124: Chapter 124 A Challenge Hanging Over the Additive Combination

On the second day after the ICM report concluded, the carnival on the Chinese internet continued to ferment.

As for Jiang Lin's remarks on the technical program at the end of the report, laymen saw the excitement and legend, while insiders saw the real methods.

No matter how shocking Jiangs Brick was, and no matter how exquisite its construction was, it was ultimately a mathematical object already placed on the table with physical boundaries.

However, the PFR Conjecture and Marton's route were completely different.

That was not a brick that could be depicted with geometric intuition, nor was it a finite construction that could be verified through clever exhaustion.

It was like a massive and intangible shadow, lying across the intersection of the four major fields of Additive Combinatorics, information theory, entropy methods, and structural theorems.

Over the past few decades, mathematicians had obtained many profound results, but the truly difficult part lay in whether the complexity bounds relying on the constant K could be compressed to the polynomial level.

This was precisely what made the PFR / Marton direction daunting.

In order to lower the logarithm on that exponent and to find that polynomial bound, countless brilliant minds had crashed and failed before this great mountain.

Many top mathematicians, including fields medal winners, had left traces of climbing on this mountain.

Various lemmas and weakened versions had been proposed.

Similarly, many more people merely stood at the foot of the mountain, glancing from afar at those tangled thorns intertwined with Fourier analysis and geometric number theory, knowing that this was definitely not a place that could be forcefully breached by youthful vigor or a momentary flash of inspiration.

Therefore, when Jiang Lin said "It's not a metaphor" in a calm tone on the international academic stage, the first reaction in the professional circle was not the excitement of witnessing history.

But rather deep alertness, or even instinctive rejection.

If this were merely bold talk temporarily uttered by a young man who had just emerged in the academic world due to a surge in adrenaline after successfully completing his report, it would have no significance in the rigorous history of mathematics.

The long river of mathematical history had never lacked geniuses who spoke wild words, nor had it lacked fantasizers who claimed to have seen the edges of truth.

But the most tricky core of the whole matter was that the person who said these words was Jiang Lin, who had just completed a forty-five-minute flawless deduction on the ICM stage.

He had just proven to the whole world with a substantive result—one officially named by the academic circle, once questioned by top scholars, and subsequently unfolded, repaired, and verified under the watchful eyes of all—that he was by no means an academic gambler who only knew how to manufacture void concepts.

He possessed the rare ability to transform intuition into rigorous logic.

Consequently, that brief remark lost the qualification of being easily thrown into the wastepaper basket.

It turned into a declaration of war hanging over the field of Additive Combinatorics.

The fact that this matter truly broke through the barriers of the academic circle and formed a global discussion storm originated from an exclusive interview in overseas mathematical news.

The original intention of the interview was a routine review of the online report arrangements and key topics of ICM 2022.

The host was a veteran reporter who had long cultivated reporting in basic science, and the guest he connected with was the globally renowned fields medal winner, Terence Tao, one of the most important representative figures in the contemporary field of Additive Combinatorics.

In the forty-minute interview, the early atmosphere was professional and mild.

Terence Tao clearly sorted out the academic veins of several important reports.

Until the interview entered the second half, the reporter keenly captured the recent hot spot and turned the sharpness of the topic to Jiang Lin.

"Professor Tao, at the end of his ICM report, Jiang Lin specifically mentioned the weak doubling phenomenon, entropy forms, and compressed language."

The reporter's voice came through the trans-oceanic optical cable, carrying an unsuppressible desire for inquiry, throwing out the question that hovered over everyone's head.

"There are many people in the academic circle currently who believe that this is almost undisguisedly implying that he is tackling the PFR / Marton direction. As an authority in this field, do you think this statement has credibility? Is it really possible for an eighteen-year-old young man to solve this core difficult problem that has plagued the academic circle for decades?"

On the screen, Terence Tao sitting in the study fell into silence.

This few seconds of silence appeared exceptionally long in the news live broadcast.

His expression did not produce any exaggerated changes due to this highly impactful question.

He neither revealed the displeasure of an authority being offended nor catered to the media's expectations to give any emotional or eye-catching assertions.

He only slightly lowered his gaze, seeming to quickly retrieve relevant mathematical structures in his mind, as well as the logical depth demonstrated in Jiang Lin's report.

A moment later, he raised his head again and spoke.

"If today, an ordinary young researcher said such words at the end of a routine report, I would sincerely advise him to settle down first and completely read all important literature in this direction over the past thirty years."

As soon as these words came out, the barrage of the live broadcast platform instantly boiled, and countless people began to frantically capture screenshots.

These words carried the majesty of a top scholar, seeming to pour cold water on this fanatical deification movement.

But the reporter knew the rhythm of the interview deeply; he kept quiet and did not make a sound to interrupt.

Terence Tao adjusted his sitting posture and continued: "But we must face a premise: Jiangs Brick has already proved one thing to the world—he possesses a rare ability to precisely organize a highly complex structural problem into a rigorous mathematical object that can be reviewed line by line by peers."

His tone still maintained the calmness and objectivity unique to mathematicians, as if analyzing an objectively existing equation.

"Of course, we must also clearly recognize that PFR / Marton is not a geometric entity that can be constructed through intuitive imagination. It is even less a topic that can be readily solved by directly translating a successful experience or a set of mature algorithms over."

The reporter keenly grasped the gap in the words and pursued: "So, your subtext is that you don't think he is currently close to solving this conjecture?"

"I will not make such an assertion."

Terence Tao's answer was crisp and neat, without the slightest hesitation.

"Mathematics is a discipline that only recognizes proofs. Before watertight logical proofs appear in the form of black and white on paper, all predictions, guesses, and judgments only stay at the level of judgment; they can never become conclusions."

In front of the screen, many scholars and media figures paying attention to this matter nodded inwardly.

They originally thought this was the final tune given by Terence Tao.

An official rhetoric that was standard, rigorous, restrained, and firmly refused to endorse unverified conjectures.

This completely matched the steady posture that a mathematician at the pinnacle of the academic pyramid should have in a public interview.

However, to everyone's surprise, Terence Tao paused for a moment, as if making a certain decision in his heart, and then added a passage.

"However, I must point out one thing: the vocabulary he used is very specific and highly directional, hiding a profound insight into the essence of the entire problem."

Terence Tao looked straight into the camera, with a deep gaze.

"If you ask me now whether he has completely solved this problem, based on the rigor of mathematics, I certainly will not give an affirmative yes."

"But if you ask me whether the technical program he proposed is worth waiting for earnestly?"

"My answer is: Yes."

"An eighteen-year-old young man who can make the academic circle have the patience to wait for his unformed ideas is, in itself, an extremely rare thing."

This segment of the interview was translated and edited in an extremely short time, and quickly transported back to major domestic content platforms.

The news headlines automatically formed almost the moment the video was uploaded, carrying an irresistible impact:

"Terence Tao Responds Directly to Jiang Lin's Research on PFR / Marton: He Is Worth Waiting for Earnestly"

"Mathematics Only Recognizes Proofs, But Jiang Lin's Bold Words Cannot Be Easily Ignored—In-Depth Interview with Terence Tao"

"Worth Waiting for Earnestly: Analyzing Terence Tao Discussing Jiang Lin's Technical Program After ICM"

...

Many people in the academic circle saw these titles and initially instinctively thought this was another case of media taking things out of context.

But when they clicked on the video and listened to the entire uncut interview, they all fell into deep thought.

This was not the media's bottomless, brainless touting.

Precisely because this was not touting, the weight it carried appeared particularly heavy.

After Terence Tao's interview segment was transmitted back home and triggered a huge shock, the media, like sharks smelling blood, quickly turned their gaze to another towering name in the domestic mathematical circle that absolutely could not be bypassed.

Shing-Tung Yau.

Mr. Qiu's own main research field was differential geometry and geometric analysis; he was not a direct technical referee in the PFR / Marton direction.

But he was one of the most internationally prestigious figures in the Chinese mathematical circle, and also the most symbolic name in the Tsinghua University mathematics system.

When Jiang Lin filled out his college application for Tsinghua University, admission was almost just a procedural issue.

In other words, although Jiang Lin had not yet officially stepped into the gates of the Tsinghua University campus in physical space, in the eyes of the outside world, the academic circle, and the media, he had already been placed in advance into this grand and expectation-filled coordinate system of Tsinghua University mathematics.

An eighteen-year-old mathematical genius about to step into a university campus publicly stated on such a sacred stage as the International Congress of Mathematicians that he was advancing the deep directions of weak doubling and entropy forms.

Facing such a landmark event that might change the status of China's basic sciences, the media could by no means avoid exploring Shing-Tung Yau's views.

Therefore, after a public event on cultivating top-tier talent in basic disciplines held at the Tsinghua University Mathematical Sciences Center, a swarm of reporters thrust their telephoto lenses and microphones like artillery pieces in front of Mr. Qiu.

"Mr. Qiu, student Jiang Lin just finished a brilliant report internationally on Jiangs Brick, and right at the end of the report, he revealed that he is deeply researching weak doubling and entropy forms."

A reporter standing in the very front row asked loudly.

"Currently, the outside world generally interprets this as him making an assault on mathematical mountains of near-Millennium Prize difficulty like PFR / Marton. What is your take on this?"

Camera flashes flickered, and all eyes focused on Shing-Tung Yau.

Shing-Tung Yau did not, as the outside world expected, start by dissecting obscure technical details.

He did not explain to the reporters what entropy forms defined using random variables were, nor did he evaluate the logical pros and cons of the Marton route.

With a slightly accented yet exceptionally steady and sonorous tone, he uttered words that leaned more toward academic philosophy.

"For a young person exploring the field of mathematics, the most precious trait is definitely not caution."

The surrounding environment instantly fell silent, save for the clicking sound of camera shutters.

"The quality of caution is something that naturally arises as a person grows older, experiences the ruthless beatings of countless failures, and bears more and more academic reputation; it does not need to be deliberately cultivated."

Shing-Tung Yau paused slightly, his gaze sweeping across the lenses before him.

"In the exploration of basic science, what is truly scarce? It is that some people, when they are eighteen or twenty years old and at the most vibrant age of their lives, can simultaneously hold two seemingly contradictory things in their minds. One is the courage to throw a punch at the hardest barriers, and the other is a delicate sense of structure for the vast mathematical system."

He sighed softly, seemingly recalling all kinds of geniuses he had encountered over his long academic career.

"I have witnessed many outstanding scholars who, after producing an important achievement that can establish their career, often choose to pause first, enjoy flowers and applause, participate in various forums, and consolidate their status and resources in the academic circle."

"This is human nature, but when you open the heavy history of mathematics and look at the great mathematicians who truly pushed the boundaries of human cognition, such as Galois and Abel, you will find that when they were young, their style of action was often not like this."

"They would not stop; instead, following their extreme inner desire for truth, they would continue to move forward recklessly."

"Moreover, they often walk to a place that bystanders and predecessors feel is too fast, too deep, too dangerous, and even seems somewhat absurd."

Taking the opportunity, a reporter followed up: "Then, based on your experience, do you think Jiang Lin really has a chance to solve that century-old problem in the near future?"

Hearing this question, Shing-Tung Yau smiled faintly, his smile carrying the wisdom of seeing through the ways of the world.

"Mathematical problems have never been judged as to whether they can be solved based on a few news reports or a few interviews."

"Whether he can finally solve that conjecture, I do not know right now, and no one in the world can know in advance."

"However, he did not choose to stagnate on the glory brought by Jiangs Brick; he chose to immediately start the next more difficult journey. This point is very important."

Shing-Tung Yau's gaze turned exceptionally firm.

"If a young person truly possesses the extraordinary ability to continuously produce in basic theory, then the first stunning achievement in his life is merely the prelude to a grand symphony, and by no means the final chapter."

After this interview footage was broadcast, it did not carry the sharp feeling of piercing through the fog like Terence Tao's phrase "worth waiting for earnestly".

But from another, grander and more far-reaching perspective, it pushed Jiang Lin's personal academic choice into a heavier historical coordinate system.

In academia, a momentary flash of inspiration achieving an isolated result is called an outburst.

Continuously producing two influential theories will truly change the academic circle's judgment on a person's underlying scientific research capabilities.

And being able to cross different subfields and maintain high-intensity, high-standard continuous output is often the growth trajectory possessed only by those truly great mathematicians in history.

With the voices of these two heavyweight figures, the outside world's expectations and doubts regarding Jiang Lin rose sharply at the same moment, like simultaneously ignited fuses, forming a torn public opinion field.

Some conservative scholars believed that Jiang Lin's pace was simply too fast, so fast that it somewhat violated objective laws.

Crossing from Jiangs Brick constructed by discrete geometry to the PFR / Marton Conjecture centered on structure theorems and information theory did not simply span a research direction in between, but instead crossed an entire vast mathematical continent.

Someone who had just completed the theoretical closed-loop of tiling theory did not even catch his breath before turning around to fiercely challenge the universally recognized mountain in the field of Additive Combinatorics. In the eyes of the traditional academic training system, this behavior was close to impractical arrogance.

Another group of people firmly believed that one could not rigidly apply the academic growth curve of ordinary scientific researchers to measure Jiang Lin.

He was only eighteen years old.

He was at the golden age when human intelligence was at its sharpest, thinking was at its most divergent, and he was completely unbound by years of academic inertia and traditional paradigms.

Looking back at those incredible geniuses in the history of mathematics, it was often precisely in this age group that they opened their era of continuous output as brilliant as a meteor shower.

On the further periphery of the internet, the vast general netizens did not understand what PFR / Marton was, nor did they care about the logical chain of Additive Combinatorics.

Using the simplest logic, they only understood one thing.

Jiang Lin, whom they regarded as a teenage genius, seemed to have raised his sword once again, aiming at a Boss even larger and more terrifying than before.

The outside world's clamor was like a violent storm, yet the School of Mathematics of Jiangda University, located in one of the sub-centers of this storm, was shrouded in an unusual silence.

Professor Gu Nanzhou sat alone in his slightly messy office.

The spacious solid wood desk was piled high with uncorrected papers and various preprints, but at this moment he had no mood at all to flip through those papers.

His gaze was tightly locked onto the computer screen, where Terence Tao's short yet massive information-packed interview video was playing on a loop.

The sky outside the window gradually darkened, and the streetlights on campus lit up one by one, but Professor Gu Nanzhou did not turn on the overhead light of the office.

Thus, in the dim light, he remained silent for a long time—so long that he did not even notice when the computer screen entered the screensaver program.

A crisp "ding" broke the dead silence of the room.

It was the notification sound of the instant messaging software.

Professor Gu Nanzhou shook his mouse to wake up the screen.

The message came from Professor Han Yanshan.

[ Have you seen it? ]

There were only three short words, yet they exuded a sense of urgency and heaviness.

Professor Gu Nanzhou placed his hands on the keyboard, paused slightly, and replied:

[ Saw it. The unedited version too. ]

The top of the dialogue box showed that the other party was typing, and a moment later, Professor Han Yanshan's question popped up.

[ What is your take on this matter? Do you think he came prepared, or is he just acting on a hot head? ]

Professor Gu Nanzhou stared at the jumping characters on the screen, his mind continuously replaying the afterimages of the last few pages of PPT from Jiang Lin's ICM report.

Those definitions regarding compressed language, that obscure yet highly directional entropy inequality...

He pondered for a long time, his fingers tapping on the keyboard a few times, deleting it again, and finally typing out a line of text.

[ This is not cross-disciplinary for him; rather, it is jumping straight from a solid brick, attempting to shake the foundation of the entire structural theory. The span is too large, but judging from the vocabulary he proposed, the logical chain is closed. ]

Professor Han Yanshan's message reply was exceptionally swift.

[ That is why back in the venue, I could not help but pick up the microphone and ask him publicly whether that was really a metaphor. I know too well how deep the water is in that direction; that is an abyss where top experts like Terence Tao and Green have to carefully grope around. ]

Professor Gu Nanzhou took a deep breath, leaned backward, and heavily rested against the ergonomic backrest.

Having struggled in academia for the better part of his life, he knew all too well the unspoken rules of how academic prestige operates.

The entire international mathematical community would certainly, absolutely not casually change the fund flows and research territories of various top laboratories just because of a sentence casually uttered by an eighteen-year-old boy at the end of a report.

Research groups at major universities would still advance their work according to their original rhythm.

However, the entire international mathematical community would definitely be forced to re-weigh the weight of the next sentence spoken by Jiang Lin, because he had just proven the incredible fact of Jiangs Brick in an impeccable manner.

This is the essence of academic prestige, and it is also an invisible privilege.

The meaning of the existence of prestige is not to make others unconditionally believe that every conjecture you propose is definitely correct.

Rather, it is to make others unable to easily ignore it when facing the direction you point out, even if their hearts are filled with doubts.

It will force one to incorporate your train of thought into the scope of consideration, deducing whether the technical program you proposed truly hides the cryptographic key to breaking the deadlock.

Meanwhile, Jiang Lin himself, who was at the absolute center of this public opinion storm, seemed to have vanished into thin air from the internet.