not just the PFR Conjecture itself.

It would change the entry conditions of the entire subsequent bridging path.

The most painful part of the traditional route lay in the fact that Marton's entropy form required objects that were clean enough, regular enough, and sufficiently non-degenerate.

Yet weak-doubling objects happened not to be such things.

They were rough, skewed, and carried a large amount of local residual structure.

The more you required them to be clean, the less they could be connected in.

But if Jiang Lin's third-layer loss recycling held true, then weak-doubling objects did not need to satisfy that set of harsh conditions right from the start.

They could first be suppressed in the covering recurrence, and then gradually reveal available approximate structures within the entropy control function.

In other words, non-degeneration turned from a premise into a result.

Professor Han Yanshan's pen fell back onto the paper, following the loss flow from just now, and drew three arrows forward.

PFR Conjecture.

Marton's entropy form.

Freiman-type structural compression.

Then it stopped beside the third word and lingered without moving.

If this chain truly held, the PFR Conjecture would not be an isolated result.

Marton covering would not be a collateral issue.

The high-dimensional compression of the Freiman-type structure theorem would also no longer just be an extension pushed hard by old tools.

A completely new set of technical language would appear here.

A set of technical language centered on residual spectrum amortization, entropy control functions, and Finite Field Model compression.

Professor Han Yanshan put his glasses back on and asked solemnly: "How many people have read the original draft of this matter?"

"Professor Lin Zhaoye knows that I have already closed this fact," Jiang Lin replied.

Professor Han Yanshan was silent for a moment.

Professor Lin Zhaoye was not in this direction.

Under the terrifying precision requirements of modern mathematics, crossing fields meant being separated by several universes.

"You said just now that your construction of the third-layer loss recycling requires even yourself to review it for three full days to relocate it," Professor Han Yanshan looked at Jiang Lin.

"Yes."

"Then do you know how many people across the entire domestic mathematics community can understand your construction within three full days?"

Jiang Lin shook his head.

He did not have much understanding of people in the mathematics community.

"Currently in this subfield domestically, the number of people who can have a substantive dialogue with you at this high-precision level—I am not talking about those general discussions or nods of understanding the gist, but those who truly read through your proof manuscript line by line, find all boundary conditions hidden in the corners, and confirm that every step of your derivation has no logical leaps—will not exceed five."

Professor Han Yanshan held up five fingers, his tone carrying an unyielding arrogance.

"People who can immediately sit down with you to check the proof line by line are even fewer; I count as one."

...

When Professor Lin Zhaoye finally found the two of them on the periphery of the empty main venue, Professor Han Yanshan's glasses had been taken off again.

He was sitting beside a table, pressing with his fingers on that A4 sheet of paper that was almost filled with writing, holding a signature pen in his other hand, continuing to fill in a loss flow incomprehensible to others at the edge of the paper.

Jiang Lin stood quietly to the side, occasionally offering a word of guidance.

"So you two were here."

Professor Lin Zhaoye walked up quickly and breathed a sigh of relief.

"I looked inside for half a circle just now and didn't see anyone."

Then he saw Professor Han Yanshan's derivation on the desktop.

"Professor Han, it seems you know too," Professor Lin Zhaoye smiled and probed with a sentence, "So, Professor Han, what do you plan to do?"

Professor Han Yanshan lowered his head to glance at the paper and said: "Jiang Lin says he still needs time to compile the technical memorandum. Then by that time, the review of the backbone structure will require peers from two completely different directions."

His brain was spinning extremely fast.

"First, we need someone doing discrete combinatorics and additive structures to perform cross-validation of the overall and detailed aspects of the Fourier support shrinkage part on the Finite Field Model. I will take this part."

"Another one must be able to start from the most fundamental logic of entropy and information theory to check his construction regarding Marton's bridging part. This requires extremely profound cross-disciplinary competence."

"Do you have a candidate in mind?" Professor Lin Zhaoye asked.

Professor Han Yanshan's finger tapped on a name on that A4 sheet of paper.

"Zhao Yufei."

"The youngest associate professor in the Department of Mathematics at MIT?" Professor Lin Zhaoye was slightly startled.

"Right."

Professor Han Yanshan's speaking speed was extremely fast.

"The third line of the research directions on his personal homepage prominently reads Additive Combinatorics. I have carefully read all his core papers on extremal combinatorics and graph regularity lemmas. He is very good at using probabilistic methods; his mathematical toolboxes—graph limits, regularity lemmas, and probabilistic scaling—are highly compatible in methodology with the underlying framework of Jiang Lin solving the PFR Conjecture this time."

"The non-degeneration construction of Marton's bridging part is extremely counter-intuitive. We need someone who can understand those perverse constraints from the perspective of discrete combinatorics, and at the same time verify the stability of the entire approximate structure from the macroscopic level of entropy and information theory."

"Zhao Yufei does not come from a traditional information theory background, but his probabilistic treatment of high-dimensional discrete structures is extremely strong, which happens to enable him to see whether this set of entropy forms holds only in symbols or actually carries combinatorial structure."

"As things stand, he is one of the most suitable candidates in the world."

Professor Han Yanshan paused for a moment.

"Regrettably, I have no personal friendship with him; perhaps we can ask Mr. Qiu to help introduce us."

Professor Lin Zhaoye frowned and thought for a moment: "But he is not in the country right now."

"It doesn't matter if he is not in the country; I will send him an email right now." Professor Han Yanshan was a typical man of action, turning to leave as he spoke.

"Actually, speaking of domestic options, Professor Ding Jian comes from a background of probability theory and statistical Physics," Professor Lin Zhaoye suddenly spoke up.

Professor Han Yanshan's footsteps paused.

Professor Lin Zhaoye continued: "When it comes to mathematical intuition for entropy forms, currently no one in the country is stronger than him. If he looks at it and is willing to help Jiang Lin vet it from the perspective of statistical Physics, there won't be any need to wait for Zhao Yufei's reply across the time difference spanning the Pacific Ocean."

"Alright, let's look for Professor Ding first." Professor Han Yanshan's words leaked an irrepressible, almost absurd excitement.