123: Chapter 123 Special Online Report

July 8.

7:40 PM.

On the computer screen, the dedicated online conference client for the 2022 International Congress of Mathematicians had already logged in ahead of time.

The camera feed was quietly paused in the preview window.

The youth in the video was wearing only the most ordinary plain black T-shirt, with his somewhat crowded bookshelf behind him, neatly and densely packed with various professional books and printed, bound papers.

On the left side of the desk sat a minimalist countdown timer, its red numbers frozen at the starting point.

45:00.

On the right was another laptop used as an auxiliary device.

Open on the screen were backup slide sequences for dealing with unexpected questions, as well as several terminal screen recordings demonstrating the finite-state verification process.

7:45 PM.

A message from the online technician popped up in the bottom right corner of the client.

[Audio and bandwidth load test, please respond.]

Jiang Lin reached out and clicked on the microphone.

"I am Jiang Lin, can you hear me?"

A clear response with a slight static sound immediately came through the earphones.

"The sound is very clear, the frequency band is stable, and the video latency is within the allowable range."

7:52 PM.

The conference host came online on schedule.

Professor Marcus Holt appeared in the main control window of the meeting.

It was this scholar who first officially used the term Jiangs Brick in a previous academic article.

And today, the ICM organizing committee specially arranged for him to host this special online presentation.

"Good evening, Mr. Jiang."

"Good morning, Professor Holt."

Night had fallen in Jiangcheng at this moment.

Meanwhile, on the other end where the ICM main server was located, it was greeting the rising morning light of another time zone.

Holt smiled on the screen, seemingly to ease the tension before the opening.

"Time zone differences are always the first test of large-scale online international conferences."

Jiang Lin nodded slightly, picking up the sentence.

"At least, it is much easier than dealing with boundary blockages."

Holt was stunned for half a second, clearly not expecting the other party to respond like that, and then he genuinely laughed out loud.

"Then, I hope the academic blockages we encounter tonight can be as few as dealing with time zones."

7:58 PM.

Permissions for the official online meeting room were opened.

The participant list on the right side of the screen began to pop up new names at a dizzying speed, and the number of attendees rose in a straight line.

Two hundred.

Four hundred.

Six hundred.

Nine hundred.

...

In the continuously scrolling list of attendees on the right side of the window, many familiar names that Jiang Lin had repeatedly seen in massive amounts of literature, classic works, and domain reviews over this period began to appear.

...

8:00 sharp.

Holt turned on the global microphone.

"Welcome everyone to this special online presentation of the 2022 International Congress of Mathematicians."

"Today, we are very honored to invite Mr. Jiang Lin. His recently published preprint provides a brand-new construction scheme for a single aperiodic monotile, a construction that is now widely known in the academic community as Jiangs Brick."

As soon as his voice fell, Jiang Lin's first slide was projected in the center of the screen.

[Jiangs Brick: Local Forcing Mechanism, Finite-State Verification, and Single Aperiodic Monotile]

Holt continued,

"Today, Mr. Jiang will personally introduce the ideas of this construction, the finite-state verification framework, and how to solve the blockage problem."

After a brief pause, he added a rule explanation as the host.

"During the presentation, we will collect your questions through the backstage and have a unified discussion after the 45-minute presentation ends. Mr. Jiang, now, the time is yours."

Jiang Lin clicked on the microphone switch.

The timer on the desk received the command and began to count down silently.

44:59.

"Thank you, Professor Holt."

His voice, transmitted from Jiangcheng, China, was converted into digital signals and accurately pushed to the earphones and speakers of top mathematicians around the world.

"What I am going to tell you about today is the construction process of a single aperiodic monotile. But I do not intend to start from its widely reported name, but rather from the constraints it must bear in order to exist."

The first slide switched rapidly.

Appearing on the screen was a locally magnified boundary segment full of jagged edges and bumps.

Jiang Lin neatly established the entire theoretical background in less than three minutes.

Then he sorted out how subsequent single-tile candidate schemes attempted to continuously compress the number of underlying objects constituting the tiling.

Finally leading to the core argument.

"Therefore, the key to evaluating this problem does not lie in whether this shape looks complex enough at first glance, but rather in where the information is actually stored."

On the screen, the overall outline of Jiangs Brick appeared completely for the first time.

There were no exaggerated red circles and arrows added for attention-seeking on the Chinese internet.

No filters in media reports rendered it as some kind of mysterious artwork.

There were only precise boundaries, local coordinates, direction markers, and a few sets of highlighted matching segments in light colors.

Like dissecting a precision instrument, Jiang Lin disassembled this brick into four logical levels.

"Here, the easiest misunderstanding for most people is to regard Jiangs Brick merely as a geometric polygon with some strange outline."

He pressed the page-turn key.

The boundary on the slide was straightened and expanded into an ultra-long sequence containing letters and numbers.

"If we want to understand its operating mechanism, a more accurate perspective is that it is a highly geometricized finite-state cryptographic text."

In the meeting room, although no sound could be heard, many people had unconsciously picked up their pens and started taking quick notes in their notebooks.

Jiang Lin did not pause.

With a set of dynamic local patterns, he clearly illustrated that when two pieces of Jiangs Brick underwent physical tiling, it was essentially not just a simple edge-to-edge adhesion.

They were conducting an information handshake, transmitting the current information of direction, phase, and the finite-state machine.

The minute bumps on certain boundary segments seemed to be just minor embellishments to increase complexity.

Their true mathematical function was to prohibit erroneous phase information from penetrating into the tiling structure of the next layer.

Certain local seemingly redundant abrupt gaps.

Actually, in the underlying logical deduction, they forced the macroscopic structure to exhibit a hierarchical separation that could not be periodic.

"Local forcing does not mean that every local pattern itself is absolutely rigid and unalterable."

"Its true meaning is that the constraints of local admissibility will continuously accumulate outward, ultimately forming an insurmountable high wall, creating a global obstacle to any tendency attempting to form periodic structures."

Hearing this, many master's and doctoral students in the Department of Mathematics in China who were auditing through various private group chats subconsciously stopped their fingers hitting the keyboards to translate the original words.

Because they understood this sentence.

But precisely because they understood, they profoundly realized what kind of exquisite logical control power and construction difficulty were contained behind this seemingly light remark.

Coming to the third part, finite-state verification, what Jiang Lin displayed was the organizational framework and logical flow of the entire computer-assisted verification chain.

For each module, he only displayed its exposed functional interfaces and the rigorous mathematical meaning behind them.

"Here, the high-speed computation of the computer is not used as a bottomless black box."

He cut to a logically clear flowchart.

"In my proof, it is used as a rigorous finite-state bookkeeping device. Every state transition accepted by the system can find a corresponding geometric interpretation on the drawings. Every possible periodic loop excluded by the program must correspond to a clear, illustratable local geometric conflict."

This passage was crucial.

Many mathematicians remained vigilant about computer-assisted proofs, and their biggest worry was black-box conclusions.

Jiang Lin did not treat the program as truth, but instead dismantled it into a reviewable state ledger.

The fourth part, Layer-7 blockage.

When this vocabulary appeared on the screen, the attention in the entire meeting room visibly concentrated highly.

Many attendees specially spared time to come online today, largely to see how this young discoverer would face and handle this problem sharply pointed out by peers on this world-class stage.

On the screen appeared the screenshot where the verification team of the University of Waterloo marked the contradiction points with eye-catching red circles.

Jiang Lin did not try to avoid or cover it up, nor did he carry out any academic mockery.

"The Layer-7 blockage query is very valuable because it accurately points out the specific location where I became overly aggressive in compressing notation in the preprint in pursuit of brevity in length."

Afterward, he directly switched to a densely packed yet well-organized large state transition table.

That red circle, which was repeatedly retweeted on academic social platforms like Twitter and considered a fatal flaw, was put back into its rightful logical position in this fully expanded state diagram.

Jiang Lin handled this storm with just three pages of diagrams.

"Therefore, the pointed-out blockage is not a counterexample to the construction mechanism."

He paused here for a moment, letting this conclusion ferment in the air.

"What it truly refutes is an equally over-compressed interpretation of the three rows of overly simplified compressed notation in my supplementary materials."

There was no applause in the meeting room.

The format of an online presentation was inherently unsuited for outbursts of applause.

However, the text chat area, which originally had occasional exchanges, experienced a vacuum lasting more than ten seconds.

For those truly in the know in this circle, this collective silence of deep thought was much more explicit than any thunderous applause.

The fifth part, subsequent perspective.

The timer on the desk showed that six minutes and forty seconds remained.

Jiang Lin calmly switched to the last set of slides.

The title was very simple, yet it exuded a faint ambition: [Beyond Jiangs Brick].

In the middle of the page, there were only two lines of text.

[Some structures are forced out by strong local rules.]

[Some structures are compressed out from weak global constraints.]

Many auditing scholars originally thought that the clarification of Layer-7 was the climax of this presentation, and the last page was just a routine concluding summary paragraph.

But when the second line of text appeared on the screen, Holt, the host far away in another time zone, unconsciously straightened his posture.

Meanwhile, Professor Han Yanshan in the offline research room suddenly sharpened his gaze.

Jiang Lin's voice came steadily through the microphone: "Jiangs Brick belongs to the world of the former. The constraints it receives are strong, and any erroneous local physical matching is absolutely not allowed by the system."

Accompanying his speaking speed, the screen displayed a seamlessly interlocking boundary of Jiangs Brick.

"However, if we broaden our horizons, many core structural problems in the field of discrete mathematics do not possess such favorable strong local rules. What they often possess are weak constraints of a global nature."

On the next page, he drew an extremely abstract yet highly heuristic schematic diagram.

On the left side of the figure was a solid lock representing strong local constraints.

On the right was a loose network representing weak global constraints.

And between the two, he drew a funnel-shaped arrow, with two words marked next to it.

Compression.

"After completing the work of finding this single aperiodic tile, I have been thinking deeply about a question. Can finite-state compression only be used as a dedicated computational method for tiling verification? Or, at a deeper level, is it itself a more generalized perspective for handling complex mathematical structures?"

"Especially recently, I am currently focusing on studying the weak doubling phenomenon in discrete structures, as well as the closely related entropy forms. Since the research is still ongoing, I have no specific detailed theorems to show you today. But I have reason to believe that the role of compression language in mathematics should not stop at solving a mere tiling problem."

The silence in the chat area on the right side of the meeting room was instantly broken, and several tentative and shocked messages popped up one after another.

[Weak doubling phenomenon?]

[Is he hinting at PFR?]

[Entropy forms — the route of Marton's conjecture?]

[Is he really referring to PFR / Marton?]

Jiang Lin did not make any response.

This was not a Q&A session, let alone the main dish he was going to serve on the table today.

He cut back to the last slide of acknowledgments.

"Today's presentation has a theme of only one brick. As for the rest of the contents, we will discuss them when they truly become flawless mathematical theorems."

On the desk, the countdown timer stopped at 00:37.

Jiang Lin looked straight into the camera and waved: "Thank you."

The presentation ended.

The public channel of the online meeting room was quiet for a second, as if everyone was digesting the huge amount of information thrown out in those last few minutes.

Afterward, the chat area began to flash out rows of brief and solemn greetings like a waterfall.

[Thank you.]

[The proof logic is very clear.]

[This is a very wonderful presentation.]

[The on-site clarification about Layer-7 is very helpful and cleared up our doubts.]

[The perspective of treating computer verification as finite-state bookkeeping is very important, I have learned a lot.]

...

Holt turned on the microphone at the right time, with obvious appreciation in his voice.

"Thank you Mr. Jiang for your explanation. Next, we officially enter the Q&A session."

The Q&A began.

The atmosphere instantly transformed from a one-way output to a tit-for-tat academic clash.

The first question came from an old professor of discrete geometry teaching at a top European institution.

The question went straight to the core, showing no mercy.

"Mr. Jiang, your verification framework relies on a finite-state machine, but as we all know, aperiodic tiling spaces are inherently infinite. Could you elaborate further on where exactly the logical leap occurs during the transition from confronting an infinite tiling plane to the elimination process of finite states?"

This question went straight for the most vulnerable bottleneck in the proof chain.

Jiang Lin operated the mouse, switching the slides back to the massive state diagram of the third part.

"Professor, please note that we are not attempting to use a computer to enumerate an infinite tiling plane."

To ensure the clear transmission of logic, his speaking speed was slightly slower than during the presentation.

"What we enumerate during verification is the set of all allowable boundary states induced by the local forcing hierarchy. If periodicity exists, it requires that a compatible closed cycle must exist within this finite state transition graph. And what my verification program eliminates is precisely the possibility of such compatible cycles under a strictly defined boundary equivalence relation."

...

The elderly professor asking the question nodded slightly on the other end of the screen, showing approval.

The second question came from a young scholar specializing in symbolic dynamical systems.

The other party asked in an exploratory manner whether this seemingly unique verification logic could be transcribed into a finite-type shift framework more standard within their field.

Jiang Lin did not force an answer just to cater to the questioner.

After a brief moment of thought, he stated that partial transcription was possible.

However, he also sharply pointed out that due to the equivalent compression effect brought about by the geometric boundary folding of Jiangs Brick, forcibly applying the standard finite-type shift formulation would not only fail to increase clarity, but would instead make the entire proof extremely bloated and unreadable.

The third question finally fell to the University of Waterloo verification team, which had recently been in the spotlight.

When the name with the Waterloo suffix appeared in the question list, all the domestic auditing chat groups broadcasting live instantly fell into pin-drop silence.

The questioner was not Professor Kaplan himself, but another core researcher in the team.

Unexpectedly, he asked very politely, even carrying an exploratory tone.

"Mr. Jiang, regarding the blocking problem of Layer-7, may I ask if the newly revised complete index table has been updated and included in the public supplementary materials?"

Jiang Lin answered flawlessly: "Yes, the revised supplementary documentation already contains the fully expanded index table, detailed decomposition diagrams of Class F mirror embeddings, and Python scripts for validating folded states. After this report concludes, I will also package and upload all of today's slides to the preprint platform."

The other party paused and continued to ask: "Then, do you agree that in the original preprint you submitted, for the sake of length, the symbol compression in this section did indeed appear excessive?"

"I agree."

Jiang Lin candidly admitted.

But pausing for less than half a second, his tone shifted as he added.

"That excessive compression was indeed a phrasing error on my part when writing the paper. But please note that the logic of the entire proof did not fail because of it; it's just that the phrasing in that section did not make one of the key state transitions easy enough for peer review."

As soon as this sentence was uttered,

Professor Gu Nanzhou, far away in his office at Jiangda University, couldn't help but silently spit out a word.

Brilliant.

Only someone with absolute confidence in the underlying logic would dare to candidly admit a writing flaw of their own on a live broadcast scene at the level of the ICM.

Because he was incomparably clear that what failed was only the typesetting and expression, while the proof itself was rock-solid.

The fourth question, the topic finally inevitably wound around to that element that made everyone curious, yet seemed somewhat sensitive in academic occasions.

Age.

The questioning scholar's choice of words was extremely euphemistic.

"Mr. Jiang, might I boldly ask, what was the origin of this massive project? To be honest, for someone at your current academic stage, being able to independently construct such an exquisite framework and single-handedly complete such a complex verification framework is truly very rare."

He used the term "academic stage".

Instead of straightforward age.

This was the most appropriate phrasing that could be used in an environment like the ICM that places extreme emphasis on academic equality.

In major domestic live broadcast chat groups, the bullet comments instantly began to flood the screen wildly.

[It's coming, it's coming.]

[Holding back for so long, someone finally asked after all.]

[Let's see how he takes it; this question is not easy to answer, as it's easy to look arrogant or seem hypocritical.]

Jiang Lin in front of the camera nodded calmly and said: "I completely understand the original intention behind your question, but I personally do not believe that my current academic stage, or my physiological age, is any part of this mathematical proof process."

The conference room once again fell into that pin-drop silence.

"Every geometric construction in my paper, every state transition table, every line of verification script code, and every correction note displayed just now—they can, and ought to be, rigorously checked by peers completely unaware of my identity and age."

He looked at the camera, as if looking at hundreds of pairs of eyes behind the screen.

"If one day in the future this proof is found to have a fatal logical error, then my youth cannot save it from the fate of being overturned. Similarly, if this proof is completely correct, then my age does not need to be brought out to endorse it."

Hearing this passage, Professor Marcus Holt's expression moved slightly, and an unmistakable look of amazement flashed in his eyes.

This response was not aggressive, but its logic was as sharp as a knife.

It pushed all external narratives about genius, identity, growth experiences, and educational miracles back into the proof itself.

Mathematics is mathematics.

It does not need the author's resume to guarantee it, nor does it ever accept the author begging for forgiveness under the excuse of youth.

The fifth question came from a veteran mathematician who had studied single-tile candidate constructions for many years.

He returned to the technology itself, asking whether Jiangs Brick was still likely to be further simplified in the future, and whether its complex boundary concavities and convexities could be effectively reduced.

This question was very realistic and utilitarian.

Jiang Lin answered extremely rigorously.

"There is such a possibility."

He generously admitted that the current construction was definitely not the simplest in an aesthetic sense, and he had never claimed that Jiangs Brick was the ultimate form of aperiodic tiling.

Its core significance lies in proving the existence of such a single tile and providing a construction path that can be rigorously scrutinized by peers.

As for whether the boundary complexity could be further compressed by algorithms, that was a subsequent optimization problem left for the future.

"We must first prove its existence, and then pursue mathematical elegance."

As soon as this sentence was thrown out, it immediately caused a short and enthusiastic response in the chat area.

[First prove existence, then pursue elegance.]

[Without a doubt, this sentence will be repeatedly cited in future reviews on tiling theory.]

The sixth question, the questioner's name lit up.

Professor Han Yanshan.

Jiang Lin was not surprised that this name appeared in the question list.

And the focus of Professor Han Yanshan's question did not fall on Jiangs Brick either.

"Student Jiang Lin, on the last slide just now, you mentioned the relationship between weak global constraints and compression. I'd like to confirm, is this merely a clever metaphor you used for the ending, or is it itself a clear technical program?"

In the conference room, quite a few scholars extremely sensitive to cutting-edge developments pricked up their ears.

Because they had also noticed the last two pages that seemed to be casually glossed over just now.

Jiang Lin was very clear about what Professor Han Yanshan was asking, and also clear about where the safety boundary of answering this question in this occasion lay.

However, he gave a firm answer.

"It's not a metaphor, but please understand, it is not yet part of the theorem we are discussing today."

Professor Han Yanshan did not turn on his microphone to interrupt, quietly waiting for the following.

Jiang Lin continued to sort out his thoughts: "In tiling theory, the local forcing mechanism produces a black-and-white hard exclusion. But in problem domains of the weak doubling type, we lose this rigid means of exclusion. The challenge we face is that we must accurately identify from a piece of chaos: exactly what core structure has been compressed? What factor is paying the price for the loss of this structure? And how does this tiny loss gradually accumulate and magnify between different scales?"

Speaking of this, his tone became extraordinarily firm.

"This is precisely the research topic I am currently advancing with all my strength."

Without showing more details, the amount of information conveyed was already shocking enough.

The chat area once again flooded with many messages that could hardly hide their excitement.

[He mentioned weak doubling once again.]

[This is definitely related to PFR, he is trying to conquer this fortress!]

[Cross-scale accumulation?]

[Could it be using an energy increment strategy? This ambition is too big.]

Marcus Holt glanced at the system time.

The discussion in the Q&A session was extremely enthusiastic and had far exceeded the scheduled twenty minutes.

But he did not immediately exercise the moderator's authority to forcibly terminate it.

The last selected question came from an American scholar studying computer-aided mathematics.

"Mr. Jiang, your verification framework is impressive. May I ask if you will make all your verification code public after the meeting?"

"Partially public."

As soon as the word 'partially' came out, it made many people who originally thought they would get a definite answer raise their heads.

He immediately gave an explanation: "All verification scripts necessary to check and recheck the correctness of this construction will be made public immediately. The tools used to assist in generating tiles and the pseudo-code of some search heuristic methods will be written in detail in the supplementary documentation. However, not all underlying tools we use internally will be open-sourced immediately today."

The questioner was obviously somewhat unwilling and pursued: "Why not choose complete open-source? This can greatly increase peers' trust in you."

Jiang Lin's answer was very direct.

"Because verifying a conclusion and discovering a conclusion are two completely different things."

He switched back to a backup diagram showing the verification process.

"I believe that all tools and data used to allow peers to review the final conclusion should be made public without reservation. But those primitive exploration tools used to generate candidates in endless space, explore failed constructions, and try to compress states—they belong to another level of work."

This sentence was filled with a strong personal imprint, very much Jiang Lin.

He was willing to let the proof undergo the most rigorous scrutiny, and the conclusion must accept the inspection of the whole world.

However, the core tool chain in his hands was not a sacrifice unconditionally offered to the world in exchange for empty fame.

Those candidate generators written after countless days and nights with hair turned white had no obligation to fully open their source code to the whole world.

He would not use selfless open-source to trade for others' recognition of his so-called sincerity.

What the mathematical community truly needed was a reviewable proof chain.

Rather than demanding that he take out all his underlying technological assets relied upon for survival to streak naked.

Marcus Holt slowly nodded on the other end of the screen, indicating understanding.

"I agree with your distinction; this is reasonable intellectual property protection. Although I also anticipate that this practice will inevitably trigger some controversy after the meeting."

Jiang Lin's expression did not change: "Discussions and controversies are always welcome."

Nine-sixteen.

The report officially concluded.

Before closing the interface, Marcus Holt sent a sentence to Jiang Lin through the private message channel.

[This was a very wonderful report, Jiang. Especially your answer regarding the age question, it was impressive.]

Jiang Lin tapped his fingers lightly on the keyboard to reply: [Thank you, Professor Holt.]

Meanwhile.

Major domestic related forums and auditing chat groups had completely exploded.

Someone spent the night organizing detailed notes of the report and sorting out the logic chain.

Someone took screenshots of the explanation process of Layer-7 and posted them on their academic blog for a lengthy interpretation.

Someone posted on professional forums, discussing in depth the profound impact of the perspective of finite state bookkeeping on future computer-aided proofs.

Someone repeatedly replayed the video, staring at the short discussion on weak global constraints on the last two pages of the report.

Even more scholars genuinely standing at the forefront extracted Professor Han Yanshan's question and Jiang Lin's answer separately, chewing on them repeatedly.

[It's not a metaphor.]

[A technical program.]

These two seemingly plain words precisely pierced into the nerve endings of the very few people who truly knew the ropes in the direction of Additive Combinatorics.

When ordinary audience members online were still enthusiastically discussing how tough Jiang Lin was when snapping back at people and how cool his attitude was.

Those brains with more sensitive noses in the mathematical circle were already trembling with fear or excitement, realizing a more terrifying thing.

Jiang Lin's footsteps did not stop at Jiangs Brick at all.

Taking advantage of the globally watched stage of the ICM report, in the last few minutes, he casually and lightly showed the entire mathematical community the entrance to his next expedition.